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Physical review. B, Condensed matter

Exact boson representation of quantum spin systems and investigation of their critical behavior

Isaac Goldhirsch, Evgeny Levich, Victor Yakhot

Published May 1, 1979• 33 citations
Publisher

Abstract

Quantum spin systems are shown to be rigorously equivalent to certain Bose systems. As a result the phase transitions occuring in such systems can be regarded as generalized Bose condensation processes. In those processes it is not the number of bosons that is kept constant but a certain function of it. The problem of degeneracy arising in Bosonizing a spin system is solved in a simple fashion. The low-temperature limit of the Bose systems reproduces the magnon gas with correct interactions. The Bloch sum rule becomes exact at all temperatures, when one replaces the magnon number by the number of bosons. The crossover to classical behavior at criticality is discussed and it is shown that at ${T}_{c}$ the quantum $S=\frac{1}{2}$ spin systems behave like the corresponding classical $S=\ensuremath{\infty}$ systems. A classical effective Hamiltonian whose corresponding partition function is equal to the partition function of the quantum spin systems is derived. Finally, possible application to dynamics are briefly discussed.

DOI
10.1103/physrevb.19.4780

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Quantum spin systems are shown to be rigorously equivalent to certain Bose systems. As a result the phase transitions occuring in such systems can be regarded as generalized Bose condensation processes. In those processes it is not the number of bosons that is kept constant but a certain function of it. The problem of degeneracy arising in Bosonizing a spin system is solved in a simple fashion. The low-temperature limit of the Bose systems reproduces the magnon gas with correct interactions. The Bloch sum rule becomes exact at all temperatures, when one replaces the magnon number by the number of bosons. The crossover to classical behavior at criticality is discussed and it is shown that at Tc{T}_{c}Tc​ the quantum S=12S=\frac{1}{2}S=21​ spin systems behave like the corresponding classical S=\ensuremath∞S=\ensuremath{\infty}S=\ensuremath∞ systems. A classical effective Hamiltonian whose corresponding partition function is equal to the partition function of the quantum spin systems is derived. Finally, possible application to dynamics are briefly discussed.
An exact Bose expansion for general spin S is presented. It is a generalisation of a previously derived expansion for S= 1 / 2 . The analysis of the resulting Bose-Hilbert space yields a new understanding of existing Bose expansions and of the meaning of the spin-boson transformations in general. Bose expansions and of the meaning of the spin-boson transformations in general. The Bloch sum rule for spin systems is proven rigorously for any value of S and every temperature. It is argued that phase transitions in quantum spin systems can be regarded as generalised Bose condensation processes. Possible applications to quantum spin systems are mentioned.
Introduction In the limit of zero temperature, the partition function is dominated by the ground state of the system, the free energy becomes equivalent to the ground state energy, and the central object of interest is the Hamiltonian of the system H (λ), dependent in general on one or more parameters λ. A ‘quantum’ Hamiltonian, as noted previously, is one which contains non-commuting operators. Here we concentrate on quantum spin Hamiltonians, of which the prime example is the Heisenberg model. Of particular interest is the possibility of a quantum phase transition , i.e. nonanalytic behaviour as a function of coupling λ. A quantum phase transition may exert a dominating influence on the system over a range of low temperatures above T = 0. Such phenomena have figured prominently in theoretical discussions of experiments on the cuprate superconductors, the heavy fermion materials, organic conductors, and related materials (e.g. Sachdev, 1999). In some cases, a correspondence can be found between a quantum system at zero temperature in 1 time and ( d – 1) space dimensions, and an equivalent classical system at finite temperature in d space dimensions, based on the Feynman path integral formalism – see Chapter 9 and Appendix 8 for further discussions. Under this mapping, the temperature kT in the classical system corresponds to a coupling λ in the equivalent quantum system. If a phase transition occurs, the same universal critical exponents are expected to apply in both cases.
The partition function for a spin \textonehalf{} in a magnetic field coupled to a set of bosons is shown to be exactly equal to that of a one-dimensional gas and also, in a delicate limit, to that of a one-dimensional Ising model. Both of these are peculiar in that they must provide a free energy which is an intensive quantity. For special values of the parameters in the Hamiltonian, the partition function is equal to that first derived by Anderson and Yuval, and by Hamann, for the Kondo problem.
Spin-orbit-coupled bosons can exhibit rich equilibrium phases at low temperature and in the presence of particle-particle interactions. In the case with a 1D synthetic spin-orbit interaction, it has been observed that the ground state of a Bose gas can be a normal phase, stripe phase, or magnetized phase in different parameter regimes. The magnetized states are doubly degenerate and consist of a many-particle two-state system. In this work, we investigate the nonequilibrium quantum dynamics by switching on a simple one-dimensional optical lattice potential as external perturbation to induce resonant couplings between the magnetized phases, and predict a quantum spin dynamics which cannot be obtained in the single-particle systems. In particular, due to particle-particle interactions, the transition of the Bose condensate from one magnetized phase to the other is forbidden when the external perturbation strength is less than a critical value, and a full transition can occur only when the perturbation exceeds such critical strength. This phenomenon manifests itself a dynamical phase transition, with the order parameter defined by the time-averaged magnetization over an oscillation period, and the critical point behavior being exactly solvable. The thermal fluctuations are also considered in detail. From numerical simulations and exact analytic studies we show that the predicted many-body effects can be well observed with the current experiments.
The dynamical critical exponent is introduced. The phase diagram and the phase transitions in the Bose–Hubbard model of interacting bosons on a lattice are determined. The concept of quantum fluctuations is introduced on the example of an interacting superfluid, and finally the special scaling of conductivity is discussed. Dynamical critical exponent The finite temperature phase transitions studied in previous chapters are the result of the competition between the entropy and the energy terms in the free energy: the weight of entropy increases with temperature, and ultimately destroys the order that may be existing in the system. A sharp phase transition between two phases exhibiting qualitatively different correlations may, however, occur even at zero temperature, by varying a coupling constant in the Hamiltonian. The transition then corresponds to a non-analyticity of the energy of the ground state. A simple example is provided by the interacting bosons, where the superfluid transition may be brought about by tuning the chemical potential at T = 0. Such T = 0 phase transitions are called quantum phase transitions , and will be the subject of the present chapter. In general, a quantum phase transition may lie at the end of a line of thermal phase transitions, as in the bosonic example mentioned above. It is possible, however, that the system may not even have an ordered state at finite temperatures, but still exhibits a quantum critical point. Two different situations are depicted in Fig. 8.1. Examples of such phase diagrams are provided by the system of interacting bosons which will be discussed in Sections 8.2 and 8.3.