arXiv subjects

Petr Braun

Publications and source records attributed to Petr Braun.

At least 19 recordsLinked to original sources

arXiv2021

Winding Number Statistics of a Parametric Chiral Unitary Random Matrix Ensemble

The winding number is a concept in complex analysis which has, in the presence of chiral symmetry, a physics interpretation as the topological index belonging to gapped phases of fermions. We study statistical properties of this topological quantity. To this end, we set up a random matrix model for a chiral unitary system with a parametric dependence. We analytically calculate the discrete probability distribution of the winding numbers, as well as the parametric correlations functions of the winding number density. Moreover, we address aspects of universality for the two-point function of the winding number density by identifying a proper unfolding procedure. We conjecture the unfolded two-point function to be universal.

math-ph↗

arXiv2021

Localization in the Kicked Ising Chain

Determining the border between ergodic and localized behavior is of central interest for interacting many-body systems. We consider here the recently very popular spin-chain model that is periodically excited. A convenient description of such a many-body system is achieved by the dual operator that evolves the system in contrast to the time-evolution operator not in time but in particle direction. We identify in this paper the largest eigenvalue of a function based on the dual operator as a convenient tool to determine if the system shows ergodic or many-body localized features. By perturbation theory in the vicinity of the noninteracting system we explain analytically the eigenvalue structure and compare it with numerics in [P. Braun, D. Waltner, M. Akila, B. Gutkin, T. Guhr, Phys. Rev. E $\bf{101}$, 052201 (2020)] for small times. Furthermore we identify a quantity that allows based on extensive large-time numerical computations of the spectral form factor to distinguish between localized and ergodic system features and to determine the Thouless time, i.e. the transition time between these regimes in the thermodynamic limit.

cond-mat.stat-mech↗

arXiv2021

A Mapping between the Spin and Fermion Algebra

We derive a formalism to express the spin algebra $\mathfrak{su}(2)$ in a spin $s$ representation in terms of the algebra of $L$ fermionic operators that obey the Canonical Anti-commutation Relations. We also give the reverse direction of expressing the fermionic operators as polynomials in the spin operators of a single spin. We extend here to further spin values the previous investigations by Dobrov [J.Phys.A: Math. Gen 36 L503, (2003)] who in turn clarified on an inconsistency within a similar formalism in the works of Batista and Ortiz [Phys.\ Rev.\ Lett. 86, 1082 (2001)]. We then consider a system of $L$ fermion flavors and apply our mapping in order to express it in terms of the spin algebra. Furthermore we investigate a possibility to simplify certain Hamiltonian operators by means of the mapping.

quant-ph↗

arXiv2020

Transition from Quantum Chaos to Localization in Spin Chains

Recent years have seen an increasing interest in quantum chaos and related aspects of spatially extended systems, such as spin chains. However, the results are strongly system dependent, generic approaches suggest the presence of many-body localization while analytical calculations for certain system classes, here referred to as the ``self-dual case'', prove adherence to universal (chaotic) spectral behavior. We address these issues studying the level statistics in the vicinity of the latter case, thereby revealing transitions to many-body localization as well as the appearance of several non-standard random-matrix universality classes.

cond-mat.stat-mech↗

arXiv2020

Exact local correlations in kicked chains at light cone edges

We show that local correlators in a wide class of kicked chains can be calculated exactly at light cone edges. Extending previous works on dual-unitary systems, the correlators between local operators are expressed through the expectation values of transfer matrices $T$ with small dimensions. Contrary to the previous studies, our results are not restricted to dual-unitary systems with spatial-temporal symmetry of the dynamics. They hold for a generic case without fine tuning of model parameters. The results are exemplified on the kicked Ising spin chain model, where we provide an explicit formula for two-point correlators near light cone edges beyond the dual-unitary regime.

cond-mat.stat-mech↗

arXiv2020

Local correlations in dual-unitary kicked chains

We show that for dual-unitary kicked chains, built upon a pair of complex Hadamard matrices, correlators of strictly local, traceless operators vanish identically for sufficiently long chains. On the other hand, operators supported at pairs of adjacent chain sites, generically, exhibit nontrivial correlations along the light cone edges. In agreement with Bertini et. al. [Phys. Rev. Lett. 123, 210601 (2019)], they can be expressed through the expectation values of a transfer matrix $T$. Furthermore, we identify a remarkable family of dual-unitary models where an explicit information on the spectrum of $T$ is available. For this class of models we provide a closed analytical formula for the corresponding two-point correlators. This result, in turn, allows an evaluation of local correlators in the vicinity of the dual-unitary regime which is exemplified on the kicked Ising spin chain.

