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Theoretical Properties of the Entanglement in a Strong Coupling Region

Chen‐Te Ma

Published Sep 15, 2016• 0 citations
OpenAlex

Abstract

Entropy is a quantity for counting physical degrees of freedom in a system. At a finite temperature, one can use thermal entropy to study thermodynamical properties. At zero temperature, entanglement entropy is expected to provide a suitable order parameter of a phase structure. Especially, the entanglement entropy exhibits an interesting codimension two area law in a strongly coupled conformal field theory. We compute thermal entropy in a non-relativistic model with an infinite fermion mass limit from an exact effective potential to obtain thermal entropy at an infinite strong coupling limit. The computational result provides vanishing thermal entropy at an infinite strong coupling limit with a finite lattice spacing. The non-trivial topological term can be included in the strongly coupled lattice system to obtain the non-trivial entropy and the topology can be marked from the entropy. We first compute the thermal entropy in a two dimensional lattice topological quantum field theory to study the lattice artifact and also argue that a theory possibly has translational invariance if a system does not have a volume law in the entanglement entropy. Finally, we show that a coefficient of a universal term of the entanglement entropy should not be affected by a choice of an entangling surface in two dimensional conformal field theory for one interval case. We also discuss a choice of the entangling surface in the entanglement entropy in two dimensional $\mathrm{CP}^{N-1}$ model at the large $N$ limit.

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Entropy is a quantity for counting physical degrees of freedom in a system. At a finite temperature, one can use thermal entropy to study thermodynamical properties. At zero temperature, entanglement entropy is expected to provide a suitable order parameter of a phase structure. Especially, the entanglement entropy exhibits an interesting codimension two area law in a strongly coupled conformal field theory. We compute thermal entropy in a non-relativistic model with an infinite fermion mass limit from an exact effective potential to obtain thermal entropy at an infinite strong coupling limit. The computational result provides vanishing thermal entropy at an infinite strong coupling limit with a finite lattice spacing. The non-trivial topological term can be included in the strongly coupled lattice system to obtain the non-trivial entropy and the topology can be marked from the entropy. We first compute the thermal entropy in a two dimensional lattice topological quantum field theory to study the lattice artifact and also argue that a theory possibly has translational invariance if a system does not have a volume law in the entanglement entropy. Finally, we show that a coefficient of a universal term of the entanglement entropy should not be affected by a choice of an entangling surface in two dimensional conformal field theory for one interval case. We also discuss a choice of the entangling surface in the entanglement entropy in two dimensional CPN−1\mathrm{CP}^{N-1}CPN−1 model at the large NNN limit.
Abstract Entropy is a quantity for counting physical degrees of freedom in a system. At a finite temperature, one can use thermal entropy to study thermodynamical properties. At zero temperature, entanglement entropy is expected to provide a suitable order parameter of a phase structure. Especially, the entanglement entropy exhibits an interesting codimension two area law in a strongly coupled conformal field theory. We compute thermal entropy in a non-relativistic model with an infinite fermion mass limit from an exact effective potential to obtain thermal entropy at an infinite strong coupling limit. The computational result provides vanishing thermal entropy at an infinite strong coupling limit with a finite lattice spacing. The non-trivial topological term can be included in the strongly coupled lattice system to obtain the non-trivial entropy and the topology can be marked from the entropy. We first compute the thermal entropy in a two-dimensional (2D) lattice topological quantum field theory to study the lattice artifact and also argue that a theory possibly has translational invariance if a system does not have a volume law in the entanglement entropy. Finally, we show that a coefficient of a universal term of the entanglement entropy should not be affected by a choice of an entangling surface in 2D conformal field theory for one interval case. We also discuss a choice of the entangling surface in the entanglement entropy in 2D model at the large N limit.
The leading asymptotic large-scale behavior of the spatially bipartite entanglement entropy (EE) of the free Fermi gas infinitely extended in multidimensionsal Euclidean space at zero absolute temperature, T=0, is by now well understood. Here, we announce and discuss the first rigorous results for the corresponding EE of thermal equilibrium states at T>0. The leading large-scale term of this thermal EE turns out to be twice the leading finite-size correction to the infinite-volume thermal entropy (density). Not surprisingly, this correction is just the thermal entropy on the boundary surface of the bipartition. However, it is given by a rather complicated analytical expression derived from semiclassical functional calculus and differs, at least at high temperature, from simpler expressions previously obtained by arguments based on conformal field theory. In the zero-temperature limit, the leading large-scale term of the thermal EE considerably simplifies and displays a ln(1/T)-singularity which one may identify with the known logarithmic correction at T=0 to the so-called area-law scaling. Our results extend to the whole one-parameter family of (quantum) R\'enyi entropies.
We study the scaling behavior of the entanglement entropy of two-dimensional conformal quantum critical systems, i.e., systems with scale-invariant wave functions. They include two-dimensional generalized quantum dimer models on bipartite lattices and quantum loop models, as well as the quantum Lifshitz model and related gauge theories. We show that under quite general conditions, the entanglement entropy of a large and simply connected subsystem of an infinite system with a smooth boundary has a universal finite contribution, as well as scale-invariant terms for special geometries. The universal finite contribution to the entanglement entropy is computable in terms of the properties of the conformal structure of the wave function of these quantum critical systems. The calculation of the universal term reduces to a problem in boundary conformal field theory
The subsystem-size dependence of ground-state entanglement entropy and its crossover to thermal entropy as a function of temperature are well understood for one-dimensional (1D) gapless systems described by conformal field theory (CFT), and for free fermions with a Fermi surface in any dimension. However, little is known about the entanglement entropy for gapless fermionic systems without quasi-particles, such as a strange metal. Here we study the entanglement entropy of fermions in a solvable large-NNN 1D lattice model akin to the Yukawa-Sachdev-Ye-Kitaev (Yukawa SYK) model. In this model, two Fermi points are coupled to scalar bosons via spatially random Yukawa interactions, providing a solvable model of a strange metal when the bosons become critical at a quantum critical point. We exactly compute the second Rényi entropy of fermions in a spatial subregion in this model. Our results unravel crucial role of intra-subregion entanglement between fermionic and bosonic degrees of freedom along with the inter-subregion entanglement in understanding the ground states of such strongly coupled fermion-boson systems. We show that the crossover from thermal entropy to entanglement entropy, is captured by a single scaling ansatz, that collapses the second Rényi entropy of fermions for different subregion sizes and temperatures into a single universal curve. We further show that the universal scaling curve for the critical strange metal is well described by the standard CFT formula, albeit with an effective central charge substantially larger than the non-interacting value.
Abstract We compute the topological entanglement entropy for a large set of lattice models in d-dimensions. It is well known that many such quantum systems can be constructed out of lattice gauge models. For dimensionality higher than two, there are generalizations going beyond gauge theories, which are called higher gauge theories and rely on higher-order generalizations of groups. Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories. In this paper, we derive a general formula for the bipartition entanglement entropy for this class of models, and from it we extract both the area law and the sub-leading terms, which explicitly depend on the topology of the entangling surface. We show that the entanglement entropy SA in a sub-region A is proportional to log (GSDA~) \left({GSD}_{\tilde{A}}\right) (GSDA~​) GSD A ˜ , where GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ is the ground state degeneracy of a particular restriction of the full model to A. The quantity GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ can be further divided into a contribution that scales with the size of the boundary ∂A and a term which depends on the topology of ∂A. There is also a topological contribution coming from A itself, that may be non-zero when A has a non-trivial homology. We present some examples and discuss how the topology of A affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension d. The result is in agreement with entanglement calculations for known topological models.
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.