We have investigated the statistical properties of an ensemble of disordered 1D spatial spin chains (SSCs) of finite length, placed in an external field, with consideration of relaxation effects. The short-range interaction complex-classical Hamiltonian was first used for solving this problem. A system of recurrent equations is obtained on the nodes of the spin-chain lattice. An efficient mathematical algorithm is developed on the basis of these equations with consideration of advanced Sylvester conditions which allows one to step by step construct a huge number of stable spin chains in parallel. The distribution functions of different parameters of spin glass system are constructed from first principles by analyzing the calculation results of the 1D SSCs ensemble. It is shown that the behaviors of different distributions parameters are quite different even at weak external fields. The ensemble energy and constants of spin-spin interactions are being changed smoothly depending on the external field in the limit of statistical equilibrium, while some of them such as the mean value of polarizations of the ensemble and parameters of its orderings are frustrated. We have also studied some critical properties of the ensemble such as catastrophes in the Clausius-Mossotti equation depending on the value of the external field. We have shown that the generalized complex-classical approach excludes these catastrophes, which allows one to organize continuous parallel computing on the whole region of values of the external field including critical points. A new representation of the partition function is suggested based on these investigations. Being opposite to the usual definition, it is a complex function and its derivatives are everywhere defined, including at critical points.
We have investigated the statistical properties of an ensemble of disordered 1D spatial spin chains (SSCs) of finite length, placed in an external field, with consideration of relaxation effects. The short-range interaction complex-classical Hamiltonian was first used for solving this problem. A system of recurrent equations is obtained on the nodes of the spin-chain lattice. An efficient mathematical algorithm is developed on the basis of these equations with consideration of advanced Sylvester conditions which allows one to step by step construct a huge number of stable spin chains in parallel. The distribution functions of different parameters of spin glass system are constructed from first principles by analyzing the calculation results of the 1D SSCs ensemble. It is shown that the behaviors of different distributions parameters are quite different even at weak external fields. The ensemble energy and constants of spin-spin interactions are being changed smoothly depending on the external field in the limit of statistical equilibrium, while some of them such as the mean value of polarizations of the ensemble and parameters of its orderings are frustrated. We have also studied some critical properties of the ensemble such as catastrophes in the Clausius-Mossotti equation depending on the value of the external field. We have shown that the generalized complex-classical approach excludes these catastrophes, which allows one to organize continuous parallel computing on the whole region of values of the external field including critical points. A new representation of the partition function is suggested based on these investigations. Being opposite to the usual definition, it is a complex function and its derivatives are everywhere defined, including at critical points.
We study the statistical properties of an ensemble of disordered 1D spatial spin-chains (SSCs) of certain length in the external field. On nodes of spin-chain lattice the recurrent equations and corresponding inequal-ity conditions are obtained for calculation of local minimum of a classical Hamiltonian. Using these equa-tions for simulation of a model of 1D spin-glass an original high-performance parallel algorithm is developed. Distributions of different parameters of unperturbed spin-glass are calculated. It is analytically proved and shown by numerical calculations that the distribution of the spin-spin interaction constant in the Heisenberg nearest-neighboring Hamiltonian model as opposed to the widely used Gauss-Edwards-Anderson distribu-tion satisfies the Lévy alpha-stable distribution law which does not have variance. We have studied critical properties of spin-glass depending on the external field amplitude and have shown that even at weak external fields in the system strong frustrations arise. It is shown that frustrations have a fractal character, they are self-similar and do not disappear at decreasing of calculations area scale. After averaging over the fractal structures the mean values of polarizations of the spin-glass on the scales of external field's space-time peri-ods are obtained. Similarly, Edwards-Anderson’s ordering parameter depending on the external field ampli-tude is calculated. It is shown that the mean values of polarizations and the ordering parameter depending on the external field demonstrate phase transitions of first-order.
