scholarly journals Paths of unitary access to exceptional points

2021 ◽  
Vol 2038 (1) ◽  
pp. 012026
Author(s):  
Miloslav Znojil
Keyword(s):  
Hilbert Space ◽  
Phase Transitions ◽  
Quantum Phase Transitions ◽  
Quantum Systems ◽  
Point Of View ◽  
Explicit Description ◽  
Direct Access ◽  
Exceptional Point ◽  
Exceptional Points ◽  
Innovative Idea

Abstract With an innovative idea of acceptability and usefulness of the non-Hermitian representations of Hamiltonians for the description of unitary quantum systems (dating back to the Dyson’s papers), the community of quantum physicists was offered a new and powerful tool for the building of models of quantum phase transitions. In this paper the mechanism of such transitions is discussed from the point of view of mathematics. The emergence of the direct access to the instant of transition (i.e., to the Kato’s exceptional point) is attributed to the underlying split of several roles played by the traditional single Hilbert space of states ℒ into a triplet (viz., in our notation, spaces K and ℋ besides the conventional ℒ ). Although this explains the abrupt, quantum-catastrophic nature of the change of phase (i.e., the loss of observability) caused by an infinitesimal change of parameters, the explicit description of the unitarity-preserving corridors of access to the phenomenologically relevant exceptional points remained unclear. In the paper some of the recent results in this direction are summarized and critically reviewed.

Quantum phase transitions without thermodynamic limits

2007 ◽  
Vol 463 (2084) ◽  
pp. 2021-2030 ◽  
Author(s):  
Dorje C Brody ◽  
Daniel W Hook ◽  
Lane P Hughston
Keyword(s):  
Phase Transitions ◽  
Equilibrium State ◽  
Density Of States ◽  
Quantum Phase Transitions ◽  
Quantum Systems ◽  
State Spaces ◽  
Expectation Value ◽  
Finite Dimensional ◽  
Finite Quantum Systems ◽  
Thermodynamic Limits

A new microcanonical equilibrium state is introduced for quantum systems with finite-dimensional state spaces. Equilibrium is characterized by a uniform distribution on a level surface of the expectation value of the Hamiltonian. The distinguishing feature of the proposed equilibrium state is that the corresponding density of states is a continuous function of the energy, and hence thermodynamic functions are well defined for finite quantum systems. The density of states, however, is not in general an analytic function. It is demonstrated that generic quantum systems therefore exhibit second-order phase transitions at finite temperatures.

scholarly journals EXCEPTIONAL POINTS IN OPEN AND PT-SYMMETRIC SYSTEMS

2014 ◽  
Vol 54 (2) ◽  
pp. 106-112 ◽  
Author(s):  
Hichem Eleuch ◽  
Ingrid Rotter
Keyword(s):  
Open Quantum Systems ◽  
Level Density ◽  
Quantum Systems ◽  
Point Of View ◽  
Eigenvalues And Eigenfunctions ◽  
Open Quantum ◽  
Exceptional Points ◽  
Characteristic Features ◽  
High Level ◽  
Symmetric Systems

Exceptional points (EPs) determine the dynamics of open quantum systems and cause also PT symmetry breaking in PT symmetric systems. From a mathematical point of view, this is caused by the fact that the phases of the wavefunctions (eigenfunctions of a non-Hermitian Hamiltonian) relative to one another are not rigid when an EP is approached. The system is therefore able to align with the environment to which it is coupled and, consequently, rigorous changes of the system properties may occur. We compare analytically as well as numerically the eigenvalues and eigenfunctions of a 2 × 2 matrix that is characteristic either of open quantum systems at high level density or of PT symmetric optical lattices. In both cases, the results show clearly the influence of the environment on the system in the neighborhood of EPs. Although the systems are very different from one another, the eigenvalues and eigenfunctions indicate the same characteristic features.

scholarly journals Exceptional dynamical quantum phase transitions in periodically driven systems

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Ryusuke Hamazaki
Keyword(s):  
Free Energy ◽  
Phase Transitions ◽  
Fundamental Problem ◽  
Quantum Phase Transitions ◽  
Quantum Phase ◽  
Exceptional Point ◽  
Many Body ◽  
Periodically Driven ◽  
Time Regime ◽  
Short Time

AbstractExtending notions of phase transitions to nonequilibrium realm is a fundamental problem for statistical mechanics. While it was discovered that critical transitions occur even for transient states before relaxation as the singularity of a dynamical version of free energy, their nature is yet to be elusive. Here, we show that spontaneous symmetry breaking can occur at a short-time regime and causes universal dynamical quantum phase transitions in periodically driven unitary dynamics. Unlike conventional phase transitions, the relevant symmetry is antiunitary: its breaking is accompanied by a many-body exceptional point of a nonunitary operator obtained by space-time duality. Using a stroboscopic Ising model, we demonstrate the existence of distinct phases and unconventional singularity of dynamical free energy, whose signature can be accessed through quasilocal operators. Our results open up research for hitherto unknown phases in short-time regimes, where time serves as another pivotal parameter, with their hidden connection to nonunitary physics.

