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arXiv:1401.2877 · Journal of Physics A Mathematical and Theoretical

A Gaudin-like determinant for overlaps of Néel and XXZ Bethe states

Michael Brockmann, Jacopo De Nardis, Bram Wouters, Jean-Sébastien Caux

Published Mar 24, 2014• 125 citations
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Abstract

We derive a determinant expression for overlaps of Bethe states of the XXZ spin chain with the Néel state, the ground state of the system in the antiferromagnetic Ising limit. Our formula, of determinant form, is valid for generic system size. Interestingly, it is remarkably similar to the well-known Gaudin formula for the norm of Bethe states, and to another recently-derived overlap formula appearing in the Lieb–Liniger model.

DOI
10.1088/1751-8113/47/14/145003

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We derive a determinant expression for overlaps of Bethe states of the XXZ spin chain with the Néel state, the ground state of the system in the antiferromagnetic Ising limit. Our formula, of determinant form, is valid for generic system size. Interestingly, it is remarkably similar to the well-known Gaudin formula for the norm of Bethe states, and to another recently-derived overlap formula appearing in the Lieb–Liniger model.
We present a ‘Gaudin-like’ determinant expression for overlaps of q -raised Néel states with Bethe states of the spin-1/2 XXZ chain in the non-zero-magnetization sector. The former is constructed by applying global U q ( sl 2 ) spin raising operators to the Néel state, the ground state of the antiferromagnetic Ising chain. The formulas presented are derived from recently-obtained results for the overlap of the Néel state with XXZ Bethe states (Brockmann et al , 2014 J. Phys. A: Math. Theor. 47 145003, Pozsgay, 2013 arXiv:1309.4593, Kozlowski and Pozsgay, 2012 J. Stat. Mech. P05021, Tsuchiya, 1998 J. Math. Phys. 39 5946). The determinants as well as their prefactors can be evaluated in the scaling limit of the XXZ spin chain to the Lieb–Liniger Bose gas. Within this limit a q -raised Néel state that contains finitely many down spins corresponds to the ground state of finitely many free bosons. This allows for a rigorous proof of the overlap formula of De Nardis et al (2014 Phys. Rev. A 89 033601) for Lieb–Liniger Bethe states and a Bose–Einstein condensate (BEC) state with an arbitrary even number of particles.
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.
Abstract We present a conjectured exact formula for overlaps between the Bethe states of the spin-1/2 XXZ chain and generic two-site states. The result takes the same form as in the previously known cases: it involves the same ratio of two Gaudin-like determinants, and a product of single-particle overlap functions, which can be fixed using a combination of the quench action and quantum transfer matrix methods. Our conjecture is confirmed by numerical data from exact diagonalization. For one-site states, the formula is found to be correct even in chains with odd length. It is also pointed out that the ratio of the Gaudin-like determinants plays a crucial role in the overlap sum rule: it guarantees that in the thermodynamic limit there remains no term in the quench action.
We present a conjectured exact formula for overlaps between the Bethe states of the spin-1/2 XXZ chain and generic two-site states. The result takes the same form as in the previously known cases: it involves the same ratio of two Gaudin-like determinants, and a product of single-particle overlap functions, which can be fixed using a combination of the quench action and quantum transfer matrix methods. Our conjecture is confirmed by numerical data from exact diagonalization. For one-site states, the formula is found to be correct even in chains with odd length. It is also pointed out that the ratio of the Gaudin-like determinants plays a crucial role in the overlap sum rule: it guarantees that in the thermodynamic limit there remains no O(1) term in the quench action.
We consider the problem of computing the overlaps between the Bethe states of the XXZ spin-1/2 chain and generic states. We derive recursive formulas for the overlaps between some simple product states and off-shell Bethe states within the framework of the algebraic Bethe ansatz. These recursive formulas can be used to prove in a simple and straightforward way the recently obtained results for the overlaps of the Bethe states with the Néel state, the dimer state, and the q -deformed dimer state. However, these recursive formulas are derived for a broader class of states and represent a concrete starting point for the computation of rather general overlaps. Our approach can be easily extended to other one-dimensional Bethe ansatz integrable models.