Quantum Spin Glass Phase Boundary in (+-)J Transverse Field Ising Systems

a r X i v :c o n d -m a t /0310381v 1 [c o n d -m a t .s t a t -m e c h ] 16 O c t 2003

Quantum Spin Glass Phase Boundary in ±J Transverse Field Ising Systems 1

Arnab Das ?,Amit Dutta ?and Bikas K.Chakrabarti ?

?

Saha Institute of Nuclear Physics,1/AF Bidhannagar,Kolkata -700064,India ?

Physics Department,Indian Institute of Technology,Kanpur -208016,India

Abstract:Here we study zero temperature quantum phase transition driven by the transverse ?eld for random ±J Ising model on chain and square lattice.We present some analytical results for one dimension and some numerical results for very small square lattice under periodic boundary condition.The numerical results are obtained employing exact diagonalization technique following Lanczos method.

Keywords:Quantum phase transition,Spin glasses,Transverse Ising model,Diagonalization techniques.

PACS Nos.:05.70.Fh,42.50.L,75.10.N 1.

Introduction

The interest in the study of transverse Ising spin glass models was revived in early 1990by the discovery of zero-temperature transition in dipolar Ising transverse ?eld magnet LiHo x Y 1?x F 4[1].Proton glasses such as mixture of ferroelectric and anti-ferroelectric materials like Rb (1?x )(NH 4)x (H 2P)4[2]also provided earlier use-ful realizations of such quantum spin glasses.

These developments initiated extensive theoretical studies in quantum spin glass models.Ising model in transverse ?eld has already been studied extensively in this context through analytical approaches using approximate renormalization techniques and real space renormalization group method,as well as using nu-merical methods like quantum Monte Carlo and exact diagonalization techniques [3].Fairly extensive studies on the quantum spin glass phases have been made

so far for transverse?eld Edward-Anderson model and Sherrington-Kirkpatrick model with random exchange distributions[3,4]and to some extent on quan-tum Heisenberg spin glass models[4].In view of the rigorous developments in the study of two dimensional nearest neighbour Ising model with random±J exchange interactions and the precise knowledge of location of the Nishimori line [5]in such classical spin glass model(driven by temperature),we consider here the quantum phase transition(at zero temperature)in the same±J Ising model. We have shown analytically that introduction of random?J impurities cannot a?ect the zero temperature phase transition in one dimensional system as they can be transformed away.We have also compared and veri?ed the result nu-merically for the small system size considered.For two dimensional systems,we present some priliminary results obtained for a square lattice using exact diag-onalization results for very small system sizes following Lanczos technique[6]. Only the behaviours of con?gurationally averaged energy gap?=(E1?E0)be-tween the?rst excited state and the ground state and the second order response functionχ=(?2E0/?Γ2),equivalent to speci?c heat,have been studied here. The variations of?andχwith respect to transverse?eldΓhave been obtained, and the phase boudary has been estimated from these results.

We work with a transverse Ising system,using only nearest neighbour interac-tions,whose Hamiltonian is given by

H=? i,j J ij S z i S z j?ΓN i=1S x i,(1)

where the transverse?eldΓis uniform through out the system and the nearest neighbour exchange constant J ij’s are chosen randomly from the binary distribu-tion

P(J ij)=pδ(J ij+J)+(1?p)δ(J ij?J).(2) Here J is taken positive and p is thus the concentration of anti-ferromagnetic?J bonds in the system.

2.Results in One Dimensional System

Here?rst we show analytically that in a one dimensional transverse Ising Hamil-tonian with uniform J andΓ,if we replace some J bonds by?J bonds randomly,

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then the resulting Hamiltonian can be gauge transformed back to one with uni-form J,and hence the critical?eld remains unchanged with randomness concen-tration.Simillar result for one dimensional system with distributed J had been obtained earlier[7].

