We consider the short-range-interaction disordered quantum spherical model with a symmetrical binary -bond distribution on the Bethe lattice (with coordination number z ). The system exhibits a quantum phase transition separating the spin-glass and disordered phases, where the quantum effects are regulated by a parameter describing the kinetic energy. The model admits, in the spherical limit, an exact solution both in the spin-glass phase and in the disordered phase. The quantum dynamics is examined via various correlation functions on the infinite tree, which are evaluated in closed form. A closed-form solution for the averaged free energy has been obtained, and various thermodynamic functions are explicitly derived in the form of low-temperature expansions.
We consider the short-range-interaction disordered quantum spherical model with a symmetrical binary -bond distribution on the Bethe lattice (with coordination number z ). The system exhibits a quantum phase transition separating the spin-glass and disordered phases, where the quantum effects are regulated by a parameter describing the kinetic energy. The model admits, in the spherical limit, an exact solution both in the spin-glass phase and in the disordered phase. The quantum dynamics is examined via various correlation functions on the infinite tree, which are evaluated in closed form. A closed-form solution for the averaged free energy has been obtained, and various thermodynamic functions are explicitly derived in the form of low-temperature expansions.
We present an exact analytic solution of the short-range interaction disordered quantum spherical model with symmetric binary $\ifmmode\pm\else\textpm\fi{}J$ bond distribution on the Bethe lattice (with coordination number z). We find the zero temperature quantum spin glass phase transition for coordinations $z\ensuremath{\ge}3$ but not for the one-dimensional system (z=2). Quantum dynamics is examined via various correlation functions on the infinite tree which are evaluated in a closed form. The relationship between the results for a fully connected lattice $(z\ensuremath{\rightarrow}\ensuremath{\infty})$ and the usual mean-field approach is established.
The infinite-range M-component models of quantum rotors and anharmonic oscillators coupled by random exchange interactions with Gaussian distributions are considered. Both models exhibit a quantum phase transition separating the spin glass and disordered phases where the quantum effects are regulated by a parameter \ensuremath{\Delta} (inverse momentum of inertia for rotor model and inverse mass for oscillator model) parametrizing the kinetic energy. The models allow, in the limit M\ensuremath{\rightarrow}\ensuremath{\infty}, for an exact solution in both the spin glass and disordered phases. Employing the quantum finite-temperature real-time description, we examine the static and dynamic properties of the systems without making use of the n-replica method. We find that the dynamical solution is marginally stable in the spin-glass phase with a constant static susceptibility below the spin-glass freezing temperature. A closed-form solution for the averaged free energy has been obtained and various thermodynamic functions are derived. In particular, the entropy is always positive in the spin-glass plane and goes to zero with a T3 dependency in the low-temperature region. This is in contrast to the classical M-component Heisenberg model in the M\ensuremath{\rightarrow}\ensuremath{\infty} limit.
We study the quenched disordered magnetic system, which is obtained from the 2D SO(3) quantum Heisenberg model, on a square lattice, with nearest neighbors interaction, by taking a Gaussian random distribution of couplings centered in an antiferromagnetic coupling, J¯>0 and with a width ΔJ. Using coherent spin states we can integrate over the random variables and map the system onto a field theory, which is a generalization of the SO(3) nonlinear sigma model with different flavors corresponding to the replicas, coupling parameter proportional to J¯ and having a quartic spin interaction proportional to the disorder (ΔJ). After deriving the CP1 version of the system, we perform a calculation of the free energy density in the limit of zero replicas, which fully includes the quantum fluctuations of the CP1 fields zi. We, thereby obtain the phase diagram of the system in terms of (T,J¯,ΔJ). This presents an ordered antiferromagnetic (AF) phase, a paramagnetic (PM) phase and a spin-glass (SG) phase. A critical curve separating the PM and SG phases ends at a quantum critical point located between the AF and SG phases, at T=0. The Edwards–Anderson order parameter, as well as the magnetic susceptibilities are explicitly obtained in each of the three phases as a function of the three control parameters. The magnetic susceptibilities show a Curie-type behavior at high temperatures and exhibit a clear cusp, characteristic of the SG transition, at the transition line. The thermodynamic stability of the phases is investigated by a careful analysis of the Hessian matrix of the free energy. We show that all principal minors of the Hessian are positive in the limit of zero replicas, implying in particular that the SG phase is stable.
In the present paper we analyze the critical properties of a quantum spherical spin-glass model with short-range random interactions. Since the model allows for rigorous detailed calculations, we can show how the effective partition function calculated with the help of the replica method for the spin-glass fluctuating fields ${Q}_{\ensuremath{\alpha}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$ separates into a mean-field contribution for ${Q}_{\ensuremath{\alpha}\ensuremath{\alpha}}(0;\ensuremath{\omega};\ensuremath{-}\ensuremath{\omega})$ and a strictly short-range partition function for the fields ${Q}_{\ensuremath{\alpha}\ensuremath{\ne}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$. Here $\ensuremath{\alpha},\ensuremath{\gamma}=1,\dots{},n$ are replica indices. The mean-field part WMF coincides with previous results. The short-range part WSR describes a phase transition in a Q3-field theory, where the fluctuating fields depend on a space variable $\stackrel{P\vec}{r}$ and times ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$. This we analyze using the renormalization group with dimensional regularization and minimal subtraction of dimensional poles. By generalizing standard field theory methods to our particular situation, we can identify the critical dimensionality as dc=5 at very low temperatures due to the dimensionality shift Dc=dc+1=6. We then perform an ${ϵ}^{\ensuremath{'}}$ expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.
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