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Fermi gases in one dimension, From Bethe ansatz to experiments

Fermi gases in one dimension:From Bethe ansatz to experiments

Xi-Wen Guan*

State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,

Wuhan Institute of Physics and Mathematics,Chinese Academy of Sciences,

Wuhan430071,China

and Department of Theoretical Physics,Research School of Physics and Engineering,

Australian National University,Canberra ACT0200,Australia

Murray T.Batchelor?

Centre for Modern Physics,Chongqing University,Chongqing400044,China

and Mathematical Sciences Institute and Department of Theoretical Physics,

Research School of Physics and Engineering,Australian National University,

Canberra ACT0200,Australia

Chaohong Lee?

State Key Laboratory of Optoelectronic Materials and Technologies,

School of Physics and Engineering,Sun Yat-Sen University,Guangzhou510275,China

and Nonlinear Physics Centre and ARC Centre of Excellence for Quantum-Atom Optics,

Research School of Physics and Engineering,Australian National University,

Canberra ACT0200,Australia

(published27November2013)

This article reviews theoretical and experimental developments for one-dimensional Fermi gases.

Speci?cally,the experimentally realized two-component delta-function interacting Fermi gas—the

Gaudin-Yang model—and its generalizations to multicomponent Fermi systems with larger spin

symmetries is discussed.The exact results obtained for Bethe ansatz integrable models of this kind

enable the study of the nature and microscopic origin of a wide range of quantum many-body

phenomena driven by spin population imbalance,dynamical interactions,and magnetic?elds.This

physics includes Bardeen-Cooper-Schrieffer-like pairing,Tomonaga-Luttinger liquids,spin-charge

separation,Fulde-Ferrel-Larkin-Ovchinnikov-like pair correlations,quantum criticality and scaling,

polarons,and the few-body physics of the trimer state(trions).The fascinating interplay between

exactly solved models and experimental developments in one dimension promises to yield further

insight into the exciting and fundamental physics of interacting Fermi systems.

DOI:10.1103/RevModPhys.85.1633PACS numbers:05.30.Fk,03.75.Ss,67.85.àd,71.10.àw

CONTENTS

I.Introduction1634

A.Exactly solved models1634

1.The virtuoso triumphs of the Bethe ansatz1634

2.Fermions in1D:A historical overview1635

B.Renewed interest in1D fermions1636

1.Novel BCS-pairing states1637

https://www.wendangku.net/doc/3b2573010.html,rge-spin ultracold atomic fermions1637

3.Quantum criticality of ultracold atoms1638

4.Experiments with ultracold atoms in1D1638

C.Outline of this review1638 II.The Gaudin-Yang Model1639

A.Bethe ansatz solution1639

B.Solutions to the discrete Bethe ansatz equations1641

1.BCS-like pairing and tightly bound molecules1641

2.Highly polarized fermions:Polaron versus

molecule1642

C.Solutions in the thermodynamic limit1643

1.BCS-BEC crossover and fermionic super

Tonks-Girardeau gas1643

2.Solutions to the Fredholm equations and

analyticity1645 III.Many-body Physics of the Gaudin-Yang Model1646

A.1D analog of the FFLO state and magnetism1646

B.Fermions in a1D harmonic trap1648

C.Tomonaga-Luttinger liquids1649

D.Universal thermodynamics and Tomonaga-Luttinger

liquids in attractive fermions1650

E.Quantum criticality and universal scaling1651

F.Spin-charge separation in repulsive fermions1654 IV.Fermi-Bose Mixtures in1D1655

A.Ground state1656

B.Universal thermodynamics1657 V.Multicomponent Fermi Gases of

Ultracold Atoms1658

A.Pairs and trions in three-component systems1658

1.Color pairing and trions1659

2.Quantum phase transitions and phase

diagrams1660

*xwe105@https://www.wendangku.net/doc/3b2573010.html,.au

?murray.batchelor@https://www.wendangku.net/doc/3b2573010.html,.au

?lichaoh2@https://www.wendangku.net/doc/3b2573010.html,

REVIEWS OF MODERN PHYSICS,VOLUME85,OCTOBER–DECEMBER2013

1633

0034-6861=2013=85(4)=1633(59)ó2013American Physical Society

3.Universal thermodynamics of the

three-component fermions1662

B.Ultracold fermions with higher-spin symmetries1663

1.Bosonization for spin-3=2fermions

with SOe5Tsymmetry1663

2.Integrable spin-3=2fermions

with SOe5Tsymmetry1664

C.Uni?ed results for SUe TFermi gases1665

1.Ground state energy1665

2.Universal thermodynamics of

SUe T-invariant fermions1667 VI.Correlation Functions1668

A.Correlation functions and the nature of

FFLO pairing1668

B.1D two-component repulsive fermions1671

C.1D multicomponent fermions1673

D.Universal contact in1D1674 VII.Experimental Progress1675

A.Realization of1D quantum atomic gases1675

1.Optical lattices1675

2.Atom chips1676

B.Tuning interaction via Feshbach resonance1677

C.Data extraction1677

1.Detecting correlation functions via

optical imaging1678

2.Detecting the dynamical structure factor

via Bragg scattering1678

D.Experiments with1D quantum atomic gases1679

1.Bose gases1679

2.Fermi gases1680 VIII.Conclusion and Outlook1682 Acknowledgments1684 References1684 I.INTRODUCTION

Fundamental quantum many-body systems involve the interaction of bosonic and/or fermionic particles.The spin of a particle makes it behave very differently at ultracold temperatures below the degeneracy temperature.There are thus fundamental differences between the properties of bo-sons and fermions.However,as bosons are not subject to the Pauli exclusion principle,they can collapse under suitable conditions into the same quantum ground state,the Bose-Einstein condensate(BEC).Remarkably,even a small attrac-tion between two fermions with opposite-spin states and momentum can lead to the formation of a Bardeen-Cooper-Schrieffer(BCS)pair that has a bosonic nature.Such BCS pairs can undergo the phenomenon of BEC as temperature tends to absolute zero.Over the past few decades,experi-mental achievements in trapping and cooling atomic gases have revealed the beautiful and subtle physics of the quantum world of ultracold atoms;see recent reviews by Dalfovo et al. (1999),Leggett(2001),Regal and Jin(2006),Lewenstein et al.(2007),Bloch,Dalibard,and Zweger(2008),Giorgini, Pitaevskii,and Stringari(2008),Zhai(2009),Chin et al. (2010),and Bloch,Dalibard,and Nascimbe′ne(2012).

In particular,recent experiments on ultracold bosonic and fermionic atoms con?ned to one dimension(1D)have provided a better understanding of the quantum statistical and dynamical effects in quantum many-body systems(Yurovsky,Olshanii, and Weiss,2008;Cazalilla et al.,2011).These atomic wave-guide particles are tightly con?ned in two transverse directions and weakly con?ned in the axial direction.The transverse excitations are fully suppressed by the tight con?nement.As a result the trapped atoms can be effectively characterized by a quasi-1D system;see Fig.1.The effective1D interparticle potential can be controlled in the whole interaction regime.In such a way,the1D many-body systems ultimately relate to previously considered exactly solved models of interacting bosons and fermions.This has led to a fascinating interplay between exactly solved models and experimental develop-ments in1D.Inspired by these developments,the study of integrable models has undergone a renaissance over the past decade.Their study has become crucial to exploring and understanding the physics of quantum many-body systems.

A.Exactly solved models

1.The virtuoso triumphs of the Bethe ansatz

The study of Bethe ansatz solvable models began when Bethe(1931)introduced a particular form of wave function—the Bethe ansatz(BA)—to obtain the energy eigenspectrum of the1D Heisenberg spin chain.After lying in obscurity for decades,the BA emerged to underpin a diverse range of physical problems,from superconductors to string theory; see,e.g.,Batchelor(2007).For such exactly solved models, the energy eigenspectrum of the model Hamiltonian is ob-tained exactly in terms of the BA equations,from which physical properties can be derived via mathematical analysis. From1931to the early1960s there were only a handful of papers on the BA,treating the passage to the thermodynamic limit and the extension to the anisotropic XXZ

Heisenberg FIG.1(color online).Experimental con?nement of two-component ultracold6Li atoms trapped in an array of1D tubes (Liao et al.,2010).The system has spin population imbalance caused by a difference in the number of spin-up and spin-down atoms.From Bloch,2010.

1634Xi-Wen Guan,Murray T.Batchelor,and Chaohong Lee:Fermi gases in one dimension:From Bethe... Rev.Mod.Phys.,V ol.85,No.4,October–December2013

spin chain(Hulthe′n,1938;Orbach,1959;Walker,1959;des Cloizeaux and Pearson,1962;Grif?ths,1964).Yang and Yang(1966a)coined the term Bethe’s hypothesis and proved that Bethe’s solution was indeed the ground state of the XXZ spin chain(Yang and Yang,1966a,1966b,1966c).

The next development was the exact solution of the 1D Bose gas with delta-function interaction by Lieb and Liniger(1963),which continues to have a tremendous impact in quantum statistical mechanics(Cazalilla et al., 2011).They diagonalized the Hamiltonian and derived the ground state energy of the model.This study was further extended to the excitations above the ground state(Lieb, 1963).McGuire(1964)considered the model in the context of quantum many-body scattering in which the condition of nondiffractive scattering appeared.

Developments for the exact solution of the1D Fermi gas with delta-function interaction(Gaudin,1967a,1967b;Yang, 1967)are discussed in Sec.I.A.2.A key point is Yang’s observation(Yang,1967)that a generalized Bethe hypothesis works for the fermion problem,subject to a set of cubic equations being satis?ed.This equation has since been re-ferred to as the Yang-Baxter equation(YBE)after the name was coined by Takhtadzhan and Faddeev(1979).Baxter’s contribution was to independently show that such relations also appear as conditions for commuting transfer matrices in two-dimensional lattice models in statistical mechanics (Baxter,1972a,1982).Moreover,the YBE was seen as a relation which can be solved to obtain new exactly solved models.The YBE thus became celebrated as the master key to integrability(Au-Yang and Perk,1989).

The study of Yang-Baxter integrable models?ourished in the1970s,1980s,and1990s in the Canberra,St.Petersburg, Stony Brook,and Kyoto schools,with far reaching implica-tions in both physics and mathematics.During this period the YBE emerged as the underlying structure behind the solvability of a number of quantum mechanical models.In addition to the XXZ spin chain,examples include the XYZ spin chain(Baxter,1972b),the t-J model at supersymmetric coupling(Essler and Korepin,1992;Foerster and Karowski, 1993a,1993b)and the Hubbard model(Lieb and Wu,1968; Shiba,1972;Shastry,1986a,1986b;Kawakami,Usuki,and Okiji,1989;Frahm and Korepin,1990,1991;Ogata and Shiba,1990;Essler et al.,2005).Three collections of key papers have been published(Jimbo,1990;Mattis,1993; Korepin and Essler,1994).

Further examples are strongly correlated electron systems (Tsvelik,1995;Takahashi,1999;Giamarchi,2004; Schollwo¨ck,2004),spin-exchange interaction(Montorsi, 1992;Sutherland,2004;Essler et al.,2005),Kondo physics of quantum impurities coupled to conduction electrons in equilibrium(Andrei,Furuya,and Lowenstein,1983; Tsvelik and Wiegmann,1983)and out of equilibrium (Mehta and Andrei,2006;Doyon,2007;Nishino and Hatano,2007;Nishino,Imamura,and Hatano,2009),the BCS model(Richardson,1963a,1963b,1965;Richardson and Sherman,1964;Cambiaggio,Rivas,and Saraceno,1997; von Delft and Ralph,2001;Links et al.,2003;Dukelsky, Pittel,and Sierra,2004;Dunning and Links,2004),models with long-range interactions(Calogero,1969;Sutherland, 1971;Gaudin,1976;Haldane,1988;Shastry,1988),two Josephson coupled BECs(Zhou et al.,2002;Zhou,Links, McKenzie,and Guan,2003),a BCS-to-BEC crossover(Ortiz and Dukelsky,2005),atomic-molecular BECs(Zhou,Links, Gould,and Mckenzie,2003;Foerster and Ragoucy,2007), and quantum degenerate gases of ultracold atoms(Korepin, Bogoliubov,and Izergin,1993;Pethick and Smith,2008; Yurovsky,Olshanii,and Weiss,2008;Cazalilla et al.,2011).

