We study the symmetry resolution of the entanglement entropy of an interval in two-dimensional conformal field theories (CFTs), by relating the bipartition to the geometry of an annulus with conformal boundary conditions. In the presence of extended symmetries such as Kac-Moody type current algebrae, symmetry resolution is possible only if the boundary conditions on the annulus preserve part of the symmetry group, i.e. if the factorization map associated with the spatial bipartition is compatible with the symmetry in question. The partition function of the boundary CFT (BCFT) is then decomposed in terms of the characters of the irreducible representations of the symmetry group preserved by the boundary conditions. We demonstrate that this decomposition already provides the symmetry resolution of the entanglement spectrum of the corresponding bipartition. Considering the various terms of the partition function associated with the same representation, or charge sector, the symmetry-resolved Rényi entropies can be derived to all orders in the UV cutoff expansion without the need to compute the charged moments. We apply this idea to the theory of a free massless boson with U(1) U ( 1 ) , \mathbb{R} ℝ and \mathbb{Z}_2 ℤ 2 symmetry.
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.
Abstract We perform a comprehensive analysis of the symmetry-resolved (SR) entanglement entropy (EE) for one single interval in the ground state of a 1 + 1D conformal field theory (CFT), that is invariant under an arbitrary finite or compact Lie group, G. We utilize the boundary CFT approach to study the total EE, which enables us to find the universal leading order behavior of the SREE and its first correction, which explicitly depends on the irreducible representation under consideration and breaks the equipartition of entanglement. We present two distinct schemes to carry out these computations. The first relies on the evaluation of the charged moments of the reduced density matrix. This involves studying the action of the defect-line, that generates the symmetry, on the boundary states of the theory. This perspective also paves the way for discussing the infeasibility of studying symmetry resolution when an anomalous symmetry is present. The second scheme draws a parallel between the SREE and the partition function of an orbifold CFT. This approach allows for the direct computation of the SREE without the need to use charged moments. From this standpoint, the infeasibility of defining the symmetry-resolved EE for an anomalous symmetry arises from the obstruction to gauging. Finally, we derive the symmetry-resolved entanglement spectra for a CFT invariant under a finite symmetry group. We revisit a similar problem for CFT with compact Lie group, explicitly deriving an improved formula for U(1) resolved entanglement spectra. Using the Tauberian formalism, we can estimate the aforementioned EE spectra rigorously by proving an optimal lower and upper bound on the same. In the abelian case, we perform numerical checks on the bound and find perfect agreement.
A comprehensive symmetry resolution of the entanglement entropy in ( 1 + 1 )-d rational conformal field theories with categorical noninvertible symmetries is presented. This amounts to symmetry resolving the entanglement with respect to the generalized twisted and anyonic charged sectors of the theory. The anyonic sectors label the irreducible representations of a modular fusion category defining the symmetry and can be understood through the ( 2 + 1 )-d symmetry topological field theory that encodes the symmetry features of the conformal field theory. Using this, we define the corresponding generalized boundary-dependent charged moments necessary for the symmetry resolution of the entanglement entropy and find that, contrary to the case of invertible symmetries, the entanglement equipartition between different charged sectors at the next-to-leading order in the ultraviolet cutoff, is broken. Published by the American Physical Society 2024
Entanglement and symmetries are two pillars of our understanding of modern Physics, but surprisingly the study of the interplay between these two fundamental concepts has become the subject of an intense research area only in the last few years. In particular, the resolution of entanglement in the various symmetry sectors of a quantum many-body system with a global conserved charge gained a lot of interest within the entanglement community. As shown in some experimental papers, the study of this symmetry resolved entanglement is necessary for a better understanding of the non-equilibrium dynamics of quantum many-body systems. This Thesis is devoted to the study of the symmetry resolution of the entanglement measures in free fermionic systems and Conformal Field Theories. It is organised as follows. In Chap. 1 we give an overview of the main definitions and concepts related to the entanglement and its measures in quantum mechanics and quantum field theory. Then, we specialize to the case of conformal field theories, to derive expressions of the entanglement entropy and negativity in all the cases that will be examined in the second part of the Thesis. We finally illustrate the main free-fermionic techniques that will be used to perform the analytical as well as numerical lattice calculations. In Chap. 2, we consider a
