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arXiv:1408.3043 · Journal of Physics A Mathematical and Theoretical

Finite temperature entanglement negativity in conformal field theory

Pasquale Calabrese, John Cardy, Erik Tonni C

Published Dec 8, 2014• 201 citations
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Abstract

We consider the logarithmic negativity of a finite interval embedded in an infinite one dimensional system at finite temperature. We focus on conformal invariant systems and we show that the naive approach based on the calculation of a two-point function of twist fields in a cylindrical geometry yields a wrong result. The correct result is obtained through a four-point function of twist fields in which two auxiliary fields are inserted far away from the interval, and they are sent to infinity only after having taken the replica limit. In this way, we find a universal scaling form for the finite temperature negativity which depends on the full operator content of the theory and not only on the central charge. In the limit of low and high temperatures, the expansion of this universal form can be obtained by means of the operator product expansion. We check our results against exact numerical computations for the critical harmonic chain.

DOI
10.1088/1751-8113/48/1/015006

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We consider the logarithmic negativity of a finite interval embedded in an infinite one dimensional system at finite temperature. We focus on conformal invariant systems and we show that the naive approach based on the calculation of a two-point function of twist fields in a cylindrical geometry yields a wrong result. The correct result is obtained through a four-point function of twist fields in which two auxiliary fields are inserted far away from the interval, and they are sent to infinity only after having taken the replica limit. In this way, we find a universal scaling form for the finite temperature negativity which depends on the full operator content of the theory and not only on the central charge. In the limit of low and high temperatures, the expansion of this universal form can be obtained by means of the operator product expansion. We check our results against exact numerical computations for the critical harmonic chain.
We report on a systematic approach for the calculation of the negativity in the ground state of a one-dimensional quantum field theory. The partial transpose of the reduced density matrix of a subsystem A = A 1 ∪ A 2 is explicitly constructed as an imaginary-time path integral and from this the replicated traces are obtained. The logarithmic negativity is then the continuation to n → 1 of the traces of the even powers. For pure states, this procedure reproduces the known results. We then apply this method to conformally invariant field theories (CFTs) in several different physical situations for infinite and finite systems and without or with boundaries. In particular, in the case of two adjacent intervals of lengths ℓ 1 ,ℓ 2 in an infinite system, we derive the result ∼ ( c /4)ln(ℓ 1 ℓ 2 /(ℓ 1 + ℓ 2 )), where c is the central charge. For the more complicated case of two disjoint intervals, we show that the negativity depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We explicitly calculate the scale invariant functions for the replicated traces in the case of the CFT for the free compactified boson, but we have not so far been able to obtain the n → 1 continuation for the negativity even in the limit of large compactification radius. We have checked all our findings against exact numerical results for the harmonic chain which is described by a non-compactified free boson.
We develop a systematic method to extract the negativity in the ground state of a 1+11+11+1 dimensional relativistic quantum field theory, using a path integral formalism to construct the partial transpose ${\ensuremath{\rho}}_{A}^{{T}_{2}}$ of the reduced density matrix of a subsystem $A={A}_{1}\ensuremath{\cup}{A}_{2}$, and introducing a replica approach to obtain its trace norm which gives the logarithmic negativity $\mathcal{E}=\mathrm{ln}\ensuremath{\Vert}{\ensuremath{\rho}}_{A}^{{T}_{2}}\ensuremath{\Vert}$. This is shown to reproduce standard results for a pure state. We then apply this method to conformal field theories, deriving the result $\mathcal{E}\ensuremath{\sim}(c/4)\mathrm{ln}[{\ensuremath{\ell}}_{1}{\ensuremath{\ell}}_{2}/({\ensuremath{\ell}}_{1}+{\ensuremath{\ell}}_{2})]$ for the case of two adjacent intervals of lengths ${\ensuremath{\ell}}_{1}$, ${\ensuremath{\ell}}_{2}$ in an infinite system, where ccc is the central charge. For two disjoint intervals it depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We check our findings against exact numerical results in the harmonic chain.
