The study of quantum entanglement involves the choice of an entanglement cut, often corresponding to a partitioning of the system in posit ion space. We generalize these partitions to a family of entanglement cuts corresponding to arbitrary directions in the classical phase space. For one dimensional free fermionic many-body wavefunctions, we analytically compute the full entanglement spectra for a family of entanglement cuts corresponding to a rotation in the classical phase space. These cuts continuously interpolate between the commonly studied position- and momentum-space cuts, as well as provide an implementation of the inversion symmetry smoothly connected to identity. The inversion symmetry of the wavefunctions is reflected in the chiral symmetry of the phase space entanglement spectra, and we use the associated chiral invariant to classify the inversion symmetric fermionic many-body wavefunctions.
The study of quantum entanglement involves the choice of an entanglement cut, often corresponding to a partitioning of the system in posit ion space. We generalize these partitions to a family of entanglement cuts corresponding to arbitrary directions in the classical phase space. For one dimensional free fermionic many-body wavefunctions, we analytically compute the full entanglement spectra for a family of entanglement cuts corresponding to a rotation in the classical phase space. These cuts continuously interpolate between the commonly studied position- and momentum-space cuts, as well as provide an implementation of the inversion symmetry smoothly connected to identity. The inversion symmetry of the wavefunctions is reflected in the chiral symmetry of the phase space entanglement spectra, and we use the associated chiral invariant to classify the inversion symmetric fermionic many-body wavefunctions.
Abstract The study of quantum entanglement involves the choice of an entanglement cut, often corresponding to a partitioning of the system in position space. We generalize these partitions to a family of entanglement cuts corresponding to arbitrary directions in the classical phase space. For one dimensional free fermionic many-body wavefunctions, we analytically compute the full entanglement spectra for a family of entanglement cuts corresponding to a rotation in the classical phase space. These cuts continuously interpolate between the commonly studied position- and momentum-space cuts, as well as provide an implementation of the inversion symmetry smoothly connected to identity. The inversion symmetry of the wavefunctions is reflected in the chiral symmetry of the phase space entanglement spectra, and we use the associated chiral invariant to classify the inversion symmetric fermionic many-body wavefunctions.
We study the properties of the entanglement spectrum in gapped non-interacting non-Hermitian systems, and its relation to the topological properties of the system Hamiltonian. Two different families of entanglement Hamiltonians can be defined in non-Hermitian systems, depending on whether we consider only right (or equivalently only left) eigenstates or a combination of both left and right eigenstates. We show that their entanglement spectra can still be computed efficiently, as in the Hermitian limit. We discuss how symmetries of the Hamiltonian map into symmetries of the entanglement spectrum depending on the choice of the many-body state. Through several examples in one and two dimensions, we show that the biorthogonal entanglement Hamiltonian directly inherits the topological properties of the Hamiltonian for line gapped phases, with characteristic singular and energy zero modes. The right (left) density matrix carries distinct information on the topological properties of the many-body right (left) eigenstates themselves. In purely point gapped phases, when the energy bands are not separable, the relation between the entanglement Hamiltonian and the system Hamiltonian breaks down.
In this dissertation, the momentum-space entanglement properties of several quantum spin chains are investigated. In the second chapter we re- view important results on many-body entanglement. In the third chapter, we numerically study the momentum-space entanglement spectra of bosonic and fermionic formulations of the spin-1/2 XXZ chain [Phys. Rev. Lett. 113, 256404 (2014)]. We investigate the behavior of the entanglement gaps, present in both formulations, across quantum phase transitions in the XXZ chain. In both cases, finite size scaling suggests that the entanglement gap closure does not occur at the physical transition points. For bosons, we find that the entan- glement gap depends on the scaling dimension of the conformal field theory as varied by the X X Z anisotropy. For fermions, the infinite entanglement gap present at the XX point persists well past the phase transition at the Heisenberg point. We elaborate on how this may support the numerical study of phase transitions. In the third chapter, we advocate that in certain critical spin chains a gap in the momentum-space entanglement spectrum separates the universal part of the spectrum, which is determined by the associated conformal field theory, from the non-universal part, which is specific to the model [arXiv:1512.09030]. We provide affirmative evidence from multicritical spin-1 chains with low energy sectors described by the SU(2)2 or the SU(3)1 Wess-Zumino-Witten model. In chapter four, we study momentum- space en- tanglement in quantum spin-half ladders [Phys. Rev. B 93, 125107 (2016)]. When the system is gapped, we analytically find the momentum-space entan- glement Hamiltonian is described by a chiral conformal field theory with a central charge of two. When the system is gapless, the entanglement Hamil- tonian consists of one gapless mode that is linear in subsystem momentum and one mode with a flat dispersion relation. We also analytically include the effect of a certain irrelevant terms (in the renormalization group sense) on the entanglement spectrum. In chapter six, we investigate the momentum- space entanglement spectrum after a quantum quench [arXiv:1603.01997]. We show that the momentum-space entanglement spectrum of the XXZ spin- half chain possesses many universal features both in equilibrium and after a quantum quench.