We study statistical properties of 3D classical spin glass layer of certain width and infinite length. The 3D spin glass is represented as an ensemble of disordered 1D spatial spin-chains (SSC) where interactions are random between spin-chains (nonideal ensemble of 1D SSCs). It is proved that at the limit of Birkhoff's ergodic hypothesis performance 3D spin glasses can be generated by Hamiltonian of disordered 1D SSC with random environment. Disordered 1D SSC is defined on a regular lattice where one randomly oriented spin is put on each node of lattice. Also it is supposed that each spin randomly interacts with six nearest-neighboring spins (two spins on lattice and four in the environment). The recurrent transcendental equations are obtained on the nodes of spin-chain lattice. These equations combined with the Silvester conditions allow step by step construct spin-chain in the ground state of energy where all spins are in minimal energy of classical Hamiltonian. On the basis of these equations an original high-performance parallel algorithm is developed for 3D spin glasses simulation. Distributions of different parameters of unperturbed spin glass are calculated. In particular, it is analytically proved and by numerical calculations shown that the distribution of spin-spin interaction constant in Heisenberg nearest-neighboring Hamiltonian model as opposed to widely used Gauss-Edwards- Anderson distribution satisfies Lévy alpha-stable distribution law which does not have variance. A new formula is proposed for construction of partition function in kind of one-dimensional integral on energy distribution of 1D SSCs
We study the classical 1D Heisenberg spin glasses in the framework of nearest-neighboring model. Based on the Hamilton equations we obtained the system of recurrence equations which allows to perform node-by-node calculations of a spin-chain. It is shown that calculations from the first principles of classical mechanics lead to ℕℙ hard problem, that however in the limit of the statistical equilibrium can be calculated by ℙ algorithm. For the partition function of the ensemble a new representation is offered in the form of one-dimensional integral of spin-chains’ energy distribution.
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
The nonequilibrium dynamics of classical random Ising spin chains with nonconserved magnetization are studied using an asymptotically exact real space renormalization group (RSRG). We focus on random field Ising model (RFIM) spin chains with and without a uniform applied field, as well as on Ising spin glass chains in an applied field. For the RFIM we consider a universal regime where the random field and the temperature are both much smaller than the exchange coupling. In this regime, the Imry-Ma length that sets the scale of the equilibrium correlations is large and the coarsening of domains from random initial conditions (e.g., a quench from high temperature) occurs over a wide range of length scales. The two types of domain walls that occur diffuse in opposite random potentials, of the form studied by Sinai, and domain walls annihilate when they meet. Using the RSRG we compute many universal asymptotic properties of both the nonequilibrium dynamics and the equilibrium limit. We find that the configurations of the domain walls converge rapidly toward a set of system-specific time-dependent positions that are independent of the initial conditions. Thus the behavior of this nonequilibrium system is pseudodeterministic at long times because of the broad distributions of barriers that occur on the long length scales involved. Specifically, we obtain the time dependence of the energy, the magnetization, and the distribution of domain sizes (found to be statistically independent). The equilibrium limits agree with known exact results. We obtain the exact scaling form of the two-point equal time correlation function <S0(t)Sx(t)> and the two-time autocorrelations <S0(t('))S0(t)>. We also compute the persistence properties of a single spin, of local magnetization, and of domains. The analogous quantities for the +/-J Ising spin glass in an applied field are obtained from the RFIM via a gauge transformation. In addition to these we compute the two-point two-time correlation function <S0(t)Sx(t)><S0(t('))Sx(t('))> which can in principle be measured by experiments on spin-glass-like systems. The thermal fluctuations are studied and found to be dominated by rare events; in particular all moments of truncated equal time correlations are computed. Physical properties which are typically measured in aging experiments are also studied, focusing on the response to a small magnetic field which is applied after waiting for the system to equilibrate for a time t(w). The nonequilibrium fluctuation-dissipation ratio X(t,t(w)) is computed. We find that for (t-t(w)) approximately t(alpha)(w) with alpha;<1, the ratio equal to its equilibrium value X=1, although time translational invariance does not hold in this regime. For t-t(w) approximately t(w) the ratio exhibits an aging regime with a nontrivial X=X(t/t(w)) not equal to 1, but the behavior is markedly different from mean field theory. Finally the distribution of the total magnetization and of the number of domains is computed for large finite size systems. General issues about convergence toward equilibrium and the possibilities of weakly history-dependent evolution in other random systems are discussed.
We investigate zero and finite-temperature properties of the one-dimensional spin-glass model for vector spins in the limit of an infinite number m of spin components where the interactions decay with a power, $\ensuremath{\sigma}$, of the distance. A diluted version of this model is also studied, but found to deviate significantly from the fully connected model. At zero temperature, defect energies are determined from the difference in ground-state energies between systems with periodic and antiperiodic boundary conditions to determine the dependence of the defect-energy exponent $\ensuremath{\theta}$ on $\ensuremath{\sigma}$. A good fit to this dependence is $\ensuremath{\theta}=3/4\ensuremath{-}\ensuremath{\sigma}$. This implies that the upper critical value of $\ensuremath{\sigma}$ is 3/4, corresponding to the lower critical dimension in the d-dimensional short-range version of the model. For finite temperatures, the large m saddle-point equations are solved self-consistently, which gives access to the correlation function, the order parameter, and the spin-glass susceptibility. Special attention is paid to the different forms of finite-size scaling effects below and above the lower critical value, $\ensuremath{\sigma}=5/8$, which corresponds to the upper critical dimension 8 of the hypercubic short-range model.