The author considers a class of spin systems whose single-site configuration space is an orbit of a representation of a compact Lie group G. For these models the author gets upper and lower bounds to the quantum partition function in terms of two classical partition functions. If a certain group-theoretic condition is satisfied, then these bounds allow one to prove the convergence of a suitable sequence of quantum partition functions to the 'corresponding' classical one. This condition is shown to be satisfied, in particular, for the D-component rotators when D is odd. The result could be useful for the extension of the Lee-Yang theorem to such models.
The author considers a class of spin systems whose single-site configuration space is an orbit of a representation of a compact Lie group G. For these models the author gets upper and lower bounds to the quantum partition function in terms of two classical partition functions. If a certain group-theoretic condition is satisfied, then these bounds allow one to prove the convergence of a suitable sequence of quantum partition functions to the 'corresponding' classical one. This condition is shown to be satisfied, in particular, for the D-component rotators when D is odd. The result could be useful for the extension of the Lee-Yang theorem to such models.
The canonical partition function Q(H, N, s) for a spin-s anisotropic Heisenberg model with N sites, in external magnetic field H, is examined in the limit s → ∞. The coupling coefficients and the magnetic moment per spin are taken to be inversely proportional to s2 and s, respectively. It is proved that lim lim s→∞s−NQ(H,N,s)=(2π)−NQ(H,N),where Q(H, N) is the partition function for a corresponding classical Heisenberg model with spins of unit magnitude. This theorem makes precise the widely believed, but heretofore unproved equivalence between the classical Heisenberg model and the infinite-spin limit of the quantum Heisenberg model. No appeal to the thermodynamic limit is necessary.
We study the relation between quantum mechanical and classical Gibbs states of spin systems with spin quantum number s. It is known that quantum states and observables can be represented by functions defined on the phase space S, which in our case is the N-fold product of unit spheres. Therefore, the classical limit s→∞ of (suitably scaled) quantum Gibbs states can be described as the limit of functions defined on S. We choose to approximate the exponential function of the Hamiltonian by a polynomial of degree n and thus have to deal with the problem of the limit of double sequences (depending on n and s) treated in the theorem of Moore-Osgood. The convergence of quantum Gibbs states to classical ones is illustrated by the example of the Heisenberg dimer. We apply our method to the explicit calculation of the phase space function describing spin monomials, and finally add some general remarks on the theory of spin coherent states.
Lee-Yang theorems are a powerful tool for studying many-body systems, with applications ranging from analyzing phase transitions to proving the efficiency of certain classical and quantum algorithms. In this work, we prove a Lee-Yang zero-freeness theorem for the partition function of a broad class of interacting fermion models, implying the existence of a provably efficient quantum algorithm for estimating their ground-state energies. This class includes several well-known models such as the attractive Hubbard model, repulsive Hubbard model on bipartite graphs, and the interacting Hofstadter model. Our results also rigorously establish the nonexistence of phase transitions in these models in the presence of a nonzero local external field.
We revisit the somewhat less studied problem of Yang-Lee zeros of the Ising antiferromagnet. For this purpose, we study two models, the nearest-neighbor model on a square lattice and the more tractable mean-field model corresponding to infinite-ranged coupling between all sites. In the high-temperature limit, we show that the logarithm of the Yang-Lee zeros can be written as a series in half odd integer powers of the inverse temperature, k, with the leading term ∼k^{1/2}. This result is true in any dimension and for arbitrary lattices. We also show that the coefficients of the expansion satisfy simple identities (akin to sum rules) for the nearest-neighbor case. These identities are verified numerically by computing the exact partition function for a two-dimensional square lattice of size 16×16. For the mean-field model, we write down the partition function (termed the mean-field polynomials) for the ferromagnetic (FM) and antiferromagnetic (AFM) cases and derive from them the mean-field equations. We analytically show that at high temperatures the zeros of the AFM mean-field polynomial scale as ∼k^{1/2} as well. Using a simple numerical method, we find the roots lie on certain curves (the root curves), in the thermodynamic limit for the mean-field polynomials for the AFM case as well as for the FM one. Our results show a new root curve that was not found earlier. Our results also clearly illustrate the phase transition expected for the FM and AFM cases, in the language of Yang-Lee zeros. Moreover, for the AFM case, we observe that the root curves separate two distinct phases of zero and nonzero complex staggered magnetization, and thus depict a complex phase boundary.
This chapter turns to the models obtained by the quantum–classical mapping QC on the D -dimensional, N -component, classical ferromagnets in (3.3) with N ≥ 2. These are the O( N ) quantum rotor models in d = D − 1 dimensions, originally written down in (1.23). The quantum Ising model studied previously had a discrete Z 2 symmetry. An important new ingredient in the rotor models will be the presence of a continuous symmetry: The physics is invariant under a uniform, global O( N ) transformation on the orientation of the rotors, which is broken in the magnetically ordered state. We will introduce the important concept of the spin stiffness , which characterizes the rigidity of the ordered state and determines the dispersion spectrum of the low energy “spin-wave” excitations. Apart from this, much of the technology and the physical ideas introduced earlier for the d = 1 Ising chain will generalize straightforwardly, although we will no longer be able to obtain exact results for crossover functions. The characterization of the physics in terms of three regions separated by smooth crossovers, the high- T and the two low- T regions on either side of the quantum critical point, will continue to be extremely useful and will again be the basis of our discussion. Because we will consider models in spatial dimensions d > 1, it will be possible to have a thermodynamic phase transition at a nonzero temperature. We shall be particularly interested in the interplay between the critical singularities of the finite-temperature transition and those of the quantum critical point.
The Lee-Yang theorem and its quantum extensions state that, for a broad class of Hamiltonians on any graph, the partition function's zeros in the complex magnetic field plane lie only on the imaginary axis. For these Hamiltonians, we prove that under a uniform Z-field of any strength h, the ground state has a spectral gap of at least h/4, independent of the system size and of the coupling strengths. The proof uses the zero-freeness of the partition function as given by Asano and Suzuki-Fisher to show exponential decay of the imaginary-time correlations for any product of Z-operators. Our result gives a polynomial-time quantum algorithm for computing the ground state energy of any Lee-Yang Hamiltonian.