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arXiv:2008.03745 · Physical Review Research

Modification of quantum many-body relaxation by perturbations exhibiting a banded matrix structure

Lennart Dabelow, Patrick Vorndamme, Peter Reimann

Published Aug 6, 2020• 7 citations
arXivHTMLPDF

Abstract

This paper explores the relation between the response of a generic isolated many-body quantum system to a weak-to-moderate perturbation on the one hand and the perturbation's matrix structure in the eigenbasis of the unperturbed system on the other hand. An analytical theory for this relationship is demonstrated and verified numerically.

DOI
10.1103/physrevresearch.2.033210

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This paper explores the relation between the response of a generic isolated many-body quantum system to a weak-to-moderate perturbation on the one hand and the perturbation's matrix structure in the eigenbasis of the unperturbed system on the other hand. An analytical theory for this relationship is demonstrated and verified numerically.
Abstract We substantially extend our relaxation theory for perturbed many-body quantum systems from ((2020) Phys. Rev. Lett. 124 120602) by establishing an analytical prediction for the time-dependent observable expectation values which depends on only two characteristic parameters of the perturbation operator: its overall strength and its range or band width. Compared to the previous theory, a significantly larger range of perturbation strengths is covered. The results are obtained within a typicality framework by solving the pertinent random matrix problem exactly for a certain class of banded perturbations and by demonstrating the (approximative) universality of these solutions, which allows us to adopt them to considerably more general classes of perturbations. We also verify the prediction by comparison with several numerical examples.
An analytical prediction is established of how an isolated many-body quantum system relaxes towards its thermal longtime limit under the action of a time-independent perturbation, but still remaining sufficiently close to a reference case whose temporal relaxation is known. This is achieved within the conceptual framework of a typicality approach by showing and exploiting that the time-dependent expectation values behave very similarly for most members of a suitably chosen ensemble of perturbations. The predictions are validated by comparison with various numerical and experimental results from the literature.
The structure of few-body Hamiltonian matrices is studied in the semiclassical regime. Given (A${^}_{1}$(q^,p^),A${^}_{2}$ (q^,p^)), a pair of operators, it can be shown that, under quite general conditions, A${^}_{2}$ takes the form of a banded matrix in the ordered eigenrepresentation of A${^}_{1}$. Moreover, the bandwidth depends only on \ensuremath{\Elzxh} and certain generalized microcanonical averages. In particular, if H=H0{H}_{0}H0​+\ensuremath{\epsilon}V, this implies that the perturbed Hamiltonian is banded in the appropriately ordered unperturbed basis.
Understanding how the dynamics of a given quantum system with many degrees of freedom is altered by the presence of a generic perturbation is a notoriously difficult question. Recent works predict that, in the overwhelming majority of cases, the unperturbed dynamics is just damped by a simple function, e.g., exponentially as expected from Fermi's golden rule. While these predictions rely on random-matrix arguments and typicality, they can only be verified for a specific physical situation by comparing to the actual solution or measurement. Crucially, it also remains unclear how frequent and under which conditions counterexamples to the typical behavior occur. In this work, we discuss this question from the perspective of projection-operator techniques, where exponential damping of a density matrix occurs in the interaction picture but not necessarily in the Schrödinger picture. We show that a nontrivial damping in the Schrödinger picture can emerge if the dynamics in the unperturbed system possesses rich features, for instance due to the presence of strong interactions. This suggestion has consequences for the time dependence of correlation functions. We substantiate our theoretical arguments by large-scale numerical simulations of charge transport in the extended Fermi-Hubbard chain, where the nearest-neighbor interactions are treated as a perturbation to the integrable reference system.
In the general frame of a modified many-body perturbation theory, the single-band Hubbard model is treated in the limit of weak electron-electron interaction. In this modification the decomposition of the Hamiltonian into an effective free system and an effective perturbation is rearranged iteratively in each order of the formalism, resulting in a successive reduction of the interaction part. In comparison with conventional perturbation techniques, this means an extended range of validity. Imposing thermodynamic self-consistency, we calculate the wave-vector-dependent electronic self-energy up to second order and analyze the influence of the Coulomb correlations. Several well-known physical properties of the Hubbard model are recovered in characteristic single-particle quantities. Therefore, the effect of the band occupation n and the electron-electron repulsion U on spectral densities, band structures, densities of states, magnetization, etc. is investigated in detail. Some of the most remarkable results are: first, a splitting of the free Bloch band into two quasiparticle subbands when the coupling strength reaches moderate values. These subbands are linked by a small damping-induced background in the density of states and are separated roughly by an energy of the order of U. In addition, the upper subband is shifted to higher energies in the spectrum proportionally to U while the lower one is nearly fixed. Second, we found a paramagnetic-ferromagnetic phase diagram that has a critical coupling strength Uc{U}_{c}Uc​ and exhibits strong indications of a lower critical occupation number nc{n}_{c}nc​. There is no spontaneous magnetization in the system for parameters smaller than these values. Compared with the Stoner model, one thus obtains striking improvements. In many cases our conclusions for the weak-coupling limit are qualitatively in good agreement with well-known results in the strong-correlation regime.