Norms of Bethe states in the Jaynes-Cummings-Gaudin model: the separated-variable integral versus the Gaudin determinant

For the quantum Jaynes-Cummings-Gaudin (JCG) model - the integrable rational Gaudin model describing n spins coupled to a single harmonic oscillator, i.e. the integrable Dicke/Tavis-Cummings model of cavity quantum electrodynamics - Babelon and Talalaev (2007) constructed the scalar product in Sklyanin's separated-variables representation: the squared norm of an eigenstate is a multiple integral over the separated variables with a Vandermonde weight and a measure rho built from Bessel functions, fixed so that the commuting Hamiltonians are Hermitian. They observed that for on-shell Bethe eigenstates this norm must also be given by the Gaudin determinant (the Jacobian of the Bethe equations), but left the exact evaluation of the integral and its relation to that determinant unexplained, calling it 'still very mysterious'. Determinant formulas for norms and overlaps of Dicke-JCG models were derived by algebraic Bethe-ansatz and eigenvalue-based methods (Tschirhart-Faribault 2014; Claeys et al. 2015, 2017), and the analogous equality between a separated-variable integral of Baxter wavefunctions and the Slavnov determinant was proven for the compact XXX spin chain (Kazama-Komatsu-Nishimura 2013), but the norm identity posed for the JCG model itself remains open: no existing work evaluates the Bessel-measure norm integral on Bethe eigenstates or establishes its equality with the Gaudin determinant.

Background

The Jaynes-Cummings-Gaudin (JCG) model describes n inequivalent quantum spins coupled to a single harmonic oscillator,

$$H=\sum_{j=0}^{n-1}2\epsilon_j s_j^z+\omega\,b^\dagger b+g\sum_{j=0}^{n-1}\left(b^\dagger s_j^-+b\,s_j^+\right),$$

where $b^\dagger,b$ are the oscillator creation and annihilation operators, $s_j^z,s_j^\pm$ are spin-$s_j$ operators, $\epsilon_j$ are the level splittings, $\omega$ the oscillator frequency and $g$ the coupling (with $[b,b^\dagger]=\hbar$ and $[s_j^+,s_j^-]=2\hbar s_j^z$, where $\hbar$ is Planck's constant, kept explicit as the quasi-classical parameter). It is the integrable member of the Dicke/Tavis-Cummings family of light-matter interaction models: as noted by Gaudin, it is a rational Gaudin model (a class of exactly solvable long-range spin systems solvable by Bethe ansatz) in which the oscillator plays the role of one infinite spin. Babelon and Talalaev (2007) gave its systematic Bethe-ansatz solution. One introduces a Lax matrix $L(\lambda)$ (a $\lambda$-dependent matrix of operators whose entries generate the commuting conserved quantities) whose operator entries $A(\lambda),B(\lambda),C(\lambda)$ obey the rational Gaudin commutation relations; $\frac{1}{2}\mathrm{Tr}\,L^2(\lambda)$ generates $n+1$ commuting Hamiltonians, of which $H$ is a linear combination. Eigenstates are created from the lowest-weight state $|0\rangle$ (the vacuum, annihilated by $b$ and by every $s_j^-$) by the operators $C(\mu)$,

$$\Omega(\mu_1,\dots,\mu_M)=C(\mu_1)\cdots C(\mu_M)|0\rangle,$$

and are joint eigenstates precisely when the Bethe roots $\mu_i$ satisfy the Bethe equations $a(\mu_i)+\sum_{j\neq i}\hbar/(\mu_i-\mu_j)=0$ with $a(\lambda)=2\lambda/g^2-\omega/g^2-\sum_j\hbar s_j/(\lambda-\epsilon_j)$.

The squared norm $\|\Omega\|^2$ of a Bethe eigenstate is the basic ingredient of all correlation functions. For Bethe-ansatz models the norm is known in determinant form (the Gaudin formula, proved for spin chains by Korepin): $\|\Omega\|^2$ is proportional to $\det J$, the determinant of the Jacobian matrix of the Bethe equations evaluated at the roots. Mukhin and Varchenko later showed, for Gaudin models on tensor products of Verma modules (the standard infinite-dimensional highest-weight representations), that an equivalent determinant is the Hessian of the logarithm of the master function (the meromorphic function of the Bethe roots whose critical points are exactly the solutions of the Bethe equations) at its critical point.

