Session Y24: Symmetry Resolved Entanglement in Strongly Correlated Systems

Symmetry Resolved Entanglement in Interacting Models

Over the last two decades, entanglement measures have been employed in probing quantum phases of matter and quantum phase transitions, in addition to being undisputably detected experimentally and efficiently calculated using quantum stochastic methods. More recently, there has been an accelerated interest in utilizing the presence of conservation laws to gain a deeper understanding of the entanglement structure in different quantum scenarios. In the presence of conservation laws in a quantum state, such as the conservation of the total charge Q, the resulting entanglement spectrum obtained from a mode-bipartite reduced density matrix can be resolved based on the local charge q and thus identify the contribution from each charge q sector to the total entanglement. From the point of view of quantum information, a quantum state of conserved charge Q contains less accessible entanglement as the unaccessible contribution to the entanglement is due to the fluctuations of the local charge q, and thus preparing a state with a large amount of accessible entanglement is an obvious goal. Motivated by this vigorous interplay between condensed matter and quantum information, we investigate the symmetry-resolved entanglement in bosonic and fermionic lattice models, namely the Bose-Hubbard model and spinless lattice fermions. We show that accessible entanglement peaks around continuous phase transitions singling a distinct scaling, suggesting that quantum states near quantum phase transitions are more resourceful in terms of quantum information. We also extend our investigation to include particle-bipartite symmetry-resolved entanglement in a translationally invariant system, where the entanglement resolution is due to the conservation of the total quasi-momentum of the lattice model. We show that the resulting symmetry-resolved particle entanglement entropy demonstrates a comparable behavior across phase transitions.   

Symmetry-resolved Entanglement in Topological Insulators and Many-body Localized Phases

In a many-body system, entanglement is typically only partially extractable and transferable to a quantum register. The fundamental idea is that in order to extract it, a measurement of the local particle numner is required. Thus, the total entanglement S can be broken down into two symmetry-resolved components: the configurational (extractable) entropy S_C and number (easily measurable) entropy S_N. In the first part of my talk, I will show that in a C₂-symmetric topological systems, lower bounds for both quantities in terms of a topological invariant can be proven.

In the second part of my talk, I will consider the number entropy as a witness for particle transport following a quantum quench. For Gaussian systems, one can show that S_N(t)~ln S(t). I will show numerical results indicating that this relation also seems to hold in the interacting t-V chain with potential disorder (equivalent to the Heisenberg chain with random fields) in a phase which was previously assumed to be many-body localized. In particular, we find that S_N(t) ~ lnln(t) for all disorder strengths which are numerically accessible, indicating an ongoing transport of particles and the absence of localization. At a minimum, these results show that all previous estimates for the critical point separating the ergodic from a putative many-body localized phase have been incorrect.

    

A probe of symmetry breaking from entanglement

Symmetries and entanglement are two of the pillars of modern quantum physics. Recently there has been a huge research activity on the interplay between these two notions. So far, it has been focused on studying entanglement in the charge sectors of a system with internal symmetries. Still, one can wonder whether entanglement can be used also to quantify how much a symmetry is broken in a system, an issue that has received little attention until now.

Hence, using methods from the theory of entanglement in many-body quantum systems, I present a subsystem measure of symmetry breaking, that we call entanglement asymmetry. As a prototypical illustration, I study the entanglement asymmetry in a quantum quench of a spin chain in which an initially broken global U(1) symmetry is restored dynamically. I will show that, expectedly, the larger is the subsystem, the slower is the restoration, but also the counterintuitive result that the more the symmetry is initially broken, the faster it is restored, a sort of quantum Mpemba effect.   

Symmetry Resolved Entanglement in Integrable Quantum Field Theory

In this talk I will review some recent results relating to a measure of entanglement known as symmetry resolved entropy (SRE). This is a measure that can be defined for theories that possess an internal symmetry and which quantifies the amount of entanglement that is contributed by each symmetry sector. In the context of integrable quantum field theory, the SRE can be computed using correlation functions of composite twist fields, extending the standard programme for entanglement measures. In my talk I will give a summary of some results I have contributed to in this direction, which deal with different models and/or states   

Equipartition of Entanglement in Quantum Hall States

Quantum Hall effects provide an ideal playground to test the equipartition of entanglement for two-dimensional non-interacting and interacting gapped systems. We will discuss the full counting statistics (FCS) and symmetry-resolved entanglement entropies of integer and fractional quantum Hall states. For the filled lowest Landau level of spin-polarized electrons on an infinite cylinder, we will show that FCS is Gaussian and entanglement spreads evenly among different charge sectors. Moreover, finite size corrections can be derived. Moving to the interacting Laughlin state, the entanglement spectroscopy can be carried out assuming the Li-Haldane conjecture. The results confirm equipartition up to small charge-dependent terms, and are then matched with numerical computations based on exact matrix product states.