Quantum dynamics of an Ising spin-chain in a random transverse field

a r X i v :c o n d -m a t /0607703v 3 [c o n d -m a t .s t r -e l ] 14 S e p 2006

Quantum dynamics of an Ising spin-chain in a random transverse ?eld

Xun Jia and Sudip Chakravarty ?

Department of Physics and Astronomy,University of California Los Angeles,Los Angeles,CA 90095-1547

(Dated:February 6,2008)

We consider an Ising spin-chain in a random transverse magnetic ?eld and compute the zero temperature wave vector and frequency dependent dynamic structure factor numerically by using Jordan-Wigner transformation.Two types of distributions of magnetic ?elds are introduced.For a rectangular distribution,a dispersing branch is observed,and disorder tends to broaden the dispersion peak and close the excitation gap.For a binary distribution,a non-dispersing branch at almost zero energy is obtained.We discuss the relationship of our work to the neutron scattering measurement in LiHoF 4.

Calculation of real-time dynamics of a correlated quan-tum system with an in?nite number of degrees of freedom are few and far between.Except for isolated examples,construction of real-time behavior from imaginary-time correlation functions (more amenable to numerical meth-ods)by analytic continuation is fraught with various in-stabilities.The theoretical challenge is particularly acute because neutron scattering experiments often provide a rather detailed map of the frequency,ω,and the wave vector,k ,dependent dynamical structure factor,S (k ,ω).The second motivation comes from the desire to study the dynamics of a quantum phase transition involving a zero temperature quantum critical point.In this respect,the Ising spin chain in a transverse ?eld 1,2,3,4constitutes a schema from which much can be learned about quan-tum criticality,5both with and without disorder.

The third motivation is to examine how the coherence of the quasiparticle excitations is modi?ed in the presence of quenched disorder and is triggered by a recent neutron scattering experiment 6in LiHoF 4,which connects the ob-served low temperature behavior of S (k ,ω)in terms of the hyper?ne coupling of the electronic spins to a nu-clear spin bath,where the Hamiltonian of the electronic spins is given by an Ising model in a transverse ?eld.Be-cause the hyper?ne splittings are small,we could imagine that,on the time scale of the electronic motion,this bath will appear essentially quenched,modulating the quan-tum ?uctuations characterized by the transverse ?eld.This is still a bit far from the experimental system,as it is three-dimensional,and the Ising couplings are long-ranged and dipolar.Nonetheless,we shall see that there are tantalizing similarities between our calculated struc-ture factor and the experimental one in a given direction of the reciprocal space (2,0,0)→(1,0,0)(in reciprocal units).A more appropriate comparison should be with quasi-one dimensional spin systems that are intentionally disordered.

Finally,the role of disorder in a quantum critical sys-tem is an important subject in itself and is certainly not fully understood.In the presence of disorder,there are rare regions with couplings stronger than the average,which results in the Gri?ths-McCoy singularities 7,8.Al-though the e?ect is weak in a classical system,it be-comes important in quantum systems,especially in low dimensions 9,10.

The Ising model in a random transverse ?eld has been studied both analytically and numerically,9,11,12,13,14,15,16,17,18,19and a great number of results of physical importance have been obtained.However,the dynamical structure factor S (k,ω)has not been computed for all k and ωin the random ?eld model,although some analytical and numerical results are available in pure systems for special values of the wave vector.Here we compute the dynamic structure factor in the presence of two types of disorder distributions:a rectangular and a binary distribution.The one-dimensional lattice Hamiltonian we study is

H =?J

i

σz i σz i +1?

i

h i σx

i ,

(1)

where the σ’s are Pauli matrices,and J is positive and

uniform.We shall choose the energy unit such that J =1.0.The ?elds h i are random variables.The ?rst model we study is the rectangular distribution with a mean h ave and a width h w .The second model is the bi-nary distribution in which h i is an independent random variable that takes two values:h S and h L with probabil-ities p and (1?p ),respectively.In particular,we choose the parameter p to be small such that the chain is almost spatially homogeneous,h S

We ?rst compute the time-dependent spin-spin corre-lation functions C (n,t )=

2 factor is,

S(k,ω)= n e?ikn dte?iωt C(n,t).(2)

The pfa?an is the square root of the determinant of an

antisymmetric matrix.Let X be an N×N(N is even)

antisymmetric matrix of the form:

X= A B

?B T C

,(3)

where A is a2×2matrix,and B,C are matrices of

appropriate dimensions.From the identity:

I20 B T A?1I(N?2) X I2?A?1B 0I(N?2)

= A0

0C+B T A?1B (4)

where I n is the unit matrix of dimension n,we have:

Det(X)=Det(A)Det(C+B T A?1B)(5) Since A is also antisymmetric,of the form

A= 0a12

?a120

,(6)

it is easy to invert A,and hence calculate C+B T A?1B. Since the matrix C+B T A?1B is also antisymmetric of dimension(N?2),the above procedure can be repeated, and the determinant of X is given by the product of 2×2determinants.Finally,since

3

though the peak is not a delta function given the?nite size of our system;see Fig.2(d).

