Asymptotic inversion, quasi-locality, and string-charge relations for the full fusion hierarchy of nested Bethe Ansatz models

Integrable lattice models solved by the nested Bethe Ansatz, such as the fundamental SU(N)-symmetric (Lai-Sutherland) spin chains with N >= 3, are expected to relax after a quantum quench to a Generalized Gibbs Ensemble (GGE) built from a complete set of local and quasi-local conserved charges. Fehér and Pozsgay (SciPost Phys. 8, 034 (2020)) constructed the candidate complete set of charges from the fusion hierarchy of transfer matrices, proved quasi-locality and the underlying space-reflected asymptotic inversion identities only for the two single-box (m = 1) SU(3) cases, derived the SU(3) string-charge relations for almost all Bethe states, and formulated the general statements as Conjectures 1-5. The open problem is to give rigorous proofs: the asymptotic inversion $\bar t_m^{(a)}(-u)\,t_m^{(a)}(u)\approx 1$ and the quasi-locality of $X_m^{(a)}(u)=(-\mathrm{i})\partial_u\log t_m^{(a)}(u)$ for all fused transfer matrices and $u$ in the physical strip, the rigorous control of dominant-term selection for all thermodynamic Bethe root distributions, the analytic diagonalization of the finite 4-site quantum transfer matrices with fused auxiliary representations that enables both, and the SU(N) string-charge relations together with a completeness proof of the GGE for nested models. No resolution of any of these strands is known in the literature to date.

Background

One-dimensional quantum integrable models possess infinitely many mutually commuting conserved charges, which prevent thermalization to an ordinary Gibbs ensemble after a quantum quench (a sudden change of Hamiltonian parameters starting from a far-from-equilibrium initial state). Instead they are expected to relax to a Generalized Gibbs Ensemble (GGE), the maximum-entropy ensemble built from all conserved charges of the model. For interacting lattice models the construction of the GGE turned out to be subtle: in the XXX (isotropic Heisenberg) and XXZ spin chains the GGE built only from the strictly local charges fails in some quench situations, and the failure was resolved by incorporating quasi-local charges. Quasi-local charges are conserved operators whose densities are supported on arbitrarily long segments but decay exponentially with the length of the segments connecting the support to the rest of the chain; they are traceless on two consecutive sites and have an extensive Hilbert-Schmidt norm (the Hilbert-Schmidt norm of an operator $A$ on a chain of $L$ sites being $\|A\|_{\rm HS}=(3^{-L}\mathrm{Tr}\,A^\dagger A)^{1/2}$ in the SU(3) chain). They were discovered for the XXX chain by Ilievski, Medenjak and Prosen (arXiv:1506.05049), and shown to complete the GGE of the interacting XXZ chain by Ilievski et al. (arXiv:1507.02993).

The quasi-local charges of the Heisenberg chain are generated by the fusion hierarchy of transfer matrices. A transfer matrix $t(u)$ is the partial trace over an auxiliary space of a monodromy matrix, a product of R-matrices (the local solutions of the Yang-Baxter equation) acting on one auxiliary space and one physical site each; transfer matrices at different spectral parameters $u$ commute, and the fused transfer matrices $t_m^{(a)}(u)$ correspond to taking the auxiliary space to be a fused representation of the symmetry group, labelled by rectangular Young diagrams of size $(a\times m)$. Their eigenvalues satisfy the T-system, a set of bilinear functional relations among the fused transfer-matrix eigenvalues. In the XXX chain the traceless operators $X_m(u)=(-\mathrm{i})\partial_u\log t_m(u)$, with $m=1,2,\dots$ labelling the symmetric fused (spin-$m/2$) representations, are quasi-local in the thermodynamic limit provided the rapidity $u$ lies in the physical strip $\mathcal{P}=\{u\in\mathbb{C}:|\Im(u)|<1/2\}$.

The Bethe-Ansatz description of these chains involves strings: groups of Bethe rapidities whose imaginary parts differ by approximately $\mathrm{i}$, which behave as composite quasiparticles. In the thermodynamic limit the Bethe root configurations are described by the densities $\rho_m^{(a)}(u)$ of $m$-strings of rapidity type $a$. The string-charge duality of Ilievski, Quinn, De Nardis and Brockmann (arXiv:1512.04454) states, for SU(2)-type chains, that the mean values of the quasi-local charges in a Bethe eigenstate determine, and are determined by, the string root densities: the ensemble built on the complete set of charges is equivalent to specifying all root densities. This is the bridge between the operator description (GGE) and the thermodynamic Bethe-Ansatz description of steady states, and it underlies the completeness assumption of Generalized Hydrodynamics, the Euler-scale transport theory of integrable systems.

For models solved by the nested Bethe Ansatz, such as the fundamental SU(N)-symmetric (Lai-Sutherland) chains with $N\ge 3$ realized by multicomponent atoms in optical lattices, the Bethe Ansatz involves $N-1$ nested sets of rapidities describing the internal degrees of freedom of the quasiparticles, and the question of whether a complete GGE exists was completely open before 2019. Fehér and Pozsgay (arXiv:1909.04470) proposed the answer: the complete set of charges is generated by the whole fusion hierarchy, i.e. by the transfer matrices $t_m^{(a)}(u)$ with $a=1,\dots,N-1$ and $m=1,2,\dots$, mirrored on the Dynkin diagram of SU(N). They rigorously proved the two $m=1$ (single-box) SU(3) cases, corresponding to the defining and the conjugate representations, by diagonalizing appropriate 4-site quantum transfer matrices (the transfer matrices acting in the crossed channel of a two-dimensional lattice-model representation of Hilbert-Schmidt traces, whose size is independent of the chain length), and derived the complete set of SU(3) string-charge relations.