A signi?cant development in the theory of quantum inte-grable systems is the algebraic BA(Sklyanin,Takhtadzhyan, and Faddeev,1979;Kulish and Sklyanin,1982;Faddeev, 1984),essential to the so-called quantum inverse scattering method(QISM),a quantized version of the classical inverse scattering method.The QISM gives a uni?ed description of the exact solution of quantum integrable models.It provides a framework to systematically construct and solve quantum many-body systems(Thacker,1981;Korepin,Bogoliubov, and Izergin,1993;Takahashi,1999;Essler et al.,2005). Other related threads are the quantum transfer matrix (QTM)(Suzuki,1985;Destri and de Vega,1992;Klu¨mper, 1992)and T systems(Kuniba,Nakanishi,and Suzuki,1994a, 1994b,2011)from which one can derive temperature-dependent properties in an exact nonperturbative fashion. Applications of this approach include the Heisenberg model (Shiroishi and Takahashi,2002),higher-spin chains(Tsuboi, 2003,2004),and integrable quantum spin ladders(Batchelor et al.,2003,2004,2007;Batchelor,Guan,and Oelkers,2004). T systems and integrability in general also play a fundamental role in the gauge and string theories of high energy physics (Kuniba,Nakanishi,and Suzuki,2011;Beisert et al.,2012). Yang-Baxter integrability has also played a crucial role in initiating and inspiring progress in mathematics,particularly to the theory of knots,links,and braids(Jones,1985; Kauffman,1987;Wadati,Deguchi,and Akutsu,1989;Wu, 1992;Yang and Ge,2006)and the development of quantum groups and representation theory(Chari and Pressley,1994; Go′mez,Ruiz-Altaba,and Sierra,1996).

2.Fermions in1D:A historical overview

In the mid-1960s many physicists worked on extending the results obtained by Lieb and Liniger(1963)and McGuire (1964)for1D bosons with delta-function interaction to the problem of1D fermions.McGuire(1965,1966)solved the eigenvalue problem of Nà1fermions of the same spin and one fermion of opposite spin and studied the low-lying excited states with repulsive and attractive potentials.The dynamics of this one spin-down Fermi problem has been studied(McGuire, 1990).The problem of Nà2fermions of the same spin with two fermions of opposite spin was solved by Flicker and Lieb (1967).A further step came when Gaudin(1967a,1967b)and Yang(1967)solved the general problem in terms of a nested BA for arbitrary spin population imbalance.1Gaudin derived the ground state energy for the balanced(fully paired)case for attractive interaction,pointing out that the result is equiva-lent to that for repulsive bosons(Lieb and Liniger,1963). 1Missing phase factors for the spin sector in Eq.(4c)of Gaudin (1967a)were corrected in Gaudin(1967b).A thorough treatment of the1D Fermi problem can be found in Gaudin’s book on the Bethe wave function(Gaudin,1983).

Xi-Wen Guan,Murray T.Batchelor,and Chaohong Lee:Fermi gases in one dimension:From Bethe (1635)

Rev.Mod.Phys.,V ol.85,No.4,October–December2013

The delta-function interacting two-component Fermi gas is commonly referred to as the Gaudin-Yang model.

Yang’s concise solution of the problem had a profound impact.As already remarked,a key point in the solution is that the matrix operators describing many-body scattering can be factorized into two-body scattering matrices,provided that a set of cubic equations—the Yang-Baxter equation—are satis?ed by the two-body scattering matrices.This in turn is equivalent to no diffraction in the outgoing waves in three-body scattering processes.In this sense Yang’s solution com-pletes McGuire’s formulation of the scattering process in the context of the1D Bose gas.Indeed,the R matrix obtained for the1D Bose gas is known as the simplest nontrivial solution of the Yang-Baxter equation(Jimbo,1989).

The solution of the1D Fermi problem triggered a series of further breakthroughs.Yang(1968)obtained the S matrix of the delta-function interacting many-body problem for Boltzmann statistics(Gu and Yang,1989).The exact solution of the1D Fermi gas with higher-spin symmetry was obtained by Sutherland(1968,1975).The1D Hubbard model solved by Lieb and Wu(1968)is a fundamental model in the theory of strongly correlated electron systems.Its solution is a signi?cant example of the factorization condition(the YBE)in which the quasimomenta of particles k are replaced by sin k.The Lieb-Wu solution thus gives a similar set of integral equations as Yang’s Fredholm equations for the continuum gas.This exactly solved model has been extensively studied in the literature.The exact results for the Hubbard model not only provide the essential physics of1D strongly correlated electronic systems(Ha,1996; Takahashi,1999;Essler et al.,2005),but also are relevant to phenomena in high T c superconductivity.Indeed,the1D Hubbard model is an archetypical many-body system featuring Fulde-Ferrel-Larkin-Ovchinnikov(FFLO)pairing,universal Tomonaga-Luttinger liquid(TLL)physics,spin-charge sepa-ration,and quantum entanglement(Gu et al.,2004;Essler et al.,2005;Larsson and Johannesson,2005).

Although further study(Takahashi,1970b;Yang,1970)of the1D Fermi gas was initiated soon after its solution,it was not until much later that this model began to receive more attention(Astrakharchik et al.,2004;Fuchs,Recati,and Zwerger,2004;Tokatly,2004;Iida and Wadati,2005; Batchelor et al.,2006a)as a result of the brilliant experimen-tal progress in ultracold-atom physics.The fundamental physics of the model is determined by the set of generalized Fredholm integral equations obtained in the thermodynamic limit.Takahashi(1970a)discussed the analyticity of the Fredholm equations in the vanishing interaction limit.A thorough study of the Fredholm equations for the Gaudin-Yang model with attractive and repulsive interactions was carried out(Guan et al.,2007;Iida and Wadati,2007,2008; Wadati and Iida,2007;Guan and Ma,2012;Zhou,Xu,and Ma,2012).The numerical solution of the Fredholm equations has also been discussed in the context of harmonic traps(Hu, Liu,and Drummond,2007;Orso,2007;Colome′-Tatche′, 2008;Kakashvili and Bolech,2009;Ma and Yang,2009, 2010a,2010b).In particular,the eigenfunction has been obtained explicitly for the Fermi gas in the in?nitely strong repulsion limit by using the hard-core contact boundary condition(Girardeau,1960)and group theoretical methods (Guan et al.,2009;Ma et al.,2009).

The next major advance with implications for the1D Fermi problem was the solution of the?nite temperature problem for1D bosons.Yang and Yang(1969)showed that the thermodynamics of the Lieb-Liniger Bose gas can be determined from the minimization conditions of the Gibbs free energy subject to the BA equations.Takahashi went on to make signi?cant contributions to Yang and Yang’s grand canonical approach to the thermodynamics of1D integrable models(Takahashi,1971a,1971b,1972,1973,1974; Takahashi and Suzuki,1972).

Takahashi gave the general name of thermodynamic Bethe ansatz(TBA)equations to the Yang-Yang type of equations for the thermodynamics.He discovered spin-string patterns of the BA equations in addition to those for the ground state of the spin chain(Takahashi,1971a).Using a similar spin-string hypothesis,Gaudin(1971)studied the thermodynamics of the Heisenberg-Ising https://www.wendangku.net/doc/3b2573010.html,i(1971,1973)independently derived the TBA equations for spin-1=2fermions in the repul-sive regime.It turns out that Takahashi’s spin-string hypothesis allows one to study the grand canonical ensemble for many1D many-body systems with internal degrees of freedom,e.g.,the 1D Fermi gas(Takahashi,1971b),the1D Hubbard model (Takahashi,1972,1974;Usuki,Kawakami,and Okiji,1990), the quantum sine-Gordon model(Fowler and Zotos,1981),and the Kondo problem(Filyov,Tsvelick,and Wiegmann,1981; Lowenstein,1981)among many other integrable models. Building on Takahashi’s spin-string hypothesis,Schlottmann derived the TBA equations for SUeNTfermions with repulsive and attractive interactions(Schlottmann,1993,1994).The Yang-Yang method has been revealed to be an elegant way to analyti-cally access not only the thermodynamics,but also correlation functions,quantum criticality,and TLL physics for a wide range of low-dimensional quantum many-body systems(Ha,1996; Takahashi,1999;Essler et al.,2005).The Yang-Yang thermo-dynamics of the1D Bose gas has been tested in recent experi-ments(van Amerongen et al.,2008;Armijo et al.,2010,2011; Kru¨ger et al.,2010;Stimming et al.,2010;Armijo,2011; Jacqmin et al.,2011;Sagi et al.,2012).

Recently,numerical schemes have been developed to solve the TBA equations of the1D two-component spinor Bose gas with delta-function interaction(Caux,Klauser,and van den Brink,2009;Klauser and Caux,2011).The QTM method has also been applied to the thermodynamics of the1D Bose and Fermi gases with repulsive delta-function interaction (Klu¨mper and Patu,2011;Patu and Klu¨mper,2013).The Canberra group and their collaborators(Zhao et al.,2009; Guan et al.,2010;He et al.,2010,2011;Guan and Batchelor, 2011;Guan and Ho,2011)developed an asymptotic method to calculate the thermodynamics of strongly interacting bosons and fermions in an analytic fashion using the polylog function in the framework of the Yang-Yang and Takahashi methods.This approach does away with the need to numerically solve the TBA equations for these systems at quantum criticality,where the temperature is very low and the interparticle interaction is strong.

B.Renewed interest in1D fermions

The renewed interest over the past decade in1D fermions has been on a number of related fronts.Here we give a brief introductory outline of these developments.

1636Xi-Wen Guan,Murray T.Batchelor,and Chaohong Lee:Fermi gases in one dimension:From Bethe... Rev.Mod.Phys.,V ol.85,No.4,October–December2013

1.Novel BCS-pairing states

Quantum matter at low temperatures has already been seen to exhibit some remarkable physical properties,such as BEC and super?uidity.Fermionic quantum matter with mis-matched Fermi surfaces has long been expected to exhibit more exotic behavior than seen in conventional materials. The two-component attractive Fermi gas is particularly inter-esting due to its connection with the exotic pairing phase (the FFLO state)involving BCS pairs with nonzero center-of-mass momenta.In this phase,where the system is partially polarized,the Fermi energies of spin-up and spin-down elec-trons become unequal.Originally,Fulde and Ferrell(1964) discovered that,under a strong external?eld,superconduct-ing electron pairs have nonzero pairing momentum and spin https://www.wendangku.net/doc/3b2573010.html,rkin and Ovchinnikov(1965)found that the formation of pairs of electrons with different momenta,i.e.,~k andà~kt~q with nonzero~q,is energetically favored over pairs of electrons with opposite momenta,i.e.,~k andà~k, when the separation between Fermi surfaces is suf?ciently large.Consequently,the density of spins and the supercon-ducting order parameter become periodic functions of the spatial coordinates.

Theoretical study of the FFLO state in1D interacting fermions was initiated by Yang(2001),who used bosoniza-tion to study the pairing correlations.The FFLO-like pair correlations and spin correlations for the attractive Hubbard model were later investigated numerically by two groups (Feiguin and Heidrich-Meisner,2007;Tezuka and Ueda, 2008).Both groups showed the power-law decay of the form n pair/cosek FFLO j x jT=j x j for the pair correlation,with spatial oscillations depending solely on the mismatch k FFLO? en"àn#Tof the Fermi surfaces.Thus the momen-tum pair distribution has peaks at the mismatch of the Fermi surfaces.The FFLO state has since been studied by various methods:density-matrix renormalization group(DMRG) (Lu¨scher,Noack,and La¨uchli,2008;Rizzi et al.,2008; Tezuka and Ueda,2010),quantum Monte Carlo(QMC) (Batrouni et al.,2008;Baur,Shumway,and Mueller,2010; Wolak et al.,2010),mean-?eld theory,and other methods (Liu,Hu,and Drummond,2007,2008b;Parish et al.,2007; Zhao and Liu,2008;Datta,2009;Edge and Cooper,2009, 2010;Pei,Dukelsky,and Nazarewicz,2010;Devreese, Klimin,and Tempere,2011;Kajala,Massel,and To¨rma¨, 2011;Chen and Gao,2012).

Recently the asymptotic correlation functions and FFLO signature were analytically studied using the dressed charge formalism in the context of the Gaudin-Yang model(Lee and Guan,2011;Schlottmann and Zvyagin,2012b).However, convincing theoretical proof for the existence of the1D FFLO state in the expansion dynamics of the1D polarized Fermi gas after its sudden release from the longitudinal con?ning potential is still rather elusive;see recent further developments by Bolech et al.(2012),Dalmonte et al.(2012), and Lu et al.(2012).So far the spatial oscillation nature of FFLO pairing has not been experimentally con?rmed.

https://www.wendangku.net/doc/3b2573010.html,rge-spin ultracold atomic fermions

It was shown(Ho and Yip,1999;Yip and Ho,1999)that large-spin atomic fermions exhibit rich pairing structures and collective modes in low-energy physics.Further progress toward understanding many-body physics with large-spin Fermi gases was made(Wu,Hu,and Zhang,2003;Wu, 2005)on spin-3=2systems which can be realized with 132Cs,9Be,and135Ba ultracold atoms(Wu,2006).Such systems exhibit a generic SO(5)symmetry[isomorphically, Spe4Tsymmetry].The spin-3=2system with SU(4)symmetry can exhibit a quartet state(four-body bound state).More generally,ultracold atoms offer an exciting opportunity to investigate spin-liquid behavior via trapped fermionic atoms with large-spin symmetry(Honerkamp and Hofstetter,2004; Zhao,Ueda,and Wang,2006;Zhou and Semenoff,2006; Cherng,Refael,and Demler,2007;Rapp et al.,2007;Tu, Zhang,and Yu,2007;Zhai,2007;Corboz et al.,2011; Szirmai and Lewenstein,2011;Krauser et al.,2012).The trimer state(‘‘trions’’)consisting of fermionic6Li atoms in the three energetically lowest substates has been reported (Huckans et al.,2009;Williams et al.,2009;Lompe et al., 2010).

