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From classical to quantum dynamics at Rokhsar-Kivelson points

Christopher L. Henley

Published Jul 26, 2013• 52 citations
OpenAlex

Abstract

Abstract. For any classical statistical-mechanics model with a discrete state space, and endowed with a dynamics satisfying detailed balance, it is possible to generalize the Rokhsar-Kivelson point for the quantum dimer model. That is, a quantum Hamiltonian can be constructed (on the same state space), such that the ground state wavefunction coincides with the classical equilibrium distribution. Furthermore the excited eigenstates correspond to classical relaxation modes, which (in cases with a symmetry or conserved quantity) permits extraction of the dispersion law of long-wavelength excitations. The mapping is natural mainly when the states have equal weight, as is typical of a highly frustrated model. Quantum and classical correlation functions are related by analytic continuation to the imaginary time axis. The quantum dimer model [1] of Rokhsar and Kivelson (RK) was inspired by the resonating-valence bond state of a quantum antiferromagnet, which had just been proposed as a starting point for the explanation of high-temperature superconductivity. That was not to be, but models of this type are of interest among

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From classical to quantum dynamics at Rokhsar–Kivelson points

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We obtain a quantum dimer model (QDM) containing a Rokhsar-Kivelson (RK) point expressed by spin-$\frac{1}{2}$ Heisenberg antiferromagnets on a diamond-like decorated square lattice. This lattice has macroscopically degenerated nonmagnetic ground states, which are equivalent to the Hilbert space of a square-lattice QDM. Then, a square-lattice QDM containing the RK point as a second-order effective Hamiltonian is obtained by introducing further neighbor couplings as perturbation. Our model can provide a new method for the experimental realization of a resonating valence bond state in the QDM.

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Abstract. For any classical statistical-mechanics model with a discrete state space, and endowed with a dynamics satisfying detailed balance, it is possible to generalize the Rokhsar-Kivelson point for the quantum dimer model. That is, a quantum Hamiltonian can be constructed (on the same state space), such that the ground state wavefunction coincides with the classical equilibrium distribution. Furthermore the excited eigenstates correspond to classical relaxation modes, which (in cases with a symmetry or conserved quantity) permits extraction of the dispersion law of long-wavelength excitations. The mapping is natural mainly when the states have equal weight, as is typical of a highly frustrated model. Quantum and classical correlation functions are related by analytic continuation to the imaginary time axis. The quantum dimer model [1] of Rokhsar and Kivelson (RK) was inspired by the resonating-valence bond state of a quantum antiferromagnet, which had just been proposed as a starting point for the explanation of high-temperature superconductivity. That was not to be, but models of this type are of interest among
For any classical statistical-mechanics model with a discrete state space, and endowed with a dynamics satisfying detailed balance, it is possible to generalize the Rokhsar–Kivelson point for the quantum dimer model. That is, a quantum Hamiltonian can be constructed (on the same state space) such that the ground state wavefunction coincides with the classical equilibrium distribution. Furthermore the excited eigenstates correspond to classical relaxation modes, which (in cases with a symmetry or conserved quantity) permits extraction of the dispersion law of long-wavelength excitations. The mapping is natural mainly when the states have equal weight, as is typical of a highly frustrated model. Quantum and classical correlation functions are related by analytic continuation to the imaginary time axis.
Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].
Abstract For present purposes, the first thing to recall about quantum mechanics is that all dynamical quantities, instead of being functions of the location of a system in phase space as in classical mechanics, are now represented by operators in a Hilbert space of quantum states. In particular, the quantum analog of the classical phase space distribution function is the quantum mechanical density matrix. Since this is seldom discussed in introductory texts on either quantum mechanics or statistical mechanics, it may be helpful to summarize some of its basic properties.
We study a general class of easy-axis spin models on a lattice of corner sharing even-sided polygons with all-to-all interactions within a plaquette. The low-energy description corresponds to a quantum dimer model on a dual lattice of even coordination number with a multidimer constraint. At an appropriately constructed frustration-free Rokhsar-Kivelson (RK) point, the ground-state wavefunction can be exactly mapped onto a classical vertex model on the dual lattice. When the dual lattice is bipartite, the vertex models are bonded and are self-dual under Wegner's duality, with the self-dual point corresponding to the RK point of the original multidimer model. We argue that the self-dual point is a critical point based on known exact solutions to some of the vertex models. When the dual lattice is nonbipartite, the vertex model is arrowed, and we use numerical methods to argue that there is no phase transition as a function of the vertex weights. Motivated by these wavefunction dualities, we construct two other distinct families of frustration-free Hamiltonians whose ground states can be mapped onto these vertex models. Many of these RK Hamiltonians provably host Z2{\mathbb{Z}}_{2}Z2​ topologically ordered phases.
A formulation of quantum-classical hybrid dynamics is presented which concerns the direct coupling of classical and quantum mechanical degrees of freedom. It is of interest for applications in quantum mechanical approximation schemes and may be relevant for the foundations of quantum mechanics, in particular, when it comes to experiments exploring the quantum-classical border. The present linear theory differs from the nonlinear ensemble theory of Hall and Reginatto but shares with it the fulfillment of all consistency requirements discussed in the literature, while earlier attempts have failed in this respect. Our work is based on the representation of quantum mechanics in the framework of classical analytical mechanics by A. Heslot, showing that notions of states in phase space, observables, Poisson brackets, and related canonical transformations can be naturally extended to quantum mechanics. This is suitably generalized for quantum-classical hybrids here.
We obtain a quantum dimer model (QDM) containing a Rokhsar-Kivelson (RK) point expressed by spin-12\frac{1}{2}21​ Heisenberg antiferromagnets on a diamond-like decorated square lattice. This lattice has macroscopically degenerated nonmagnetic ground states, which are equivalent to the Hilbert space of a square-lattice QDM. Then, a square-lattice QDM containing the RK point as a second-order effective Hamiltonian is obtained by introducing further neighbor couplings as perturbation. Our model can provide a new method for the experimental realization of a resonating valence bond state in the QDM.