Exact spectrum of the Floquet evolution operator of star-triangle integrable quantum circuits

Miao and Vernier (Quantum 7, 1160 (2023)) constructed two families of integrable discrete-time quantum circuits acting on chains of $Q$-level qudits — the $Q$-state Potts circuits and the $\mathbb{Z}_Q$ clock circuits — directly from solutions of the two-dimensional classical star-triangle relation, proving a conjecture of Lotkov et al. and deriving infinite families of local conserved charges from two-parameter commuting transfer matrices. The construction establishes integrability but not solvability: the exact finite-size spectrum of the Floquet evolution operator $\mathbf{U}_F(\phi)$ is not known, and the authors designate its computation as the first natural open question of their work, suggesting Bethe-ansatz routes via the Temperley–Lieb/six-vertex embedding for the Potts circuits and via Albertini's Bethe equations for Fateev–Zamolodchikov models for the $\mathbb{Z}_Q$ circuits. No exact or partial spectral solution for either family is known in the published literature; all nearby exact results concern the inequivalent vertex-model brick-wall circuits.

Background

An integrable quantum circuit is the discrete-time analogue of an integrable spin chain: a chain of $L$ qudits ($Q$-level quantum systems) is evolved stroboscopically by a fixed Floquet operator, and integrability means that this operator commutes with an extensive family of independent local conserved charges. All previously known interacting examples were built from the Yang–Baxter equation, the solvability condition of two-dimensional vertex models, by arranging the resulting two-site gates in a brick-wall (Trotterized) pattern; such circuits inherit commuting one-parameter transfer matrices, and several of them have been diagonalized by Bethe ansatz (coordinate Bethe ansatz for the XXZ circuit, free-fermion techniques for special cases).

The star-triangle relation (STR) is the alternative solvability condition, governing two-dimensional classical spin models with edge Boltzmann weights; it is the mechanism behind Onsager's and Baxter's exact solutions of the Ising model and its relatives, including the $Q$-state Potts model and the Fateev–Zamolodchikov self-dual $\mathbb{Z}_N$ models. Miao and Vernier showed that STR solutions also generate integrable quantum circuits, through a mechanism genuinely different from the brick-wall construction: from an STR solution one builds a two-parameter family of mutually commuting two-row transfer matrices $\mathbf{T}(\lambda,\mu,\phi)$ (here $\phi$ labels the STR solution, generalizing an anisotropy, while $\lambda,\mu$ are spectral-type parameters), and at the special point $(\lambda,\mu)=(0,\phi)$ the transfer matrix factorizes into a product of commuting one-site and two-site gates,

$$\mathbf{U}_F(\phi)=\mathbf{U}_1(\phi)\,\mathbf{U}_2(\phi)=\mathbf{G}\,\mathbf{T}(0,\phi,\phi),$$

where $\mathbf{G}$ is the one-site translation operator. The Floquet operator $\mathbf{U}_F(\phi)$ commutes with the whole family $\mathbf{T}(\lambda,\mu,\phi)$ and hence with infinitely many local charges. Two families were constructed: (a) $Q$-state Potts circuits, with gates built from the Potts STR solution and symmetric under the permutation group $S_Q$; their integrability had been conjectured by Lotkov, Gritsev, Fedorov and Kurlov and was proven in this work, with the charges expressible through generators of the affine Temperley–Lieb algebra (the algebra of gluing operators $e_j$ obeying $e_j e_{j\pm1} e_j=e_j$ and $e_j^2\propto e_j$, which underlies loop and Potts-type models); at $Q=3$ the circuit is connected to the Zamolodchikov–Fateev 19-vertex model (a solvable three-state two-dimensional vertex model). (b) $\mathbb{Z}_Q$ clock circuits from the Fateev–Zamolodchikov self-dual STR solution, invariant under the clock symmetry $\mathbb{Z}_Q$ (for $Q=3$ the two families coincide up to normalization, since $\mathbb{Z}_3$ plus charge conjugation generates $S_3$); at $Q=4$ this family is closely related to the critical Ashkin–Teller chain. Unlike the Potts circuits, however, the clock circuits are not unitary for general $Q>3$: the source shows that, except for trivial cases, no value of $\phi$ makes them unitary, so their Floquet eigenvalues are general complex numbers rather than unimodular phases. A crucial structural feature is that the staggering of parameters producing the circuit geometry takes place in the internal parameters of each STR Boltzmann weight rather than in site inhomogeneities as in brick-wall circuits, so the transfer matrices are not of the standard single-row vertex-model form and the known Bethe ansätze for brick-wall or staggered vertex circuits do not apply directly.

