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Anomalous Enhancement of Entanglement Entropy in Nonequilibrium Steady States

Hideaki Hakoshima, Akira Shimizu

Published Jun 12, 2017• 0 citations
OpenAlex

Abstract

We investigate the size scaling of the entanglement entropy (EE) in nonequilibrium steady states (NESSs) of a one-dimensional open quantum system with a random potential. It models a mesoscopic conductor, composed of a long quantum wire (QWR) with impurities and two electron reservoirs. The difference of the chemical potentials of the reservoirs induces a steady current, and a NESS is realized in the QWR. Owing to multiple scatterings by the impurities, the translational invariance is broken and the NESS becomes much different from a boost of an equilibrium state. We assume zero temperature for the reservoirs, and the corresponding pure quantum state for the total system. The EE is given by the von Neumann entropy $S_L$ of a subsystem of length $L$ in the QWR. $S_L$ at equilibrium, $S_L^{\rm eq}$, obeys the logarithmic law $S_L^{\rm eq}=O(\ln L)$. In NESSs far from equilibrium, however, $S_L$ grows anomalously fast obeying the `quasi volume law,' $S_L = \eta(L)L |\Delta k_F| + O(\ln L)$. Here, $\Delta k_F$ is the difference of the Fermi wavenumbers of the reservoirs, and $\eta(L)$ satisfies $a \le \eta(L) \lesssim 2a$, where $a$ is a positive constant independent of $L$ or $\Delta k_F$. This anomalous behavior arises from both nonequilibrium and multiple scatterings by impurities.

