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arXiv:1008.5191 · Physical Review B

Entanglement entropy from charge statistics: Exact relations for noninteracting many-body systems

Hao Song, Christian Flindt, Stephan Rachel, Israel Klich, Karyn Le Hur

Published Apr 25, 2011• 161 citations
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Abstract

We present an exact expression for the entanglement entropy generated at a quantum point contact between noninteracting electronic leads in terms of the full counting statistics of charge fluctuations, which we illustrate with examples from both equilibrium and nonequilibrium transport. The formula is also applicable to ground-state entanglement entropy in systems described by noninteracting fermions in any dimension, which in one dimension include the critical spin-1/2 $\mathit{XX}$ and Ising models where conformal field theory predictions for the entanglement entropy are reproduced from the full counting statistics. These results may play an important role in experimental measurements of entanglement entropy in mesoscopic structures and cold atoms in optical lattices.

DOI
10.1103/physrevb.83.161408

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We present an exact expression for the entanglement entropy generated at a quantum point contact between noninteracting electronic leads in terms of the full counting statistics of charge fluctuations, which we illustrate with examples from both equilibrium and nonequilibrium transport. The formula is also applicable to ground-state entanglement entropy in systems described by noninteracting fermions in any dimension, which in one dimension include the critical spin-1/2 XX\mathit{XX}XX and Ising models where conformal field theory predictions for the entanglement entropy are reproduced from the full counting statistics. These results may play an important role in experimental measurements of entanglement entropy in mesoscopic structures and cold atoms in optical lattices.
Entanglement entropy is a measure of quantum correlations between separate parts of a many‐body system, which plays an important role in many areas of physics. Here we review recent work in which a relation between this quantity and the Full Counting Statistics description of electron transport was established for noninteracting fermion systems. Using this relation, which is of a completely general character, we discuss how the entanglement entropy can be directly measured by detecting current fluctuations in a driven quantum system such as quantum point contact.
The full counting statistics is studied for a one-dimensional system of noninteracting fermions with and without disorder. For two unbiased LLL site lattices connected at time t=0t=0t=0, the charge variance increases as the natural logarithm of ttt, following the universal expression $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\approx}\frac{1}{{\ensuremath{\pi}}^{2}}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$. Since the static charge variance for a length lll region is given by $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\approx}\frac{1}{{\ensuremath{\pi}}^{2}}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}l$, this result reflects the underlying relativistic or conformal invariance and dynamical exponent z=1z=1z=1 of the disordered free lattice. With disorder and strongly localized fermions, we have compared our results to a model with a dynamical exponent $z\ensuremath{\ne}1$ and a model for entanglement entropy based upon dynamical scaling at the infinite disorder fixed point (IDFP). The latter scaling, which predicts $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\propto}\mathrm{ln}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$, appears to better describe the charge variance of disordered one-dimensional fermions. When a bias voltage is introduced, the behavior changes dramatically, and the charge and variance become proportional to ${(\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t)}^{1/\ensuremath{\psi}}$ and lnt\mathrm{ln}\phantom{\rule{0.16em}{0ex}}tlnt, respectively. The exponent $\ensuremath{\psi}$ may be related to the critical exponent, characterizing spatial/energy fluctuations at the IDFP.
We investigate the entanglement and the R\'enyi entropies of two electronic leads connected by a quantum point contact. For noninteracting electrons, the entropies can be related to the cumulants of the full counting statistics of transferred charge which in principle are measurable. We consider the entanglement entropy generated by operating the quantum point contact as a quantum switch which is opened and closed in a periodic manner. Using a numerically exact approach we analyze the conditions under which a logarithmic growth of the entanglement entropy predicted by conformal field theory should be observable in an electronic conductor. In addition, we consider clean single-particle excitations on top of the Fermi sea (levitons) generated by applying designed pulses to the leads. We identify a Hong-Ou-Mandel-like suppression of the entanglement entropy by interfering two levitons on a quantum point contact tuned to half transmission.
We investigate in detail the behavior of the bipartite fluctuations of particle number $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}$ and spin ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{z}$ in many-body quantum systems, focusing on systems where such U(1) charges are both conserved and fluctuate within subsystems due to exchange of charges between subsystems. We propose that the bipartite fluctuations are an effective tool for studying many-body physics, particularly its entanglement properties, in the same way that noise and full counting statistics have been used in mesoscopic transport and cold-atomic gases. For systems that can be mapped to a problem of noninteracting fermions, we show that the fluctuations and higher-order cumulants fully encode the information needed to determine the entanglement entropy as well as the full entanglement spectrum through the R\'enyi entropies. In this connection, we derive a simple formula that explicitly relates the eigenvalues of the reduced density matrix to the R\'enyi entropies of integer order for any finite density matrix. In other systems, particularly in one dimension, the fluctuations are in many ways similar but not equivalent to the entanglement entropy. Fluctuations are tractable analytically, computable numerically in both density matrix renormalization-group and quantum Monte Carlo calculations, and in principle accessible in condensed-matter and cold-atom experiments. In the context of quantum point contacts, measurement of the second charge cumulant showing a logarithmic dependence on time would constitute a strong indication of many-body entanglement.
Entanglement entropy encodes important features of strongly interacting quantum many-body systems and gauge theories, but its analytical study is still limited to systems with high level of symmetry. This motivates the search for efficient techniques to investigate this quantity numerically, through Monte Carlo calculations on the lattice. In this contribution, we discuss the computation of the entropic c-function by means of an algorithm based on Jarzynski's equality, which is an exact theorem from non-equilibrium statistical mechanics. After presenting benchmark results for the Ising model in two dimensions, where our algorithm successfully reproduces the analytical predictions from conformal field theory, we discuss its generalization to the three-dimensional Ising model, for which we were able to extract universal terms beyond the area law. Finally we point out some future generalizations of this calculation.
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.