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arXiv:2207.01020 · Physical review. B./Physical review. B

Exact nonequilibrium hole dynamics, magnetic polarons, and string excitations in antiferromagnetic Bethe lattices

K. Knakkergaard Nielsen

Published Sep 27, 2022• 7 citations
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Abstract

We investigate a rare instance of an exactly solvable nonequilibrium many-body problem. In particular, we derive an exact solution for the nonequilibrium dynamics of an initially localized single hole in a fully anisotropic antiferromagnetic Bethe lattice, described by the $t\ensuremath{-}{J}_{z}$ model. The solvability of the model relies on the fractal self-similarity of Bethe lattices, making it possible to compute the full motion of the hole as it moves through the lattice, as well as exactly characterizing the resulting effect on spin-spin correlation functions. We find that the hole remains bound to its initial position with large aperiodic oscillations in the hole density distribution. We track this back to the irregular pattern of the eigenenergies of the magnetic polaron ground state and string excitations, which we also determine exactly.

DOI
10.1103/physrevb.106.115144

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New Journal of Physics•Nov 16, 2021•20 citations
We investigate a rare instance of an exactly solvable nonequilibrium many-body problem. In particular, we derive an exact solution for the nonequilibrium dynamics of an initially localized single hole in a fully anisotropic antiferromagnetic Bethe lattice, described by the t\ensuremath−Jzt\ensuremath{-}{J}_{z}t\ensuremath−Jz​ model. The solvability of the model relies on the fractal self-similarity of Bethe lattices, making it possible to compute the full motion of the hole as it moves through the lattice, as well as exactly characterizing the resulting effect on spin-spin correlation functions. We find that the hole remains bound to its initial position with large aperiodic oscillations in the hole density distribution. We track this back to the irregular pattern of the eigenenergies of the magnetic polaron ground state and string excitations, which we also determine exactly.
We develop a nonperturbative theory for hole dynamics in antiferromagnetic spin lattices, as described by the t-J model. This is achieved by generalizing the self-consistent Born approximation to nonequilibrium systems, making it possible to calculate the full time-dependent many-body wave function. Our approach reveals three distinct dynamical regimes, ultimately leading to the formation of magnetic polarons. Following the initial ballistic stage of the hole dynamics, coherent formation of string excitations gives rise to characteristic oscillations in the hole density. Their damping eventually leaves behind magnetic polarons that undergo ballistic motion with a greatly reduced velocity. The developed theory provides a rigorous framework for understanding nonequilibrium physics of defects in quantum magnets and quantitatively explains recent observations from cold-atom quantum simulations in the strong coupling regime.
The dynamics of charge carriers in lattices of quantum spins is a long standing and fundamental problem. Recently, a new generation of quantum simulation experiments based on atoms in optical lattices has emerged that gives unprecedented insights into the detailed spatial and temporal dynamics of this problem, which compliments earlier results from condensed matter experiments. Focusing on observables accessible in these new experiments, we explore here the equilibrium as well as nonequilibrium dynamics of a mobile hole in two coupled antiferromagnetic spin lattices. Using a self-consistent Born approximation, we calculate the spectral properties of the hole in the bilayer and extract the energy bands of the quasiparticles, corresponding to magnetic polarons that are either symmetric or antisymmetric under layer exchange. These two kinds of polarons are degenerate at certain momenta due to the antiferromagnetic symmetry, and we, furthermore, examine how the momentum of the ground-state polaron depends on the interlayer coupling strength. The long time dynamics of a hole initially created in one layer is shown to be characterized by oscillations between the two layers with a frequency given by the energy difference between the symmetric and the antisymmetric polaron. We finally demonstrate that the expansion velocity of a hole initially created at a given lattice site is governed by the ballistic motion of polarons for long times, and that it increases as a quantum phase transition to a disordered state is approached. While our linear spin wave theory does not include interlayer dimer correlations important close to the phase transition, this speedup is a physically robust prediction as it reflects the decreasing energy cost of hopping when the magnetic order decreases.