On the other hand,fermionic alkaline-earth atoms display an exact SUeNTspin symmetry with N?2It1,where I is the nuclear spin(Cazalilla,Ho,and Ueda,2009;Gorshkov et al.,2010;Xu,2010).Such fermionic systems with enlarged SUeNTspin symmetry are expected to display a remarkable diversity of new quantum phases and quantum critical phe-nomena due to the existence of multiple charge bound states. De Salvo et al.(2010)have reported quantum degeneracy in a gas of ultracold fermionic87Sr atoms with I?9=2in an optical dipole trap.An experiment by Taie et al.(2010) dramatically realized the model of fermionic atoms with SUe2T SUe6Tsymmetry where electron spin decouples from its nuclear spin I?5=2for173Yb together with atoms of its spin-1=2isotope.This group also successfully realized the SU(6)Mott-insulator state with ultracold fermions of 173Yb atoms in a3D optical lattice(Taie et al.,2012).

In the context of large-spin ultracold atomic fermions, Lecheminant,Boulat,and Azaria(2005)considered1D ultra-cold atomic systems of fermions with general half-integer spins.The instabilities of the BCS-pairing phase and molecu-lar super?uid phase in these systems have been studied by a low-energy approach.The low-energy physics and competing orders in large-spin fermionic systems in a1D lattice were further investigated(Capponi et al.,2007;Azaria,Capponi, and Lecheminant,2009;Nonne et al.,2010,2011).In this scenario,a new class of integrable models of ultracold fermi-ons and bosons with large-spin symmetries was found(Cao, Jiang,and Wang,2007;Jiang,Cao,and Wang,2009,2011). They derived the BA solutions for spin-3=2fermions with SO (5)symmetry and the Spe2st1T-invariant model of fermions. From the integrable model perspective,the study of mul-ticomponent attractive fermions was initiated long ago by Takahashi(1970b)and Yang(1970).In light of ultracold higher-spin atoms,Controzzi and Tsvelik(2006)proposed an exact solution of a model describing the low-energy physics of spin-3=2fermionic atoms in a1D lattice.The exact results obtained from1D many-body systems with higher-spin symmetries provided insight into understanding the few-body physics of trions(Guan et al.,2008;Liu,Hu, and Drummond,2008a;He et al.,2010),quartet states(four-body charge bound states)(Guan et al.,2009;Schlottmann

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and Zvyagin,2012a,2012b,2012c),and an arbitrary large-spin-neutral bound state of different sizes(Schlottmann, 1993,1994;Guan et al.,2010;Lee and Guan,2011;Yang and You,2011;Yin,Guan,Batchelor,and Chen,2011).The study of critical phenomena and universal TLL physics in low-dimensional ultracold atomic Fermi gases with large pseudospin symmetries is a rapidly developing frontier in ultracold-atom physics.

3.Quantum criticality of ultracold atoms

Quantum criticality describes a V-shaped phase of quan-tum critical matter fanning out to?nite temperatures from the quantum critical point(QCP).It is associated with competi-tion between the two distinct ground states near the QCP. Near a QCP,the quantum critical behavior is characterized by the energy gapá$ àz and a diverging length scale $j à c jà ,where c is the critical value of the driving parameter .The universality class of quantum criticality is characterized by the dynamic critical exponent z and the correlation exponent (Wilson,1975;Fisher et al.,1989; Sachdev,1999).The many-body system is expected to show universal scaling behavior in the thermodynamic quantities at quantum criticality due to the collective nature of many-body effects.Thus a universal and scale-invariant description of the system is expected through the power-law scaling of thermo-dynamic properties.However,understanding the various as-pects of quantum criticality in quantum systems represents a major challenge to our knowledge of many-body physics (Sondhi et al.,1997;V ojta,2003;Coleman and Scho?eld, 2005;Lo¨hneysen et al.,2007;Gegenwart,Si,and Steglich, 2008;Sachdev and Keimer,2011).

Ultracold atoms have become the tool of choice to simulate and test universal quantum critical phenomena.The study of quantum criticality and?nite-size scaling in trapped atomic systems is thus attracting considerable interest(Kato et al., 2008;Campostrini and Vicari,2009,2010a,2010b;Pollet, Prokof’ev,and Svistunov,2010;S.Fang et al.,2011;Hazzard and Mueller,2011;Ceccarelli,Torrero,and Vicari,2012). The experimental study of critical behavior in a trapped Bose gas was initiated by Donner et al.(2007).In particular, signi?cant experimental progress on quantum criticality and quantum phase transitions in2D Bose atomic gases has been made(Gemelke et al.,2009;Huang et al.,2010,2011; Huang,Zhang,Gemelke,and Chin,2011;Zhang et al., 2011,2012).

Some of the remarkable features of criticality in general are the notions of universality class and https://www.wendangku.net/doc/3b2573010.html,ing integrable quantum?eld theory,Zamolodichikov(1989)was able to show that the2D Ising model in a magnetic?eld,or equivalently the quantum Ising chain with a transverse?eld (Henkel and Saleur,1989),displays E8symmetry close to the critical point.Such exotic quantum symmetry in the excitation spectrum was observed in a recent experiment in the traditional setting of condensed matter physics(Coldea et al.,2010).

Zhou and Ho(2010)proposed a precise theoretical scheme for mapping out quantum criticality of ultracold atoms.In this framework,exactly solvable models of ultracold atoms, exhibiting quantum phase transitions,provide a rigorous way to explore quantum criticality in many-body systems.The equation of state has been obtained for a number of key integrable models,allowing the exploration of TLL physics and quantum criticality.These include the Gaudin-Yang Fermi gas(Zhao et al.,2009;Guan and Ho,2011;Yin, Guan,Batchelor,and Chen,2011),the Lieb-Liniger Bose gas(Guan and Batchelor,2011),the Fermi-Bose mixture (Yin,Guan,Chen,and Batchelor,2011),and the spin-1spinor Bose gas with antiferromagnetic spin–spin exchange interac-tion(Kuhn et al.,2012a,2012b).The exact results for the scaling forms of thermodynamic properties in these systems near the critical point illustrate the physical origin of quantum criticality in many-body systems.

4.Experiments with ultracold atoms in1D

Many remarkable1D quantum phenomena have been experimentally observed due to recent rapid progress in material synthesis and tunable manipulation of ultracold atoms.These developments have provided a better under-standing of signi?cant quantum statistical effects and strong correlation effects in low-dimensional quantum many-body systems.The observed results to date are seen to be in excellent agreement with results obtained using the mathe-matical methods and analysis of exactly solved models. These include the experimental realization of the Tonks-Girardeau gas(Kinoshita,Wenger,and Weiss,2004; Paredes et al.,2004)and a quantum Newton’s cradle,i.e.,a demonstration of out-of-equilibrium physics in arrays of trapped1D Bose gases(Kinoshita,Wenger,and Weiss, 2006)and quantum correlations(Tolra et al.,2004; Kinoshita,Wenger,and Weiss,2005;Betz et al.,2011; Endres et al.,2011;Haller et al.,2011;Guarrera et al., 2012).Haller et al.(2009)made a further experimental breakthrough by realizing a stable highly excited gaslike phase,called the super Tonks-Girardeau gas,in the strongly attractive regime of bosonic cesium atoms.

The Yang-Yang thermodynamics and thermal?uctuations of an ultracold Bose gas of87Rb atoms were further tested in a series of publications(van Amerongen et al.,2008;Armijo et al.,2010,2011;Kru¨ger et al.,2010;Stimming et al.,2010; Armijo,2011;Jacqmin et al.,2011,2012;Sagi et al.,2012). The universal low-energy physics was demonstrated as host-ing a TLL(Haller,Hart et al.,2010;Blumenstein et al.,2011). The experimental research using ultracold Fermi gases to explore pairing phenomena in a1D Fermi gas was?rst reported by Moritz et al.(2005).In a major breakthrough toward understanding the exotic pairing signature and quan-tum phase diagram of the attractive Fermi gas,Liao et al. (2010)measured the?nite temperature density pro?les of trapped fermionic6Li atoms;see Fig.1.They con?rmed the key features of the T?0phase diagram predicted from the exact solution(Batchelor et al.,2006a;Feiguin and Heidrich-Meisner,2007;Guan et al.,2007;Hu,Liu,and Drummond, 2007;Iida and Wadati,2007;Orso,2007;Parish et al.,2007; Kakashvili and Bolech,2009).

C.Outline of this review

In light of these recent developments we review the BA solution of the Gaudin-Yang model in Sec.II and discuss the physical understanding of the solution in terms of BCS

1638Xi-Wen Guan,Murray T.Batchelor,and Chaohong Lee:Fermi gases in one dimension:From Bethe... Rev.Mod.Phys.,V ol.85,No.4,October–December2013

pairing,the polaron problem,molecule states,and the super Tonks-Girardeau gas.In Sec.III we further discuss many-body phenomena in the Gaudin-Yang model.Especially,we discuss1D fermions in a harmonic trap and review the universal features of1D interaction,including magnetism, FFLO-like pairing,TLL physics,spin-charge separation,uni-versal thermodynamics,and quantum criticality.In Sec.IV, we review recent progress on mixtures of ultracold atoms and the exact solution of the1D Fermi-Bose mixture.

Section V reviews the exotic many-body physics of1D multicomponent interacting fermions,including the three-component Fermi gas,the SUeNTinvariant Fermi gases, and spin-3=2fermions with SO(5)symmetry.The discussion in this section covers magnetism for systems of large-spin fermions,trions,molecular states of different sizes,multi-component TLL phases,universal low-temperature thermo-dynamics,and critical behavior caused by population imbalance.In Sec.VI,we focus on the asymptotics of various relevant correlation functions for the Gaudin-Yang model and multicomponent Fermi gases.The characteristics of the FFLO-like pairing correlations and spin-charge separation correlation functions are discussed in the framework of conformal?eld theory(CFT).

The experimental breakthroughs with quasi-1D ultracold atoms and tests of1D many-body physics are reviewed in Sec.VII.A brief conclusion and an outlook on future developments are given in Sec.VIII.

II.THE GAUDIN-YANG MODEL

The Hamiltonian

?#;"Z

y exT

à tVexT

exTdx

tg1D Z

y#exT y"exT "exT #exTdx

à1

? y

exT "exTà y

exT #exT dx(1)

describes a1D function interacting two-component (spin-1=2)Fermi gas of N fermions with mass m and an external magnetic?eld H constrained by periodic boundary conditions to a line of length L.The function VexTis the trapping potential.The?eld operators #and "describe the fermionic atoms in the states j#i and j"i,respectively.The -type interaction between fermions with opposite hyper?ne states preserves the spin states such that the Zeeman term in the Hamiltonian(1)is a conserved quantity.

The experimental realization(Moritz et al.,2005;Liao et al.,2010)of this system of interacting fermions is de-scribed in Sec.VI.The coupling constant g1D??2c=m, where c?à2=a1D can be tuned by Feshbach resonance (Olshanii,1998;Bergeman,Moore,and Olshanii,2003). For repulsive interaction c>0and for attractive interaction c<0.Where appropriate,we use units of??2m?1.

A dimensionless coupling constant ?c=n is used to char-acterize physical regimes,i.e., )1for the strong coupling regime and (1for the weak coupling regime.Here n is the linear number density.A.Bethe ansatz solution

For a homogeneous gas,i.e.,VexT?0,the eigenvalue problem for Hamiltonian(1)reduces to the1D N-body delta-function interaction problem

H?à

X N

i?1

@x i

tg1D

ex iàx jT(2)

solved by Gaudin and Yang.Bethe’s hypothesis states that the wave function of such a many-body system is a superpo-sition of plane waves.In the domain0

?P;Q exp iek P1x Q1tááátk PN x QNT;(3)

where both P?P1;...;P N and Q?Q1;...;Q N are permu-tations of the integers f1;2;...;N g.The sum runs over all N! permutations P.The N!?N!coef?cients?P;Q of the exponentials can be arranged as an N!?N!matrix.The columns are denoted by N!?N!dimensional vectors P

;...;P N

(Yang,1967;Takahashi,1999).For example,for two fermions with one spin up and one spin down,the wave function is written as

c? 12e?12;12 e iek1x1tk2x2Tt?21;12 e iek2x1tk1x2T

t 21e?12;21 e iek2x1tk1x2Tt?21;21 e iek2x2tk1x1T; where ij denotes the step function ijex jàx iT.A plane wave

repeatedly re?ected from the hyperplanes x Q

?x Q

gives a total of N!plane waves.The idea in setting up such an ansatz is an attempt at a hypothetical solution followed by demon-strating that it gives the eigenfunction of the many-body problem,rather than solving the problem directly.

The derivative of the wave function is discontinuous when two particles are in?nitesimally close to one another. This property can be derived by considering the eigenvalue problem H c?E c in the center of mass coordinate X?ex Q

tx Q

T=2and the relative coordinate Y?x Q

àx Q

of the two adjacent particles involved.This discontinuity in the ?rst derivative of the wave function and the continuity of the wave function at x Q

?x Q

give a two-body scattering relation between the adjacent vector coef?cients áááijááá?Y ij P

P i

ááájiááá.

