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Quantum Critical Behavior of the Infinite-range Transverse Ising Spin Glass An Exact Numer


Quantum Critical Behavior of the In?nite-range Transverse Ising Spin Glass : An Exact Numerical Diagonalization Study

arXiv:cond-mat/9705297v1 [cond-mat.stat-mech] 29 May 1997

Parongama Sen
Institute of Theoretical Physics, University of Cologne, Zulpicher Strasse 77, 50937 Cologne, Germany e-mail: paro@thp.uni-koeln.de

Purusattam Ray
The Institute of Mathematical Sciences, C. I. T. Campus, Chennai 600 113, India e-mail: ray@imsc.ernet.in

Bikas K. Chakrabarti
Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700064, India e-mail: bikas@saha.ernet.in

Abstract
We report exact numerical diagonalization results of the in?nite-range Ising spin glass in a transverse ?eld Γ at zero temperature. Eigenvalues and eigenvectors are determined for various strengths of Γ and for system sizes N ≤ 16. We obtain the moments of the distribution of the spin-glass order parameter, the spin-glass susceptibility and the mass gap at di?erent values of Γ. The disorder averaging is done typically over 1000 con?gurations. Our ?nite size scaling analysis indicates a spin glass transition at Γc ? 1.5. Our estimates for the exponents at the transition are in agreement with those known from other approaches. For the dynamic exponent, we get z = 2.1 ± 0.1 which is in contradiction with a recent estimate (z = 4). Our cumulant analysis indicates the existence of a replica symmetric spin glass phase for Γ < Γc .

PACS nos : 75.10.Jm, 75.10 Nr, 64.60.Cn, 64.60.Fr

0

Quantum phase transitions in disordered systems have been studied intensively in recent years [1]. Of particular interest is the Ising spin glass model in a transverse ?eld which provides a rather simple model where it can be shown [2] that at zero temperature, the spinglass ordered ground state is destabilised by quantum ?uctuations. As a result, if one varies the strength Γ of the transverse ?eld, there can be a phase transition between the spin-glass ordered and disordered ground states at a critical value of Γ = Γc . The case of in?nite range model is specially interesting since in the absence of the transverse ?eld, the model reduces to the usual Sherrington-Kirkpatrick (SK) model [3]. The classical SK-model is very well studied. It has a ?nite temperature transition to a low temperature spin glass phase where replica symmetry is broken [4]. In presence of the transverse ?eld Γ, the spin glass ordering occurs at lower temperatures and above a critical value of the ?eld, the quantum ?uctuations destroy the order. At non-zero temperatures, however, quantum ?uctuations are unimportant in determining the critical behavior. In this letter, we are interested in the SK-model in a transverse ?eld at zero temperature, where the transition between the spin glass to disordered ground state is driven by quantum ?uctuations alone through the tuning of the strength of the transverse ?eld Γ. There has been quite a number of studies on this model at recent times. The perturbation expansion of the free energy by Ishii and Yamamoto [5] gives the value of the critical ?eld Γ ? 1.5 and the susceptibility exponent ≈ 0.5. The non-perturbative analysis [6], the mean?eld theory of quantum rotors [7] and the numerical techniques such as quantum Monte Carlo method [8] give consistent estimates of the critical ?eld and the critical exponents except the value of the dynamic exponent z. The Monte Carlo study suggests the value of z = 4, much too higher than the value z = 2 obtained in other studies. In a recent work, the Schr¨dinger equation for the model has been solved numerically [9] and the interaction o energy and longitudinal susceptibility are determined. The estimates of the critical ?eld and the exponents from this study match with the analytical results. But, in this method the exponents are obtained not directly individually, but rather through a scaling relation. An important observation that comes out of the mean ?eld theory is the possibility of a replica symmetric spin glass phase in quantum transition below Γ << Γc [10,7]. This point certainly deserves further attention. It may be noted that most of these results are obtained using the approximate classical mapping of the quantum systems. Here we report the results of exact diagonalization of the in?nite range Ising spin glass model in transverse ?eld, using the Lanczos algorithm for system sizes N ≤ 16. Con?guration averaging is done typically over 1000 realizations of bond values (for low N). We obtain the various moments of the order parameter distribution P (q) and the mass gap. The cumulant analysis of P (q) provides an estimate of the critical ?eld Γc ? 1.5 in the limit of large N. This value of Γc is comparable to the values obtained from other studies. For spin glass susceptibility and the mass gap, our ?nite size scaling analysis follows the idea of the phenomenological renormalization group transformation [11]. Our estimate of the correlation length exponent ν = 0.252 ± 0.004, the susceptibility exponent γ = 0.51 ± 0.02 and the dynamic exponent z = 2.1 ± 0.1 (obtained from the scaling of the mass gap [12]) compares well with the values obtained from other studies. Our estimate of z does not match with the quantum Monte Carlo result [8] which we think is too high and may be an artifact of the size e?ect along the Trotter direction. Our cumulant analysis shows that for Γ << Γc , P (q) attains a two peak structure (related by the spin inversion symmetry), the 1