Babelon and Talalaev also solved the model by separation of variables (SoV), Sklyanin's method in which the commuting operators $\lambda_1,\dots,\lambda_n$ defined as the zeros of the operator $C(\lambda)$ are used as coordinates: the eigenstate factorizes as $\prod_i\psi(\lambda_i)$, where $\psi(z)=\prod_{j=1}^{M}(z-\mu_j)$ is the polynomial whose zeros are the Bethe roots, and the commuting Hamiltonians become second-order differential operators (their eq. (17); the derivation assumes completeness of the Bethe ansatz, which is not proven in this bosonic setting). They then found the scalar product in this representation (their eq. (18)),

$$\|\Omega\|^2=\int\prod_{i=1}^{n}d\lambda_i\,d\bar{\lambda}_i\;\;W\bar{W}\;\rho(x_1,\dots,x_n)\prod_{i=1}^{n}|\psi(\lambda_i)|^2,$$

where $W=\prod_{i<j}(\lambda_i-\lambda_j)$ is the Vandermonde determinant, $x_i=\hbar^{-1}\prod_j|\epsilon_i-\lambda_j|^2/\prod_{k\neq i}(\epsilon_i-\epsilon_k)^2$, and the measure $\rho$ (their eq. (19)) is the Bessel-function kernel $\rho=\int_0^\infty dy\,e^{-y}y^{M+n-\sum_i(s_i+1/2)}\prod_i J_{2s_i+1}(2\sqrt{yx_i})/x_i^{s_i+1/2}$ ($J_\nu$ denotes a Bessel function), determined by requiring that the differential-operator Hamiltonians be Hermitian; for $n=1$ it collapses to a Laguerre polynomial, and an equivalent series form is their eq. (24). Both the integral and $\det J$ compute the same norm, yet their relation is not understood: the authors state that 'this important formula should be further studied', that for $\psi(z)$ a Bethe state 'one should be able to compute it exactly' because of the Gaudin formula $\|\Omega(\mu_1,\dots,\mu_M)\|^2\simeq\det J$, and that this is 'still very mysterious'. This is the problem addressed here. It is sharpened by neighbouring results: determinant expressions for norms and overlaps of Dicke-JCG models are known from algebraic Bethe ansatz and eigenvalue-based (dual-representation) constructions (Tschirhart-Faribault; Claeys et al.), and for the compact XXX spin chain the analogous equality between an SoV multiple-contour integral of Baxter wavefunctions (the SoV counterpart of the polynomial $\psi(z)$ above) and the Slavnov determinant (the standard determinant formula for the scalar product of an on-shell Bethe state with an arbitrary off-shell one) has been proven (Kazama-Komatsu-Nishimura) - but none of these evaluates the Bessel-measure integral of the JCG model.

Problem Statement

Prove and make explicit - or refute - the identity between the two known expressions for the squared norm of on-shell Bethe eigenstates (joint eigenstates, i.e. states whose Bethe roots satisfy the Bethe equations stated in (b) below) of the quantum Jaynes-Cummings-Gaudin model (n spins $s_j$ with splittings $\epsilon_j$ coupled to one oscillator, $H=\sum_j 2\epsilon_j s_j^z+\omega b^\dagger b+g\sum_j(b^\dagger s_j^-+b s_j^+)$):

(a) the separated-variable norm integral of Babelon and Talalaev, their eq. (18) with the measure of eq. (19):

$$\|\Omega\|^2=\int\prod_{i=1}^{n}d\lambda_i\,d\bar{\lambda}_i\;\;|W|^2\;\rho(x_1,\dots,x_n)\prod_{i=1}^{n}|\psi(\lambda_i)|^2,$$

where $\lambda_i$ are the zeros of the operator $C(\lambda)$ (the Bethe-ansatz creation operator; these zeros are the separated variables), $W=\prod_{i<j}(\lambda_i-\lambda_j)$ is the Vandermonde, $x_i=\hbar^{-1}\prod_j|\epsilon_i-\lambda_j|^2/\prod_{k\neq i}(\epsilon_i-\epsilon_k)^2$, $\rho$ is the Bessel-kernel measure $\rho=\int_0^\infty dy\,e^{-y}y^{M+n-\sum_i(s_i+1/2)}\prod_i J_{2s_i+1}(2\sqrt{yx_i})/x_i^{s_i+1/2}$ that makes the separated-variable Hamiltonians Hermitian, and $\psi(z)=\prod_{j=1}^{M}(z-\mu_j)$ is the polynomial whose zeros are the Bethe roots;

(b) the Gaudin formula $\|\Omega(\mu_1,\dots,\mu_M)\|^2\propto\det J$, with $J$ the Jacobian matrix of the Bethe equations $a(\mu_i)+\sum_{j\neq i}\hbar/(\mu_i-\mu_j)=0$, $a(\lambda)=2\lambda/g^2-\omega/g^2-\sum_j\hbar s_j/(\lambda-\epsilon_j)$, evaluated at the Bethe roots of the same state.