The proofs rest on two mechanisms. First, a space-reflected asymptotic inversion $\bar t(-u)\,t(u)\approx 1$ (where $\bar t$ is built from the monodromy with the Lax operators in reverse site order), which converts the formal logarithmic derivatives into manifestly local expressions; second, the dominance of a single term in the semi-standard Young-tableaux expansion of the fused-transfer-matrix eigenvalues for $u\in\mathcal{P}$ in the thermodynamic limit. The paper established the dominance only for almost all Bethe states (in the infinite-temperature Hilbert-Schmidt sense) and neighborhoods reached by continuity, and noted that even for the SU(2) chain complete analytic proofs of quasi-locality exist only up to spin $s=3/2$. The general statements were left as Conjectures 1-5.

Problem Statement

Prove, or refute, the conjectured rigorous foundation of the string-charge relations and of the complete Generalized Gibbs Ensemble for the fundamental SU(N)-symmetric spin chains with $N\ge 3$ (nested Bethe Ansatz), as formulated in Conjectures 1-5 of Fehér and Pozsgay (arXiv:1909.04470). The statements concern the fused transfer matrices $t_m^{(a)}(u)$, whose auxiliary spaces carry the rectangular Young-diagram representations $(a\times m)$ with $a=1,\dots,N-1$ and $m=1,2,\dots$, the space-reflected transfer matrices $\bar t_m^{(a)}(u)$ (built from the monodromy with the Lax operators in reverse site order), and the traceless generating functions $X_m^{(a)}(u)=(-\mathrm{i})\partial_u\log t_m^{(a)}(u)$.

(1) Operator statements. For every $a$, $m$ and every rapidity $u$ in the physical strip $\mathcal{P}=\{u\in\mathbb{C}:|\Im(u)|<1/2\}$, prove the asymptotic inversion identities $\bar t_m^{(a)}(-u)\,t_m^{(a)}(u)\approx 1$, and prove that the operators $X_m^{(a)}(u)$ are quasi-local (here $\approx$ denotes asymptotic equality in the thermodynamic limit, for example in the sense that the Hilbert-Schmidt distance from the identity is exponentially small in the chain length $L$, as established for the two $m=1$ SU(3) cases in the source).

(2) Enabling computation. Provide the exact analytic diagonalization (or an equivalent exact spectral control) of the finite 4-site quantum transfer matrices whose auxiliary spaces carry the fused representations, and establish that their dominant eigenvalues are non-degenerate throughout $\mathcal{P}$; this is the route of proof identified in the source, and it is open for general fused representations.

(3) Dominant-term selection. The eigenvalue of $t_m^{(a)}(u)$ is a finite sum over semi-standard Young tableaux; the string-charge relations follow if the single tableau whose row $k$ (counted from the top) is filled entirely with the number $k$ dominates in the thermodynamic limit for $u\in\mathcal{P}$. Currently this dominance is established only for almost all Bethe states (in the infinite-temperature Hilbert-Schmidt sense) and for root distributions connected to these by continuity. Classify the thermodynamic root-density regimes for which the dominant-term identification holds, or exhibit a Bethe root distribution for which it fails.

(4) String-charge relations and completeness. Derive, as rigorous statements, the SU(N) string-charge relations

$$\rho_m^{(a)}(u)=\frac{1}{2\pi L}\left(X_m^{(a)[+]}(u)+X_m^{(a)[-]}(u)-X_{m+1}^{(a)}(u)-X_{m-1}^{(a)}(u)\right),\qquad a=1,\dots,N-1,\; m=1,2,\dots$$

where $X_m^{(a)[\pm]}(u)$ denotes the shifted evaluation $f^{[\pm]}(u)=\lim_{\epsilon\to0}f(u\pm\mathrm{i}/2\mp\epsilon)$ in the convention of the source, so that the relations tie the charge mean values to the densities $\rho_m^{(a)}(u)$ of the $m$-strings of rapidity type $a$. Prove that the charges built on the full fusion hierarchy form a complete set for the Generalized Gibbs Ensemble of the nested models, i.e. that in the thermodynamic limit they determine all nested root densities and hence the post-quench steady state for general initial states.

These strands form one program with a single objective: converting the conjectured complete-GGE construction for nested Bethe-Ansatz models into proven theorems. Statements (1) are the operator-level claims, (2) is their proposed route of proof, (3) is the eigenvalue-level ingredient, and (4) is the resulting thermodynamic statement. A solution may establish them in another order or by different exact techniques, in which case the enabling computation (2) is superseded by whatever exact method replaces it; the problem is settled only when the claims of (1), (3) and (4) are proven for the fundamental SU(N) chains with $N\ge 3$ beyond the two proven single-box SU(3) cases, or when a specific claim among them is refuted (for instance by an explicit thermodynamic Bethe root distribution that selects a different dominant tableau term).

References

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