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arXiv:2404.02206 · Journal of High Energy Physics

Symmetry resolution in non-Lorentzian field theories

Aritra Banerjee, Rudranil Basu, Arpan Bhattacharyya, Nilachal Chakrabarti

Published Jun 19, 2024• 12 citations
arXivHTMLPDF

Abstract

A bstract Starting from the computation of Symmetry Resolved Entanglement Entropy (SREE) for boosted intervals in a two dimensional Conformal Field Theory, we compute the same in various non-Lorentzian limits, viz, Galilean and Carrollian Conformal Field Theory in same number of dimensions. We approach the problem both from a limiting perspective and by using intrinsic symmetries of respective non-Lorentzian conformal algebras. In particular, we calculate the leading order terms, logarithmic terms, and the $$ \mathcal{O}(1) $$ O 1 terms and explicitly show exact compliance with equipartition of entanglement , even in the non-Lorentzian system. Keeping in mind the holographic origin of SREE for the Carrollian limit, we further compute SREE for BMS 3 -Kac-Moody, which couples a U(1) × U(1) theory with bulk gravity.

DOI
10.1007/jhep06(2024)121

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A bstract Starting from the computation of Symmetry Resolved Entanglement Entropy (SREE) for boosted intervals in a two dimensional Conformal Field Theory, we compute the same in various non-Lorentzian limits, viz, Galilean and Carrollian Conformal Field Theory in same number of dimensions. We approach the problem both from a limiting perspective and by using intrinsic symmetries of respective non-Lorentzian conformal algebras. In particular, we calculate the leading order terms, logarithmic terms, and the O(1)\mathcal{O}(1)O(1) O 1 terms and explicitly show exact compliance with equipartition of entanglement , even in the non-Lorentzian system. Keeping in mind the holographic origin of SREE for the Carrollian limit, we further compute SREE for BMS 3 -Kac-Moody, which couples a U(1) × U(1) theory with bulk gravity.
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.
We propose a symmetry resolution of entanglement for categorical noninvertible symmetries (CaT-SREE) in ( 1+1 )-dimensional conformal field theories. The definition parallels that of grouplike invertible symmetries, employing the concept of symmetric boundary states with respect to a categorical symmetry. Our examination extends to rational conformal field theories, where the behavior of CaT-SREE mirrors that of grouplike invertible symmetries. We find that CaT-SREE can be defined if there is no obstruction to gauging the categorical symmetry, as happens in the case of grouplike symmetries. We also provide instances of the breakdown of entanglement equipartition at the next-to-leading order in the cutoff expansion. Our findings shed light on how the interplay between conformal boundary conditions and categorical symmetries lead to specific patterns in the entanglement entropy. 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 111-dimensional system of free fermions hopping on a lattice, in which the conserved U(1)U(1)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 111-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.
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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 We present a method to compute the symmetry-resolved entanglement entropy of spherical regions in higher-dimensional conformal field theories. By employing Casini-Huerta-Myers mapping, we transform the entanglement problem into thermodynamic calculations in hyperbolic space. This method is demonstrated through computations in both free field theories and holographic field theories. For large hyperbolic space volume, our results reveal a universal expansion structure of symmetry-resolved entanglement entropy, with the equipartition property holding up to the constant order. Using asymptotic analysis techniques, we prove this expansion structure and the equipartition property in arbitrary dimensions.