peaks becoming narrower with N. This is similar to what one gets in the classical SK-model above the Almeida-Thouless line (see [4]) where the glass phase is replica symmetric. The extended width of P (q), characteristic of replica broken phase, is clearly absent in our study. The SK model in a transverse ?eld is described by the Hamiltonian
N N z Jij Siz Sj ? Γ

H=
<ij>

Six
i

(1)

where the Jij ’s are long ranged and follow the Gaussian distribution D(Jij ) = (
2 ?NJij N 1/2 ) exp( ) 2πJ 2 2J 2

and < ij > in the summation denotes that each pair of spins is taken only once. Siz and Six are Pauli spin matrices. We have performed exact diagonalisation of the Hamiltonian (1) using Lanczos algorithm for N = 4, 6, 8, 10, 12, 14 and 16 and repeated for various realisations of Jij values. For each realization of Jij , all the relevant quantities like the order parameter etc. are computed and then averaged over all the realizations. For the diagonalisation precedure, the basis states |φα > are chosen to be the eigenstates of the spin operators {Siz , i = 1, N}. In the absence of the transverse ?eld term, the Hamiltonian (1) reduces to the Hamiltonian of the S-K model and is diagonal in this representation. With the transverse ?eld present, each eigenstate of (1) is obtained as a superposition of the basis states which are 2N in number (corresponding to Siz = ±1). The nth eigenstate for the quantum Hamiltonian is written as
2N

|ψn >=

α=1

an |φα > α

and the ground state is denoted by |ψ0 >. We calculate the moments of the distribution P (q) of the order parameter q = (1/N) i < Siz >2 . The overhead bar indicates the con?guration average and the < ... > denotes the expectation values at the ground state. P (q) is not directly obtained here, rather the moments mk = qk = 01 q k P (q)dq are calculated using [4], 1 qk = k N
N N

....
i1 ik

< Siz1 ...Sizk >2 .

(2)

It can be easily shown that the RHS of the above equation gives the kth moment of the distribution. We take for example q2 = 1 N2
N i1 N i2

[< Siz1 Siz2 >< Siz1 Siz2 >]

Now, < Siz1 Siz2 > in the ground state is given by < ψ0 |Siz1 Siz2 |ψ0 >. The latter quantity N is again given in terms of the basis states, i.e., < ψ0 |Siz1 Siz2 |ψ0 >= 2 |a0 |2 < Siz1 Siz2 >α , α α where < ... >α denotes the product of the i1 th and i2 th spin in the αth basis state. Writing |a0 |2 = ωα , the RHS of (2) is therefore given by α 2

=

1 N2

N i1

N

2N

2N

[
i2 α

ωα < Siz1 Siz2 >α
β

ωβ < Siz1 Siz2 >β ].

After con?gurational averaging, it gives m2 = dqPJ (q)q 2 . The exact diagonalisation procedure gives us the ωα ’s. Using the above, we have calculated the Binder cumulant [13] for a value of N and Γ as 1 m4 gN (Γ) = [3 ? ] 2 (m2 )2 and the nonlinear susceptibility
N

(3)

χ = (1/N)
ij

z < Siz Sj >2 .

