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arXiv:cond-mat/0004235 · Physical review. B, Condensed matter

Phase diagram of quantum systems: Heisenberg antiferromagnets

Pietro Gianinetti, Alberto Parola

Published Feb 15, 2001• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.63.104414

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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 \ensuremathσ\ensuremath{\sigma}\ensuremathσ 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.
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.
A variational approach based on Bogoliubov inequality for the free energy is employed in order to treat the quantum spin-1 anisotropic ferromagnetic Heisenberg model in the presence of a crystal field. Within the Bogoliubov scheme an improved pair approximation has been used. The temperature-dependent thermodynamic functions have been obtained and provide much better results than the previous simple mean-field scheme. In one dimension, which is still nonintegrable for quantum spin-1, we get the exact results in the classical limit, or near-exact results in the quantum case, for the free energy, magnetization, and quadrupole moment, as well for the transition temperature. In two and three dimensions the corresponding global phase diagrams have been obtained as a function of the parameters of the Hamiltonian. First-order transition lines, second-order transition lines, tricritical and tetracritical points, and critical endpoints have been located through the analysis of the minimum of the Helmholtz free energy and a Landau-like expansion in the approximated free energy. Only first-order quantum transitions have been found at zero temperature. Limiting cases, such as isotropic Heisenberg, Blume-Capel, and Ising models, have been analyzed and compared to previous results obtained from other analytical approaches as well as from Monte Carlo simulations.
These lecture notes introduce quantum spin systems and several computational methods for studying their ground-state and finite-temperature properties. Symmetry-breaking and critical phenomena are first discussed in the simpler setting of Monte Carlo studies of classical spin systems, to illustrate finite-size scaling at continuous and first-order phase transitions. Exact diagonalization and quantum Monte Carlo (stochastic series expansion) algorithms and their computer implementations are then discussed in detail. Applications of the methods are illustrated by results for some of the most essential models in quantum magnetism, such as the S=1/2 Heisenberg antiferromagnet in one and two dimensions, as well as extended models useful for studying quantum phase transitions between antiferromagnetic and magnetically disordered states.
The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest (J1)({J}_{1})(J1​)- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state (T=0)(T=0)(T=0) and the finite-TTT phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.
We study quantum ferrimagnets in one, two, and three dimensions by using a variety of methods and approximations. These include: (i) a treatment based on the spin coherent state path-integral formulation of quantum ferrimagnets by taking into account the leading order quantum and thermal fluctuations (ii) a field-theoretical (non-linear σσσ-model type) formulation of the special case of one-dimensional quantum ferrimagnets at zero temperature (iii) an effective description in terms of dimers and quantum rotors, and (iv) a quantum renormalization group study of ferrimagnetic Heisenberg chains. Some of the formalism discussed here can be used for a unified treatment of both ferromagnets and antiferromagnets in the semiclassical limit. We show that the low (high) energy effective Hamiltonian of a (S_1, S_2) Heisenberg ferrimagnet is a ferromagnetic (antiferromagnetic) Heisenberg model. We also study the phase diagram of quantum ferrimagnets in the presence of an external magnetic field h (hc1<h<hc2h_{c1} < h < h_{c2}hc1​<h<hc2​) and show that the low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states. We also calculate the transition temperature for the Berezinskii-Kosterlitz-Thouless phase transition in the special case of two-dimensional quantum ferrimagnets.
Since its introduction, the Renormalization Group has been the theoretical tool of choice for understanding and describing collective behavior characterized by an emergent scale invariance as in the vicinity of a continuous phase transition. It provides a powerful conceptual framework but, exact results being scarce, the search for generic and efficient approximation schemes has been very active from the very beginning. One recent line of research starts from an exact formulation of the Renormalization Group, in the form of a functional Renormalization Group (FRG) for scale-dependent generating functionals of correlation functions, and introduces potentially {it nonperturbative} approximations through ansatzes for the scale-dependent generating functional under study. The question that we address is to what extent such generic approximation schemes are able to describe specific problems in which the long-distance behavior involves strongly nonuniform configurations with, {it e.g.}, localized excitations.An example of such an approximation scheme within the FRG is the so-called derivative expansion of the effective average action (coarse-grained Gibbs free energy in the language of magnetic systems) which amounts to truncating the functional form of the latter in powers of the external momenta or equivalently in gradients of the fields. The derivative-expansion approximation focuses on the long wavelength properties and, in terms of coarse-grained configurations of the system, works as an expansion about uniform configurations. One may therefore wonder if such a scheme is able to capture the physics associated with nonuniform configurations containing, {it e.g.}, domain walls, spin waves, or localized defects such as the kinks and anti-kinks found in the instanton analysis of the 1-dimensional Ising model.In this work we assess the ability of the derivative expansion of the FRG to quantitatively describe the long-distance physics of a model across the whole range of space dimensions {it without a priori knowledge} of the relevant real-space coarse-grained configurations. To do so we first investigate how low orders of the derivative expansion describe the approach to the lower critical dimension d→dlc of the Ising-like scalar φ4 theory which is expected to be controlled by the proliferation of localized excitations (instantons). We show that the convergence of the fixed-point effective potential describing the critical point of the model is nonuniform in the field when the dimension d approaches dlcdlc, with the emergence of a boundary layer around the minimum of the potential. At the lowest nontrivial approximation level (known as LPA’) this allows us to make analytical predictions for the value of the lower critical dimension dlcdlc and for the behavior of the critical temperature as d→dlc, which are both found in fair agreement with the known exact results. We also consider for the same Ising-like scalar φ4 model the ordered phase in the limit where d→dlc. We focus on the way the effective potential in the FRG becomes convex as the proper spatial fluctuations, which now involve nonuniform configurations with a domain wall separating two phases, are included in the calculation.