An effective expansion for the free energy of quantum spin glass models with infinite-range interactions

Abstract

An expansion for the free energy functional of quantum spin models is build around diagonalisable single-ion term of the system’s Hamiltonian. This expansion is obtained within the Matsubara time formalism by introducing recursive relations involving the spin correlation functions. Introduction Spin glass systems (SG) are collections of atoms or molecules with randomly distributed magnetic moments and frustrated interactions. They have gained much attentions of the scientific community since the seminal work by Edwards and Anderson [1]. Physical concepts and mathematical tools developed in the contest of SG are applicable not only in physics but also in biology, economy, informatics, e.g. in the study of optimization theory and neural networks. In quantum spin glass models with infinite-range interactions attention has been paid to description of the transition between disordered (paramagnetic) and ordered spin glass phases. However, even an approximate description of the phase change involves an expansion of the free energy functional in terms of the powers of the order parameters. In this work we present a method to built such an expansion in a systematic way. The general formulation requires that the Hamiltonian H of the system of interacting spins is composed of two parts: a single-site part H0 and an exchange, two-site part H1. The single-site term may represent an external magnetic field or single-ion anisotropies. The exchange two-site term we use is the quantum Sherrington-Kirkpatrick (SK) model [2]: ∑∑ ≠ − = α α α N j i j i ij S S J H 2 1 1 (1) where α i S are the spin operators Pauli matrices referred to the spin i of an ensemble of N spins z y x , , = α and . , , 1 N i = The random exchange interactions Please cite this article as: Zbigniew Domanski, An effective expansion for the free energy of quantum spin glass models with infinite-range interactions, Scientific Research of the Institute of Mathematics and Computer Science, 2009, Volume 8, Issue 2, pages 37-42. The website: http://www.amcm.pcz.pl/ Z. Domanski 38 { } ij J are governed by independent probability distributions taken to be Gaussian with zero mean and variance . / 1 N 1. Free energy In order to obtain the free energy averaged over the distribution of { } ij J we apply the replica trick and Matsubara ”time” representation [3]. The replica trick is used to overcome the difficulties of averaging a logarithm and it employs the identity [ ] [ ] ( ) n Z Z J n n J 1 lim ln 0 − = → (2) where [ ]J means an averaging over the random exchange interactions ij J . The Matsubara ”time” representation is based on the operator identity ( ) ( ) ( )( ) [ ] ( ) A t B A t t tA B d T t t 0 0 exp exp exp exp 0 ⋅ + − ⋅ − =     ∫ τ τ τ (3) ( ) ( ) ( ) tA B tA t B exp exp ⋅ ⋅ − = (4) Here the time ordering operator τ T rearranges the operator in the expansion of the exponential, in order of decreasing time arguments. This representation enables us to avoid the problem of the non-commutativity of the spin operators α S and to treat them as numbers. Applying (2)-(4) to the Hamiltonian (1) the replica free energy has the following form [4, 5] ( ) [ ] ( ) ∫ ∫ = Φ ⋅ = eff 0 Tr ln exp Tr ln H q H q e Dz T e Dz F τ (5) 3 2 1 Φ + Φ + Φ = Φ q z (6) ( ) ∫ ∑ = Φ 1 0 1 2 n n S d τ τ β (7) ( ) ( ) ∫ ∫ ∑ ′ ⋅ ′ − = Φ 1 0 1 0 , 2 m n m n S S d d τ τ τ τ β (8) ( ) ( ) ( ) ∫ ∫ ∑ ′ ′ ′ = Φ 1 0 1 0 3 , n n n S R S d d τ τ τ τ τ τ β (9) An effective expansion for the free energy of quantum spin glass models with infinite-range interactions 39 where T / 1 = β is the inverse temperature and the integration in Eq. (5) is over the field: ( ) ( )    − = 2 / / exp / 2 1 2 q z q z d Dzq π (10) In Equations (5)-(10), the quantity q is the Edwards-Anderson spin glass order parameter and ( ) τ τ ′ , R is the dynamic spin self-interaction. ( ) τ τ ′ , R describes the quantum fluctuations. The spin glass order parameter vanishes in the paramagnetic (disordered) phase whereas the dynamic spin self-interaction persists in both phases. In order to locate the phase change one needs to expand the free energy functional as a series in power of q. The phase transition takes place when the coefficient of 2 q becomes zero as a function of the temperature T and thus, one of the most useful information can be retrieve from an expansion of the functional F, Eqs. (5-9). 2. An expansion Consider the free energy F given by Equations (5)-(10). Using these equations we can compute the thermal average with respect to the effective Hamiltonian eff H defined in the exponent of Eq. (5). We denote this thermal average by the angular bracket, namely: ( ) ( ) ( ) ( ) ( ) ( ) Φ Φ τ τ τ τ τ τ τ exp Tr exp Tr 0 1 0 1 1 eff 1 T e S S T e S S T H p n n H p n n p p ⋅ ⋅ ⋅ = (11) From Eq. (10) we see that ( ) ( ) q q Dz z Dz q ν ν ν ν ν 2 2 2 ∂ ∂ = ∂ ∂ − (12) Using Eqs. (11) and (12) we can compute the first derivative of F with respect to the order parameter q. First, we do this by direct computation. Direct computation involves the straightforward evaluation of multiple thermal averages of Eq. (11). ( ) ( ) [ ] ( ) [ ] ∫ ∫ + + ⋅ ∂ + ⋅ ∂ = ∂ 3 2 1 exp Tr ln exp Tr ln