scholarly journals Non-Equilibrium Phenomena in Quantum Systems, Criticality and Metastability

Proceedings ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 43
Author(s):  
Angelo Carollo ◽  
Bernardo Spagnolo ◽  
Davide Valenti
Keyword(s):  
Phase Transitions ◽  
Josephson Junctions ◽  
Quantum Particle ◽  
Quantum Phase Transitions ◽  
Quantum Systems ◽  
Quantum Phase ◽  
Topological Properties ◽  
Novel Approach ◽  
Bistable Dynamics ◽  
Non Equilibrium

We summarise here some relevant results related to non-equilibrium quantum systems. We characterise quantum phase transitions (QPT) in out-of-equilibrium quantum systems through a novel approach based on geometrical and topological properties of mixed quantum systems. We briefly describe results related to non-perturbative studies of the bistable dynamics of a quantum particle coupled to an environment. Finally, we shortly summarise recent studies on the generation of solitons in current-biased long Josephson junctions.

scholarly journals Multiply Degenerate Exceptional Points and Quantum Phase Transitions

2015 ◽  
Vol 54 (12) ◽  
pp. 4293-4305 ◽  
Author(s):  
Denis I. Borisov ◽  
František Ružička ◽  
Miloslav Znojil
Keyword(s):  
Phase Transitions ◽  
Quantum Phase Transitions ◽  
Quantum Phase ◽  
Exceptional Points

scholarly journals Supersymmetry and Exceptional Points

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 892 ◽  
Author(s):  
Miloslav Znojil
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
Energy Levels ◽  
Harmonic Oscillators ◽  
Exceptional Point ◽  
Basic Principles ◽  
Exceptional Points ◽  
Super Symmetry ◽  
Physical Literature

The phenomenon of degeneracy of energy levels is often attributed either to an underlying (super)symmetry (SUSY), or to the presence of a Kato exceptional point (EP). In our paper a conceptual bridge between the two notions is proposed to be provided by the recent upgrade of the basic principles of quantum theory called, equivalently, PT − symmetric or three-Hilbert-space (3HS) or quasi-Hermitian formulation in the current physical literature. Although the original purpose of the 3HS approach laid in the mere simplification of technicalities, it is shown here to serve also as a natural theoretical link between the apparently remote concepts of EPs and SUSY. An explicit illustration of their close mutual interplay is provided by the description of infinitely many supersymmetric, mutually non-equivalent and EP-separated regularized spiked harmonic oscillators.

PASSIVITY OF GROUND STATES OF QUANTUM SYSTEMS

2005 ◽  
Vol 17 (01) ◽  
pp. 1-14 ◽  
Author(s):  
WALTER F. WRESZINSKI
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Ground State ◽  
Quantum Systems ◽  
Point Of View ◽  
Cyclic Vector ◽  
Vector State ◽  
Unitary Evolution ◽  
C Algebra ◽  
Local Perturbations

We consider a quantum system described by a concrete C*-algebra acting on a Hilbert space ℋ with a vector state ω induced by a cyclic vector Ω and a unitary evolution Ut such that UtΩ = Ω, ∀t ∈ ℝ. It is proved that this vector state is a ground state if and only if it is non-faithful and completely passive. This version of a result of Pusz and Woronowicz is reviewed, emphasizing other related aspects: passivity from the point of view of moving observers and stability with respect to local perturbations of the dynamics.

scholarly journals Exceptional Points and Dynamical Phase Transitions

10.14311/1273 ◽  
2010 ◽  
Vol 50 (5) ◽  
Author(s):  
I. Rotter
Keyword(s):  
Phase Transitions ◽  
Quantum Physics ◽  
Open Quantum Systems ◽  
Level Density ◽  
Experimental Studies ◽  
Quantum Systems ◽  
Microwave Cavity ◽  
Exceptional Points ◽  
Dynamical Phase ◽  
Dynamical Phase Transitions

In the framework of non-Hermitian quantum physics, the relation between exceptional points,dynamical phase transitions and the counter intuitive behavior of quantum systems at high level density is considered. The theoretical results obtained for open quantum systems and proven experimentally some years ago on a microwave cavity, may explain environmentally induce deffects (including dynamical phase transitions), which have been observed in various experimental studies. They also agree(qualitatively) with the experimental results reported recently in PT symmetric optical lattices.

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