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We investigate the size scaling of the entanglement entropy (EE) in nonequilibrium steady states (NESSs) of a one-dimensional open quantum system with a random potential. It models a mesoscopic conductor, composed of a long quantum wire (QWR) with impurities and two electron reservoirs. The difference of the chemical potentials of the reservoirs induces a steady current, and a NESS is realized in the QWR. Owing to multiple scatterings by the impurities, the translational invariance is broken and the NESS becomes much different from a boost of an equilibrium state. We assume zero temperature for the reservoirs, and the corresponding pure quantum state for the total system. The EE is given by the von Neumann entropy SLS_LSL​ of a subsystem of length LLL in the QWR. SLS_LSL​ at equilibrium, SLeqS_L^{\rm eq}SLeq​, obeys the logarithmic law SLeq=O(ln⁡L)S_L^{\rm eq}=O(\ln L)SLeq​=O(lnL). In NESSs far from equilibrium, however, SLS_LSL​ grows anomalously fast obeying the `quasi volume law,' SL=η(L)L∣ΔkF∣+O(ln⁡L)S_L = \eta(L)L |\Delta k_F| + O(\ln L)SL​=η(L)L∣ΔkF​∣+O(lnL). Here, ΔkF\Delta k_FΔkF​ is the difference of the Fermi wavenumbers of the reservoirs, and η(L)\eta(L)η(L) satisfies a≤η(L)≲2aa \le \eta(L) \lesssim 2aa≤η(L)≲2a, where aaa is a positive constant independent of LLL or ΔkF\Delta k_FΔkF​. This anomalous behavior arises from both nonequilibrium and multiple scatterings by impurities.
We investigate the size scaling of the entanglement entropy (EE) in nonequilibrium steady states (NESSs) of a one-dimensional open quantum system with a random potential. It models a mesoscopic conductor, composed of a long quantum wire (QWR) with impurities and two electron reservoirs at zero temperature. The EE at equilibrium obeys the logarithmic law. However, in NESSs far from equilibrium the EE grows anomalously fast, obeying the `quasi volume law,' although the conductor is driven by the zero-temperature reservoirs. This anomalous behavior arises from both the far from equilibrium condition and multiple scatterings due to impurities.
Entanglement plays a prominent role in the study of condensed matter many-body systems: Entanglement measures not only quantify the possible use of these systems in quantum information protocols, but also shed light on their physics. However, exact analytical results remain scarce, especially for systems out of equilibrium. In this work we examine a paradigmatic one-dimensional fermionic system that consists of a uniform tight-binding chain with an arbitrary scattering region near its center, which is subject to a DC bias voltage at zero temperature. The system is thus held in a current-carrying nonequilibrium steady state, which can nevertheless be described by a pure quantum state. Using a generalization of the Fisher-Hartwig conjecture, we present an exact calculation of the bipartite entanglement entropy of a subsystem with its complement, and show that the scaling of entanglement with the length of the subsystem is highly unusual, containing both a volume-law linear term and a logarithmic term. The linear term is related to imperfect transmission due to scattering, and provides a generalization of the Levitov-Lesovik full counting statistics formula. The logarithmic term arises from the Fermi discontinuities in the distribution function. Our analysis also produces an exact expression for the particle-number-resolved entanglement. We find that although to leading order entanglement equipartition applies, the first term breaking it grows with the size of the subsystem, a novel behavior not observed in previously studied systems. We apply our general results to a concrete model of a tight-binding chain with a single impurity site, and show that the analytical expressions are in good agreement with numerical calculations. The analytical results are further generalized to accommodate the case of multiple scattering regions.
Abstract Out-of-equilibrium states of many-body systems tend to evade a description by standard statistical mechanics, and their uniqueness is epitomized by the possibility of certain long-range correlations that cannot occur in equilibrium. In quantum many-body systems, coherent correlations of this sort may lead to the emergence of remarkable entanglement structures. In this work, we analytically study the asymptotic scaling of quantum correlation measures—the mutual information (MI) and the fermionic negativity—within the zero-temperature steady state of voltage-biased free fermions on a one-dimensional lattice containing a non-interacting impurity. Previously, we have shown that two subsystems on opposite sides of the impurity exhibit volume-law entanglement, which is independent of the absolute distances of the subsystems from the impurity. Here, we go beyond that result and derive the exact form of the subleading logarithmic corrections to the extensive terms of correlation measures, in excellent agreement with numerical calculations. In particular, the logarithmic term of the MI asymptotics can be encapsulated in a concise formula, depending only on simple four-point ratios of subsystem length scales and on the impurity scattering probabilities at the Fermi energies. This echoes the case of equilibrium states, where such logarithmic terms may convey universal information about the physical system. To compute these exact results, we devise a hybrid method that relies on Toeplitz determinant asymptotics for correlation matrices in both real space and momentum space, successfully circumventing the inhomogeneity of the system. This method could potentially find wider use for analytical calculations of entanglement measures in similar scenarios.
Entanglement measures constitute powerful tools in the quantitative description of quantum many-body systems out of equilibrium. We study entanglement in the current-carrying steady state of a paradigmatic one-dimensional model of noninteracting fermions at zero temperature in the presence of a scatterer. We show that disjoint intervals located on opposite sides of the scatterer, and within similar distances from it, maintain volume-law entanglement regardless of their separation, as measured by their fermionic negativity and coherent information. The mutual information of the intervals, which quantifies the total correlations between them, follows a similar scaling. Interestingly, this scaling entails in particular that if the position of one of the intervals is kept fixed, then the correlation measures depend non-monotonically on the distance between the intervals. By deriving exact expressions for the extensive terms of these quantities, we prove their simple functional dependence on the scattering probabilities, and demonstrate that the strong long-range entanglement is generated by the coherence between the transmitted and reflected parts of propagating particles within the bias-voltage window. The generality and simplicity of the model suggest that this behavior should characterize a large class of nonequilibrium steady states.
Thermal equilibrium states of local quantum many-body systems are notorious for their spatially decaying correlations, which place severe restrictions on the types of many-body entanglement structures that may be observed at finite temperatures. These restrictions may however be defied when an out-of-equilibrium steady state is considered instead. In this paper, we study the entanglement properties of free fermions on a one-dimensional lattice that contains a generic charge- and energy-conserving noninteracting impurity, and that is connected at its edges to two reservoirs with different equilibrium energy distributions. These distributions may differ in either temperature, chemical potential, or both, thereby inducing an external bias. We analytically derive exact asymptotic expressions for several quantum information measures -- the mutual information, its Rényi generalizations, and the fermionic negativity -- that quantify the correlation and entanglement between two subsystems located on opposite sides of the impurity. We show that all these measures scale (to a leading order) linearly with the overlap between one subsystem and the mirror image of the other (upon reflection of the latter about the impurity), independently of the distance between the subsystems. While a simple proportionality relation between the negativity and Rényi versions of the mutual information is observed to hold at zero temperature, it breaks down at finite temperatures, suggesting that these quantities represent strong long-range correlations of different origins. Our results generalize previous findings that were limited to the case of a chemical-potential bias at zero temperature, rigorously demonstrating that the effect of long-range volume-law entanglement is robust at finite temperatures.
The current-carrying steady state that arises in the middle of a metallic wire connected to macroscopic leads is characterized regarding its response functions, correlations, and entanglement entropy. The spectral function and the dynamical structure factor show clear nonequilibrium signatures accessible by state-of-the-art techniques. In contrast with the equilibrium case, the entanglement entropy is extensive with logarithmic corrections at zero temperature that depend on the lead-wire coupling and, in a nonanalytic way, on voltage. This shows that some robust universal quantities found in gapless equilibrium phases do not persist away from equilibrium.