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arXiv:2308.04505 · Annalen der Physik

Spectral Structure and Doublon Dissociation in the Two‐Particle Non‐Hermitian Hubbard Model

Stefano Longhi

Published Oct 2, 2023• 16 citations

Abstract

Abstract Strongly‐correlated systems in non‐Hermitian models are an emergent area of research. Herein, a non‐Hermitian Hubbard model is considered, where the single‐particle hopping amplitudes on the lattice are not reciprocal, and provide exact analytical results of the spectral structure in the two‐particle sector of Hilbert space under different boundary conditions. The analysis unveils some interesting spectral and dynamical effects of purely non‐Hermitian nature and that deviate from the usual scenario found in the single‐particle regime. Specifically, a spectral phase transition of the Mott‐Hubbard band on the infinite lattice is predicted as the interaction energy is increased above a critical value, from an open to a closed loop in complex energy plane, and the dynamical dissociation of doublons, i.e., instability of two‐particle bound states, in the bulk of the lattice, with a sudden revival of the doublon state when the two particles reach the lattice edge. Particle dissociation observed in the bulk of the lattice is a clear manifestation of non‐Hermitian dynamics arising from the different lifetimes of single‐particle and two‐particle states, whereas the sudden revival of the doublon state at the boundaries is a striking burst edge dynamical effect peculiar to non‐Hermitian systems with boundary‐dependent energy spectra, here predicted for the first time for correlated particles.

DOI: 10.1002/andp.202300291

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Abstract Strongly‐correlated systems in non‐Hermitian models are an emergent area of research. Herein, a non‐Hermitian Hubbard model is considered, where the single‐particle hopping amplitudes on the lattice are not reciprocal, and provide exact analytical results of the spectral structure in the two‐particle sector of Hilbert space under different boundary conditions. The analysis unveils some interesting spectral and dynamical effects of purely non‐Hermitian nature and that deviate from the usual scenario found in the single‐particle regime. Specifically, a spectral phase transition of the Mott‐Hubbard band on the infinite lattice is predicted as the interaction energy is increased above a critical value, from an open to a closed loop in complex energy plane, and the dynamical dissociation of doublons, i.e., instability of two‐particle bound states, in the bulk of the lattice, with a sudden revival of the doublon state when the two particles reach the lattice edge. Particle dissociation observed in the bulk of the lattice is a clear manifestation of non‐Hermitian dynamics arising from the different lifetimes of single‐particle and two‐particle states, whereas the sudden revival of the doublon state at the boundaries is a striking burst edge dynamical effect peculiar to non‐Hermitian systems with boundary‐dependent energy spectra, here predicted for the first time for correlated particles.

We investigate features in the single-particle spectral function beyond the Hubbard bands in the strongly correlated normal phase of the Bose-Hubbard model. There are two distinct classes of additional peaks generated by the bosonic statistics. The first type is thermally activated Hubbard ``sidebands'', with the same physical origin as the zero-temperature Hubbard bands, but generated by excitations from thermally activated local occupation number states. The second class are two-particle fluctuation resonances driven by the lattice dynamics. In the unity filling Mott insulator, this takes the form of a localized triplon combined with a dispersing holon. Both types of resonances also manifest themselves in the structure factor and the interaction modulation spectra obtained from nonequilibrium bosonic dynamical mean-field theory calculations. Our findings explain experimental lattice modulation and Bragg spectroscopy results, and they predict a strong temperature dependence of the first sideband, thereby opening the door to precise thermometry of strongly correlated lattice bosons.

The interband dynamics of a two-band Bose–Hubbard (BH) model is studied with strongly correlated bosons forming single-site double occupancies referred to as doublons. Our model for resonant doublon interband coupling exhibits interesting dynamical features such as quantum Zeno effect, the generation of states such as a two-band Bell-like state and an upper-band Mott-like state. The evolution of the asymptotic state is controlled here by the effective opening of one or both of the two bands, which models decay channels.

Spectral properties of the Bose-Hubbard model and a recently proposed coupled-cavity model are studied by means of quantum Monte Carlo simulations in one dimension. Both models exhibit a quantum phase transition from a Mott insulator to a superfluid phase. The dynamic structure factor $S(k,\ensuremath{\omega})$ and the single-particle spectrum $A(k,\ensuremath{\omega})$ are calculated, focusing on the parameter region around the phase transition from the Mott insulator with density one to the superfluid phase, where correlations are important. The strongly interacting nature of the superfluid phase manifests itself in terms of additional gapped modes in the spectra. Comparison is made to recent analytical work on the Bose-Hubbard model. Despite some subtle differences due to the polaritonic particles in the cavity model, the gross features are found to be very similar to the Bose-Hubbard case. For the polariton model, emergent particle-hole symmetry near the Mott lobe tip is demonstrated and temperature and detuning effects are analyzed. A scaling analysis for the generic transition suggests mean-field exponents, in accordance with field-theory results.

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Dynamical mean-field theory (DMFT) studies frequently observe a fine structure in the local spectral function of the SU(2) Fermi-Hubbard model at half filling: In the metallic phase close to the Mott transition, subpeaks emerge at the inner edges of the Hubbard bands. Here we demonstrate that these subpeaks originate from the low-energy effective interaction of doublon-holon pairs, by investigating how the correlation functions of doublon and holon operators contribute to the subpeaks. A mean-field analysis of the low-energy effective Hamiltonian provides results consistent with our DMFT calculation using the numerical renormalization group as an impurity solver. In the SU(3) and SU(4) Hubbard models, the subpeaks become more pronounced due to the increased degeneracy of doublon-holon pair excitations.

We investigate the dynamical properties of one-dimensional dissipative Fermi-Hubbard models, which are described by the Lindblad master equations with site-dependent jump operators. The corresponding non-Hermitian effective Hamiltonians with pure loss terms possess parity-time (

\mathcal{PT}

) symmetry if we compensate the system additionally with an overall gain term. By solving exactly the two-site Lindblad equation with fixed dissipation, we find that the dynamics of the rescaled density matrix shows an instability as the interaction increases over a threshold, which can be equivalently described in the scheme of non-Hermitian effective Hamiltonians. This instability is also observed in multisite systems and closely related to the

\mathcal{PT}

-symmetry breaking accompanied by appearance of complex eigenvalues of the effective Hamiltonian. Moreover, we unveil that the dynamical instability of the antiferromagnetic Mott phase comes from the

\mathcal{PT}

-symmetry breaking in highly excited bands, although the low-energy effective model of the non-Hermitian Hubbard model in the strongly interacting regime is always Hermitian. We also provide a quantitative estimation of the time for the observation of dynamical

\mathcal{PT}

-symmetry breaking which could be probed in experiments.