The matrix operator Y ab ij is de?ned by

Y ab ij?

àiek iàk jTP abtcI

iek iàk jTàc

;(4)

where I is the identity and P ab is the permutation operator acting on the vector áááijááá.Because of the Fermi statistics, P ab?à1for all a and b.Yang denoted the unequal indices a,b,and c in the three particle scattering process as the interchanges with coordinates x a,x b,and x c under the per-mutation Q a,Q b,and Q c.The consistency condition for factorizing the many-body scattering matrix into the product of two-body scattering matrices Y ab ij leads to the celebrated YBE:

Y ab

Y bc

Y ab ij?Y bc ij Y ab ik Y bc jk;(5)

Xi-Wen Guan,Murray T.Batchelor,and Chaohong Lee:Fermi gases in one dimension:From Bethe (1639)

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where Y ab ij Y ab ji?1.De?ning the R matrix by R ij?P ij Y ij ij and spectral parameters u?k2àk1and v?k3àk2the YBE is often written in the form

R12euTR23eutvTR12evT?R23evTR12eutvTR23euT:(6) Returning to solving the problem of N particles in a periodic box of length L,the second step is to apply the periodic boundary condition cex1;...;x i;...;x NT?

cex1;...;x itL;...;x NTon the wave function with period L for every1i N.The two-column Young tableau ?2N#;1N"àN# (Yang,1967;Oelkers et al.,2006)encodes the spin symmetry,where N"and N#are the numbers of fermions in the hyper?ne levels j"i and j#i such that N"!N#.This gives the second eigenvalue problem

R iek iTA EeP j QT?expeik i LTA EeP j QT;(7) where A EeP j QTis an abbreviation of the amplitude of the wave function which provides the eigenvector of the N operators R i with i?1;...;N:

R iek iT?R it1;iek it1àk iTáááR N;iek Nàk iT

?R1;iek1àk iTáááR ià1;iek ià1àk iT:(8) Using Bethe’s hypothesis again,Yang solved the eigen-value problem(7)by the ansatz

A EeP j QT?X

áááP M

Fe P

;y1TáááFe P

;y MT;(9)

where y1

Fe ;yT?

Y yà1

j?1k jà tic0 jt1

à àic0

By the symmetry of the Young tableau?2N#;1N"àN# ,the vector A E describes a spin system with a number of N#spins on an N-site lattice.

The generalized ansatz(9)plays an important role in solving multicomponent many-body problems(Sutherland, 1968).Alternatively,the eigenvalue problem(7)can be worked out in a straightforward way in terms of the QISM, where the operator R iek iTcan be written in terms of the quantum transfer matrix(Korepin,Bogoliubov,and Izergin, 1993;Ma,1993;Li et al.,2003;Oelkers et al.,2006;Jiang, Cao,and Wang,2009).This approach was introduced in the study of the1D Hubbard model(Ramos and Martins,1997; Martins and Ramos,1998;Essler et al.,2005).

The energy eigenspectrum is given in terms of the quasimomenta f k i g of the fermions via

E??2

X N

j?1

k2j(10)

subject to the BA equations which in terms of the function

e bexT?xtibc=2 xàibc=2

are

expeik i LT?

Y N#

e1ek ià T;

Y N

j?1

e1e àk jT?à

Y N#

e2e à T;

(11)

for i?1;2;...;N and ?1;2;...;N#.All wave numbers k i are distinct and uniquely de?ne the wave function(3) (Gu and Yang,1989).

The fundamental physics of the model is determined by the BA equations(11).For repulsive interaction,the quasimo-menta f k i g are real,but the rapidities f g are real only for the ground state.The complex roots are the spin strings for excited states.In the thermodynamic limit,i.e.,L,N!1, where N=L is?nite,the BA equations(11)can be written as generalized Fredholm equations

r1ekT?

Z B

àB2

K1ekàk0Tr2ek0Tdk0;

r2ekT?

Z B

àB1

K1ekàk0Tr1ek0Tdk

Z B

àB2

K2ekàk0Tr2ek0Tdk0;(12)

where the integration boundaries B1and B2are determined by n?N=L?

R B

àB1

r1ekTdk,n#?N#=L?

R B

àB2

r2ek0Tdk0. In the above equations,the kernel function

K‘exT?

1‘c

e‘c=2T2tx2

The functions r mekTdenote the Bethe root distributions,with r1ekTthe quasimomenta distribution function and r2ekTthe spin rapidity parameter distribution function.The ground state energy per unit length is given by E?

R B

àB1

k2r1ekTdk. For the attractive regime,the quasimomenta f k i g of fermi-ons with different spins form two-body bound states,i.e.,the wave numbers are complex with k i? 0i?i c=2in the thermodynamic limit(Yang,1970;Takahashi,1971b).Here i?1;...;N#.The excess fermions have real quasimomenta f k j g with j?1;...;Nà2N#.In the thermodynamic limit, the density of unpaired fermions 1ekTand the density of pairs 2ekTsatisfy the Fredholm equations

1ekT?

Z A

àA2

K1ekàk0T 2ek0Tdk0;

2ekT?

Z A

àA1

K1ekàk0T 1ek0Tdk0

Z A

àA2

K2ekàk0T 2ek0Tdk0:(13)

The distribution 2ekTcoincides with the distribution function of the real parts of the bound states.The linear densities are de?ned by N=L?2

R A

àA2

2ekTdkt

R A

àA1

1ekTdk and N#=L?R A

àA2

2ekTdk.The ground state energy per unit length is given by E?

R A

àA2

e2k2àc2=2T 2ekTdkt

R A

àA1

k2 1ekTdk. In the context of magnetism,the magnetization per unit length is de?ned by M z?enà2n#T=2.By de?nition,the ground state energy can be expressed as a function of total particle density n and magnetization M z.In the grand

1640Xi-Wen Guan,Murray T.Batchelor,and Chaohong Lee:Fermi gases in one dimension:From Bethe... Rev.Mod.Phys.,V ol.85,No.4,October–December2013

canonical ensemble,the magnetic ?eld H and the chemical potential can be obtained via

H ?2@E en;M z Tz ; ?

@E en;M z T

;(14)which have been used to work out the phase diagrams of the attractive Fermi gas (Hu,Liu,and Drummond,2007;Orso,

2007)and the repulsive Fermi gas (Colome

′-Tatche ′,2008;Guan and Ma,2012).We now turn to extracting such infor-mation from both the discrete and continuum versions of the BA solution.

B.Solutions to the discrete Bethe ansatz equations

The 1D Fermi gas (1)with spin population imbalance exhibits an unconventional pairing order that presents a major subtlety of many-body correlations in the Gaudin-Yang model.Starting with the discrete BA equations (11),we review how the exact solution enables us to precisely understand such subtle many-body physics driven by the interaction of quan-tum statistics and dynamics.In particular,we see that for the weakly and strongly attractive coupling regimes 1D interact-ing fermions give signi?cantly different phenomena:weakly bound BCS-like pairs versus tightly bound molecules.

1.BCS-like pairing and tightly bound molecules

For weakly attractive interaction,i.e.,L j c j (1,two fer-mions with spin up and spin down form a weakly bound pair

with a small binding energy b ?à?2j c j =mL ,where the two-body binding energy is less than the kinetic energy (Batchelor et al.,2006a ).The complex conjugate pair leads to an exponential decay of the wave function with a factor e àj c jj x i àx j j =2.Thus the balanced case has a BCS-like fully paired state where the size of the Cooper pairs is much larger than the mean average distance between the fermions.In this weakly attractive regime,the energy gap separating the ?rst triplet excited state from the ground state is found to have

an asymptotic behavior á%2n 2??????????

j j p exp eà 2=2j jTas j j !0(Krivnov and Ovchinnikov,1975;Fuchs,Recati,and Zwerger,2004).In fact,the BA equations (11)for weak attraction give an explicit relation H %e?2n 2=2m T?e2 2m z t4j j m z Tbetween the external ?eld and magnetiza-tion in the thermodynamic limit.The lower critical ?eld gives the energy gap at m z ?0.This relation indicates a vanishing energy gap á?H c !0for !0.Here the magnetization is de?ned by m z :?M z =n ?eN "àN #T=2N (Iida and Wadati,2007;He et al.,2009).

For a polarized gas with weak attraction,Fermi statistics lead to segmentation in quasimomentum space,i.e.,the excess fermions are located at the two outer wings in quasi-momentum space;see Fig.2.For a ?nite-size system with arbitrary polarization P ?eN "àN #T=N ,the BA equations (11)determine N #weakly bound BCS pairs with k p % ?i ????????????

j c j =L p and N à2N #unpaired fermions with real k i (Batchelor et al.,2006b ).In this case,f g and f k j g are symmetrically distributed around zero in the quasimomentum parameter space;see Fig.2.

Assuming that N #is odd and N is even,the ?rst few leading orders of the positive roots f g and f k j g are determined by the equations

k j %2n j L tc Lk j tc

eM #à1T=2 ?1

2k j

k j ; %2n L t3c 2L tc L X eM #à1T=2 ?1 T

tc 2L

eN à2M #T=2j ?1

2 àk 2

;(15)

where n j ?eM #t1T=2;eM t3T=2;...;eN àM #à1T=2,and n ?1;2;...;M #=2.This case ? is excluded in Eq.(15).The root patterns reveal the cooperative nature of many-body effects,i.e.,an individual quasimomentum depends on that of all the particles.Here the momenta of unpaired fermions and bound pairs depend on the scattering energies between pair and between paired and unpaired fermions.This indicates that the quantum statistics of the weakly interacting fermions is mutual according to exclusion statis-tics (Haldane,1991).From Eq.(15),the ground state energy per unit length is given by (Batchelor et al.,2006b )

E L ?13n 3" 2t13

n 3# 2

t2cn "n #tO ec 2T:(16)

Here the ground state energy (16)is also valid for weakly repulsive interaction,i.e.,for Lc (1.This leading order correction to the interaction energy indicates a mean-?eld effect.

On the other hand,for strong attraction,i.e.,L j c j )1(or c )k F ),the discrete BA equations (11)give the root

patterns k b i % i ?i 12c for bound pairs and real k u

j for un-paired fermions (Yang,1970;Takahashi,1971b ).Here i ?1;...;N #and j ?1;...;N 1,the number of unpaired fermions N 1?N à2N #.The binding energy "b ?àc 2=2is the larg-est energy scale than the kinetic energies of pairs or excess fermions.For a strong attraction [up to order O e1=c 3T],N fermions have root patterns (Batchelor et al.,2006b )

k u

e2n u t1T

u ; %

e2n b t1T

b ;where the effective statistical parameters are given by

b %1

2 1à2N à2N #L j c j à1; u %

1à4N #L j c j

à1

;

FIG.2(color online).Schematic BA root con?gurations for pair-ing and depairing in the Gaudin-Yang model.For weakly attractive interaction,the unpaired roots sit in the outer wings due to Fermi statistics.For strongly attractive interaction,the unpaired roots can penetrate into the central region,occupied by the bound pairs (Batchelor et al.,2006b ;Iida and Wadati,2007).

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and n u ?àN 1=2;àN 1=2t1;...;N 1=2à1with n b ?àN #=2;àN #=2t1;...;N #=2à1.In this strong attraction limit,the ground state energy per unit length is given by

E=L ?E u 0t2E b 0tn #"b ,where the energies of excess

fermions and pairs are given by

E u 0?13n 31 2 2

u ;

E b 0?13n 32 2 2

b :

(17)

The bound states behave like hard-core bosons which can

be viewed as ideal particles with fractional exclusion statis-tics.However,the bound pairs have tails and they interfere with each other.It is impossible to separate the intermolecular forces from the interference between molecules and single fermions.From this explicit form of the ground state energy,we see that for n #)x ?n "àn #the single atoms are repelled by the molecules,i.e.,

E en #;x TàE 0%16n 3# 2 4x j c j t12x ex tn #T

c >0:(18)

Here E 0is the ground state energy per unit length of the

balanced gas.This result indicates that the single atoms are repelled by the molecules on the tightly bound dimer limit.The atom-dimer scattering problem of three fermions in a quasi-one-dimensional trap has been studied (Mora et al.,2004;Mora,Egger,and Gogolin,2005).

2.Highly polarized fermions:Polaron versus molecule

In higher dimensions,a spin-down fermion immersed in a fully polarized spin-up Fermi sea gives rise to the quasipar-ticle phenomenon called Fermi polaron (Combescot et al.,2007;Combescot and Giraud,2008;Prokof’ev and Svistunov,2008a ,2008b ;Bruun and Massignan,2010;Klawunn and Recati,2011;Mathy,Parish,and Huse,2011;Parish,2011;Schmidt and Enss,2011).The Fermi polaron is a spin-down impurity fermion dressed by the surrounding scattered fermions in the spin-up Fermi sea.In particular,recent observations of Fermi polarons in a 3D or 2D tunable

Fermi liquid of ultracold atoms (Nascimbe

`ne et al.,2009;Schirotzek et al.,2009;Kohstall et al.,2012;Koschorreck et al.,2012)provide insightful understanding of quasiparticle physics in many-body systems.For an attractive polaron,with increasing attraction,the single spin-down fermion undergoes a polaron-molecule transition in the fermionic medium

(Nascimbe

`ne et al.,2009;Schirotzek et al.,2009).For repulsive interaction,theoretical studies suggested the existence of such novel quasiparticles—repulsive polarons (Pilati et al.,2010;Massignan and Bruun,2011;Schmidt and Enss,2011;Ngampruetikorn,Levinsen,and Parish,2012;Schmidt et al.,2012).The properties of repulsive polarons,such as the energy,lifetime,and quasiparticle residue,give a fundamental understanding of the coherent nature of the quasiparticle.The repulsive polaron is metastable and eventually decays to either a molecule state or an attractive polaron with particle-hole excitations in the majority Fermi sea.This quasiparticle phenomenon was experimentally ob-served by a magnetically tuned Feshbach resonance on the BEC side with positive scattering length (Kohstall et al.,2012;Koschorreck et al.,2012).