(4)

We calculate the order parameter and the spin glass susceptibility which are given by the ?rst and second moments of the distribution P (q). While calculating the moments of q, we take a set of half of the basis states, ignoring the other half, which consists of the trivially degenerate ones and obtained by simply ?ipping the spins in the states of this set. This is necessary as otherwise, even in the ferromagnetic case one will get < Siz >= 0, as the two degenerate states are equally probable. The quantities have to be suitably rescaled to get the proper values. The energy eigenvalues of the nth state is given by En =< ψn |H|ψn >. The mass gap is ?E = E1 ? E0 (4)

where E0 is the ground state energy and E1 is the energy of the ?rst excited state. We determine the average mass gap ?EN (Γ) for di?erent values of of Γ and N. We estimate ?rst the cooperative energy per spin, which is the expectation value of the cooperative part of the Hamiltonian in the ground state (in this case, it is simply obtained by operating the ?rst term of the Hamiltonian (1) on |ψ0 >). The asymptotic value of this quantity, which gives the classical ground state energy per spin in the S-K model (in the limit Γ → 0), is found to be equal to ?0.74 ± 0.01 (in units of J). This agrees very well with the ground state energy (?0.76) of the classical SK model [4]. The Binder cumulant gN (Γ) as a function of Γ is shown in Fig. 1 for di?erent values of N. The form of gN (Γ) changes with the system size and near Γc is expected to scale with N in the following way [8] gN (Γ) ? f ((Γ ? Γc )N x ) (5) where f (X) is a scaling function and x is related to the mean ?eld correlation length exponent ν. gN (Γ) versus Γ for various N then yield a family of curves. These curves intersect at a common point which gives the value of Γc . Due to the corrections to ?nite size scaling, the intersections of the gN (Γ) curves for all N values may not coincide at a common point (see [4] for discussion). We determine the point of intersection Γc (N, N ′ ) of the g(Γ) curves for two system sizes N and N ′ and plot it against 1/(NN ′ )1/2 in Fig. 2. The value of Γc (N, N ′ ) converges to Γc ? 1.5 ± 0.1 in the limit of N, N ′ → ∞. This value of Γc is comparable to 3

the value obtained from the non-perturbative analysis [6], the Pad? treatment [14] and the e solution of the Schr¨dinger equation [9]. o The ?nite size scaling concept of Fisher and Barber extended to the long-ranged systems suggests that the exponent x = 1/(νmf dc ), where νmf is the mean ?eld correlation length exponent and dc is the upper critical dimensionality of the corresponding short range system [15]. For any pair of systems with total spins N and N ′ , we ?rst ?nd out the point of intersection Γc (N, N ′ ). With Γc = Γc (N, N ′ ) we try to scale gN (q) and gN ′ (q) around Γc with a suitable value of x, so that all the data points fall on the same scaling curve f (X). √ Fig. 3 shows the value of x against 1/ NN ′ . The value of x is almost independent of N and we get x = 0.66 ± 0.01. If we take dc = 6 which is the upper critical dimension of the classical Ising spin glass we get ν = νmf = 0.252 ± 0.004 in agreement with the predicted value of ν = 1/4 [7,8]. The ?nite size scaling form for the spin glass susceptibility χ can be written as χN (Γ) ? N y Φ((Γ ? Γc )N x ) (6)

where the exponent y = xγ, γ being the exponent which characterizes the divergence of the susceptibility at Γc in the thermodynamic limit. Φ(X) is a scaling function. The ?nite size analysis can be done invoking the idea of the phenomenological renormalization group transformation (see [11] for a review). If Φ(X) is a power function of X, then the above scaling form is satis?ed exactly for any two ?nite systems of spin numbers N and N ′ in the sense that χN (Γ)/N y = χN ′ (Γ′ )/N ′y with the recursion relation Γ ? Γc = (Γ′ ? Γc )(N ′ /N)x . The ?xed point Γ? of the transformation satis?es χN (Γ? ) N = ?) χN ′ (Γ N′
y

