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Infinite Range Correlations in Non-Equilibrium Quantum Systems and Their Possible Experimental Realizations

Zohar Nussinov

Published Oct 18, 2017• 1 citation
OpenAlex

Abstract

We consider systems that start from and/or end in thermodynamic equilibrium while experiencing a finite rate of change of their energy density or of other intensive quantities $q$ at intermediate times. We demonstrate that at these times, during which the global intensive quantities $q$ vary, the size of the associated covariance, the connected pair correlator $|G_{ij}| = |\langle q_{i} q_{j} \rangle - \langle q_{i} \rangle \langle q_{j} \rangle|$, between any two (arbitrarily far separated) sites $i$ and $j$ is, on average, finite. This non-vanishing character of the connected correlations for asymptotically distant sites also applies to theories with purely local interactions. In simple models, these correlations may be traced to the generic volume law entanglement of finite temperature states. Once the global mean of $q$ no longer changes, the average of $|G_{ij}|$ over all spatial separations $|i-j|$ may tend to zero. However, when the equilibration times are significant (e.g., as in a glass that is not in true thermodynamic equilibrium yet in which the energy density (or temperature) reaches a final steady state value), these long range correlations may persist also long after $q$ ceases to change. We briefly discuss possible experimental implications of our findings and speculate on their potential realization in glasses (where a prediction of a theory based on the effect that we describe here suggests a universal collapse of the viscosity that agrees with all published viscosity measurements) and non-Fermi liquids.

