Condensation in the quantum spherical spin-glass

Physics Letters A 375 (2011) 1493–1495

Contents lists available at ScienceDirect

Physics Letters A www.elsevier.com/locate/pla

Condensation in the quantum spherical spin-glass Alba Theumann ∗ , Vilarbo da Silva Jr. Instituto de Fisica, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, 91501-970, Porto Alegre, RS, Brazil

a r t i c l e

i n f o

Article history: Received 6 December 2010 Accepted 23 February 2011 Available online 26 February 2011 Communicated by A.R. Bishop Keywords: Quantum spin-glass Quantum transition Condensation Quantum spherical model

a b s t r a c t The quantum spherical spin-glass is analyzed by using a replica symmetric theory (RS) and the physical properties of the system can be explained by the existence of a zero frequency condensate below the critical temperature. The free energy coincides with previous results obtained with an annealed average of the partition function. A calculation with one step replica symmetry breaking (1S-RSB) shows that the only solution is the replica symmetric one. © 2011 Elsevier B.V. All rights reserved.

1. The model and results The classical spherical spin-glass was first solved [1] by using Wigner’s semicircular law, thus making unnecessary the use of the replica method that is the standard procedure in spin-glass theory [2] to evaluate the quenched average of the free energy. Later it was pointed out an anomaly in this model [3], by using supersymmetry methods, in the sense that the order parameter q measuring the replica overlap vanishes identically, thus the exact result of Ref. [1] is obtained through the annealed average of the partition function. The basic assumption in the solution of the classical mean spherical model is that the system of N continuous interacting spins satisfy a mean spherical constraint that is insured by a Lagrange multiplier μ. The equation for μ( T ) is solved for all T  T c but for T < T c the spherical constraint is no longer satisfied [4] and the value of μ “sticks” at the critical value. Recent investigations [5] in the m-component vector spin-glass have shown the existence of a generalized Bose–Einstein (BE) condensation phase for large values of m. Correspondingly, as the spherical model is obtained from the m → ∞ limit of the m-vector model while the same equivalence holds in the spherical spin-glass model [6], it has been discussed [2] that in the spherical spin glass model it may be defined an order parameter q that represents a BE condensation in the highest eigenvalue of the random interaction matrix, introduced to insure the validity of the spherical constraint for T < T c . The quantum spherical spin-glass model was first solved by using Wigner’s semicircular law [7] and later shown to be invariant under a Becchi–Rouet–Stora–Tyutin (BRST) supersymmetry [8] that

leads to the vanishing of the replica overlap q and justifies the annealed average of the partition function. The purpose of the present work is to show that the alternative procedure of performing a quenched average by using a replica symmetric theory describes below T c a phase analogous to a generalized BE condensation [5] and a time dependent spin-glass order parameter emerges as in the classical case [2] to insure the validity of the spherical constraint. Our calculation follows from those in Ref. [8], then we present here only the necessary details. The replicated Hamiltonian for a quantum spherical spin-glass is

HSG + μ

 i ,α

=

  1 2I

i , j ,α

Corresponding author. Tel.: +55 51 3308 6454; fax: +55 51 3308 7111. E-mail address: [email protected] (A. Theumann).

0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.02.051

 P i2α + μ S 2i α δi , j −

1 2

 J i j S iα S jα

(1)

where the first term is the kinetic energy of quantum rotors with moment of inertia I while the S i α are continuous spin variables and we introduced the canonical momenta P i α with commutation rules [ S kα , P j γ ] = i δk, j δα ,γ . The sites i , j = 1, . . . , N, the replica indices α , γ = 1, . . . , n and the couplings J i , j are independent random variables with Gaussian distribution of zero mean and variance J 2 / N. The replicated grand partition function Z n can be expressed as a functional integral [8] n

Z =

  iα

*

S 2i α

+

1 2

β D S i α exp

 I  ∂ S i ,α 2

dτ 0

i, jα

2

∂τ

 + μ S 2i ,α (τ ) δi , j



J i j S i ,α (τ ) S j ,α (τ )

.