So far most studies concerning the ?rst-order nature of the polaron-molecule transition in a 3D fermionic medium

(Combescot et al.,2007;Combescot and Giraud,2008;Bruun and Massignan,2010;Mathy,Parish,and Huse,2011)involve a variational ansatz with some approximations that are ultimately not justi?ed in low dimensions (Giraud and Combescot,2009;Parish,2011).It is generally accepted that quasiparticle excitations actually do not exist in 1D systems due to the collective nature of the 1D many-body effect.The elementary excitations in 1D are still eigenstates,where all particles are involved in a low-energy nature.Therefore,we cannot ?nd a simple operator,acting on the ground state,to get a quasiparticle excitation,unlike for higher dimensional systems.But this does not rule out a well-de?ned quasiparticle-like behavior, e.g.,a polaron,which is a typical example of the collective nature of the 1D many-body effect.The quantum impurity problem in 1D trapped ultracold atoms has shed new insight on the collective nature of particles (Palzer et al.,2009;Catani et al.,2012).The BA solvable models are likely to provide a rigorous treatment of polaronlike phenomena in different mediums (McGuire,1966;Leskinen et al.,2010;Guan,2012;Li et al.,2012).In particular,a 1D attractive polaronic phe-nomenon does occur if one (or a few)spin-down fermion (fermions)is (are)immersed into a large spin-up Fermi sea (McGuire,1966;Giraud and Combescot,2009;Leskinen et al.,2010;Klawunn and Recati,2011;Parish,2011;Guan,2012;Massel et al.,2012).The excitation energy of a system with one spin-down fermion has a certain momentum-dependent relation,which includes a mean-?eld attractive binding energy plus a classical kinetic energy of polaron with effective mass m ?;see Fig.3.

McGuire (1965,1966)studied the exact eigenvalue prob-lem of N à1fermions of the same spin and one fermion of the opposite spin.He calculated the energy shift caused by this extra spin-down fermion by solving the equation az i t1=tan z i ?const for the quasimomentum k i ?2z i =L with i ?1;...;N and a ?4=gL .Here g >0for an attractive interaction strength.McGuire found a Hermitian conjugate pair z 1;2? ?i and N à2real roots z i .The energy is

given by E ?e2=L 2TP N i ?1z 2i .This single impurity problem

was recently studied (Guan,2012)by means of the

spin-up spin-down

Polaron

Molecule

FIG.3(color online).Schematic BA root con?guration of polaron-molecule crossover in the 1D attractive Fermi gas.The upper panel shows the free fermion distribution.In the weakly attractive limit (middle panel),the single impurity fermion dressed by the surround-ing scattered spin-up fermions from the medium behave like a polaron (dashed oval)with a mean-?eld binding energy and an effective mass m ?%m .For strong attraction (lower panel),the single impurity fermion binds with one spin-up fermion from the Fermi sea to form a tightly bound molecule of a two-atom with a sole molecule binding energy and an effective mass m ?%2m .

1642Xi-Wen Guan,Murray T.Batchelor,and Chaohong Lee:Fermi gases in one dimension:From Bethe ...

Rev.Mod.Phys.,V ol.85,No.4,October–December 2013

equations (11).For an attractive interaction,the quasimo-menta k #;"?p ?i of a pair and N à2real roots f k i g with i ?1;...;N à2are determined by Eqs.(11)with N #?1.It was found (Guan,2012)that the imaginary part in the pair is determined by the equation

L ?tanh à1 j c j

tc 2=4

:For an excited state with total momentum of the system q ,the spin-down fermion associated with the weakly bound pair in the fully polarized Fermi sea thus has a nonzero momentum

p %q 1à2j c j

X N "

2à1i ?1

L ek i

(19)

which depends on all individual momenta of the spin-up fermions.This gives a collective signature of the 1D many-body effect.Thus the energy shift is explicitly given by

E eq;N;N #?1TàE "eN ";0T% p àb t

?2q 2

2m ?

(20)

which behaves like a polaron quasiparticle.Here the attrac-tive mean-?eld binding energy of the polaron is given by p àb %àe6= 2Te F j j for weak attraction.The Fermi energy is e F ?e?2=2m Te1=3Tn 2 2.We see that this binding energy depends solely on the Fermi energy of the medium and interaction strength in 1D.In Eq.(20),the polaronlike state has an effective mass m ?%m ?1tO ec 2T which is almost the same as the actual mass of the fermions due to the decoupling from the bound pair in the weak coupling limit.We point out that the polaronlike state occurs only for few impurity fermions immersed into a fully polarized Fermi sea.

For a weakly repulsive interaction and in the thermodynamic limit,the low-energy physics of the 1D Fermi gas is described by a spin-charge separation theory.The spin rapidity parame-ters decouple from the quaismomenta of the fermions.However,using the BA equations (11),a single spin-down fermion immersed into the 1D fully polarized Fermi sea with weak repulsion can form a repulsive Fermi polaron,with energy of the form (20)and an effective mass m ?%m ?1tO ec 2T .But here the impurity fermion receives a positive mean-?eld shift p àb %e6= 2Te F j j from the fermionic medium.

In the opposite limit,a spin-down fermion immersed into a fully polarized spin-up medium with strong attraction,i.e.,with L j c j )1,the bound pair has k #;"?p ?i and the N à2real roots f k i g with i ?1;...;N à2are given by (Guan,2012)

k i %

n j L à4p L j c j 1à4L j c j à1

;(21)with n j ??1;?3;...;?eN "à1T.For an excited state with

total system momentum q ,the relation between the center-of-mass pair quasimomentum p and the total momentum of the system q is given by

p %q 2 1à

2eN "à2T

L j c j

;which is independent of the individual quasimomenta of the spin-up fermions.The energy shift is given by áE ?E M à

with the chemical potential ?n 2 2,where the molecule energy is given by

E M %E b t?2q 2

2m ?

:(22)

The binding energy of the bound state is

E b %?2n 22m à 22t

8 2

3j j

(23)

which tends to the binding energy of a sole molecule "b ?

àe?2=2m Tc 2=2in the strongly attractive regime L j c j !1.The effective mass of the molecule

m ?%2m

1à

j j (24)becomes twice the actual mass of the fermions in this limit.From the shift energies (20)and (22),we see that as the attractive interaction grows,the spin-down fermion binds only with one spin-up fermion from the medium to gradually form a tightly bound molecule.The polaron-molecule cross-over is regarded as a change from a mean-?eld attractive binding energy of a polaron with an effective mass m ??m to the binding energy of a single molecule with an effective mass m ??2m as the attraction grows from zero to in?nity.The nonequilibrium dynamics of an impurity in a 1D lattice within a harmonic trap have been studied using numerical methods and the BA solution (Massel et al.,2012).The numerical simulation of an impurity injected into a 1D quantum liquid has been reported (Knap et al.,2013).

C.Solutions in the thermodynamic limit

In Sec.II.B.2we discussed the solutions to the discrete BA equation (11)in the limits j c j !0,1.They give rise to different phenomena in the two extreme https://www.wendangku.net/doc/3b2573010.html,ually,the many-body phenomena of interest refers to the physics of the system in the thermodynamic limit,where N ,L !1keeping N=L ?nite.This entails considering the solutions to the two sets of Fredholm equations (12)and (13)for the repulsive and attractive regimes.

1.BCS-BEC crossover and fermionic super Tonks-Girardeau gas

In order to conceive the physical nature of the super Tonks-Girardeau gas,we ?rst recall the Lieb-Liniger Bose gas with zero-range delta-function interaction,where the Tonks-Girardeau gas was determined by a Fermi-Bose mapping to an ideal Fermi gas (Girardeau,1960).For a strong attractive interaction,McGuire (1964)predicted that the quasimomenta of the bosons are given in terms of a bound state of N particles,with k ?j %?12c ?N à2j t1 where j ?1;...;N=2.In this case,the wave function is given by

éex 1;...;x N T%N exp c 2X

1 i

where N ?e?????????????????en à1T!p =???????

2 p Tj c j en à1T=2is a normalization constant.The energy of the McGuire cluster state is given

by E 0?à112

c 2

N eN 2à1T.However,if the interaction strength is abruptly changed from strongly repulsive to

Rev.Mod.Phys.,V ol.85,No.4,October–December 2013

strongly attractive,the highly excited gaslike state may be metastable against this clusterlike state due to the Fermi pressure inherited from the repulsive Tonks-Girardeau gas. This gaslike state exhibits a more exclusive quantum statistics than the free-Fermi gas and is called super Tonks-Girardeau gas.The super Tonks-Girardeau gaslike state was?rst pre-dicted by Astrakharchik et al.(2005)and further proved by Batchelor et al.(2005b)using the exact BA solution of the Lieb-Liniger Bose gas.Remarkably,such a highly excited state was realized in a breakthrough experiment (Haller et al.,2009).

The equally populated components in an attractive Fermi gas give rise to physics related to the crossover between a BCS super?uid and a BEC(Fuchs,Recati,and Zwerger,2004; Tokatly,2004;Iida and Wadati,2005;Wadati and Iida,2007; Chen,Cao,and Gu,2010;Chen et al.,2010;Feiguin et al., 2012).In this context,for a balanced attractive Fermi gas with N#?N=2,the discrete BA equations(11)reduce to

expe2i LT?à

Y N#

?1 à tic F

à àic F

;(26)

with c F?à2=a F1D.This is equivalent to

expeik j LT?à

Y N B

l?1k jàk ltic B

k jàk làic B

;(27)

with c B?à2=a B1D for the Lieb-Liniger gas in the super Tonks-Girardeau phase under the identi?cation c B?2c F, N B?N=2,and m B?2m F(Wadati and Iida,2007;Chen, Cao,and Gu,2010;Chen et al.,2010).Since the bound pair formed by two fermions with opposite spin has a mass m B?2m F,the M bound pairs are equivalently described by the super Tonks-Girardeau phase of the interacting Bose gas with effective1D scattering length a B1D?1a F1D.This relation is also obtained by an exact mapping based on the two-body scattering problem associated with BEC-BCS crossover(Mora et al.,2005).

For the balanced Fermi gas,the binding energy is sub-tracted from the energy that gives the energy of the bosonic pairs of a two atom,with result

E F 0?EtN# b?

2m F

X N#

2 2 :

The energy eigenvalues of the bosons are given by

2m B

X N B

j?1

k2j:

In this regard,the ground state of the balanced attractive Fermi gas can be viewed as the fermionic super Tonks-Girardeau gas(Chen,Cao,and Gu,2010;Chen et al., 2010).The identi?cation between the balanced attractive Fermi gas and the attractive Lieb-Liniger Bose gas suggests an effective attraction between pairs.

In the thermodynamic limit,the BA equation(26)gives a particular Fredholm equation which can be deduced from Eq.(13),i.e.,

2ekT?1

Z Q

àQ2

K2ekàk0T 2ek0Tdk0;(28)

where the Fermi pair momentum Q2is determined by n?

R Q

àQ2

2ekT.It turns out that(Iida and Wadati,2005;Wadati

and Iida,2007;Chen,Cao,and Gu,2010)the reduced

Fredholm equation(28)maps to the Lieb-Liniger integral

equation for1D spinless bosons on identifying m B?2m F,

N B?N F=2,and B?4 F.For weak attraction,the ground

state is the BCS-like pairing state with a pairing correlation

length larger than the average interparticle spacing and the

energy is given by E?1

n3 2à1

n2j c jtOec2T.In particu-

lar,for strong attraction,the ground state of bound pairs

determined by Eq.(28)can be regarded as a particular super

Tonks-Girardeau gas of hard-core bosons(Chen,Cao,and

Gu,2010).The distribution function of the pair density

plotted in Fig.4provides an understanding of the subtle

BEC-BCS crossover in the balanced Fermi gas.In the weak

coupling regime,the single quasimomentum essentially de-

pends on that of the other particles.This gives a signature of

mutual statistics(Wilczek,1982;Aneziris,Balachandran,and

Sen,1991;Haldane,1991;Wu,1994).However,in the limit

j j!1,the quasimomentum distribution becomes an

equally spaced separation.This indicates free-Fermi nonmu-

tual statistics.Further study on the dimer-dimer scattering

properties in the con?nement-induced-resonance has been

reported(Mora et al.,2005;Mora,Egger,and Gogolin,

2005);see also Feiguin et al.(2012).

