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arXiv:2305.16046 · Journal of High Energy Physics

Integrable crosscap states: from spin chains to 1D Bose gas

Miao He, Yunfeng Jiang

Published Aug 16, 2023• 7 citations
arXivHTMLPDF

Abstract

Abstract The notion of a crosscap state, a special conformal boundary state first defined in 2d CFT, was recently generalized to 2d massive integrable quantum field theories and integrable spin chains. It has been shown that the crosscap states preserve integrability. In this work, we first generalize this notion to the Lieb-Liniger model, which is a prototype of integrable non-relativistic many-body systems. We then show that the defined crosscap state preserves integrability. We derive the exact overlap formula of the crosscap state and the on-shell Bethe states. As a byproduct, we prove the conjectured overlap formula for integrable spin chains rigorously by coordinate Bethe ansatz. It turns out that the overlap formula for both models take the same form as a ratio of Gaudin-like determinants with a trivial prefactor. Finally we study quench dynamics of the crosscap state, which turns out to be surprisingly simple. The stationary density distribution is simply a constant. We also derive the analytic formula for dynamical correlation functions in the Tonks-Girardeau limit.

DOI
10.1007/jhep08(2023)079

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Abstract The notion of a crosscap state, a special conformal boundary state first defined in 2d CFT, was recently generalized to 2d massive integrable quantum field theories and integrable spin chains. It has been shown that the crosscap states preserve integrability. In this work, we first generalize this notion to the Lieb-Liniger model, which is a prototype of integrable non-relativistic many-body systems. We then show that the defined crosscap state preserves integrability. We derive the exact overlap formula of the crosscap state and the on-shell Bethe states. As a byproduct, we prove the conjectured overlap formula for integrable spin chains rigorously by coordinate Bethe ansatz. It turns out that the overlap formula for both models take the same form as a ratio of Gaudin-like determinants with a trivial prefactor. Finally we study quench dynamics of the crosscap state, which turns out to be surprisingly simple. The stationary density distribution is simply a constant. We also derive the analytic formula for dynamical correlation functions in the Tonks-Girardeau limit.
Integrable crosscap states in gl \mathfrak{gl} gl(N) spin chains
Abstract We study the integrable crosscap states of the integrable quantum spin chains and we classify them for the gl \mathfrak{gl} gl gl (N) symmetric models. We also give a derivation for the exact overlaps between the integrable crosscap states and the Bethe states. The first part of the derivation is to calculate sum formula for the off-shell overlap. Using this formula we prove that the normalized overlaps of the multi-particle states are ratios of the Gaudin-like determinants. Furthermore we collect the integrable crosscap states which can be relevant in the AdS/CFT correspondence.
In the paper [1], the authors developed a new method to compute the exact overlap formulas between integrable boundary states and on-shell Bethe states in integrable spin chains. This method utilizes the coordinate Bethe ansatz representation of wave functions and singularity property of the off-shell overlaps. In this paper, we use this new method to derive the formula for overlaps between the Lieb-Liniger Bethe states and the Bose-Einstein condensate (BEC) state. As a simple application this method, we obtained the overlaps between the Lieb-Liniger eigenstates and the free particle states with pair structure.
Abstract We study crosscap states in integrable field theories and spin chains in 1+11+11+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.
We consider integrable boundary states in the XXX spin-1/2 spin chain. We begin by briefly reviewing the algebraic Bethe Ansatz as well as integrable boundary states in spin chains. Then a recently discovered class of integrable states known as crosscap states is described and expanded. In these states each spin is entangled with its antipodal spin. We present a novel proof of the integrability of both a crosscap state that is known in the literature and one that has not previously been studied. We then use the machinery of the algebraic Bethe Ansatz to derive the overlaps between the crosscap states and off-shell Bethe states.
A bstract We derive a universal formula for the overlaps between integrable matrix product states (MPS) and Bethe eigenstates in glN {\mathfrak{gl}}_N glN​ gl N symmetric spin chains. This formula expresses the normalized overlap as a product of a MPS-independent Gaudin-determinant ratio and a MPS-dependent scalar factor constructed from eigenvalues of commuting operators, defined via the K -matrix associated with the MPS. Our proof is fully representation-independent and relies solely on algebraic Bethe Ansatz techniques and the KT -relation. We also propose a generalization of the overlap formula to soN {\mathfrak{so}}_N soN​ so N and spN {\mathfrak{sp}}_N spN​ sp N spin chains, supported by algebra embeddings and low-rank isomorphisms. These results significantly broaden the class of integrable initial states for which exact overlap formulas are available, with implications for quantum quenches and defect CFTs.
We specialize a recently-proposed determinant formula (Brockmann, De Nardis, Wouters and Caux 2014 J. Phys. A: Math. Theor. 47 145003 ) for the overlap of the zero-momentum Néel state with Bethe states of the spin-1/2 XXZ chain to the case of an odd number of downturned spins, showing that it is still of 'Gaudin-like' form, similar to the case of an even number of down spins. We generalize this result to the overlap of q -raised Néel states with parity-invariant Bethe states lying in a nonzero magnetization sector. The generalized determinant expression can then be used to derive the corresponding determinants and their prefactors in the scaling limit to the Lieb–Liniger (LL) Bose gas. The odd number of down spins directly translates to an odd number of bosons. We furthermore give a proof that the Néel state has no overlap with non-parity-invariant Bethe states. This is based on a determinant expression for overlaps with general Bethe states that was obtained in the context of the XXZ chain with open boundary conditions (Pozsgay 2013 arXiv: 1309.4593 , Kozlowski and Pozsgay 2012 J. Stat. Mech. P05021 , Tsuchiya 1998 J. Math. Phys. 39 5946 ). The statement that overlaps with non-parity-invariant Bethe states vanish is still valid in the scaling limit to LL which means that the Bose–Einstein condensate state (De Nardis, Wouters, Brockmann and Caux 2014 Phys. Rev. A 89 033601 ) has zero overlap with non-parity-invariant LL Bethe states.