(2)

1494

A. Theumann, V. da Silva Jr. / Physics Letters A 375 (2011) 1493–1495

The chemical potential μ is a Lagrange multiplier that ensures the mean spherical condition

β

∂lnZ   − = ∂(μ)





dτ S 2i = β N

i

(3)

0

where ln Z = limn→0 Z n−1 , while β = 1/ T is the inverse temperature in units k B = h¯ = 1. Looking for a condensed phase we make the assumption of replica symmetry to write n





S α ,γ (ω) S α ,γ (−ω) = qα ,γ (ω)

= q0 (ω)δα ,γ + q(ω)(1 − δα ,γ ) where

(4)

ω = 2π mT are boson Matsubara frequencies and

s 0 = β I ω 2 + 2β μ

(5)

and we get for the free energy in the limit n → 0

β F RS =

 1 4

ω

+ −

1 2

the annealed method in Ref. [8]. This is the correct solution at high temperature and gives μ( T ) decreasing monotonously with decreasing temperature until it reaches the value μ = J at the critical temperature T c . For T < T c the value of μ “sticks” at μ = J and we obtain from Eqs. (7) and (9) the second solution

  (β J )2 q0 (ω)2 − q(ω)2 

(β J )2 

ln s0 /2 −



q(ω) = qδω,0



2

1



q0 (ω) − q(ω)

2

(β J ) q

− β μ.

2 s0 − (β J )2 [q0 (ω) − q(ω)]

(6)

  2  =0 q(ω) 1 − (β J )2 q0 (ω) − q(ω)

(7)

and



−1

q0 (ω) − q(ω) = s0 − (β J )2 q0 (ω) − q(ω) with solution

q0 (ω) − q(ω) =

s0 −



s20 − 4(β J )2

2(β J )2

The chemical potential straint in Eq. (2)

(9)

.

μ( T ) is the solution of the spherical con-



  q0 (ω) − q(ω) = 1 − q(ω)

ω

(8)

(10)

ω

and from Eqs. (5) and (9), converting frequency sums into integrals, we obtain

L + dy L−



L 2+ − y 2







y 2 − L 2− coth

βy √

2



I



  √ (q(ω) = 2π J 2 I 1 −

where L 2± = 2(μ ± J ). Inserting Eq. (8) in Eq. (6) we obtain for the free energy at the saddle point

1



4

− βμ −

2

q0 (ω) − q(ω)

ω

1 2

ln 2

√

2  J

1 2π J 2

√





dy y 4 J − y 2 coth

I

0

βy √

2 I

 .

(14)

The critical temperature T c is obtained by setting μ = J and q = 0 in Eq. (11) and we recover the result of Ref. [8], where T c vanishes at the critical value I c = 9π162 J . At T = 0 we obtain from Eq. (14) q( T = 0) = 1 −



Ic I

, I  I C , while q( T ) is shown in Fig. 1 for sev-

eral values of I . The de Almeida–Thouless instability [9] may be analyzed for every frequency and we obtain for the frequency dependent replicon eigenvalue

λ(ω) =







s20 − 4β 2 J 2 q0 (ω) − q(ω)  0.

(15)

Then λ(ω = 0) vanishes first at the onset of the transition to the condensed phase of the ω = 0 mode with order parameter q in Eq. (14). We also investigated the one step replica symmetry breaking (1S-RSB) solution [10] by dividing the n × n matrix q(ω) in ( n−nm )2 blocks of dimension n − m × n − m where we parametrize

qα ,α (ω) = q0 (ω), qα ,β (ω) = q1 (ω)

if |β − α | < n − m,

qα ,β (ω) = q2 (ω)

if |β − α | > n − m.

(16)

We show below an example for m = n/2

(11)

ω

β F SP = (β J )2

(13)

while the spherical condition in Eq. (11) determines q:

q=1−

The saddle point equations obtained by extremizing F RS are



Fig. 1. Order parameter q(ω = 0) as a function of temperature for different values of the rotor moment of inertia. (a) solid: I J = ∞ (classical); (b) dots: I J = 1; (c) dash: I J = 0.2.