Let us take the one dimensional random bond tranverse Ising Hamiltonian

H=? i J i S z i S z i+1? ΓS x i,(3)

where the transverse?eldΓis uniform throughout the system,and J i’s are ran-domly chosen from the same distribution as given in(2).Since the J i’s have same magnitude J all through,and their randomness is only in their sign,we may write J i=J sgn(J i),and thus Hamiltonian(3)takes the form

H=?J sgn(J i)S z i S z i+1?Γ S x i.(4) Now let us de?ne a new set of spin variables as below

?S z i =S z i

i?1 k=1sgn(J k)

?S x

i

=S x i

?S y i =S y i

i?1

k=1sgn(J k).

It is easy to see that?S’s satisfy the same commutation and anti-commutation relations as those of S’s and hence will exhibit exactly the same dynamical be-haviour.Now,

?S z i ?S z

i+1

=S z i S z i+1 i?1 k=1[sgn(J k)]2 sgn(J i),

or,

?S z i ?S z

i+1

=S z i S z i+1sgn(J i),

since[sgn(J k)]2=1.Thus in terms of new spin variables,Hamiltonian(4)be-comes

H=?J i?S z i?S z i+1?Γ i?S x i.(5)

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The above Hamiltonian describes the same random system in terms of new vari-ables,and yet,as one can see,it has in itself no randomness at all.One can use Jordon-Wigner transformation in terms of?S’s and see that here also quan-tum phase transition occurs only atΓ≥Γc(=J)as it occurs in a non-random Hamiltonian in S’s.In Fig.1,we present some data computed for a chain of size N=9,which shows that the gap?vanishes atΓc≈1(the?eld being scaled by J).These data for?=E1?E0is obtained from the computed average value of E0and E1,each one averaged over about10con?gurations for p=0.For in?nite system,?is a linear function ofΓforΓ≥Γc.In our case,linearity is observed at high values ofΓ,andΓc is determined by backward linear extrapolation from the linear region.

?

Γ

Figure1:A numerical estimate of the energy gap?for a chain with N=9. Phase boundary is obtained from the location of?(Γ)=0,and is shown in

the inset.

From the numerical data in Fig.1,we see that there is a slight variation ofΓc with p(Γc varies between0.9and1.0).This variation can be attributed to the very small size of the system.However,it may be noted that with even number of?J bonds in the chain,with periodic boundary condition,there is no prob-

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lem of incommensuration and E0(Γ)or E1(Γ)become strictly identical for such values of p.Similarly,in every case of odd number of?J bonds in the chain, incommensuration problem always occurs for one spin only,rendering identical values(but di?erent from the even?J case)for E0(Γ)and E1(Γ)in all such cases.

3.Results for Square Lattice

We consider now the same system(represented by Hamiltonian(1))on a square lattice of size3×3with periodic boundary condition.We again calculate the ground state and the?rst excited state energy E0and E1respectievely as func-tions of the transverse?eldΓ,for di?erent values of p.Each value of E0and E1 is averaged over at least10con?gurations for each p=0.Apart from?,we also calculateχ=?2E0/?Γ2and their variations withΓas shown in Figs.2and3 respectively.

?

Γ

Figure2:A numerical estimate of the con?gurational avrage of the energy gap?for a square lattice of size3×3.Phase boundary obtained from?(Γ)=

0is outlined in the inset.

Our results here are severely constrained by the system size.The value of pure ferromagnetic critical?eldΓc(p=0)is found here to be around2.2,while the

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χ

Γ

Figure3:Here variation ofχ=?2E0/?Γ2withΓis shown.The transition point occurs atΓ=Γc whereχdiverges;for?nite system one gets only a peak inχatΓ=Γc(p).We have outlined the corrosponding phase boundary

in the inset.

series results[3]or cluster algorithms[8]give the value to be around3.0.This dis-crepancy is attributed to the smallness of our system size(N=32).However the qualitative behaviour of the order-disorder phase boundary(between ferro/spin glass and para)seems to be reasonable:Γc(p)decreases with p initially,and then increases again as p approaches unity(pure anti-ferromagnet).The use of peri-odic boundary condition here(to avoid some numerical errors)also restricts the domain features and thereby a?ects our results.The absence of the knowledge of the ground-state wave function(and the correlation functions)in this method also forbids us to analyse the structure of the ordered phases. Acknowledgement:BKC is grateful to Hidetoshi Nishimori for useful discus-sions.

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