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NON RELATIVISTIC QUANTUM FIELD THEORY: DYNAMICS AND IRREVERSIBILITY

Instituto Balseiro, UNCuyo and CNEA S.C. de Bariloche, Río Negro, R8402AGP, Argentina, Lucas Daguerre, Gonzalo Torroba, Centro Atómico Bariloche and CONICET S.C. de Bariloche, Río Negro, R8402AGP, Argentina, Raimel A. Medina, IST Austria Am Campus 1, 3400 Klosterneuburg, Austria, Mario Solís, Instituto Balseiro, UNCuyo and CNEA S.C. de Bariloche, Río Negro, R8402AGP, Argentina

Published Jan 15, 2022• 0 citations
Publisher

Abstract

We study aspects of quantum field theory at finite density using techniques and concepts from quantum information theory. We focus on massive Dirac fermions with chemical potential in 1+1 space-time dimensions. Using the entanglement entropy on an interval, we construct an entropic c-function that is finite. This c-function is not monotonous,and incorporates the long-range entanglement from the Fermi surface. Motivated by previous works on lattice models,we next compute the Renyi entropies numerically, and find Friedel-type oscillations. Next, we analyze the mutual in-formation as a measure of correlation functions between different regions. Using a long-distance expansion developed by Cardy, we show how the mutual information detects the Fermi surface correlations already at leading order in the expansion. Finally, we analyze the relative entropy and its Renyi generalizations in order to distinguish states with different charge. We find that states in different superselection sectors give rise to a super-extensive behavior in the relative entropy.