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We consider systems that start from and/or end in thermodynamic equilibrium while experiencing a finite rate of change of their energy density or of other intensive quantities qqq at intermediate times. We demonstrate that at these times, during which the global intensive quantities qqq vary, the size of the associated covariance, the connected pair correlator ∣Gij∣=∣⟨qiqj⟩−⟨qi⟩⟨qj⟩∣|G_{ij}| = |\langle q_{i} q_{j} \rangle - \langle q_{i} \rangle \langle q_{j} \rangle|∣Gij​∣=∣⟨qi​qj​⟩−⟨qi​⟩⟨qj​⟩∣, between any two (arbitrarily far separated) sites iii and jjj is, on average, finite. This non-vanishing character of the connected correlations for asymptotically distant sites also applies to theories with purely local interactions. In simple models, these correlations may be traced to the generic volume law entanglement of finite temperature states. Once the global mean of qqq no longer changes, the average of ∣Gij∣|G_{ij}|∣Gij​∣ over all spatial separations ∣i−j∣|i-j|∣i−j∣ may tend to zero. However, when the equilibration times are significant (e.g., as in a glass that is not in true thermodynamic equilibrium yet in which the energy density (or temperature) reaches a final steady state value), these long range correlations may persist also long after qqq ceases to change. We briefly discuss possible experimental implications of our findings and speculate on their potential realization in glasses (where a prediction of a theory based on the effect that we describe here suggests a universal collapse of the viscosity that agrees with all published viscosity measurements) and non-Fermi liquids.
We consider general systems that start from and/or end in thermodynamic equilibrium while experiencing a finite rate of change of their energy density or other intensive quantities q at intermediate times. We demonstrate that at these times, during which q varies at a finite rate, the associated covariance, the connected pair correlator Gij=〈qiqj〉−〈qi〉〈qj〉, between any two (far separated) sites i and j in a macroscopic system may, on average, become finite. Once the global mean q no longer changes, the average of Gij over all site pairs i and j may tend to zero. However, when the equilibration times are significant (e.g., as in a glass that is not in true thermodynamic equilibrium yet in which the energy density (or temperature) reaches a final steady state value), these long range correlations may persist also long after q ceases to change. We explore viable experimental implications of our findings and speculate on their potential realization in glasses (where a prediction of a theory based on the effect that we describe here suggests a universal collapse of the viscosity that agrees with all published viscosity measurements over sixteen decades) and non-Fermi liquids. We discuss effective equilibrium in driven systems and derive uncertainty relation based inequalities that connect the heat capacity to the dynamics in general open thermal systems. These rigorous thermalization inequalities suggest the shortest possible fluctuation times scales in open equilibrated systems at a temperature T are typically “Planckian” (i.e., O(ħ/(kBT))). We briefly comment on parallels between quantum measurements, unitary quantum evolution, and thermalization and on how Gaussian distributions may generically emerge.
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.
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.
Quantum Mechanics was established as the theory of the microscopic world, which allowed to understand processes in atoms and molecules. Its emergence led to a new scientific paradigm that quickly spread to different research fields. Two relevant examples are Quantum Thermodynamics and Quantum Many-Body Theory, where the former aims to characterize thermodynamic processes in quantum systems and the latter intends to understand the properties of quantum many-body systems. In this thesis, we tackle some of the questions in the overlap between these disciplines, focusing on the concepts of temperature and correlations. Specifically, it contains results on the following topics: locality of temperature, correlations in long-range interacting systems and thermometry at low temperature. The problem of locality of temperature is considered for a system at thermal equilibrium and consists in studying whether it is possible to assign a temperature to any of the subsystems of the global system such that both local and global temperatures are equal. We tackle this problem in two different settings, for generic one-dimensional spin chains and for a bosonic system with a phase transition at non-zero temperature. In the first case, we consider generic one-dimensional translation-invariant spin systems with short-range interactions and prove that it is always possible to assign a local temperature equal to the global one for any temperature, including at criticality. For the second case, we consider a three-dimensional discretized version of the Bose-Einstein model at the grand canonical ensemble for some temperature and particle density, and characterize its non-zero-temperature phase transition. Then, we show that temperature is locally well-defined at any temperature and at any particle density, including at the phase transition. Additionally, we observe a qualitative relation between correlations and locality of temperature in the system. Moving to correlations, we consider fermionic two-site long-range interacting systems at thermal equilibrium. We show that correlations between anti-commutative operators at non-zero temperature are upper bounded by a function that decays polynomially with the distance and with an exponent that is equal to the interaction exponent, which characterizes the interactions in the Hamiltonian. Moreover, we show that our bound is asymptotically tight and that the results extend to density-density correlations as well as other types of correlations for quadratic and fermionic Hamiltonians with long-range interactions. Regarding the results on thermometry, we consider a bosonic model and prove that strong coupling between the probe and the system can boost the thermal sensitivity for low temperature. Furthermore, we provide a feasible measurement scheme capable of producing optimal estimates at the considered regime.
Non-equilibrium phase transitions of open quantum systems generically exhibit diverging classical but not quantum correlations. Still entanglement -- characterizing the latter correlations -- can be sensitive to the phase transition. Furthermore, mutual information, bounding the total correlations, should exhibit critical scaling at the transition. In this work, we study these quantities in the steady state of open quantum Ising chains with power-law interactions (with the exponent 0≤α≤30\le α\le 30≤α≤3) where spins are subject to spontaneous emission. The bulk of this paper is dedicated to a detailed analytical as well as numerical analysis of the infinite-range model (α=0α=0α=0), a model that is closely related to the paradigmatic open Dicke model. Our main findings are that the entanglement, while being finite, peaks, exhibits a kink and takes a universal value at the transition, while the mutual information exhibits critical scaling not only at the transition but well into the ordered phase, underscoring a hidden criticality that is not captured by (two-point) correlations. We consider three distinct entanglement measures: logarithmic negativity; quantum Fisher information; and, spin squeezing. Specifically, we show that the collective spin operator that maximizes the quantum Fisher information can be identified with the \textit{gapless} mode of the phase transition, while the squeezed direction is that of the \textit{gapped} mode. Finally, we investigate power-law interacting models using matrix product states where we find comparable bounds on squeezing even when no phase transition is expected (for larger ααα), thus the connection to the phase transition does not appear to hold for shorter-range interactions.
We study correlations in fermionic lattice systems with long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anti-commuting operators and generalize a long-range Lieb-Robinson type bound. Our results show that in these systems of spatial dimension DDD with, not necessarily translation invariant, two-site interactions decaying algebraically with the distance with an exponent α≥2 D\alpha \geq 2\,Dα≥2D, correlations between such operators decay at least algebraically with an exponent arbitrarily close to α\alphaα at any non-zero temperature. Our bound is asymptotically tight, which we demonstrate by a high temperature expansion and by numerically analyzing density-density correlations in the 1D quadratic (free, exactly solvable) Kitaev chain with long-range pairing.