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arXiv:2211.05437 · Journal of High Energy Physics

Non-Hermitian Hamiltonian deformations in quantum mechanics

Apollonas S. Matsoukas-Roubeas, Federico Roccati, Julien Cornelius, Zhenyu Xu, Aurélia Chenu, Adolfo del Campo

Published Jan 13, 2023• 40 citations
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Abstract

A bstract The construction of exactly-solvable models has recently been advanced by considering integrable $$ T\overline{T} $$ T T ¯ deformations and related Hamiltonian deformations in quantum mechanics. We introduce a broader class of non-Hermitian Hamiltonian deformations in a nonrelativistic setting, to account for the description of a large class of open quantum systems, which includes, e.g., arbitrary Markovian evolutions conditioned to the absence of quantum jumps. We relate the time evolution operator and the time-evolving density matrix in the undeformed and deformed theories in terms of integral transforms with a specific kernel. Non-Hermitian Hamiltonian deformations naturally arise in the description of energy diffusion that emerges in quantum systems from time-keeping errors in a real clock used to track time evolution. We show that the latter can be related to an inverse $$ T\overline{T} $$ T T ¯ deformation with a purely imaginary deformation parameter. In this case, the integral transforms take a particularly simple form when the initial state is a coherent Gibbs state or a thermofield double state, as we illustrate by characterizing the purity, Rényi entropies, logarithmic negativity, and the spectral form factor. As the dissipative evolution of a quantum system can be conveniently described in Liouville space, we further study the spectral properties of the Liouvillians, i.e., the dynamical generators associated with the deformed theories. As an application, we discuss the interplay between decoherence and quantum chaos in non-Hermitian deformations of random matrix Hamiltonians and the Sachdev-Ye-Kitaev model.

DOI
10.1007/jhep01(2023)060

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A bstract The construction of exactly-solvable models has recently been advanced by considering integrable TT‾T\overline{T}TT T T ¯ deformations and related Hamiltonian deformations in quantum mechanics. We introduce a broader class of non-Hermitian Hamiltonian deformations in a nonrelativistic setting, to account for the description of a large class of open quantum systems, which includes, e.g., arbitrary Markovian evolutions conditioned to the absence of quantum jumps. We relate the time evolution operator and the time-evolving density matrix in the undeformed and deformed theories in terms of integral transforms with a specific kernel. Non-Hermitian Hamiltonian deformations naturally arise in the description of energy diffusion that emerges in quantum systems from time-keeping errors in a real clock used to track time evolution. We show that the latter can be related to an inverse TT‾T\overline{T}TT T T ¯ deformation with a purely imaginary deformation parameter. In this case, the integral transforms take a particularly simple form when the initial state is a coherent Gibbs state or a thermofield double state, as we illustrate by characterizing the purity, Rényi entropies, logarithmic negativity, and the spectral form factor. As the dissipative evolution of a quantum system can be conveniently described in Liouville space, we further study the spectral properties of the Liouvillians, i.e., the dynamical generators associated with the deformed theories. As an application, we discuss the interplay between decoherence and quantum chaos in non-Hermitian deformations of random matrix Hamiltonians and the Sachdev-Ye-Kitaev model.
This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachistochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of infinite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT -symmetric way.
This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachistochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of infinite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT -symmetric way.
Motivated by TT‾T\overline{T}TT, we introduce and study a wide class of solvable deformations of quantum-mechanical theories. These deformations map the Hamiltonian to a function of itself. We solve these theories by computing all finite-temperature correlation functions of the deformed theory in terms of the correlators of the undeformed theory. Applications to AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT, Sachdev-Ye-Kitaev, and the Schwarzian theory are considered. We write down the deformed Schwarzian action for an arbitrary Hamiltonian deformation and find that the maximal Lyapunov exponent is unchanged.
The purpose of this dissertation is to review the salient features of non-Hermitian quantum mechanics. An introduction to Hermitian quantum mechanics is included to make this review as accessible as possible. Attempts at formulating a consistent physical theory are introduced, before examining non-Hermitian theories’ uses as convenient computational frameworks. Particular emphasis is placed on recent developments in open quantum systems that utilise non-Hermitian Hamiltonians. Chapter four introduces a logic that maps a non-Hermitian Hamiltonian onto a non-Hamiltonian algebra that has a Hermitian Hamiltonian. This was put forward by Sergi, who then goes on to show its application to a two level system. The time evolution is then derived in terms of the density matrix model. This system can then be used to analyse different types of decay such as coherence and population difference. This serves to illustrate the usefulness of the approach.
We consider evolution of dynamical systems described by non-Hermitian Hamiltonians, using the density operator approach.The latter is formulated both at the level of the Hilbert space and the phase space, and adapted for applications to open quantum systems.We illustrate the formalism using a family of non-Hermitian system, which generators are quadratic with respect to both momentum and position.Despite the initial simplicity of a Hamiltonian, the structure of its solutions and spectral characteristics are nontrivial, and they can drastically change depending on parameters of the model and its symmetry in phase space.We present analytical solutions in L 2 (R) and in phase space, and an explicit form of the similarity transformation changing these generators into the corresponding normal operators.
Non-Hermitian disordered systems have emerged as a central arena in modern physics, with ramifications spanning condensed matter, quantum, statistical, and high energy contexts. The same principles also underlie phenomena beyond physics, such as network science, complex systems, and biophysics, where dissipation, nonreciprocity, and stochasticity are ubiquitous. Here, we review the physics and mathematics of non-Hermitian disordered systems, with particular emphasis on non-Hermitian random matrix theory. We begin by presenting the 38-fold symmetry classification of non-Hermitian systems, contrasting it with the 10-fold way for Hermitian systems. After introducing the classic Ginibre ensembles of non-Hermitian random matrices, we survey various diagnostics for complex-spectral statistics and distinct universality classes realized by symmetry. As a key application to physics, we discuss how non-Hermitian random matrix theory characterizes chaos and integrability in open quantum systems. We then turn to the criticality due to the interplay of disorder and non-Hermiticity, including Anderson transitions in the Hatano-Nelson model and its higher-dimensional extensions. We also discuss the effective field theory description of non-Hermitian disordered systems in terms of nonlinear sigma models.