Furthermore,it was demonstrated(Girardeau,2010;Guan

and Chen,2010)that another metastable highly excited gas-

like state without bound pairs in the strongly attractive regime

can be realized through a sudden switch of the interaction

from strongly repulsive to strongly attractive.In the limit

c!à1,this gaslike state is still an eigenstate of the system

with the energy per particle

n2 2

4ln2

j j

12ln2

;

but it is a highly excited state.From the experimental point of

view,these different quantum states can be tested from

1.2

1.4

1.6

1.8

-0.50-0.250.000.250.500.75

(

)

q/D

11 1 γ=?0.075

11 =?∞

2γ=?0.075

3γ=?0.280

4γ=?0.280

5γ=?1.136

6γ=?2.239

7γ=?4.414

8γ=?4.414

9γ=?17.78

=?17.78

FIG.4.The normalized pair quasimomentum distribution function

feqTfor different values of interaction strength .The analytic

result for the distribution function matches the numerical solution.

The quasimomentum distribution indicates the fermionization from

mutual statistics to nonmutual generalized exclusion statistics as

increases.From Iida and Wadati,2005.

Rev.Mod.Phys.,V ol.85,No.4,October–December2013

measuring the frequencies of the lowest breathing mode from the mean square radius of the1D trapped gases in a harmonic potential(Menotti and Stringari,2002;Astrakharchik et al., 2004),e.g.,in the super Tonks-Girardeau Bose gas(Haller et al.,2009).The low breathing mode featuring different states of the strongly repulsive and attractive Fermi gas can be analyzed via the local density approximation(LDA).The lowest breathing mode is given by the mean square radius of the trapped fermionic Tonks-Girardeau gas!2?à2h x2i=ed h x2i=d!2xT;see Fig.5.Here h x2i?

exTx2dx=N.The frequency ratio!2=!2x exhibits a peak which is a typical characteristic of the super Tonks-Girardeau phase.Further evidence for this fermionic super Tonks-Girardeau gaslike state has been seen in the experimental observation of the fermionization of two distinguishable fermions(Zu¨rn et al., 2012).It is also particularly interesting that a ferromagnetic transition is likely to occur in1D strongly interacting fermi-ons across the resonance from in?nite repulsion to?nite attraction(Cui and Ho,2013a,2013b).

2.Solutions to the Fredholm equations and analyticity Despite the two sets of Fredholm integral equations(12) and(13)for the homogeneous gas being derived long ago (Yang,1967,1970;Takahashi,1971b),their analytical studies are still restricted to particular regimes,e.g., )1,j j(1, and j j(à1.The?rst few terms in the asymptotic expan-sions of the ground state energy for the1D attractive Fermi gas for both the strong and weak coupling cases has been calculated in terms of a power series(Iida and Wadati, 2005,2007;Guan et al.,2007;He et al.,2009)and in terms of Legendre polynomials(Zhou,Xu,and Ma,2012).The ?rst few terms of the ground state energy have been derived recently(Guan and Ma,2012)by an asymptotic expansion for (a)strong repulsion,(b)weak repulsion,(c)weak attraction, and(d)strong attraction.

For strong repulsion,the ground state energy of the Gaudin-Yang model is given by(Guan and Ma,2012)

E L %

8 ><

>:1

n3 2

1à4ln2

t12eln2T2

à32eln2T3

t 2 e3T

;for P?0;

n3 2

1à8n#

t48n2#

à1

256n3#à32

2n2n#

;for P!0:5:

(29)

Here ezTis the Riemann zeta function.The leading order(1= )correction was also found in Fuchs,Recati,and Zwerger (2004)and Batchelor et al.(2006b).Figure6shows that this ground state energy is a good approximation for the balanced and imbalanced Fermi gas with a strongly repulsive interaction.

For strong attraction,the ground state energy is given by E=L?E u0t2E b0tn#"b,where

E u 0%

en"àn#T3 2

8n#

j c j

48n2#

8n#

15j c j

e12 2en"àn#T2à480n2

t5n2

;(30)

E b 0%

n3# 2

2e2n"àn#T

j c j

3e2n"àn#T2

15j c j

e180n#n2

t20 2n3

à90n"n2

à22 2n3

t15n3

à120n3

t63 2n2

n"à60 2n#n2"T

(31)

This energy is highly accurate for arbitrary polarization as can be seen in Fig.6.The high precision of expansions for the ground state energy of the attractive Fermi gas was also studied(Iida and Wadati,2007;He et al.,2009;Zhou,Xu, and Ma,2012).

In contrast to the strong coupling case,it is more dif?cult to proceed with asymptotic expansion for the two sets of Fredholm equations(12)and(13)at vanishing interaction strength.In terms of the polarization P,the ground state energy in weak attraction limit was found to be(Iida and Wadati,2007)

2n3

e1t3P2Tà

e1àP2Tj jàB2 2

:(32)

The coef?cient B2is a complicated function obtained from the power series expansions with respect to .However,it contains divergent sums and the coef?cients are singular as

FIG.5(color online).The square of the lowest breathing mode frequency vs the ground state energy per unit length vs the rescaled interaction strength a1D=!x.The quantum gases are trapped in a1D

harmonic potential V x?1

2m!2x x2.Here GS and FSTG stand for the

frequency ratio!2=!2x for the ground state and fermionic super Tonks-Girardeau gas,respectively.From Guan and Chen,2010. Rev.Mod.Phys.,V ol.85,No.4,October–December2013

!0(Iida and Wadati,2007).So far only the leading order correction to the interaction energy is mathematically convincing and consistent with the result (16)obtained from the discrete BA equations (11).Beyond the mean-?eld term,?nding the next leading term in the ground state energy is still an open problem.For zero polarization,Krivnov and Ovchinnikov (1975)found

O ec 2T%à

2n 3

2eln j jT2obtained from the 1D Hubbard model in a dilute limit.Iida

and Wadati (2007)found the term O ec 2T%à 2n 3=12.This difference reveals a subtlety of the vanishing interaction limit,i.e.,the two limits c !0and the thermodynamic limit (N ,L !1with N=L ?nite)do not commute.In fact,the ground state energy (32)counts only the density distributions away from the integration boundaries in the Fredholm equa-tions (12)and (13),i.e.,j B i àk j $c and j A i àk j $c with i ?1,2.At the integration edges,these distribution functions are not analytically extractable as !0(Guan and Ma,2012).The analyticity of the ground state energy at ?0is still in question (Takahashi,1970c ;Guan and Ma,2012).Takahashi showed that (a)the ground state energy function f en ";n #;c Tis analytic on the real c axis when n "Tn #,and (b)f en ";n #;c Tis analytic on the real c axis except for c ?0when n "?n #.However,the Fredhom equations for weakly repulsive and attractive interactions are identical as long as B 1>B 2and A 1>A 2,where the integration boundaries match each other between the two sides (Guan and Ma,2012).In this identical region,the asymptotic expansions of the energies of the repulsive and attractive fermions are identical to all orders as c !0.But the identity of the asymptotic expansions may not mean that the energy analytically connects due to the divergence of the Fredholm equations in the region c !i 0.

III.MANY-BODY PHYSICS OF THE GAUDIN-YANG MODEL

So far we have discussed only the ground state properties of the Gaudin-Yang model.We now survey the wide range of fundamental many-body physics exhibited in the model.

A.1D analog of the FFLO state and magnetism

The particularly interesting feature of the attractive Fermi gas is the exotic FFLO-like pairing,where the system is gapless with mismatched Fermi points between the two Fermi seas.In the gapped phase,it is well understood that the correlation function for the single-particle Green’s function decays exponentially (Bogoliubov and Korepin,

1988,1989,1990)h c y x;s c 1;s i !e

àx=

with ?v F =áand s ?",#,whereas the singlet pair correlation function de-cays as a power of distance,i.e.,h c y x;"c y x;#c 1;"c 1;#i !x à

.Here áis the energy gap,and the critical exponents and are both greater than zero.However,once the external ?eld exceeds the lower critical ?eld,the system has a gapless phase where both of these correlation functions decay as a power of distance and the pairs lose their dominance.The molecule and excess fermions form the polarized FFLO-like pairing state,where the spatial oscillations of the pairing correlation are caused by an imbalance in the densities of spin-up and spin-down fermions,i.e.,n "àn #.In Sec.VI ,we further discuss the pair and spin correlations with the spatial oscillation signature in the context of conformal ?eld theory.

In terms of the polarization,the Gaudin-Yang model with attractive interaction exhibits three quantum phases at zero temperature:the fully paired phase which is a quasiconden-sate with zero polarization,the fully polarized normal Fermi gas with P ?1,and the partially polarized FFLO-like phase with polarization 0

12H

?12 b t u à b ;

(33)

where b ? t b =2and u ? tH=2are given by

Eq.(14).This relation reveals an important energy transfer relation among the binding energy,the variation of Fermi surfaces,and the external ?eld.

For ?xed density and strong attraction,the paired phase with magnetization M z ?0is stable when the ?eld H

H c 1%?2n 22m 22à 28 1à34j j 2à

j j 3 :(34)When the external ?eld exceeds the upper critical ?eld

H c 2%?2n 22m 22

t2 2 1à43j j t16

215j j ;

(35)FIG.6(color online).The ground state energy as a function of ?cL=N in units of ?2N 3=2mL 2.The comparison between the asymptotic solutions and numerical solutions of the Fredholm equations for different polarization is shown.In the attractive regime,the binding energy "b ?àc 2=2was subtracted.From Guan and Ma,2012.

Rev.Mod.Phys.,V ol.85,No.4,October–December 2013

a phase transition from the FFLO-like phase into the normal gas phase occurs;see Fig.7.The lower critical ?eld gives the energy gap in the spin sector.

The magnetization can be obtained from Eq.(33);see Fig.8.It was found (Woynarovich and Penc,1991;Guan et al.,2007;Iida and Wadati,2007;He et al.,2009)that in the vicinity of the critical ?elds H c 1and H c 2,the system exhibits a linear ?eld-dependent magnetization

M z %8

><>:2eH àH c 1T2

1t2j j t112t81à 23 ;n 2h 1àH c 2àH 4n 2 2 1t4j j t12 2à16e 2à6T3j j 3 i ;(36)with a ?nite susceptibility.For a ?xed total number of particles,or say in a canonical ensemble,the magnetic ?eld driven phase transitions in the 1D Fermi gases with an attractive interaction are linear ?eld dependent,which was also found in the SU eN Tattractive Fermi gas (Guan et al.,2010;Lee,Guan,and Batchelor,2011).

The magnetism of the attractive Fermi gases was dis-cussed by Schlottmann (Schlottmann,1993,1994,1997;Schlottmann and Zvyagin,2012a ,2012b ).However,the argu-ment that was made on the initial slope of the magnetization

in these papers (Schlottmann,1993,1994,1997)does not appear to be correct for a ?xed total number of particles.The reason has been discussed (Woynarovich,1991):‘‘The bound pairs which have to be broken up to yield the particles with uncompensated spins form a Fermi sea,their density of states is ?nite at the Fermi level,and that keeps the initial suscep-tibility ?nite.’’It was also shown (Vekua,Matveenko,and Shlyapnikov,2009)that the curvature of free dispersion at the Fermi points couples the spin and change modes and leads to a linear critical behavior and ?nite susceptibility for a wide range of models.They showed that when the magnetic ?eld H !H c ,the magnetization m z

$?????????????????H àH c p for a ?xed chemical potential.However,for ?xed density,the magneti-zation m z $eH àH c T= v b N as H ?H c t0t.This leads to a

?nite onset susceptibility given by ?1= v b N with the pair

density stiffness v b

N ?v F =4in the strong attraction limit !1.Here we further remark that for ?nitely strong attraction the onset susceptibility ?K eb T= v b N ,where

K eb T%

3j j t334 2 is the Tomonaga-Luttinger liquid parameter at the critical point and

v b N ?v F

4 1à2j j à32 2 is the stiffness of bound pairs in the limit H !H c t0t.

The magnetization in the Hubbard model with a half-?lled band gives rise to the square-root dependence on the ?eld (Takahashi,1969),where low density solitons appearing in the spin sector above the critical ?eld behave like free fermions (Japaridze and Nersesyan,1978,1981;Pokrosvsky and Talapov,1979).More rigorously speaking,the linear ?eld-dependent magnetization is clearly seen from the energy trans-fer relation (33),where the effective chemical potentials u /e2m z T2and b /en à2m z T2.Thus the linear term m z in Eq.(33)gives a ?nite susceptibility at the onset of

magnetization.

p o l a r i z

a t i o n

H /b

εFIG.7(color online).Upper panel:Phase diagram of the Gaudin-Yang model in the -H plane.The phase boundaries are obtained from Eq.(14)in terms of the numerical solution of the BA equations (12).From Orso,2007.Lower panel:Phase diagram of the model in the H -n plane with j j ?10and density n ?1.The dashed lines denote the two critical lines [Eqs.(34)and (35)].The colored phases are obtained by numerical solution of the energy magnetization (33).From He et al.,2009.