Logarithmic negativity is a proper measure of quantum entanglement in mixed states. Originally proposed in the context of quantum information, in recent years it is attracting interest in quantum field theory, condensed matter physics, and statistical physics. The key object to construct the negativity is the partial transpose of the reduced density matrix. Here, focusing on two adjacent intervals, the authors provide a complete study of the distribution of the eigenvalues of the partial transpose (negativity spectrum) for gapless one-dimensional systems described by a conformal field theory (CFT). Similar to the entanglement spectrum, the distribution of the negativity spectrum is universal, and it depends only on the central charge of the underlying CFT. The precise form of the spectrum depends on whether the two intervals are in a pure or mixed state, and in both cases, a dependence on the sign of the eigenvalues is found. This dependence is weak for bulk eigenvalues, whereas it is strong at the spectrum edges. The authors also investigate the scaling of the smallest (negative) and largest (positive) eigenvalues of the partial transpose. The analytical results are thoroughly checked against numerical DMRG simulations as well as in exactly solvable models.
We consider single interval Rényi and entanglement entropies for a two dimensional conformal field theory on a circle at nonzero temperature. Assuming that the finite size of the system introduces a unique ground state with a nonzero mass gap, we calculate the leading corrections to the Rényi and entanglement entropy in a low temperature expansion. These corrections have a universal form for any two dimensional conformal field theory that depends only on the size of the mass gap and its degeneracy. We analyze the limits where the size of the interval becomes small and where it becomes close to the size of the spatial circle.
Abstract Defects in two-dimensional conformal field theories (CFTs) contain signatures of their characteristics. In this work, we analyze entanglement properties of subsystems in the presence of energy and duality defects in the Ising CFT using the density matrix renormalization group (DMRG) technique. In particular, we compute the entanglement entropy (EE) and the entanglement negativity (EN) in the presence of defects. For the EE, we consider the cases when the defect lies within the subsystem and at the edge of the subsystem. We show that the EE for the duality defect exhibits fundamentally different characteristics compared to the energy defect due to the existence of localized and delocalized zero energy modes. Of special interest is the nontrivial ‘finite-size correction’ in the EE obtained recently using free fermion computations [1]. These corrections arise when the subsystem size is appreciable compared to the total system size and lead to a deviation from the usual logarithmic scaling characteristic of one-dimensional quantum-critical systems. Using matrix product states with open and infinite boundary conditions, we numerically demonstrate the disappearance of the zero mode contribution for finite subsystem sizes in the thermodynamic limit. Our results provide further support to the recent free fermion computations, but clearly contradict earlier analytical field theory calculations based on twisted torus partition functions. Subsequently, we compute the logarithm of the EN (log-EN) between two disjoint subsystems separated by a defect. We show that the log-EN scales logarithmically with the separation of the subsystems. However, the coefficient of this logarithmic scaling yields a continuously-varying effective central charge that is different from that obtained from analogous computations of the EE. The defects leave their fingerprints in the subleading term of the scaling of the log-EN. Furthermore, the log-EN receives similar ‘finite size corrections’ like the EE which leads to deviations from its characteristic logarithmic scaling.
Here we show that the Rényi entanglement entropy of a region of large size ℓ in a one-dimensional critical model whose ground state breaks conformal invariance (such as in those described by non-unitary conformal field theories), behaves as , where is the effective central charge, c (which may be negative) is the central charge of the conformal field theory and is the lowest holomorphic conformal dimension in the theory. We also obtain results for models with boundaries, and with a large but finite correlation length, and we show that if the lowest conformal eigenspace is logarithmic ( with N nilpotent), then there is an additional term proportional to . These results generalize the well known expressions for unitary models. We provide a general proof, and report on numerical evidence for a non-unitary spin chain and an analytical computation using the corner transfer matrix method for a non-unitary lattice model. We use a new algebraic technique for studying the branching that arises within the replica approach, and find a new expression for the entanglement entropy in terms of correlation functions of twist fields for non-unitary models.