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Quantum Frontiers

Luttinger liquid under non-Hermitian quantum sensing

Jingcheng Liang, Jiangping Hu

Published Jan 22, 2025• 1 citation
Publisher

Abstract

Abstract The interplay between dissipation and interactions could give rise to exotic phenomena. In this work, we apply the non-Hermitian (NH) linear response theory to investigate the density response of a spinless Luttinger liquid under local NH perturbations. We show that NH backward scattering results in interference between correlation functions arising from dissipation and Langevin noise terms, leading to fluctuations in the induced density rate of change as the Luttinger parameter K varies. For odd integer values of K, this rate vanishes, indicating immunity to dissipation. We demonstrate that the steady-state induced density is predominantly determined by Langevin noise, underscoring the necessity of including Langevin noise terms in NH Hamiltonians. For attractive interactions, the response to NH perturbations grows exponentially and exhibits a $\pi /2$ π / 2 phase shift, indicating stronger sensitivity to NH perturbations compared to Hermitian ones. Additionally, we show that NH forward scattering leads to a decay of the induced density. We refer to these peculiar behaviors of density fluctuations under NH perturbations as anomalous Friedel oscillations. Our findings offer new prospects for sensing and understanding dissipation and interactions.

DOI
10.1007/s44214-024-00074-0

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Abstract The interplay between dissipation and interactions could give rise to exotic phenomena. In this work, we apply the non-Hermitian (NH) linear response theory to investigate the density response of a spinless Luttinger liquid under local NH perturbations. We show that NH backward scattering results in interference between correlation functions arising from dissipation and Langevin noise terms, leading to fluctuations in the induced density rate of change as the Luttinger parameter K varies. For odd integer values of K, this rate vanishes, indicating immunity to dissipation. We demonstrate that the steady-state induced density is predominantly determined by Langevin noise, underscoring the necessity of including Langevin noise terms in NH Hamiltonians. For attractive interactions, the response to NH perturbations grows exponentially and exhibits a π/2\pi /2π/2 π / 2 phase shift, indicating stronger sensitivity to NH perturbations compared to Hermitian ones. Additionally, we show that NH forward scattering leads to a decay of the induced density. We refer to these peculiar behaviors of density fluctuations under NH perturbations as anomalous Friedel oscillations. Our findings offer new prospects for sensing and understanding dissipation and interactions.
We demonstrate the universal properties of dissipative Tomonaga-Luttinger (TL) liquids by calculating correlation functions and performing finite-size scaling analysis of a non-Hermitian XXZ spin chain as a prototypical model in one-dimensional open quantum many-body systems. Our analytic calculation is based on effective field theory with bosonization, finite-size scaling approach in conformal field theory, and the Bethe-ansatz solution. Our numerical analysis is based on the density-matrix renormalization group generalized to non-Hermitian systems (NH-DMRG). We uncover that the model in the massless regime with weak dissipation belongs to the universality class characterized by the complex-valued TL parameter, which is related to the complex generalization of the c=1c=1c=1 conformal field theory. As the dissipation strength increases, the values of the TL parameter obtained by the NH-DMRG begin to deviate from those obtained by the Bethe-ansatz analysis, indicating that the model becomes massive for strong dissipation. Our results can be tested with the two-component Bose-Hubbard system of ultracold atoms subject to two-body loss.
We study a one-dimensional Fermi gas in the presence of dissipative coupling to environment through the Lindblad equation. The dissipation involves energy exchange with the environment and favours the relaxation of electrons to excitations. After switching on the dissipation, the system approaches a steady state, which is described by a generalized Gibbs ensemble. The fermionic single particle density matrix resembles deceivingly to that in a hermitian interaction quench. It decays inversely with the distance for short times due to the fermionic correlations in the initial state, which changes into a noninteger power law decay for late times, representing dissipation-induced Luttinger liquid behavior. However, the crossover between the two regions occurs due to dissipation-induced damping, and is unrelated to the propagation of excitations. The velocity of information spreading is set by the dissipative coupling, and differs significantly from the original sound velocity. The thermodynamic entropy grows as -t ln t initially, and saturates to an extensive value. Our results can be tested experimentally in one-dimensional Dirac systems.
Linear response theory is the theoretical foundation of modern experiments. In particular, it plays a vital role in measuring quantum matters. Its main idea is to take the external probe signal of the physical system as a perturbation and use the correlation function in the unperturbed equilibrium state to depict the response to the observable in system. In recent half century, the linear response theory for the closed quantum system has achieved great success in experiments on quantum matters. In recent years, with the tremendous progress of the precise manipulation of the light-matter interaction, the ultracold atom experiments can precisely control dissipative quantum many-body systems. With the discovery of many novel phenomena, dissipative quantum systems and non-Hermitian physics have attracted extensive attention in theory and experiment. We developed a linear response theory, named non-Hermitian linear response theory, to deal with open quantum systems. This theory takes the non-Hermitian term and quantum noise, which are induced by dissipation, as an external perturbative input, to detect the properties of the quantum system, and relates the time evolution of the observable with the correlation function in the unperturbed state of the system. The non-Hermitian linear response theory provides a new method for distinguishing the exotic quantum phase from the normal phase. The theoretical predictions are highly consistent with the recent experimental results of cold atom systems. This paper will review the non-Hermitian linear response theory and discuss its applications in quantum many-body and time-reversal symmetric quantum systems.
We study the density disturbance of a correlated 1D electron liquid in the presence of a scatterer or a barrier. The 2k_F-periodic density profile away from the barrier (Friedel oscillation) is computed for arbitrary electron--electron interaction and arbitrary impurity strength. We find that in presence of correlations, the Friedel oscillation decays slower than predicted by Fermi liquid theory. In the case of a spinless Luttinger liquid characterized by an interaction constant g<1, the asymptotic decay of the Friedel oscillation is x^-g. For a weak scatterer, the decay is even slower at small-to-intermediate distances from the impurity, with a crossover to the asymptotic x^-g law.
A Luttinger liquid (LL) describes low energy excitations of many interacting one dimensional systems, and exhibits universal response both in and out of equilibrium. We analyze its behavior in the non-Hermitian realm after quantum quenching to a PT-symmetric LL by focusing on the fermionic single particle density matrix. For short times, we demonstrate the emergence of unique phenomena, characteristic to non-Hermitian systems, that correlations propagate faster than the conventional maximal speed, known as the Lieb-Robinson bound. These emergent supersonic modes travel with velocities that are multiples of the conventional light cone velocity. This behavior is argued to be generic for correlators in non-Hermitian systems. In the long time limit, we find typical LL behavior, extending the LL universality to the nonequilibrium, non-Hermitian case. Our analytical results are benchmarked numerically and indicate that the dispersal of quantum information is much faster in non-Hermitian systems.
We study operator correlations of a spinful Luttinger liquid after introducing a non-Hermitian interaction quench, yielding supersonic modes and dominant superconducting correlations as signatures of the nonunitary dynamics as well as spin-charge separation. A comparative analysis with the Hermitian counterpart, i.e., when the quench is Hermitian, shows a significant difference in the behavior of the model. We derive exact expressions for different operator correlations and show that the superconducting correlations decay slower than the charge and spin density wave correlations, especially within the short-time limit, and at the long-time limit all the operator correlations merge differed only by phase factors in the case of non-Hermitian interaction quench whereas they do not merge in the case of Hermitian interaction quench. In both cases known Luttinger liquid universality is retained at the long-time limit. We also analyze how the dynamics of operator correlations vary in the presence of anisotropy in the quenching parameters.