cond-mat.stat-mech↗

arXiv2018

New approach to periodic orbit theory of spectral correlations

The existing periodic orbit theory of spectral correlations for classically chaotic systems relies on the Riemann-Siegel-like representation of the spectral determinants which is still largely hypothetical. We suggest a simpler derivation using analytic continuation of the periodic-orbit expansion of the pertinent generating function from the convergence border to physically important real values of its arguments. As examples we consider chaotic systems without time reversal as well as the Riemann zeta function and Dirichlet L-functions zeros.

nlin.CD↗

arXiv2017

Collectivity and Periodic Orbits in a Chain of Interacting, Kicked Spins

The field of quantum chaos originated in the study of spectral statistics for interacting many-body systems, but this heritage was almost forgotten when single-particle systems moved into the focus. In recent years new interest emerged in many-body aspects of quantum chaos. We study a chain of interacting, kicked spins and carry out a semiclassical analysis that is capable of identifying all kinds of genuin many-body periodic orbits. We show that the collective many-body periodic orbits can fully dominate the spectra in certain cases.

nlin.CD↗

arXiv2016

Semiclassical Identification of Periodic Orbits in a Quantum Many-Body System

While a wealth of results has been obtained for chaos in single-particle quantum systems, much less is known about chaos in quantum many-body systems. We contribute to recent efforts to make a semiclassical analysis of such systems feasible, which is nontrivial due to the exponential proliferation of orbits with increasing particle number. Employing a recently discovered duality relation, we focus on the collective, coherent motion that together with the also present incoherent one typically leads to a mixture of regular and chaotic dynamics. We investigate a kicked spin chain as an example of a presently experimentally and theoretically much studied class of systems.

nlin.CD↗

arXiv2016

Trace formula for spin chains

While detailed information about the semiclassics for single-particle systems is available, much less is known about the connection between quantum and classical dynamics for many-body systems. As an example, we focus on spin chains which are of considerable conceptual and practical importance. We derive a trace formula for coupled spin $j$ particles which relates the quantum energy levels to the classical dynamics. Our derivation is valid in the limit $j\rightarrow\infty$ with $j\hbar={\rm const.}$ and applies to time-continuous as well as to periodically driven dynamics. We provide a simple explanation why the Solari-Kochetov phase can be omitted if the correct classical Hamiltonian is chosen.

quant-ph↗

arXiv2015

Arithmetic and pseudo-arithmetic billiards

The arithmetic triangular billiards are classically chaotic but have Poissonian energy level statistics, in ostensible violation of the BGS conjecture. We show that the length spectra of their periodic orbits divides into subspectra differing by the parity of the number of reflections from the triangle sides; in the quantum treatment that parity defines the reflection phase of the orbit contribution to the Gutzwiller formula for the energy level density. We apply these results to all 85 arithmetic triangles and establish the boundary conditions under which the quantum billiard is \textquotedblleft genuinely arithmetic\textquotedblright, i. e., has Poissonian level statistics; otherwise the billiard is "pseudo-arithmetic" and belongs to the GOE universality class

nlin.CD↗

arXiv2014

Self-averaging characteristics of spectral fluctuations

The spectral form factor as well as the two-point correlator of the density of (quasi-)energy levels of individual quantum dynamics are not self-averaging. Only suitable smoothing turns them into useful characteristics of spectra. We present numerical data for a fully chaotic kicked top, employing two types of smoothing: one involves primitives of the spectral correlator, the second a small imaginary part of the quasi-energy. Self-averaging universal (like the CUE average) behavior is found for the smoothed correlator, apart from noise which shrinks like $1\over\sqrt N$ as the dimension $N$ of the quantum Hilbert space grows. There are periodically repeated quasi-energy windows of correlation decay and revival wherein the smoothed correlation remains finite as $N\to\infty$ such that the noise is negligible. In between those windows (where the CUE averaged correlator takes on values of the order ${1\over N^2}$) the noise becomes dominant and self-averaging is lost. We conclude that the noise forbids distinction of CUE and GUE type behavior. Surprisingly, the underlying smoothed generating function does not enjoy any self-averaging outside the range of its variables relevant for determining the two-point correlator (and certain higher-order ones). --- We corroborate our numerical findings for the noise by analytically determining the CUE variance of the smoothed single-matrix correlator.