,

where Γ? → Γc as N, N ′ → ∞. For any two sizes N and N ′ , we take Γ? from the intersection of the curves gN (Γ) and gN ′ (Γ) and apply the above ?xed point equation to get the value of y. The resulting estimate of y depends on N and N ′ and approaches the exact value as N, N ′ → ∞ (see [16] for discussion). Fig. 4 shows the value of y plotted against 1/Nm, where Nm = (NN ′ ), from which we determine the asymptotic value of y = 0.33 ± 0.01. From y and x, we get γ = 0.51 which agrees with the value 1/2 obtained in other studies [7,8]. The scaling form for the mass gap ?EN (Γ) can be written as [12] ?EN (Γ) ? N 1?z Ψ((Γ ? Γc )N x ) where Ψ is a scaling function. Employing the same scaling analysis as above, for the mass gap, we obtain the value of the exponent z. Fig. 5 shows the value of z plotted against 1/Nm . We ?nd the asymptotic value of value z = 2.1 ± 0.1. This value of z agrees with its predicted value in [7] but disagrees with the quantum Monte Carlo simulation result z = 4. Proceeding in the same way, we ?nd β = 1.0 ± 0.1 where β is the order parameter exponent, i.e., in the thermodynamic limit q ? (Γ ? Γc )β . This is in agreement with the previous studies [1]. ?From Fig. 1, we ?nd that the behavior of the Binder cumulant is very similar to what happens in normal spin system. If g(T ) is the value of the cumulant in the thermodynamic 4

limit at any temperature T , then for a normal ferromagnetic system it is expected that g(T ) = 1 for T << Tc and g(T ) = 0 for T >> Tc . The value of g(TC ) at TC depends on the dimensionality of the system. At low T -values, the value attained by the Binder cumulant depends on N and extrapolates to the value unity in the thermodynamic limit. This coincides with the fact that the order parameter distribution (P (m) of magnetisation m for a ferromagnet) reduces to a delta function (modulo symmetric operation) in the limit N → ∞ and the system goes to a de?nite symmetry broken state. This is not the case where replica symmetry is broken like in classical SK model at low temperatures and zero external ?eld or spin glass models in higher dimensions. Here P (q) does not reduce to a delta function in the ordered phase even at N → ∞. Instead, it consists of a delta function plus a continuous, almost size independent part going right down to q = 0. As a result, in the spin-glass phase, the Binder cumulant attains a value which is signi?cantly lower than unity and does not show any change with the system size [17]. Our g(Γ) curves in Fig. 1 for di?erent N values show a crossing point (corresponding to a phase transition) at a ?nite value of Γ = Γc , below which the curves splay out for di?erent N and then saturate to values which tend to unity with increasing N. This suggests that for Γ << Γc , the system goes to a de?nite replica symmetric spin-glass ordered ground state. In summary, all our estimates for the critical tunnelling ?eld (critical point) and exponents agree with the previous estimates, our estimate for z agrees with that estimated by Miller and Huse [6] and also Ye et al [7], while it disagrees that obtained by Alvarez and Ritort [8]. We believe, the high value of z obtained in the quantum Monte Carlo study (of Ritort et al) is due to an artifact of the size anisotropy in the Trotter direction. We also emphasize that the variation of gN (Γ) functions with N for small Γ indicates the quantum spin glass phase (for Γ below Γc ) is replica symmetric in this model. We are grateful to M. Acharyya, S. M. Bhattacharjee, A. Dutta, H. Rieger and R. R. dos Santos for many discussions and suggestions. We also thank D. Dhar for a careful reading of the manuscriot and comments. PS acknowledges support from SFB 341.

