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arXiv:2211.00368

Classical limit of Gibbs states for quantum spin systems

Heinz–Jürgen Schmidt

Published Nov 1, 2022• 0 citations
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Abstract

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 ${\mathcal S}$, which in our case is the $N$-fold product of unit spheres. Therefore, the classical limit $s\to\infty$ of (suitably scaled) quantum Gibbs states can be described as the limit of functions defined on ${\mathcal S}$. 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.

DOI
10.48550/arxiv.2211.00368

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We study the relation between quantum mechanical and classical Gibbs states of spin systems with spin quantum number sss. It is known that quantum states and observables can be represented by functions defined on the phase space S{\mathcal S}S, which in our case is the NNN-fold product of unit spheres. Therefore, the classical limit s→∞s\to\inftys→∞ of (suitably scaled) quantum Gibbs states can be described as the limit of functions defined on S{\mathcal S}S. We choose to approximate the exponential function of the Hamiltonian by a polynomial of degree nnn and thus have to deal with the problem of the limit of double sequences (depending on nnn and sss) 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.
For a quantum observable AℏA_\hbarAℏ​ depending on a parameter ℏ\hbarℏ we define the notion ``AℏA_\hbarAℏ​ converges in the classical limit''. The limit is a function on phase space. Convergence is in norm in the sense that Aℏ→0A_\hbar\to0Aℏ​→0 is equivalent with ∥Aℏ∥→0\Vert A_\hbar\Vert\to0∥Aℏ​∥→0. The ℏ\hbarℏ-wise product of convergent observables converges to the product of the limiting phase space functions. ℏ−1\hbar^{-1}ℏ−1 times the commutator of suitable observables converges to the Poisson bracket of the limits. For a large class of convergent Hamiltonians the ℏ\hbarℏ-wise action of the corresponding dynamics converges to the classical Hamiltonian dynamics. The connections with earlier approaches, based on the WKB method, or on Wigner distribution functions, or on the limits of coherent states are reviewed.
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The topic of this Bachelor's thesis is the quantum-to-classical crossover using generalized coherent states (GCS). In other words, the thesis consists of a mathematically rigorous consideration of the transition between quantum theory and classical theory, and how this can be carried out using the theory of GCS. The necessary mathematical background for this is also reviewed. For the theory of generalized coherent states, Lie algebras and Lie groups are first reviewed. Lie groups are groups with a structure of a differentiable manifold. For the Lie groups we consider, there exist coresponding Lie algebras, whose elements generate the Lie groups using the exponential map. Generalized coherent states are a generalization of Glauber states, which are continuous quantum states used in describing the harmonic oscillator. The generalization is carried through using some Lie group other than the Heisenberg-Weyl group used in defining the Glauber states. A special property of GCS is their semiclassical structure: a structure resembling the classical phase space may be defined in the differentiable manifold used in generating the GCS. An example of GCS are the spin coherent states, generated using the Lie group SU(2). For the coherent states to be applicable in the quantum-to-classical crossover, one has to also find some symmetry so that the system under consideration may be described using a single GCS. In the reference literature it is assumed that a global symmetry enables the description of a system of N elements by a coherent state corresponding to a coherent state of a single element. In the spin chain example however, the symmetry can be explicitly applied on the system and it turns out that this assumption is valid: using suitable symmetries, a spin chain may be represented by a single spin coherent state. The quantum-to-classical crossover can be carried out by defining some classical operators whose expectation value with respect to the GCS corresponds to the functions of classical mechanics. In addition to this, the coherent states in question have to fulfill some requirements. In the case of spin coherent states these requirements are fulfilled and the quantum-to-classical crossover may be carried through.
We introduce a classical limit of the dynamics of quantum spin systems based on coherent states of SU(N)\mathrm{SU}(N)SU(N), where NNN is the dimension of the local Hilbert space. This approach, which generalizes the well-known Landau-Lifshitz dynamics from SU(2) to SU(N)\mathrm{SU}(N)SU(N), provides a better approximation to the exact quantum dynamics for a large class of realistic spin Hamiltonians, including $S\ensuremath{\ge}1$ systems with large single-ion anisotropy and weakly coupled multispin units, such as dimers or trimers. We illustrate this idea by comparing the spin structure factors of a single-ion S=1S=1S=1 model that are obtained with the SU(2) and SU(3) classical spin dynamics against the exact solution.
It is well-known in physics that the limit of large quantum spin SSS should be understood as a semiclassical limit. This raises the question of whether such emergent classicality facilitates the approximation of computationally hard quantum optimization problems, such as the local Hamiltonian problem. We demonstrate this explicitly for spin-SSS generalizations of Quantum Max-Cut (QMaxCutS\mathrm{QMaxCut}_SQMaxCutS​), equivalent to the problem of finding the ground state energy of an arbitrary spin-SSS quantum Heisenberg antiferromagnet (QHAS\mathrm{QHA}_SQHAS​). We prove that approximating the value of QHAS\mathrm{QHA}_SQHAS​ to inverse polynomial accuracy is QMA-complete for all SSS, extending previous results for S=1/2S=1/2S=1/2. We also present two distinct families of classical approximation algorithms for QMaxCutS\mathrm{QMaxCut}_SQMaxCutS​ based on rounding the output of a semidefinite program to a product of Bloch coherent states. The approximation ratios for both our proposed algorithms strictly increase with SSS and converge to the Briët-Oliveira-Vallentin approximation ratio αBOV≈0.956α_{\mathrm{BOV}} \approx 0.956αBOV​≈0.956 from below as S→∞S \to \inftyS→∞.
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