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Physical review. B, Condensed matter

Relevance of the imaginary-time branch in real-time formalisms for thermodynamic equilibrium: Study of the Heisenberg model

Vitor Rocha Vieira, Iveta R. Pimentel

Published Apr 1, 1989• 9 citations
Publisher

Abstract

The real-time formalisms for quantum spin-(1/2 systems in thermodynamic equilibrium are derived. The need for the consideration of a three-component formalism instead of the usual duplication of degrees of freedom of the real-time theories is shown in general, and also taking the Heisenberg model, treated using the random-phase approximation, as a specific example.

DOI
10.1103/physrevb.39.7196

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Physical review. B, Condensed matter•Apr 1, 1989•6 citations
The real-time formalisms for quantum spin-(1/2 systems in thermodynamic equilibrium are derived. The need for the consideration of a three-component formalism instead of the usual duplication of degrees of freedom of the real-time theories is shown in general, and also taking the Heisenberg model, treated using the random-phase approximation, as a specific example.
In this work we present a new theoretical approach for the study of the phase diagram of interacting quantum particles: bosons, fermions or spins. Starting from the microscopic Hamiltonian of a quantum model, we first derive a set of exact differential equations for the free energy and the correlation functions describing the effects of fluctuations on the thermodynamics of the system. These equations reproduce the full renormalization group structure in the neighbourhood of a critical point keeping, at the same time, fuli information on the non universal properties of the model. A simple approximation to our formally exact equations is studied for the spin-S Heisenberg model where explicit results for critical exponents, critical temperature and coexistence curve are obtained. In three dimensions the results are in good agreement with available quantum Monte Carlo simulations and series expansions.
An approach for studying phase transitions in systems with quantum degrees of freedom is discussed. Starting from the microscopic Hamiltonian of a quantum model, we first derive a set of exact differential equations for the free energy and the correlation functions describing the effects of fluctuations on the thermodynamics of the system. These equations reproduce the full renormalization group structure in the neighborhood of a critical point keeping, at the same time, full information on the nonuniversal properties of the model. As a concrete application we investigate the phase diagram of a Heisenberg antiferromagnet in a staggered external magnetic field. At long wavelengths the known relationship to the quantum nonlinear $\ensuremath{\sigma}$ model naturally emerges from our approach. By representing the two point function in an approximate analytical form, we obtain a closed partial differential equation which is then solved numerically. The results in three dimensions are in good agreement with available quantum Monte Carlo simulations and series expansions. More refined approximations to the general framework presented here and few applications to other models are briefly discussed.
We discuss the formal relationship between the real-time Keldysh and imaginary-time theory for nonequilibrium in quantum dot systems. The latter can be reformulated using the recently proposed Matsubara-voltage approach. We establish general conditions for correct analytic continuation procedure on physical observables, and apply the technique to the calculation of static quantities in steady-state nonequilibrium for a quantum dot subject to a finite bias voltage and external magnetic field. Limitations of the Matsubara voltage approach are also pointed out.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
The finite-temperature real-time field theories for spin (1/2 are derived with use of a functional-integral approach, and their application to problems of equilibrium and nonequilibrium statistical mechanics discussed. These formalisms have a standard Wick's theorem and linked-graph theorem, leading to the use of Feynman diagrams. Similar to the problems with bosons or fermions, the calculation of equilibrium quantities requires the consideration of the imaginary-time branch, leading to a three-component formalism, instead of the usual two-component formalism, typical of real-time field theories.