We investigate the interplay between the formation of a lattice and magnetic polaron in the case of a single hole in the antiferromagnetic background. We present an exact analytical solution of the Holstein $t\ensuremath{-}J$ model in infinite dimensions. The ground-state energy, electron-lattice correlation function, spin bag dimension as well as spectral properties are calculated. The magnetic and hole-lattice correlations sustain each other, i.e., the presence of antiferromagnetic correlations favors the formation of the lattice polaron at lower values of the electron-phonon coupling while the polaronic effect contributes to reduce the number of spin defects in the antiferromagnetic background. The crossover towards a spin-lattice small polaron region of the phase diagram becomes a discontinuous transition in the adiabatic limit.
We solved numerically the integral equation for the selfenergy which describes the motion of a single hole in a two-dimensional quantum antiferromagnet (AF) within the selfconsistent Born approximation. This formulation stresses the similarity of the AF-spin polaron with the standard polaron problem. We confine our calculation to finite cluster geometries and compare with results from previous exact diagonalization studies. The spectral function is characterized by a narrow quasiparticle (qp) peak at the low energy side of the spectra, which appears to be well separated from the incoherent band part for large enough J values. For small J we find a reduced width of ~7t for the incoherent band. The bottom of the coherent qp band always occurs at (±π/2, ±π/2). Its bandwidth initially increases with J until J≃t and then decreases as 2t2/J. A complete characterization of our solution is given, including the dispersion relation and effective masses of this quasiparticle. The comparison with exact diagonalization studies for a 4×4 cluster is remarkably good. From our results we see that a lattice of 16×16 sites describes adequately the thermodynamic limit. We conclude that the simple Born approximation is a valuable scheme to characterize the dynamics of one hole in the t-J model. both in the perturbative and the strong coupling regimes.
Abstract Tremendous recent progress in the quantum simulation of the Hubbard model paves the way to controllably study doped antiferromagnetic Mott insulators. Motivated by these experimental advancements, we numerically study the real-time dynamics of a single hole created in an antiferromagnet on a square lattice, as described by the t–J model. Initially, the hole spreads ballistically with a velocity proportional to the hopping matrix element. At intermediate to long times, the dimensionality as well as the spin background determine the hole dynamics. A hole created in the ground state of a two dimensional (2D) quantum antiferromagnet propagates again ballistically at long times but with a velocity proportional to the spin exchange coupling, showing the formation of a magnetic polaron. We provide an intuitive explanation of this dynamics in terms of a parton construction, which leads to a good quantitative agreement with the numerical tensor network state simulations. In the limit of infinite temperature and no spin exchange couplings, the dynamics can be approximated by a quantum random walk on a Bethe lattice with coordination number z ̃ = 4 . Adding Ising interactions corresponds to an effective disordered potential, which can dramatically slow down the hole propagation, consistent with subdiffusive dynamics. The study of the hole dynamics paves the way for understanding the microscopic constituents of this strongly correlated quantum state.
Abstract The dynamics of a hole motion in a quantum antiferromagnet has been studied in the past three decades because of its relationship to models related to superconductivity in cuprates. The same problem has received significant attention because of its connection to very recent experiments of the dynamics of ultra-cold atoms in optical lattices where models of strongly correlated electrons can be simulated. In this paper we apply the diagrammatic Monte Carlo method to calculate the single-hole Green’s function in the t–J model, where the J term is linearized, in a wide range of imaginary-time with the aim to examine the polaron formation and in particular the details of the contribution of the so-called string excitations found in such recent experiments. We calculate the single-hole spectral function by analytic continuation from imaginary to real time and study the various aspects that constitute the string picture, such as, the energy–momentum dependence of the main quasiparticle peak and its residue, the internal excitations of the string which appear as multiple peaks in the spectral function as well as their momentum dependence. We find that the earlier analysis of the spectral function based on a mobile-hole connected with a string of overturn spins and the contribution of the internal string excitations as obtained from the non-crossing approximation is accurate.