0.0

0.1

0.2

0.3

0.4

0.5

M a g n e t i z a t i o n M Z

Magnetic Field H/

FIG.8(color online).Magnetization vs the external ?eld H= b for

c ?à10in the units 2m ???1for different densities n .The dashe

d lines ar

e plotted from the analytic result (36).The solid curves are obtained from numerical solutions o

f the dressed energy equations.The inset shows a similar comparison between analytic and numerical results for the susceptibility vs external ?eld H= b .From He et al.,2009.

Rev.Mod.Phys.,V ol.85,No.4,October–December 2013

B.Fermions in a 1D harmonic trap

In experiments,1D quantum atomic gases are prepared by loading ultracold atoms in an anisotropic harmonic trap with strong transverse con?nement and weak longitudinal con?ne-ment.In general,interacting many-body systems trapped in a harmonic potential is a rather complicated problem.The problem of the 1D Hamiltonian (1)trapped in a harmonic

potential 12m!2x x 2has been studied by various methods

(Girardeau and Minguzzi,2007;Hu,Liu,and Drummond,

2007;Orso,2007;Colome

′-Tatche ′,2008;Gao and Asgari,2008;Ma and Yang,2009;Yang,2009;Girardeau,2010;Yin,Guan,Chen,and Batchelor,2011;Cui,2012b ).For N ?2,the eigenvalue problem of the trapped gas has been studied analytically by Busch et al.(1998)and Idziaszek and Calarco (2006).The energy shift for N ?2(Busch et al.,1998)is given by

???2

p àeàE=2t3=4T

àeàE=2t1=4T

?1=a 1D ;where a 1D is a scattering length and àex Tis the Euler gamma function (Busch et al.,1998).The system of two fermions with arbitrary interaction in a 1D harmonic potential has been

experimentally investigated (Zu

¨rn et al.,2012).In this experiment,the Tonks-Girardeau state and the metastable super Tonks-Girardeau state have been observed.

This problem for an arbitrary number of particles was studied analytically (Guan et al.,2009;Ma and Yang,2009;Yang,2009),where the limiting cases c !?1and c ?0have been studied using group theory.In particular,Yang (2009)gave an analysis of the ground state energy of fermi-ons in a 1D trap with delta-function interaction.In light of Yang’s argument,for any value of interaction strength,the eigenvalue problems of (a)the trapped Hamiltonian with symmetry Y ??N àM;M in full 1N space,and (b)the Hamiltonian in region R Y with the boundary condition that the wave function vanishes on its surface are equivalent.Here

the region R Y is bounded by C 2N àM ?C 2

M planes at which the wave function éY vanishes.For any value of g ,the ground state wave function for problem (b)has no zeros in the interior of R Y and is not degenerate.Thus this suggests that the ground state energy of the system with total spin J ?N=2àM increases monotonically and approaches to the energy E J ?N=2.The Lieb and Mattis theorem (Lieb and Mattis,1962)further suggests E J >E J 0if J >J 0.For c !1,the ground state energy of the trapped gas with total spin J is given by E J ?P N à1n ?0e12tn T?12N 2,which is independent of the total spin J .For c ?0and J ?

N=2àM ,the energy is given by E J ?12e?N=2tJ 2t

?N=2àJ 2T.Ma and Yang (2009)argued that E J =N 2!f J eg=????

N p Twith

f J et T?8>>><

>>>:1=2for t !1;1=4teJ=N T2for t ?0;

àe1=2àJ=N Tt 2=4for t !à1;where t ?g=????

N p .In particular,for c !1,the exact wave function of the system é?c A c J where the spatial wave function and symmetric spin-wave function have been derived explicitly (Guan et al.,2009)

c A ex 1;...;x N T?

1eN !T

det ? j ex i T j ?1;...;N

i ?1;...;N ;c J ?

N !=eeN àM T!M !T ?1

f Y ?N àM;M

Q g Z ;

for the symmetry R ??N àM;M .Here Q ?P Q 1with Q 1?Q ‘i ?1Q N

j ?M t1sgn ex i àx j T.The basis tensor function Y ?M;M was constructed explicitly from group theory (Guan et al.,2009).

The fermion density distribution for 1D interacting fermi-ons with harmonic trapping has strong oscillations on top of a uniform density cloud (Rigol et al.,2003;Gao et al.,2006;Gao and Asgari,2008;Guan et al.,2009;Ma and Yang,2009).These oscillations can be described by an analytical form of the density distribution (Butts and Rokhsar,1997;

Gleisberg et al.,2000;So

¨f?ng,Bortz,and Eggert,2011)n ex T%n 0ex Tà

eà1TN=2F cos ?2k F ex Tx

1àx 2=L 2F

(37)

for x L F ,where the density cloud is given by the Thomas-Fermi pro?le,i.e.,

n 0ex T?2!L F

??????????????????????

1àx 2=L 2F

q with a Thomas-Fermi radius L F ????????????

N=!p .If the longitudinal con?nement is weak enough,the atomic density varies smoothly along the longitudinal direction and so the atomic gases can be treated as locally homogeneous systems (Kheruntsyan et al.,2005;Hu,Liu,and Drummond,2007;Orso,2007).This type of approximate treatment is known as the LDA.In this way density functional theory has been used to study 1D interacting fermions (Magyar and Burke,2004;Gao et al.,2006,2007;Hu et al.,2010).

To ensure the validity of the LDA,the correlation length ez Tshould be much smaller than the characteristic inhomogeneity length

inh ?

n ez T

j dn ez T=dz j

;

i.e., ez T( inh .The two length scales ez Tand inh are determined by the local chemical potential ez Tand the local density n ez T.From the de?nition of inh ,the LDA becomes invalid near the edge of an atomic cloud where the density drops rapidly.However,in real measurements,almost all signal strengths are proportional to the density.Therefore,due to the very small density at the edge,the central region of large density dominates the measurement signals.For a large number of particles,N !1,the density pro?les of the trapped gas can be precisely analyzed within the LDA.

In a harmonic trap,the equation of state (14)can be reformulated within the LDA by the replacement ex T?

e0Tà12m!2x x 2in which x is the position and !x is the

frequency within the https://www.wendangku.net/doc/3b2573010.html,ing the LDA for the 1D Bose gas in a harmonic trap,its global chemical potential reads (Kheruntsyan et al.,2005)

g ? 0?n ez T àV ez T? 0?n ez T à12m!2z z 2;

(38)

where the local chemical potential 0?n ez T at position z is

given by the chemical potential for a homogeneous system of

Rev.Mod.Phys.,V ol.85,No.4,October–December 2013

a uniform density n ?n ez T.The total number of atoms N is given as N ?R n ez Tdz .

Similarly,for a 1D two-component Fermi gas in a harmonic trap,the global chemical potential is (Hu,Liu,and Drummond,2007;Orso,2007;Heidrich-Meisner,Orso,and Feiguin,2010;Ma and Yang,2010a )

hom ?n ex T;P ex T ? 0à12m!2x x 2

;

where the chemical potential hom ?n;P can be obtained from the homogenous gas (14).n ex Tis the total linear number density and P ex Tis the local spin polarization.They can be determined from restriction on the total particle number N ?R 1

à1n ex Tdx and polarization P ?R 1à1n u

ex Tdx=N which are rewritten as (Hu,Liu,and Drummond,2007;Orso,2007;Gao and Asgari,2008;Yin,Guan,Chen,and Batchelor,2011)

Na 21D =a 2x ?4Z 1

à1~n

ex Td ~x ;eNa 21D TP ?4Z 1

à1

~n u ex Td ~x a 2x :(39)Here ~n

ex T?1=j ex Tj ,a x ????????????????????

?=em!x Tp ,and ~n u ex T?n u ex T=j c j .If the trapping potentials are the same for the two spin components,calculations for the integrable homogenous attrac-tive gas con?ned to a 1D trapping potential thus lead to a two-shell structure composed of a partially polarized 1D FFLO-like state in the trapping center surrounded by wings composed of either a fully paired state or a fully polarized Fermi gas (Hu,Liu,and Drummond,2007;Orso,2007;Gao and Asgari,2008);see Fig.9.This prediction was veri?ed by Liao et al.(2010)with the observation of three distinct phases in experimental measure-ments of ultracold 6Li atoms in an array of 1D tubes.The analytical study of the phase diagram of the 1D attractive Fermi gas has been presented by Guan et al.(2007),Iida and Wadati (2007),and Guan and Ho (2011).

C.Tomonaga-Luttinger liquids

The TLL (Tomonaga,1950;Luttinger,1963),describing the collective motion of bosons,has played an important role in the novel description of universal low-energy physics for low-dimensional many-body physics (Gogolin,Nersesyan,and Tsvelik,1998;Giamarchi,2004).In 1D systems of

interacting bosons,fermions,or spin systems,the effect of

quantum ?uctuations is strong enough to yield striking anoma-lous quantum phenomena.In this approach,for example,the low-energy physics of a 1D interacting fermion system can be described by a bilinear form of bosonic creation and annihilation operators.The TLL is phenomenologically treated by bosonization techniques (Tsvelik and Wiegmann,1983;Gogolin,Nersesyan,and Tsvelik,1998;Giamarchi,2004;Cazalilla et al.,2011)based on a linearization of the dispersion relation of the particles in the collective motion,i.e.,!eq T?v s j q j ;here v s is the sound velocity of the collective motion.In contrast to the Fermi liquid,this thus leads to a power-law density of states for the TLL at the Fermi energy E F ,i.e.,j E àE F j ,where the exponent ?eK t1=K à2T=2depending on the so-called TLL parameter K .

In general,the correlation functions of such 1D systems at zero temperature show a power-law decay determined by the TLL parameter K and the velocity v s .These critical systems not only have global scale invariance but exhibit local con-formal invariance.With the help of exact BA solutions,a wide class of 1D interacting systems can be mapped onto TLLs in the low-energy limit,including the electronic systems with spin degrees of freedom such as spin-charge

separation (So

′lyom,1979;Kawakami and Yang,1990;Schulz,1990,1991;V oit,1995;Giamarchi,2004;Essler et al.,2005).Moreover,progress in treating such collective motion of particles beyond the low-energy limit was made by Imambekov and Glazman (2009a ,2009b),and Imambekov,Schmidt,and Glazman (2011).This method can be applied to a wide variety of 1D systems with collective motion of particles.This generalized TLL theory could be possibly justi?ed through exact BA results for 1D integrable models in ultracold atoms and correlated electronic systems.

In contrast to the conventional quasiparticles carrying both spin and charge degrees,the elementary excitations form spin and charge waves that propagate with different velocities in 1D (Gogolin,Nersesyan,and Tsvelik,1998).The relativistic dispersion relation for each one of these excitations is written

as ! ep T??????????????????????????

á2 tv 2 p 2q ,where á is the energy gap and

v is the velocity.For a gapless excitation with vanishing energy gap v ?@p ! ep T.For 1D interacting systems,this gives a phonon dispersion that leads to conformal invariance in the excitation spectrum.However,for a large energy gap,

the dispersion can be rewritten as ! ep T?á tv 2 p 2=

e2á T:?á tp 2=e2m ? Twhich is the classical dispersion of a free particle with an effective mass m ? .

From the BA solution (11)with N "?N #,the charge and

spin velocities are v c;s ?12v F e1? = 2

Tfor the weak cou-pling regime (Fuchs,Recati,and Zwerger,2004;Batchelor et al.,2006a ,2006b ).Here the Fermi velocity v F ?? n=m .For strong attraction,the charge and spin velocities are given by v c ?1

4v F e1à1= Tand v s ?????áp e1à2= T(Fuchs,Recati,and Zwerger,2004;Batchelor et al.,2006a ,2006b )with an energy gap á%e?2=2m Tc 2=2.This gap increases with increasing interaction strength so that the spin velocity is divergent in the strongly attractive limit.However,for strong repulsion the charge velocity v c ?v F e1à4ln2= Ttends to the Fermi velocity and the spin velocity goes to zero v s ?v F 2=3 e1à6ln2= T(Lee et al.,2012)due to

00.20.4

0.60.81P

0.6

0.8

1.2

1.4

R /a z N

1/2

00.20.4

0.60.81

0.20.4

0.6

0.8

1R i n /a z N

1/2

FIG.9.The cloud radii of outer shell R "and inner shell R in are theoretically predicted by means of the BA equations (12)within the LDA.The radii vs polarization P for the values of the parameter

Na 21D =a 2x ?1;10,1,0.1,and 0are shown.From Orso,2007.

Rev.Mod.Phys.,V ol.85,No.4,October–December 2013

suppression of spin transportation due to the strong repulsion;see Fig.10.We discuss TLLs and spin-charge separation phenomena in the attractive Fermi gas in Secs.III.E and III.F .

D.Universal thermodynamics and Tomonaga-Luttinger liquids in attractive fermions

The Yang-Yang formalism with its generalization for the study of thermodynamics of BA integrable systems (Takahashi,1999)is a convenient tool for the study of universal thermodynamics and quantum criticality in the presence of external ?elds.At ?nite temperatures and in the thermodynamic limit,the densities in the Fredholm equations (13)evolve into occupied and unoccupied roots in the whole parameter spaces,namely,A 1,A 2!1.In particular,the roots in the spin sector form complicated string patterns that characterize the spin excitations,i.e.,spin-wave bound states.The density distribution functions of pairs,unpaired fermions,and spin strings involve the densities of ‘‘particles’’ i ek Tand ‘‘holes’’ h i ek T(i ?1,2).Following the Yang-Yang grand canonical ensemble method,the grand partition func-tion is written as Z ?tr ee àH=T T?e àG=T ,in terms of the Gibbs free energy G ?E àHM z à n àTS and the mag-netic ?eld H ,the chemical potential ,and the entropy S (Takahashi,1999).In terms of the dressed energies b ek T:?