−

1 2 ω





ln q0 (ω) − q(ω)

(12)

that together with Eq. (9) reproduces exactly the free energy of the annealed calculation in Ref. [8] for all temperatures. The trivial solution for the order parameter in Eq. (7) is q(ω) = 0, that inserted in Eq. (11) recuperates the equation for μ( T ) obtained before with

⎡

q0 ⎢ q1 ⎢ ⎢ q1 ⎢ ⎢ q2 ⎣q 2 q2

q1 q0 q1 q2 q2 q2

... ... ... ... ... ...

q1 q1 q0 q2 q2 q2

q2 q2 q2 q0 q1 q1

q2 q2 q2 q1 q0 q1

... ... ... ... ... ...

⎤

q2 q2 ⎥ ⎥ q2 ⎥ ⎥. q1 ⎥ ⎦ q1 q0

The eigenvalues of this matrix are discussed in Ref. [11] and we obtain for β F

 1  β2 J 2  2 β F RSB = q0 + (n − m − 1)q21 + mq22 2 ω 2 +

n−m−1 n−m

ln B +

1 n−m

ln C −

β 2 J 2 q2 2

C

 (17)

A. Theumann, V. da Silva Jr. / Physics Letters A 375 (2011) 1493–1495

where we omitted the argument in Eq. (5) and

B = s0 − C=B−

β2 J 2 2

β2 J 2 2

ω in the q-variables, s0 is given

(q0 − q1 ),

1 2B

(18) We acknowledge financial support from FAPERGS. Vilarbo da Silva Jr. acknowledges a fellowship from CNPq.

mq2

References

,

q0 − q1 + m(q1 − q2 ) =

frequencies contribute to the definition of the critical line in the T vs. I phase diagram. Quantum effects prevent the spin glass condensation as shown in Fig. 1. A model for multispin interactions but a different temporal dependence has been also analyzed [12]. Acknowledgements

(q1 − q2 )(n − m).

The saddle point equations in the limit n → 0 are

q0 − q1 =

1495

1 2C

,

(β J )2 1− [q0 − q1 + m(q1 − q2 )]2

 = 0.

(19)

The equations are satisfied if m = 0 or q1 = q2 , that is the RS solution found before. We conclude that the properties of the quantum spherical spin glass model may be described either by means of an annealed average as in Ref. [8] or by means of a condensed phase [5] in a RS theory. A study of one step replica symmetry breaking [10] shows that the only solution is the replica symmetric one. In this low temperature phase only the ω = 0 mode condensates, while all the

[1] J.M. Kosterlitz, D.J. Thouless, R.C. Jones, Phys. Rev. Lett. 36 (1976) 1217. [2] Cirano De Dominicis, Irene Giardina, Random Fields and Spin Glasses, Cambridge University Press, 2006. [3] Alba Theumann, J. Phys. A 20 (1987) 25. [4] G.S. Joyce, Critical properties of the spherical model, in: C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena, vol. 2, Academic Press, London/New York, 1972. [5] T. Aspelmeier, M.A. Moore, Phys. Rev. Lett. 92 (2004) 077201. [6] J.R.L. de Almeida, R.C. Jones, J.M. Kosterlitz, D.J. Thouless, J. Phys. C 11 (1978) L871. [7] Prabodh Shukla, Surjit Singh, Phys. Lett. A 81 (1981) 477. [8] Pedro Castro Menezes, Alba Theumann, Phys. Rev. B 75 (2007) 024433. [9] J.R.L. de Almeida, D.J. Thouless, J. Phys. A: Math. Gen. 11 (1978) 983. [10] G. Parisi, J. Phys. A: Math. Gen. 13 (1980) 1101. [11] A. Crisanti, H.-J. Sommers, Z. Phys. B: Condensed Matter 87 (1992) 341. [12] Th.M. Nieuwenhuizen, Phys. Rev. Lett. 74 (1995) 4289.