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Register a new userRelating bulk to boundary entanglement
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William Witczak-Krempa
Publication date
2019
fields
Physics
and research's language is
English
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Quantum field theories have a rich structure in the presence of boundaries. We study the groundstates of conformal field theories (CFTs) and Lifshitz field theories in the presence of a boundary through the lens of the entanglement entropy. For a family of theories in general dimensions, we relate the universal terms in the entanglement entropy of the bulk theory with the corresponding terms for the theory with a boundary. This relation imposes a condition on certain boundary central charges. For example, in 2 + 1 dimensions, we show that the corner-induced logarithmic terms of free CFTs and certain Lifshitz theories are simply related to those that arise when the corner touches the boundary. We test our findings on the lattice, including a numerical implementation of Neumann boundary conditions. We also propose an ansatz, the boundary Extensive Mutual Information model, for a CFT with a boundary whose entanglement entropy is purely geometrical. This model shows the same bulk-boundary connection as Dirac fermions and certain supersymmetric CFTs that have a holographic dual. Finally, we discuss how our results can be generalized to all dimensions as well as to massive quantum field theories.
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Li and Haldane conjectured and numerically substantiated that the entanglement spectrum of the reduced density matrix of ground-states of time-reversal breaking topological phases (fractional quantum Hall states) contains information about the counting of their edge modes when the ground-state is cut in two spatially distinct regions and one of the regions is traced out. We analytically substantiate this conjecture for a series of FQH states defined as unique zero modes of pseudopotential Hamiltonians by finding a one to one map between the thermodynamic limit counting of two different entanglement spectra: the particle entanglement spectrum, whose counting of eigenvalues for each good quantum number is identical (up to accidental degeneracies) to the counting of bulk quasiholes, and the orbital entanglement spectrum (the Li-Haldane spectrum). As the particle entanglement spectrum is related to bulk quasihole physics and the orbital entanglement spectrum is related to edge physics, our map can be thought of as a mathematically sound microscopic description of bulk-edge correspondence in entanglement spectra. By using a set of clustering operators which have their origin in conformal field theory (CFT) operator expansions, we show that the counting of the orbital entanglement spectrum eigenvalues in the thermodynamic limit must be identical to the counting of quasiholes in the bulk. The latter equals the counting of edge modes at a hard-wall boundary placed on the sample. Moreover, we show this to be true even for CFT states which are likely bulk gapless, such as the Gaffnian wavefunction.
The entanglement entropy (EE) can measure the entanglement between a spatial subregion and its complement, which provides key information about quantum states. Here, rather than focusing on specific regions, we study how the entanglement entropy changes with small deformations of the entangling surface. This leads to the notion of entanglement susceptibilities. These relate the variation of the EE to the geometric variation of the subregion. We determine the form of the leading entanglement susceptibilities for a large class of scale invariant states, such as groundstates of conformal field theories, and systems with Lifshitz scaling, which includes fixed points governed by disorder. We then use the susceptibilities to derive the universal contributions that arise due to non-smooth features in the entangling surface: corners in 2d, as well as cones and trihedral vertices in 3d. We finally discuss the generalization to Renyi entropies.
We study the growth of entanglement in quantum systems with a conserved quantity exhibiting diffusive transport, focusing on how initial inhomogeneities are imprinted on the entropy. We propose a simple effective model, which generalizes the minimal cut picture of textit{Jonay et al.} in such a way that the `line tension of the cut depends on the local entropy density. In the case of noisy dynamics, this is described by a Kardar-Parisi-Zhang (KPZ) equation coupled to a diffusing field. We investigate the resulting dynamics and find that initial inhomogeneities of the conserved charge give rise to features in the entanglement profile, whose width and height both grow in time as $proptosqrt{t}$. In particular, for a domain wall quench, diffusion restricts entanglement growth to be $S_text{vN} lesssim sqrt{t}$. We find that for charge density wave initial states, these features in the entanglement profile are present even after the charge density has equilibrated. Our conclusions are supported by numerical results on random circuits and deterministic spin chains.
We propose that the properties of the capacity of entanglement (COE) in gapless systems can efficiently be investigated through the use of the distribution of eigenvalues of the reduced density matrix (RDM). The COE is defined as the fictitious heat capacity calculated from the entanglement spectrum. Its dependence on the fictitious temperature can reflect the low-temperature behavior of the physical heat capacity, and thus provide a useful probe of gapless bulk or edge excitations of the system. Assuming a power-law scaling of the COE with an exponent $alpha$ at low fictitious temperatures, we derive an analytical formula for the distribution function of the RDM eigenvalues. We numerically test the effectiveness of the formula in relativistic free scalar boson in two spatial dimensions, and find that the distribution function can detect the expected $alpha=3$ scaling of the COE much more efficiently than the raw data of the COE. We also calculate the distribution function in the ground state of the half-filled Landau level with short-range interactions, and find a better agreement with the $alpha=2/3$ formula than with the $alpha=1$ one, which indicates a non-Fermi-liquid nature of the system.
Topological phases of quantum matter defy characterization by conventional order parameters but can exhibit quantized electro-magnetic response and/or protected surface states. We examine such phenomena in a model for three-dimensional correlated complex oxides, the pyrochlore iridates. The model realizes interacting topological insulators with and without time-reversal symmetry, and topological Weyl semimetals. We use cellular dynamical mean field theory, a method that incorporates quantum-many-body effects and allows us to evaluate the magneto-electric topological response coefficient in correlated systems. This invariant is used to unravel the presence of an interacting axion insulator absent within a simple mean field study. We corroborate our bulk results by studying the evolution of the topological boundary states in the presence of interactions. Consequences for experiments and for the search for correlated materials with symmetry-protected topological order are given.
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