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Physical Review Letters

Quantal Ising Liquid

Richard M. Stratt

Published Oct 1, 1984• 32 citations
Publisher

Abstract

An example is presented of a model of an amorphous quantum mechanical system, a liquid of quantal Ising spins, which can be solved exactly within certain many-body theories. Analytical solutions of the model in mean-field theory are shown to reveal a decrease in the extent of the ferromagnetic region (compared to an equivalent classical system) and the occurrence of some degree of quantum localization. Both phenomena are analyzed as a competition between quantum mechanics and the condensed phase.

DOI
10.1103/physrevlett.53.1305

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Recent developments in the chemical literature have shown how discretized path integrals may be used to treat the quantum-mechanical internal degrees of freedom of molecules in liquids. This paper suggests that analogous methods might also be useful for quantum-mechanical solid-state problems, such as those posed by lattice-spin systems. The idea, illustrated in this paper by a common model for tunneling in solids, the transverse Ising model, is to transform the quantal spins into classical spins with ``internal structure.'' The resulting classical system can then be treated by analytical methods. For the transverse Ising model, both the traditional mean-field theory and a new approximation (which includes a fluctuation correction) are derived in this way. The effects of thermal and quantum-mechanical fluctuations are shown to be largely identical for the example considered.

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•Aug 27, 2009•0 citations
An example is presented of a model of an amorphous quantum mechanical system, a liquid of quantal Ising spins, which can be solved exactly within certain many-body theories. Analytical solutions of the model in mean-field theory are shown to reveal a decrease in the extent of the ferromagnetic region (compared to an equivalent classical system) and the occurrence of some degree of quantum localization. Both phenomena are analyzed as a competition between quantum mechanics and the condensed phase.
Recent developments in the chemical literature have shown how discretized path integrals may be used to treat the quantum-mechanical internal degrees of freedom of molecules in liquids. This paper suggests that analogous methods might also be useful for quantum-mechanical solid-state problems, such as those posed by lattice-spin systems. The idea, illustrated in this paper by a common model for tunneling in solids, the transverse Ising model, is to transform the quantal spins into classical spins with ``internal structure.'' The resulting classical system can then be treated by analytical methods. For the transverse Ising model, both the traditional mean-field theory and a new approximation (which includes a fluctuation correction) are derived in this way. The effects of thermal and quantum-mechanical fluctuations are shown to be largely identical for the example considered.
The observation that deuterons tunnel much less than protons in tunneling mediated phase transitions suggests that it might be useful to look at disordered quantum-spin models in which the disorder resides entirely in the size of the quantal fluctuations. As an example, we consider a system of randomly mixed classical and quantum-mechanical (tunneling) spins. The mean-field-theory phase diagram shows that sufficiently dilute quantal spins cannot lead to zero-temperature disorder. In addition, our analysis of the infinite transverse field limit suggests that the critical behavior must differ from that of the pure classical problem.
Abstract Methods of statistical mechanics have been enormously successful in clarifying the macroscopic properties of many-body systems. Typical examples are found in magnetic systems, which have been a test bed for a variety of techniques. This chapter introduces the Ising model of magnetic systems and explains its mean-field treatment, a very useful technique of analysis of many-body systems by statistical mechanics. Mean-field theory explained here forms the basis of the methods used repeatedly throughout this book. The arguments in the present chapter represent a general mean-field theory of phase transitions in the Ising model with uniform ferromagnetic interactions. Special features of spin glasses and related disordered systems are taken into account in subsequent chapters.
A quantum Ising-like spin-1 system with both dipole and quadrupole interactions is analyzed. Such model is a generalized quantum version of the Blume-Emery-Griffiths model in an external anisotropic field. Models of this type are interesting candidates to describe phase transitions in liquid crystals. The phase diagram as well as the thermodynamic potentials of the model are constructed by a recently formulated improved version of mean field theory. The latter, which has the usual mean field theory as its zero-th order approximation, takes into account fluctuations up to the fourth-order. The phase diagram is found to exhibit phase transition manifolds of both the first and the second order, as well as multicritical lines. Four phases can be recognized: one ferromagnetic and three paramagnetic. The phases are identified by the requirements of minimum free energy constrained by the condition of self consistency of the order parameters p and q
ABSTRACT Quantum spin liquids (QSLs) are magnetically frustrated phases characterized by spin fractionalization, emergent gauge fields, and long‐range entanglement. Quantum spin–orbital liquids (QSOLs) form a subset of QSLs with fluctuating orbital degrees of freedom, normally adding a layer of complexity to an already involved research field. This topical review provides guidelines for understanding a specific type of QSOL in which the orbital operators facilitate the analysis of quantum liquids, thereby providing adequate starting points for /exploring these phases. Such models are extensions of the spin‐1/2 Kitaev honeycomb model (KHM), in the sense that their exact solutions depend on an extensive number of conserved quantities, combined with a mapping to a problem of Majorana fermions hopping on a static gauge field. The starting points to understand such models are classical Hamiltonians characterized by bond‐dependent Ising interactions. Such classical models are exactly solvable in spin basis thanks to their extensive symmetries and can be directly connected to classical spin ice (CSI) systems that satisfy a Gauss law. QSOLs are easily stabilized in these models by applying a transverse field or introducing other exchange mechanisms that preserve the conserved local operators. The theory of such Ising QSOLs bridges the KHM to specific types of CSIs, thus providing further insight into paradigmatic forms of spin liquids. Furthermore, bond‐dependent Ising models are related to minimal Hamiltonians describing Rydberg‐atom simulations, which provide experimental grounds for investigating them.
Abstract In this chapter we will begin our study of the Ising model on the two-dimensional lattice. In two dimensions the model has a phase transition, with critical exponents that have different values from those obtained in the mean field approximation. For this reason, it provides an important example of critical phenomena. As we will see in great detail in this chapter and in the next, among all exactly solved models of statistical mechanics, the two-dimensional Ising model is not only the one that has been most studied but it is also the model that has given a series of deep mathematical and physical results. Many solutions of the model stand out for the ingenious methods used, such as the theory of determinants, combinatorial approaches, Grassmann variables, or elliptic functions. Many results have deeply influenced the understanding of critical phenomena and have strongly stimulated new fields of research. Ideas that have matured within the study of the two-dimensional Ising model, such as the duality between its highand low-temperature phases, have been readily generalized to other systems of statistical mechanics and have also found important and fundamental applications in other important areas such as, for instance, quantum field theory. Equally fundamental is the discovery that in the vicinity of the critical point, the dynamics of the model can be described from the relativistic Dirac equation for Majorana fermions.