In contrast,the binary distribution is quite remark-able.We?rst choose h L=1.4so that the system is in the paramagnetic regime,and set h S=0.1,and p=0.05. The density of states in Fig.3shows zero energy states separated by a gap from the states at higher energy. The calculated S(k,ω)is shown in Figs.4(a)through (f),where h L=1.1,1.2,...,1.6.While the dispersing branch is broadened due to disorder,the weight of the central peak around(k,ω)=(0,0)is so high that a non-dispersing branch can extend quite far away from the origin.

0.0 2.0 4.0

FIG.3:(Color online)The density of statesρ(ω)for the binary distribution at h L=1.4.Clearly,a few new states are allowed at zero energy,as the disorder is turned on.

For Fig.4,see Graph4.jpg.

FIG.4:(Color online)(a)～(f)Dynamic structure factor S(k,ω)for the binary distribution in the paramagnetic phase. Parameters are J=1.0,h S=0.1,p=0.05,and h L= 1.1,1.2,...,1.6from(a)through(f).Both dispersing and non-dispersing branches exist,and the excitation is gapped. In Fig.5,we show a cut of S(k,ω)at k=0,to illustrate that there is hardly any di?erence between the results averaged over20realizations and those averaged over100 realizations.

Let us de?ne the weight of the central peak as

I h= ??S2(k,ω)d k dω

2for (k,ω)∈??.Note that in(7)we integrate S2(k,ω) rather than S(k,ω).This is

because small numerical er-rors can result in slightly negative values of S(k,ω)for some(k,ω),which is obviously unphysical.The depen-dences of I h and??on h L are plotted in Fig.6.As we FIG.5:(Color online)A cut of S(0,ω)corresponding to Fig.4(d)for20and100realizations of disorder.

tune h L from1.6to1.1,I h increases monotonically,while the region??shrinks.We conclude that,as the quan-tum phase transition is approached,the weight is trans-ferred from the dispersing branch to the non-dispersing peak.

I

n t

e

n

s i

t

y

1.0 1.2 1.4 1.6

A r

e

a

h

FIG.6:The binary distribution:as the quantum phase transi-tion is approached from the paramagnetic regime,the central peak intensity I h increases monotonically,and the area??corresponding to the half the value of the peak decreases.

In addition to the singularity due to the quantum phase transition,in disordered systems there is also the Gri?ths-McCoy singularity:the disorder will drive some rare regions into a phase di?erent from the rest.For the pure Ising chain when h i=h,the dispersion relation is2

ω=2

4 spins are strongly coupled,such that h i

thus the cluster is ferromagnetic.Calculations,similar to

that given in Ref.5,21show that these clusters give rise

to the non-dispersing peak at zero energy,as we describe

below.

For the binary distribution,the system is almost ho-

mogeneous except for the rare regions of strongly coupled

clusters where h i=h S for all sites inside the cluster.At

shorter length scales,the behavior of the pure system

dominates,and this leads to the dispersing branch.At

longer scales the e?ects of disorder become important,

resulting in the zero energy peak.The autocorrelation

function S(ω),which is the integral of S(k,ω)over k can

be approximated in the following manner.The normal-

ized probability that a given site belongs to a ferromag-

netic cluster of length L-sites is P(L)=Lp L?1(1?p)2.

When h i=0the two-fold degenerate ground state within

a cluster is far away from the excited states of energy of

order2J;the perturbation h i=h S,will split the ground

state by?g e?cL,where?g and c are unknown positive con-

stants determined by the details of the Hamiltonian.The

form of the splitting results from large order in pertur-

bation theory,however.If we treat these clusters as in-

dependent and average over disorder,or equivalently in-

tegrate over the cluster size L,we get

S(ω)～ d LP(L)δ(ω??g e?cL)

～(1?p)2ω(?g

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