A complete solution must: (i) specify the integration domain/contours for the $\lambda_i$ integrations consistently with the Hermiticity requirement (the oscillator site makes the realization non-compact, so this is not automatic); (ii) show that for every on-shell Bethe state at generic parameters with non-degenerate roots, the integral (a) evaluates in closed form - exactly, e.g. as a determinant or hypergeometric-type expression in the Bethe roots - and coincides with (b) up to a state-independent normalization constant, which must be fixed explicitly; and (iii) identify the mechanism of the equality (for example an Andreief-type determinant identity (an identity reducing multi-dimensional integrals of products of one-variable functions to determinants), an orthogonality or duality argument, or an equivalent algebraic derivation), thereby replacing the authors' 'mysterious' relation by an explicated one. A valid refutation would exhibit explicit on-shell states of a concrete JCG system for which the ratio of (a) to (b) is not constant across states, together with an analysis showing the discrepancy is not an artifact of contour, domain, or normalization choices.

Two further issues touched by the source paper - a proof of completeness of the Bethe ansatz in this bosonic setting, and a rigorous theory of the semiclassical condensation of Bethe roots onto the cuts of the spectral curve - are separate problems and lie outside this problem's completion boundary.

References

1. O. Babelon and D. Talalaev, On the Bethe ansatz for the Jaynes-Cummings-Gaudin model, J. Stat. Mech. (2007) P06013, doi:10.1088/1742-5468/2007/06/P06013, arXiv:hep-th/0703124, https://arxiv.org/abs/hep-th/0703124
2. V. E. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys. 86 (1982) 391-418, doi:10.1007/BF01212176, https://doi.org/10.1007/BF01212176
3. E. Mukhin and A. Varchenko, Remarks on critical points of phase functions and norms of Bethe vectors, in Arrangements – Tokyo 1998, Advanced Studies in Pure Mathematics 27, Mathematical Society of Japan, pp. 239-246, doi:10.2969/ASPM/02710239, arXiv:math/9810087, https://arxiv.org/abs/math/9810087
4. A. Faribault and D. Schuricht, On the determinant representations of Gaudin models' scalar products and form factors, J. Phys. A 45 (2012) 485202, doi:10.1088/1751-8113/45/48/485202, arXiv:1207.2352, https://arxiv.org/abs/1207.2352
5. H. Tschirhart and A. Faribault, Algebraic Bethe ansätze and eigenvalue-based determinants for Dicke-Jaynes-Cummings-Gaudin quantum integrable models, J. Phys. A 47 (2014) 405204, doi:10.1088/1751-8113/47/40/405204, arXiv:1406.0965, https://arxiv.org/abs/1406.0965
6. P. W. Claeys, S. De Baerdemacker, M. Van Raemdonck and D. Van Neck, Eigenvalue-based determinants for scalar products and form factors in Richardson-Gaudin integrable models coupled to a bosonic mode, J. Phys. A 48 (2015) 425201, doi:10.1088/1751-8113/48/42/425201, arXiv:1506.03702, https://arxiv.org/abs/1506.03702
7. P. W. Claeys, D. Van Neck and S. De Baerdemacker, Inner products in integrable Richardson-Gaudin models, SciPost Phys. 3 (2017) 028, doi:10.21468/SciPostPhys.3.4.028, arXiv:1706.05511, https://arxiv.org/abs/1706.05511
8. Y. Kazama, S. Komatsu and T. Nishimura, A new integral representation for the scalar products of Bethe states for the XXX spin chain, J. High Energy Phys. 2013 (2013) 013, doi:10.1007/JHEP09(2013)013, arXiv:1304.5011, https://arxiv.org/abs/1304.5011
9. N. M. Bogoliubov, I. Ermakov and A. Rybin, Dynamic correlation funtions [sic] of the generalized Tavis-Cummings model, arXiv:1702.03740 (2017), https://arxiv.org/abs/1702.03740
10. J. Links, Completeness of the Bethe states for the rational, spin-1/2 Richardson-Gaudin system, SciPost Phys. 3 (2017) 007, doi:10.21468/SciPostPhys.3.1.007, arXiv:1603.03542, https://arxiv.org/abs/1603.03542
11. E. Mukhin, V. Tarasov and A. Varchenko, On separation of variables and completeness of the Bethe ansatz for quantum gl_N Gaudin model, arXiv:0712.0981 (2007), https://arxiv.org/abs/0712.0981
12. O. Babelon and B. Douçot, Classical Bethe ansatz and normal forms in an integrable version of the Dicke model, Physica D 241 (2012) 2095-2108, doi:10.1016/j.physd.2012.03.009, preprint arXiv:1106.3274 (titled 'Classical Bethe Ansatz and Normal Forms in the Jaynes-Cummings Model'), https://arxiv.org/abs/1106.3274
13. Yu. V. Bezvershenko and P. I. Holod, Extended state space of the rational sl(2) Gaudin model in terms of Laguerre polynomials, Ukr. J. Phys. 58 (2013) 1084-1091, doi:10.15407/ujpe58.11.1084, https://doi.org/10.15407/ujpe58.11.1084