We study the single-particle entanglement spectrum in 2D topological insulators which possess n-fold rotation symmetry. By defining a series of special choices of subsystems on which the entanglement is calculated, or real space cuts, we find that the number of protected in-gap states for each type of these real space cuts is a quantum number indexing (if any) nontrivial topology in these insulators. We explicitly show that the number of protected in-gap states is determined by a Zn index (z1,...,zn), where zm is the number of occupied states that transform according to mth one-dimensional representation of the Cn point group. We find that for a space cut separating 1/pth of the system, the entanglement spectrum contains in-gap states pinned in an interval of entanglement eigenvalues $[1/p,1\ensuremath{-}1/p]$. We determine the number of such in-gap states for an exhaustive variety of cuts, in terms of the Zn index. Furthermore, we show that in a homogeneous system, the Zn index can be determined through an evaluation of the eigenvalues of point-group symmetry operators at all high-symmetry points in the Brillouin zone. When disordered n-fold rotationally symmetric systems are considered, we find that the number of protected in-gap states is identical to that in the clean limit as long as the disorder preserves the underlying point-group symmetry and does not close the bulk insulating gap.
A bstract In this work, we study the universal total and symmetry-resolved entanglement spectra for a single interval of some 2d Fermionic CFTs using the Boundary Conformal Field theory (BCFT) approach. In this approach, the partition of Hilbert space is achieved by cutting out discs around the entangling boundary points and imposing boundary conditions preserving the extended symmetry under scrutiny. The reduced density moments are then related to the BCFT partition functions and are also found to be diagonal in the symmetry charge sectors. In particular, we first study the entanglement spectra of massless Dirac fermion and modular invariant Z 2 -gauged Dirac fermion by considering the boundary conditions preserving either the axial or the vector U(1) symmetry. The total entanglement spectra of the modular invariant Z 2 -gauged Dirac fermion are shown to match with the compact boson result at the compactification radius where the Bose-Fermi duality holds, while for the massless Dirac fermion, it is found that the boundary entropy term doesn’t match with the self-dual compact boson. The symmetry-resolved entanglement is found to be the same in all cases, except for the charge spectrum which is dependent on both the symmetry and the theory. We also study the entanglement spectra of N massless Dirac fermions by considering boundary conditions preserving different chiral U(1) N symmetries. Entanglement spectra are studied for U(1) M subgroups, where M ≤ N , by imposing boundary conditions preserving different chiral symmetries. The total entanglement spectra are found to be sensitive to the representations of the U(1) M symmetry in the boundary theory among other behaviours at O (1). Similar results are also found for the Symmetry resolved entanglement entropies. The characteristic log log ( ℓ / ϵ ) term of the U(1) symmetry is found to be proportional to M in the symmetry-resolved entanglement spectra.
Bosons and fermions are defined by their exchange properties and the underlying symmetries determine the structure of the corresponding state spaces. For two particles there are two possible exchange symmetries, resulting in symmetric or antisymmetric behaviour, but when exploring multiparticle systems also quantum states with chiral symmetries appear. In this work we demonstrate that chiral symmetries lead to extremal forms of quantum entanglement. More precisely, we show that subspaces with this symmetry are highly entangled with respect to the geometric measure of entanglement, leading to observables which can be useful for entanglement characterization. Along the way, we develop a simple method to solve the problem of genuine multiparticle entanglement for unitarily invariant three-particle states and use it to identify genuine multipartite entangled states whose partial transposes with respect to all bipartitions are positive. Finally, we consider generalizations with less symmetry and discuss potential applications.