5

REFERENCES
[1] See e.g., B. K. Chakrabarti, A. Dutta and P.Sen, Quantum Ising phases and Transitions in Transverse Ising Models, Lecture Notes in Physics M41, Springer Verlag, Heidelberg (1996); H. Rieger and A. P. Young, Review article for XIV Sitges Conference: Complex Behavior of Glassy Systems, Lecture Notes in Physics, Springer Verlag, Heidelberg (1997; in press) [2] A. J. Bray and M. A. Moore, J. Phys. C13, L655 (1980) [3] D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett 35, 1792 (1980) [4] K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986) ; M. Mezard, G. Parisi and M. A. Virasoro, Spin Glass Theory and Beyond, World Scienti?c, Singapore (1987). [5] H. Ishii and T. Yamamoto, J. Phys. C 18 6225 (1985). [6] J. Miller and D. Huse, Phys. Rev. Lett 70, 3147 (1993) [7] J. Ye, S. Sachdev and N. Read, Phys. Rev. Lett 70, 4011 (1993) ; N. Read, S. Sachdev and J. Ye, Phys. Rev. B52, 384 (1995) [8] J. V. Alvarez and F. Ritort, J. Phys. A29, 7355 (1996) [9] D. Lancaster and F. Ritort, J. Phys. A30, L41 (1997) [10] P. Ray, B. K. Chakrabarti and A. Chakrabarti, Phys. Rev. B 39 11828 (1989). [11] M. N. Barber, in Phase Transitions and Critical Phenomena, edt. C. Domb and J. L. Lebowitz, vol. 8, pp 146 (1983) [12] J. Hamer and M. N. Barber, J. Phys. A 14 241; 259 (1981). [13] K. Binder, Z. Phys. B43, 119 (1981) [14] T. Yamamoto and H. Ishii, J. Phys. C20, 6053 (1987) [15] R. Botet, R. Jullien and P. Pfeuty, Phys. Rev. Lett. 49 478 (1982). [16] R. R. dossantos and L. Sneddon, Phys. Rev. B23, 3541 (1981) [17] E. Marinari, G. Parisi, J. Ruiz-Lorenzo and F. Ritort, Phys Rev. Lett. 76, 843 (1996)

6

FIGURES
FIG. 1. The variation of gN (Γ) is shown against the transverse ?eld Γ for di?erent system sizes N . Note that the intersection of the curves shift towards larger Γ values as N is increased. FIG. 2. The e?ective critical transverse ?eld Γc (N, N + 2) against 1/Nm = 1/ N (N + 2) are shown. Γc for the in?nite system is found to be at 1.5. The best ?t line is shown. FIG. 3. The e?ective values of x against 1/Nm = 1/ N (N + 2) are shown to be fairly independent of the system sizes. FIG. 4. The e?ective values of y are shown against 1/Nm = 1/ N (N + 2). The best ?t curve with an asymptotic value y = 0.33 is also shown. FIG. 5. The e?ective values of z against 1/Nm = 1/ N (N + 2) are shown. The best ?t curve with an asymptotic value z = 2.1 is also shown.

7

1 0.9 0.8 0.7 0.6 gN (q ) 0.5 0.4 0.3 0.2 0.1 0

? ? 444 2 2 ? + + 2 4 3 3 + ? ? 3 2 4 + ? 3 2 4 + 3 2 ? + 3 4 2 3 +

Fig. 1 N=6 N=8 N = 10 N = 12 N = 14 N = 16

3 + 2 4
?

? 4 3 3 2 ? 3 + 4 3 3 3 3 3 + 2 + 3 3 3 3 3 2 2 ? 4+ + + 442 + + + + + + + ? 2 2 ? 4 4 2 2 2 2 2 2 ? ? 4 4 4 4 4 ? ? 4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1

1.6 1.4 1.2 1
c (N; N + 2) 0.8

Fig. 2

3

3 3

3 3 3

0.6 0.4 0.2 0 0 0.05 0.1 1=Nm

0.15

0.2

0.25

1

Fig. 3 0.7

0.69

0.68

0.67

x

0.66

3 3

3

3

3

3

0.65

0.64

0.63

0.62 0.06 0.08 0.1 0.12 0.14 1

=Nm

0.16

0.18

0.2

0.22

1

Fig. 4 0.8

0.7

0.6

0.5

y

0.4

3 3 3 3

3

3

0.3

0.2

0.1

0 0 0.05 0.1 1

=Nm

0.15

0.2

0.25

1

Fig. 5 4

3.5

3

2.5

z

2

3

3

3

3

3

1.5

1

0.5

0 0.05 0.06 0.07 0.08 0.09 1

=Nm

0.1

0.11

0.12

0.13

0.14

0.15

1



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