1-dimensional system of free fermions hopping on a lattice, in which the conserved
U(1) charge is the total particle number. We evaluate the symmetry resolved R'enyi and entanglement entropies determining exactly all the non-universal constants and logarithmic corrections to the scaling, thanks to the so-called Fisher-Hartwig conjecture. In Chap. 3, we study the symmetry resolved entanglement entropies in
1-dimensional systems in presence of boundaries, considering the cases of an interval starting from the boundary and disconnected from it. We evaluate the charged and symmetry resolved entropies for conformal invariant theories and a semi-infinite chain of free fermions, finding entanglement equipartition. Using a generalization of the lattice calculations performed in Chap. 2, we also evaluate the first term breaking equipartition for the free fermionic chain. In Chap. 4, we study the symmetry resolution of the entanglement negativity in free fermionic systems, which admits a decomposition in terms of the charge imbalance between the two subsystems. We use conformal field theory to derive the universal behavior of the charge imbalance resolved negativity for a free Dirac field at finite temperature and size. For the lattice version of the Dirac filed, we also determine exactly the model-dependent terms using the Fisher-Hartwig conjecture. We find that at leading order the charge imbalance resolved negativity verifies equipartition. For the lattice model, we also evaluate the first corrections to this equipartition. In Chap. 5, we study the time evolution of the symmetry resolved entanglement entropy and a suitably defined symmetry resolved mutual information in a system of free fermions after a quench. In the context of free fermions and conformal field theories, we are able to provide analytical results for the time-dependent charged and resolved entropies, and shed light on two new physical phenomena. First, the symmetry resolved R'enyi entropies start evolving with a calculable time delay that depends on the charge sector. Second, there is an effective equipartition of the entanglement in the limit of large subsystem size. For free fermions, we also find that effective equipartition is verified by the mutual information. We expect that these phenomena occur in more general quantum systems, as they can be understood from the quasiparticle picture for the spreading of entanglement. In the Appendix, we give an overview of conformal field theory, reporting the main concepts and formulas that are needed for a more complete understanding of the previous Chapters.
A bstract Entanglement is analyzed in the Majorana fermion conformal field theory (CFT) in the vacuum, in the fermion state, and in states built from conformal interfaces. In the boundary-state approach, the Hilbert space admits two factorizations for a single interval, producing distinct entanglement spectra determined by spin structures. Although Rényi and relative entropies are shown to be insensitive to these structures, symmetry-resolved entanglement naturally reveals their differences. The Majorana fermion’s
Z2F ℤ 2 F symmetry, generated by the fermion-parity operator (−1) F , distinguishes bosonic from fermionic sectors, motivating the notion of fermion-parity resolution. While
Z2F ℤ 2 F is naturally a symmetry of the vacuum and fermion reduced density matrices, the Hilbert space factorization is shown to stabilize this symmetry in conformal interface states. When an unpaired Majorana zero mode is present, fermion-parity-resolved entropies display equipartition at all orders in the UV cutoff; in its absence, the breaking of equipartition is quantified by Ramond-sector data. This behavior persists across all states considered. Connections with symmetry-protected topological phases of matter are outlined. All results are compared with twist field computations.
Entanglement is resolved in conformal field theory (CFT) with respect to conformal families to all orders in the UV cutoff. To leading order, symmetry-resolved entanglement is connected to the quantum dimension of a conformal family, while to all orders it depends on null vectors. Criteria for equipartition between sectors are provided in both cases. This analysis exhausts all unitary conformal families. Furthermore, topological entanglement entropy is shown to symmetry-resolve the Affleck-Ludwig boundary entropy. Configuration and fluctuation entropy are analyzed on grounds of conformal symmetry.