For the related continuous-time chains much is known: the Fateev–Zamolodchikov spin chains were diagonalized through Bethe-ansatz-type coupled equations derived by Albertini, and the superintegrable 3-state Potts transfer matrix has Bethe-ansatz zeroes with a delicate completeness structure classified by Albertini, Dasmahapatra and McCoy, including many solutions violating the traditional string hypothesis (the assumption, used in standard Bethe-ansatz counting, that roots organize into regular strings of nearly equally spaced values). Temperley–Lieb representations can be mapped to the six-vertex model, whose Bethe ansatz is classical. None of these results, however, covers the circuit point of the two-parameter STR transfer-matrix family, which is a fine-tuned member of that family. The source paper proves integrability — the existence of commuting charges — but explicitly leaves open whether the exact spectrum of $\mathbf{U}_F(\phi)$ can be computed.

Problem Statement

Derive the exact finite-size spectrum of the Floquet evolution operator of the star-triangle integrable quantum circuits constructed by Miao and Vernier. For a chain of $L$ qudits with periodic boundary conditions, the circuit is

$$\mathbf{U}_F(\phi)=\mathbf{U}_1(\phi)\,\mathbf{U}_2(\phi)=\mathbf{G}\,\mathbf{T}(0,\phi,\phi),$$

with $\mathbf{G}$ the one-site translation operator, the gates $\mathbf{U}_1(\phi)$, $\mathbf{U}_2(\phi)$ fixed by the star-triangle solution at anisotropy $\phi$, and $\mathbf{T}(\lambda,\mu,\phi)$ the commuting two-parameter transfer matrix family of the source (its Potts-family and $\mathbb{Z}_Q$-family gate coefficients are given explicitly in Sections 4 and 5 of the source). The problem is to determine the exact eigenvalues of $\mathbf{U}_F(\phi)$ — equivalently of the two-row transfer matrix $\mathbf{T}(0,\phi,\phi)$, up to the momentum phases contributed by $\mathbf{G}$ — for both families: (a) the $Q$-state Potts circuits and (b) the $\mathbb{Z}_Q$ clock circuits (the two coincide at $Q=3$ up to normalization), for general $Q\ge2$ and generic $\phi$. For family (b) at general $Q>3$ the circuit is non-unitary — the source shows that no value of $\phi$ makes it unitary except trivial cases — so the requested spectrum there consists of general complex eigenvalues rather than unimodular phases. A complete answer determines all eigenvalues of $\mathbf{U}_F(\phi)$ — canonically through closed functional (Bethe-ansatz-type) equations whose solutions parametrize them — establishes completeness, i.e. a counting showing that the resulting solutions account for the full Hilbert space in every symmetry sector, including the exceptional and root-of-unity sectors where Temperley–Lieb-type spectra are known to contain non-string solutions, and identifies the quantum numbers labelling the eigenstates. The source suggests two routes (the Temperley–Lieb representation of the Potts circuits and its six-vertex realization; adaptation of Albertini's Bethe equations for Fateev–Zamolodchikov models to the $\mathbb{Z}_Q$ circuits), but any exact method — functional relations, Onsager-algebra techniques (the operator algebra underlying Onsager's solution of the two-dimensional Ising model), $Q$-operators, separation of variables, or an exact operator identity to an already-solved transfer matrix — is admissible, provided it yields the finite-size spectrum of the stated circuits rather than of a limiting Hamiltonian chain.

References

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