nlin.CD↗

arXiv2011

Beyond the Heisenberg time: Semiclassical treatment of spectral correlations in chaotic systems with spin 1/2

The two-point correlation function of chaotic systems with spin 1/2 is evaluated using periodic orbits. The spectral form factor for all times thus becomes accessible. Equivalence with the predictions of random matrix theory for the Gaussian symplectic ensemble is demonstrated. A duality between the underlying generating functions of the orthogonal and symplectic symmetry classes is semiclassically established.

nlin.CD↗

arXiv2011

Parametrization of spin-1 classical states

We give an explicit parametrization of the set of mixed quantum states and of the set of mixed classical states for a spin--1. Classical states are defined as states with a positive Glauber-Sudarshan P-function. They are at the same time the separable symmetric states of two qubits. We explore the geometry of this set, and show that its boundary consists of a two-parameter family of ellipsoids. The boundary does not contain any facets, but includes straight-lines corresponding to mixtures of pure classical states.

quant-ph↗

arXiv2010

Level statistics in arithmetical and pseudo-arithmetical chaos

We resolve a long-standing riddle in quantum chaos, posed by certain fully chaotic billiards with constant negative curvature whose periodic orbits are highly degenerate in length. Depending on the boundary conditions for the quantum wave functions, the energy spectra either have uncorrelated levels usually associated with classical integrability or conform to the "universal" Wigner-Dyson type although the classical dynamics in both cases is the same. The resolution turns out surprisingly simple. The Maslov indices of orbits within multiplets of degenerate length either yield equal phases for the respective Feynman amplitudes (and thus Poissonian level statistics) or give rise to amplitudes with uncorrelated phases (leading to Wigner-Dyson level correlations). The recent semiclassical explanation of spectral universality in quantum chaos is thus extended to the latter case of "pseudo-arithmetical" chaos.

nlin.CD↗

arXiv2010

Semiclassical Theory for Universality in Quantum Chaos with Symmetry Crossover

We address the quantum-classical correspondence for chaotic systems with a crossover between symmetry classes. We consider the energy level statistics of a classically chaotic system in a weak magnetic field. The generating function of spectral correlations is calculated by using the semiclassical periodic-orbit theory. An explicit calculation up to the second order, including the non-oscillatory and oscillatory terms, agrees with the prediction of random matrix theory. Formal expressions of the higher order terms are also presented. The nonlinear sigma (NLS) model of random matrix theory, in the variant of the Bosonic replica trick, is also analyzed for the crossover between the Gaussian orthogonal ensemble and Gaussian unitary ensemble. The diagrammatic expansion of the NLS model is interpreted in terms of the periodic orbit theory.

nlin.CD↗

arXiv2009

Periodic-orbit theory of universal level correlations in quantum chaos

Using Gutzwiller's semiclassical periodic-orbit theory we demonstrate universal behaviour of the two-point correlator of the density of levels for quantum systems whose classical limit is fully chaotic. We go beyond previous work in establishing the full correlator such that its Fourier transform, the spectral form factor, is determined for all times, below and above the Heisenberg time. We cover dynamics with and without time reversal invariance (from the orthogonal and unitary symmetry classes). A key step in our reasoning is to sum the periodic-orbit expansion in terms of a matrix integral, like the one known from the sigma model of random-matrix theory.

nlin.CD↗

arXiv2009

Near action-degenerate periodic-orbit bunches: A skeleton of chaos

Long periodic orbits of hyperbolic dynamics do not exist as independent individuals but rather come in closely packed bunches. Under weak resolution a bunch looks like a single orbit in configuration space, but close inspection reveals topological orbit-to-orbit differences. The construction principle of bunches involves close self-"encounters" of an orbit wherein two or more stretches stay close. A certain duality of encounters and the intervening "links" reveals an infinite hierarchical structure of orbit bunches. -- The orbit-to-orbit action differences $ΔS$ within a bunch can be arbitrarily small. Bunches with $ΔS$ of the order of Planck's constant have constructively interfering Feynman amplitudes for quantum observables, and this is why the classical bunching phenomenon could yield the semiclassical explanation of universal fluctuations in quantum spectra and transport.

nlin.CD↗