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arXiv:2111.09901 · Journal of Statistical Physics

Crosscap States in Integrable Field Theories and Spin Chains

João Caetano, Shota Komatsu

Published Apr 22, 2022• 24 citations
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Abstract

Abstract We study crosscap states in integrable field theories and spin chains in $$1+1$$ 1 + 1 dimensions. We derive an exact formula for overlaps between the crosscap state and any excited state in integrable field theories with diagonal scattering. We then compute the crosscap entropy, i.e. the overlap for the ground state, in some examples. In the examples we analyzed, the result turns out to decrease monotonically along the renormalization group flow except in cases where the discrete symmetry is spontaneously broken in the infrared. We next introduce crosscap states in integrable spin chains, and obtain determinant expressions for the overlaps with energy eigenstates. These states are long-range entangled and their entanglement entropy grows linearly until the size of the subregion reaches half the system size. This property is reminiscent of pure-state black holes in holography and makes them interesting for use as initial conditions of quantum quench. As side results, we propose a generalization of Zamolodchikov’s staircase model to flows between D-series minimal models, and discuss the relation to fermionic minimal models and the GSO projection.

DOI
10.1007/s10955-022-02914-6

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Integrable crosscap states in gl \mathfrak{gl} gl(N) spin chains
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The linear growth of entanglement after a quench from a state with short-range correlations is a universal feature of many body dynamics. It has been shown to occur in integrable and chaotic systems undergoing either Hamiltonian, Floquet or circuit dynamics and has also been observed in experiments. The entanglement dynamics emerging from long-range correlated states is far less studied, although no less viable using modern quantum simulation experiments. In this work, we investigate the dynamics of the bipartite entanglement entropy and mutual information from initial states which have long-range entanglement with correlation between antipodal points of a finite and periodic system. Starting from these crosscap states, we study both brickwork quantum circuits and Hamiltonian dynamics and find distinct patterns of behaviour depending on the type of dynamics and whether the system is integrable or chaotic. Specifically, we study both dual unitary and random unitary quantum circuits as well as free and interacting fermion Hamiltonians. For integrable systems, we find that after a time delay the entanglement experiences a linear in time decrease followed by a series of revivals, while, in contrast, chaotic systems exhibit constant entanglement entropy. On the other hand, both types of systems experience an immediate linear decrease of the mutual information in time. In chaotic systems this then vanishes, whereas integrable systems instead experience a series of revivals. We show how the quasiparticle and membrane pictures of entanglement dynamics can be modified to describe this behaviour, and derive explicitly the quasiparticle picture in the case of free fermion models which we then extend to all integrable systems.
Abstract It has been observed recently that various spin chain Hamiltonians admit special zero energy ‘crosscap’ eigenstates. These states are made up of maximally entangled Bell pairs prepared on antipodal sites of a periodic chain. We generalize the states by allowing the antipodal pairs to have non-maximal, tunable entanglement. We give sufficient conditions for such states to be exact zero energy eigenstates of a local Hamiltonian. The conditions are naturally satisfied in many models which have a global U (1) symmetry. These models include well known integrable models such as the XX model, the Bariev model, the folded XXZ model, and also a variety of non-integrable models. Using the zero-energy crosscap states we also derive a family of exact zero modes with sub-volume law entanglement.
With increasing subsystem size and energy, bipartite entanglement entropies of energy eigenstates cross over from the groundstate scaling to a volume law. In previous work, we pointed out that, when strong or weak eigenstate thermalization (ETH) applies, the entanglement entropies of all or, respectively, almost all eigenstates follow a single crossover function. The crossover functions are determined by the subsystem entropy of thermal states and assume universal scaling forms in quantum-critical regimes. This was demonstrated by field-theoretical arguments and the analysis of large systems of non-interacting fermions and bosons. Here, we substantiate such scaling properties for integrable and non-integrable interacting spin-1/2 chains at criticality using exact diagonalization. In particular, we analyze XXZ and transverse-field Ising models with and without next-nearest-neighbor interactions. Indeed, the crossover of thermal subsystem entropies can be described by a universal scaling function following from conformal field theory. Furthermore, we analyze the validity of ETH for entanglement in these models. Even for the relatively small system sizes that can be simulated, the distributions of eigenstate entanglement entropies are sharply peaked around the subsystem entropies of the corresponding thermal ensembles.
A bstract Understanding the mechanisms by which complex correlations emerge through the dynamics of quantum many-body systems remains a fundamental challenge in modern physics. To address this, quench dynamics starting from nonthermal states have been extensively studied, leading to significant progress. In this paper, we propose a novel quench protocol, termed the “crosscap quench”, to investigate how highly structured thermal pure states relax into typical ones. We begin by analyzing conformal field theories (CFTs) and derive universal features in the time evolution of the entanglement entropy. Furthermore, leveraging the AdS/CFT correspondence, we study holographic CFTs, providing an analytically tractable example in chaotic CFTs. Finally, we validate these findings through numerical simulations in both nonintegrable and integrable quantum spin systems.