DOI
10.31527/analesafa.2021.32.4.93

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We study aspects of quantum field theory at finite density using techniques and concepts from quantum information theory. We focus on massive Dirac fermions with chemical potential in 1+1 space-time dimensions. Using the entanglement entropy on an interval, we construct an entropic c-function that is finite. This c-function is not monotonous,and incorporates the long-range entanglement from the Fermi surface. Motivated by previous works on lattice models,we next compute the Renyi entropies numerically, and find Friedel-type oscillations. Next, we analyze the mutual in-formation as a measure of correlation functions between different regions. Using a long-distance expansion developed by Cardy, we show how the mutual information detects the Fermi surface correlations already at leading order in the expansion. Finally, we analyze the relative entropy and its Renyi generalizations in order to distinguish states with different charge. We find that states in different superselection sectors give rise to a super-extensive behavior in the relative entropy.
Abstract We study different aspects of quantum field theory at finite density using methods from quantum information theory. For simplicity we focus on massive Dirac fermions with nonzero chemical potential, and work in 1 + 1 space-time dimensions. Using the entanglement entropy on an interval, we construct an entropic c-function that is finite. Unlike what happens in Lorentz-invariant theories, this c-function exhibits a strong violation of monotonicity; it also encodes the creation of long-range entanglement from the Fermi surface. Motivated by previous works on lattice models, we next calculate numerically the Renyi entropies and find Friedel-type oscillations; these are understood in terms of a defect operator product expansion. Furthermore, we consider the mutual information as a measure of correlation functions between different regions. Using a long-distance expansion previously developed by Cardy, we argue that the mutual information detects Fermi surface correlations already at leading order in the expansion. We also analyze the relative entropy and its Renyi generalizations in order to distinguish states with different charge and/or mass. In particular, we show that states in different superselection sectors give rise to a super-extensive behavior in the relative entropy. Finally, we discuss possible extensions to interacting theories, and argue for the relevance of some of these measures for probing non-Fermi liquids.
We compute the leading contribution to the ground-state R\'enyi entropy ${S}_{\ensuremath{\alpha}}$ for a region of linear size LLL in a Fermi liquid in ddd dimensions. The result contains a universal boundary law violating term simply related to the more familiar entanglement entropy. We also obtain a universal crossover function that smoothly interpolates between the zero-temperature result and the ordinary thermal R\'enyi entropy of a Fermi liquid. Formulas for the entanglement entropy of more complicated regions, including nonconvex and disconnected regions, are obtained from the conformal field theory formulation of Fermi surface dynamics. These results permit an evaluation of the quantum mutual information between different regions in a Fermi liquid. We also study the number fluctuations in a Fermi liquid. Taken together, these results give a reasonably complete characterization of the low-energy quantum information content of Fermi liquids.
We use holographic methods to characterize the RG flow of quantum information in a Chern-Simons theory coupled to massive fermions. First, we use entanglement entropy and mutual information between strips to derive the dimension of the RG-driving operator and a monotonic c-function. We then display a scaling regime where, unlike in a CFT, the mutual information between strips changes non-monotonically with strip width, vanishing in both IR and UV but rising to a maximum at intermediate scales. The associated information transitions also contribute to non-monotonicity in the conditional mutual information which characterizes the independence of neighboring strips after conditioning on a third. Finally, we construct a measure of extensivity which tests to what extent information that region A shares with regions B and C is additive. In general, mutual information is super-extensive in holographic theories, and we might expect super-extensivity to be maximized in CFTs since they are scale-free. Surprisingly, our massive theory is more super-extensive than a CFT in a range of scales near the UV limit, although it is less super-extensive than a CFT at all lower scales. Our analysis requires the full ten-dimensional dual gravity background, and the extremal surfaces computing entanglement entropy explore all of these dimensions.
We study nonperturbative aspects of quantum field theory (QFT) in rigid anti de Sitter (AdS) spacetime using quantum information theoretic methods. While irreversibility of renormalization group (RG) flows is well established in flat space, it is not obvious whether it persists in AdS, where negative curvature and an asymptotic timelike boundary significantly modify infrared dynamics. Using strong subadditivity and AdS invariance, we derive an entropic second-order differential inequality for the difference between the vacuum entanglement entropy of a QFT and that of its ultraviolet fixed point, evaluated for spherical bulk regions. This inequality allows us to define RG charges that measure the relevant number of degrees of freedom, and we prove the irreversibility of the RG in 2, 3,2,\,3,2,3, and 444 spacetime dimensions. We further analyze free scalar and fermion theories in AdS, developing lattice formulations adapted to the geometry and computing entanglement entropies and RG charges. In AdS2_22​, we obtain analytic results for a massive Dirac fermion and compare them with numerical lattice calculations. These examples illustrate the general irreversibility theorem and clarify the distinction between conformal and massive theories in AdS.
Relative entropy is a measure of distinguishability for quantum states, and it plays a central role in quantum information theory. The family of Renyi entropies generalizes to Renyi relative entropies that include, as special cases, most entropy measures used in quantum information theory. We construct a Euclidean path-integral approach to Renyi relative entropies in conformal field theory, then compute the fidelity and the relative entropy of states in one spatial dimension at zero and finite temperature using a replica trick. In contrast to the entanglement entropy, the relative entropy is free of ultraviolet divergences, and is obtained as a limit of certain correlation functions. The relative entropy of two states provides an upper bound on their trace distance.
The large-scale behaviour of entanglement entropy in finite-density states, in and out of equilibrium, can be understood using the physical picture of particle pairs. However, the full theoretical origin of this picture is not fully established yet. In this work, we clarify this picture by investigating entanglement entropy using its connection with the large-deviation theory for thermodynamic and hydrodynamic fluctuations. We apply the universal framework of Ballistic Fluctuation Theory (BFT), based the Euler hydrodynamics of the model, to correlation functions of branch-point twist fields, the starting point for computing Rényi entanglement entropies within the replica approach. Focusing on free fermionic systems in order to illustrate the ideas, we show that both the equilibrium behavior and the dynamics of Rényi entanglement entropies can be fully derived from the BFT. In particular, we emphasise that long-range correlations develop after quantum quenches, and accounting for these explain the structure of the entanglement growth. We further show that this growth is related to fluctuations of charge transport, generalising to quantum quenches the relation between charge fluctuations and entanglement observed earlier. The general ideas we introduce suggest that the large-scale behaviour of entanglement has its origin within hydrodynamic fluctuations.