T ln ? h 2ek T= 2ek T and u ek T:?T ln ? h 1ek T= 1ek T for paired

and unpaired fermions,the equilibrium states are determined by the minimization condition of the Gibbs free energy,which gives rise to a set of coupled nonlinear integral equations—the TBA equations (Takahashi,1999).For the attractive Gaudin-Yang model,these equations are

b ek T?2

k 2

à à14

c 2 tTK 2?ln e1te à b ek T=T T

tTK 1?ln e1te à

u ek T=T

u ek T?k 2à à12H tTK 1?ln e1te à b

ek T=T T

àT X

1‘?1

K ‘?ln ?1t à1

‘ek T ;

(40)

ln ‘e T?

‘H T tK ‘?ln e1te à u

e T=T TtX

1m ?1

T ‘m ?ln ?1t à1

m e T :

(41)

The function ‘e T:? h ‘e T= ‘e Tis the ratio of the

string densities.Here ?denotes the convolution integral ef ?g Te T?R 1à1f e à 0Tg e 0

Td 0.The function T ‘m ek Tis given by Takahashi (1999)and Guan et al.(2007).The Gibbs free energy per unit length is given by G ?p b tp u where the effective pressures of the bound pairs and unpaired fermions are given by

p b

?à

T Z 1à1dk ln e1te à b

ek T=T T;p u

?àT Z 1à1

dk ln e1te à u ek T=T T:

(42)

In the grand canonical ensemble,the total number of

particles associated with the chemical potential can be changed.The Fermi sea of unpaired fermions can be lifted by the external ?eld.The entropy S is a measure of the thermal disorder.The spin ?uctuations (spin strings)are ferromagnetically coupled to the Fermi sea of unpaired fermions.The direct numerical computation of the TBA equations was presented by Kakashvili and Bolech (2009).The TBA equations for this model involve an in?nite number of coupled nonlinear integral equations that impose a number of challenges to accessing the physics of the model.

For zero external ?eld,the lowest excitations split into collective excitations carrying charge and collective excita-tions carrying spin.This leads to the phenomenon of spin-charge separation.The charge excitations are described by sound modes with a linear dispersion.However,for the external ?eld in excesses of the lower critical ?eld the spin gap vanishes.In contrast to the spin-charge separation for-malism,the spin-charge coupling drastically changes the critical behavior in the attractive regime of the Fermi gas.The TBA equations (41)indicate that the spin ?uctuations (the spin-wave bound states)are ferromagnetically coupled to the Fermi sea of unpaired fermions (Zhao et al.,2009).In contrast to the antiferromagnetic coupling J AF ?àe2=j c jTp u eT;H Tfor the repulsive regime (Guan,Batchelor,and Lee,2008),the spin-spin exchange interaction in the spin sector is described by an effective spin-1=2ferromagnetic chain with a coupling constant J F %e2=j c jTp u eT;H T>0in the strong coupling regime j c j )1.The ferromagnetic spin-wave ?uctuations are produced due to the thermal ?uctuation in the Fermi sea of unpaired fermions.However,J F tends to zero for !1.Therefore the spin transportation becomes weaker and weaker until it vanishes as j j !1.At zero temperature all unpaired fer-mions are polarized and spin strings are fully suppressed.In this gapless phase,excitations involve particle-hole excita-tions and spin-string excitations.The TBA equations (41)can be greatly simpli?ed in the strong coupling regime due to the

suppression of spin ?uctuations,where à1‘$e

à‘H=T

!0as T !0.Thus one can extract the universal TLL physics using Sommerfeld expansion for temperatures less than chemical potential and magnetic ?eld.

0.20.40.60.81

FIG.10.Charge and spin velocities vs dimensionless interaction strength for the 1D balanced Fermi gas.The solid lines are the velocities obtained by numerically solving the BA equations (11).The dotted lines denote the analytical result for the velocity in the weak coupling regimes.From Batchelor et al.,2006a .

Rev.Mod.Phys.,V ol.85,No.4,October–December 2013

In fact,in this spinless phase,the spin?uctuation is sup-pressed in the limit T!0and j j)1.Thus the bound pairs and unpaired fermions form a two-component TLL. Conformal invariance predicts that the energy per unit length has a universal?nite-size scaling form that is characterized by the dimensionless number C,which is the central charge of the underlying Virasoro algebra(Af?eck,1986;Blo¨te, Cardy,and Nightingale,1986;Cardy,1986).The?nite-size corrections to the ground state energy have been analytically derived(Lee and Guan,2011)

"0?"10àC

6L2

?u;b

v ;(43)

where C?1with v u and v b the velocities of unpaired fermions and bound pairs,respectively.For strong interaction, they are given explicitly by

v b%

2A2

j c j

3A22

;

v u%

2 n1

2A1

j c j

3A21

;

(44)

where A1?4n2,A2?2n1tn2,and n2?n#.We describe universal behavior of the macroscopic properties of this Fermi gas in Secs.III.E and III.F.

Although a phase transition in1D many-body systems at ?nite temperatures does not exist,the system does exhibit universal crossover from relativistic dispersions to quadratic dispersions.Thus at low temperatures,the bound pairs,nor-mal Fermi gas,and the FFLO phase become relativistic TLLs of bound pairs(TLL P),unpaired fermions(TLL F),and a two-component TLL(TLL PP),respectively;see Fig.11.A de-tailed discussion has been given(Zhao et al.,2009;Yi,Guan, and Batchelor,2012).For the temperature k B T(E F,where E F is the Fermi energy,the leading low-temperature correc-tion to the free energy of the polarized gas can be calculated

(42),namely(Guan et al.,2007;He et al.,2009;Zhao et al.,2009;Batchelor et al.,2010)

FeT;HT%

>>>

E0eHTà Ck2B T2

v b

v u

T;for TLL PP;

E0eHTà Ck2B T2

v b

;for TLL P;

E0eHTà Ck2B T2

v F

;for TLL F;

(45)

which belongs to the universality class of the Gaussian model with central charge C?1.For strong attraction,the veloc-ities are given in Eq.(44).In Eq.(44),the ground state energy E0eHTis as given in Sec.II.B.1.In fact,from the TBA equations(41),the universal thermodynamics(45)can be shown to be valid for arbitrary interaction strength.

The two branches of gapless excitations in the1D FFLO-like phase form collective motions of particles.The low-energy(long wavelength)physics of the strongly attractive Fermi gas is described by an effective Hamiltonian

H eff?

v u

?e@x uT2te@x uT2 t

v b

?e@x bT2te@x bT2 à

@x u

????

pà

@x ut2@x b

????

p(46)

as long as the spin?uctuation is frozen out(Vekua, Matveenko,and Shlyapnikov,2009;Zhao et al.,2009). Here the?elds@x i,@x i with i?b,u are the density and current?uctuations for the pairs and unpaired fermions. However,in the spin gapped phase,i.e.,for H

E.Quantum criticality and universal scaling

As seen,the1D attractive Fermi gas exhibits various phases of strongly correlated quantum liquids and is thus particularly valuable to investigate quantum criticality.Near a quantum critical point,the many-body system is expected to show universal scaling behavior in the thermodynamic quan-tities due to the collective nature of the many-body effects.In the framework of Yang-Yang TBA thermodynamics,exactly solvable models of ultracold atoms,exhibiting quantum phase transitions,provide a rigorous way to treat quantum criticality in archetypical quantum many-body systems,such as the Gaudin-Yang Fermi gas(Guan and Ho,2011),the Lieb-Liniger Bose gas(Guan and Batchelor,2011),a mixture of bosons and fermions(Yin et al.,2012),and the spin-1 Bose gas with both delta-function and antiferromagnetic interactions(Kuhn et al.,2012a,2012b).

At zero temperature,the quantum phase diagram in the grand canonical ensemble can be analytically determined from the so-called dressed energy equations(Takahashi, 1999;Guan and Batchelor,2011;Guan and Ho,2011)

FIG.11(color online).Quantum phase diagram of the Gaudin-Yang model in the T-H plane showing a contour plot of the entropy in the strong interaction regime.The dashed lines are determined from the deviation from linear-temperature-dependent entropy ob-tained from the result(45).The universal crossover temperatures separate the TLLs from quantum critical regimes.From Yi,Guan, and Batchelor,2012.

Rev.Mod.Phys.,V ol.85,No.4,October–December2013

be?T?2

?2à à

àZ A

àA2

K2e?à?0T be?0Td?0

àZ A

àA1

K1e?àkT uekTdk;

uekT?k2à àH

Z A

àA2

K1ekà?T be?Td?;(47)

which are obtained from the TBA equations in the limit T!0.The integration boundaries A2and A1characterize the Fermi surfaces for bound pairs and unpaired fermions, respectively.It is convenient to use dimensionless quantities where energy and length are measured in units of binding energy"b and cà1,respectively.In terms of the dimensionless quantities~ :? ="b,h:?H="b,t:?T="b,~n:?n=j c j? à1,and~p:?P=j c"b j,the phase boundaries have been determined analytically from Eq.(47);see Guan and Ho (2011).There are four phases denoted by vacuumeVT,fully paired phaseePT,ferromagnetic phaseeFT,and partially pairedePPTor(FFLO-like)phases presenting the same phase diagram as in Fig.9.

The low density and strong coupling limits are particularly important to study quantum criticality.In fact,the TBA equations(41)can be converted into a dimensionless form with the above rescaling.Following the notation used by Guan and Ho(2011),the phase boundaries between VàF, VàP,FàPP,and PàPP are denoted by c1to c4, respectively.The closed forms of the critical?elds

c1?àh

; c2?à

;

c3?à1

1à

ehà1T3=2à

3 2

ehà1T2

;

c4?àh

e1àhT3=2t

2 2

e1àhT2

(48)

are needed to determine scaling functions of thermodynamic properties.Here c1, c2applies to all regimes and c3, c4 are expressions in the strongly interacting regime.The above critical?elds c3, c4correspond to the upper and lower critical?elds in the h-n plane,which can be found in Guan et al.(2007),Iida and Wadati(2007),and He et al.(2009). The TBA equations(40)and(41)encode the microscopic roles of each single particle that lead to a global coherent state—quantum criticality.Quantum criticality is manifested by universal scaling of thermodynamic properties near the critical points.The key input to obtain critical scaling behavior is to derive the form of the equation of state which takes full thermal and quantum?uctuations at low tempera-tures into account.The dimensionless form of the pressure (Guan and Ho,2011)

~pet;~ ;hT:?p=j c j"b?~p bt~p u(49) serves as the equation of state,where to Oec4Tthe pressures of the bound pairs and unpaired fermions are given by

~p b?à

t3=2

????

p F b

3=2

~p b

t2~p u

tOec4T;

~p u?à

t3=2

???????

p F u

3=2

?1t2~p b tOec4T;

(50) with in addition

X b

~p b

4~p u

t3=2

????

f b

5=2

???

f u

5=2

;

X u

2~p b

t3=2

????

p f b

5=2

teàh=t eàK I0eKT:

In these equations the functions F b n,F u n,f b n,and f u n are de?ned by F b;u n:?Li neàe X b;u=tTand f b;u n:?Li neàe b;u=tT,with the notation b?2~ t1, u?~ th=2.The function

Li sezT?

k?1

z k=k s

is the polylog function and

I0exT?

k?0

Despite the equation of state(49)having only a few leading terms in expansions with respect to the interaction strength, it contains thermal?uctuations in contrast to the TLL thermodynamics(45).

The TLL thermodynamics(45)has been derived from low-temperature expansion along T(j à c j.This universal thermodynamics is a consequence of the linearly dispersing phonon modes(Maeda,Hotta,and Oshikawa,2007),i.e.,the long wavelength density?uctuations of two weakly coupled gases or a gas of bound pairs or single fermions.The quantum critical regime lies beyond T)j à c j.In this limit,the equation of state(49)provides closed forms for the scaling functions of thermodynamic quantities,such as density,mag-netization,and the compressibility.Near the critical point,the thermodynamic functions can be cast into a universal scaling form(Fisher et al.,1989;Sachdev,1999).The explicit universal scaling form of the density for T)j à c j is

eVàFT~n%à

???????

p Li

1=2

eàee~ à c1T=tT;

eFàPPT~n%n o3à 1

Li1=2eàe2e~ à c3T=tT;

eVàPT~n%à

????

p Li

1=2

eàe2e~ à c2T=tT;

ePàPPT~n%n o4à 2

Li1=2eàee~ à c4T=tT:

(51)

Here the constants n o3and n o4are the background densities near the critical points 3and 4.These constants,together with a and b,are known explicitly in terms of h(Guan and Ho,2011).

Rev.Mod.Phys.,V ol.85,No.4,October–December2013