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arXiv:1706.00515 · Journal of Physics A Mathematical and Theoretical

Scale invariant transfer matrices and Hamiltionians

Vaughan F. R. Jones

Published Jan 3, 2018• 20 citations
arXivHTMLPDF

Abstract

Abstract Given a direct system of Hilbert spaces s ↦ H s (with isometric inclusion maps ι s t : H s → H t for s ⩽ t ) corresponding to quantum systems on scales s , we define notions of scale invariant and weakly scale invariant operators. In some cases of quantum spin chains we find conditions for transfer matrices and nearest neighbour Hamiltonians to be scale invariant or weakly so. Scale invariance forces spatial inhomogeneity of the spectral parameter. But weakly scale invariant transfer matrices may be spatially homogeneous in which case the change of spectral parameter from one scale to another is governed by a classical dynamical system exhibiting fractal behaviour.

DOI
10.1088/1751-8121/aaa4dd

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Abstract Given a direct system of Hilbert spaces s ↦ H s (with isometric inclusion maps ι s t : H s → H t for s ⩽ t ) corresponding to quantum systems on scales s , we define notions of scale invariant and weakly scale invariant operators. In some cases of quantum spin chains we find conditions for transfer matrices and nearest neighbour Hamiltonians to be scale invariant or weakly so. Scale invariance forces spatial inhomogeneity of the spectral parameter. But weakly scale invariant transfer matrices may be spatially homogeneous in which case the change of spectral parameter from one scale to another is governed by a classical dynamical system exhibiting fractal behaviour.
This book presents a number of basic concepts and mathematical techniques for analyzing quantum and classical dynamical systems on an equal basis, mainly but not solely aimed at quantifying randomizing dynamical behaviour. One of the central topics is the introduction of a quantum dynamical entropy based on successive observations of the system. This leads to a symbolic model in terms of a state on a quantum spin chain and its associated entropy that quantifies the complexity of the time evolution. Several model systems are analyzed, but more realistic dynamics and scaling limits are considered as well. Gradually the key concepts and mathematical results are presented and special attention is paid to their relevance for quantum dynamics. The book starts out with a reminder of standard Hilbert space quantum mechanics and then gently introduces the need for the more general and abstract C*-dynamical system setup. Basic examples like quantum spin chains, CAR- and CCR-formalism, and the irrational rotation algebra are described. Next the essential notions from ergodic theory and quantum irreversibility are provided along with the properties and use of von Neumann entropy as quantifiers for randomness. The last part of the book introduces a quantum dynamical entropy based on operational partitions: for classical dynamics the well-established KS-invariant is recovered, while for quantum dynamics a variety of systems are analyzed. The book ends with some open problems and speculations on the footprints of microscopic quantum dynamics on transport phenomena at the macroscopic scale.
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The authors initiate the study of ``operator spreading'' in quantum maps on a torus, systems which do not have a tensor product Hilbert space or a notion of spatial locality. The evolution of functions on classical phase space and quantum operator evolutions are compared and contrasted using the perturbed Arnold cat map as an example. Dynamics of operators in quantum chaotic maps are characterized by distinct time scales that can be related to the quasienergy spectrum of the unitary via the spectral form factor.
The spectral statistics and entanglement within the eigenstates of generic spin chain Hamiltonians are analysed. A class of random matrix ensembles is defined which include the most general nearest-neighbour qubit chain Hamiltonians. For these ensembles, and their generalisations, it is seen that the long chain limiting spectral density is a Gaussian and that this convergence holds on the level of individual Hamiltonians. The rate of this convergence is numerically seen to be slow. Higher eigenvalue correlation statistics are also considered, the canonical nearest-neighbour level spacing statistics being numerically observed and linked with ensemble symmetries. A heuristic argument is given for a conjectured form of the full joint probability density function for the eigenvalues of a wide class of such ensembles. This is numerically verified in a particular case. For many translationally-invariant nearest-neighbour qubit Hamiltonians it is shown that there exists a complete orthonormal set of eigenstates for which the entanglement present in a generic member, between a fixed length block of qubits and the rest of the chain, approaches its maximal value as the chain length increases. Many such Hamiltonians are seen to exhibit a simple spectrum so that their eigenstates are unique up to phase. The entanglement within the eigenstates contrasts the spectral density for such Hamiltonians, which is that seen for a non-interacting chain of qubits. For such non-interacting chains, their always exists a basis of eigenstates for which there is no entanglement present.
Abstract This chapter is a review of the recently established quantum-classical correspondence for integrable systems based on the construction of the master T-operator. For integrable inhomogeneous quantum spin chains with gl(N)-invariant R-matrices in finite-dimensional representations, the master T-operator is a sort of generating function for the family of commuting quantum transfer matrices depending on an infinite number of parameters. Any eigenvalue of the master T-operator is the tau-function of the classical modified KP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0th time of the hierarchy. This implies a remarkable relation between the quantum spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, a system of algebraic equations can be obtained for the spectrum of the spin chain Hamiltonians.