Proofs and exact thermodynamics for the quantum-dilogarithm 3D integrable lattice models of Bazhanov–Kashaev–Mangazeev–Sergeev

A 2025 preprint by Bazhanov, Kashaev, Mangazeev and Sergeev announced a new class of integrable lattice models on the three-dimensional cubic lattice, built from any 'quantum dilogarithm' — a function on a Pontryagin self-dual locally compact abelian group satisfying an inversion relation and a five-term (pentagon) identity. Three instances give models with continuous spins on the real line (Faddeev dilogarithm), spins in the product of the real line with a cyclic group (Andersen–Kashaev dilogarithm), and spins in the product of the unit circle with the integers (Woronowicz dilogarithm). The paper announces universal vertex-type and interaction-round-a-cube solutions of the Zamolodchikov tetrahedron equation, the exact equivalence of the two formulations, and closed-form partition functions per site (Lobachevsky-function expressions) for the Faddeev-dilogarithm models — but contains no proofs or derivations: all are deferred to a companion paper that has not appeared, and the later literature citing the construction treats it as claimed rather than established. The open problem is to place this construction on a rigorous footing and complete it: prove the tetrahedron-equation identities and the vertex–IRC equivalence from the inversion and pentagon identities alone, derive the announced partition functions, compute the thermodynamics of the two remaining dilogarithm families, and establish the announced quasi-classical ($b\to0$) correspondence with circular quadrilateral lattices in 3D.

Background

Exactly solvable lattice models in two-dimensional statistical mechanics owe their solvability to the Yang–Baxter equation (YBE), a functional relation for the local Boltzmann weights (the factors multiplying each spin configuration in the partition function) which guarantees that the row-to-row transfer matrices commute and can be diagonalized, allowing exact evaluation of thermodynamic quantities. The natural three-dimensional generalization replaces the row-to-row transfer matrix by a layer-to-layer transfer matrix, which adds one entire two-dimensional layer of the cubic lattice at a time, and replaces the YBE by the Zamolodchikov tetrahedron equation (ZTE), a nonlinear identity for an operator $\mathbf{R}$ acting on the tensor product of three spin spaces, interpreted as a local Boltzmann weight attached to a vertex where three lattice edges meet. The first solution was found by Zamolodchikov in 1980–81 and proven by Baxter, who also computed its partition function per site $z_\infty=\lim_{M\to\infty}Z^{1/M^3}$ (the exponential of the free energy per unit volume) on the infinite cubic lattice — to this day one of very few exactly computable free energies of a genuinely three-dimensional model. The $N$-state Bazhanov–Baxter models generalized this family, revealing the underlying mechanism to be a restricted star-triangle relation (RSTR), an integral identity relating two- and three-spin Boltzmann weights of an interaction-round-a-cube (IRC) model, in which spins sit on lattice sites and the weights are attached to the eight corner spins of each elementary cube.

Bazhanov, Kashaev, Mangazeev and Sergeev (arXiv:2512.23338) announced a construction that makes this mechanism purely algebraic. They define a quantum dilogarithm abstractly as a function $\varphi$ on a Pontryagin self-dual locally compact abelian group $\mathcal{S}$ — a group isomorphic to its own character group (the group of continuous homomorphisms into the unit circle), so that a symmetric Fourier transform $\mathsf{f}(x,y)$ exists; examples are the real line $\mathbb{R}$, the product $\mathbb{R}\times\mathbb{Z}_N$ of the real line with the cyclic group of integers modulo $N$, and $\mathbb{T}\times\mathbb{Z}$ (unit circle times integers) — satisfying (i) an inversion relation $\varphi(x)\varphi(-x)=\varphi(0)^2\mathsf{G}(x)$ with a Gaussian exponent $\mathsf{G}$, and (ii) a Fourier-transform form of the five-term (pentagon) identity. The prototype is Faddeev's modular quantum dilogarithm $\Phi_{\mathsf{b}}(x)$ — a meromorphic deformation of Euler's classical dilogarithm depending on a complex parameter $\mathsf{b}$ — whose pentagon identity coincides with the RSTR of the Bazhanov–Baxter model; two further instances are the Andersen–Kashaev dilogarithm (arising in complex Chern–Simons theory, a topological quantum field theory of complex-valued gauge fields; spin set $\mathbb{R}\times\mathbb{Z}_N$) and the Woronowicz dilogarithm (arising from the quantum $E(2)$ group, a deformation of the Euclidean group of motions of the plane; spin set $\mathbb{T}\times\mathbb{Z}$). For any such $\varphi$ the authors write universal formulae — a vertex-type R-matrix $\langle x_1,x_2,x_3|\mathbf{R}|x_1',x_2',x_3'\rangle$ (spins on lattice edges, three spectral parameters — auxiliary variables labelling a continuous family of weights) and IRC weights $W(a|e,f,g|b,c,d|h)$ built from a bilateral hypergeometric series (a hypergeometric-type series summed over all integer values of its summation index) — and announce that these satisfy the vertex-type and IRC-form tetrahedron equations, implying continuous commuting families of layer-to-layer transfer matrices, and hence integrability, for all three model families.

For the Faddeev case they further announce exact results for the partition function per site of the infinite lattice, expressed through the Lobachevsky function $\Lambda(\beta)=-\int_0^\beta\log(2\sin x)\,\mathrm{d}x$ and angles $\beta_0,\ldots,\beta_3$ built from a spherical triangle that encodes the three spectral parameters: $\log z_\infty=\frac{4\eta^2}{\pi}\sum_j\big[\Lambda(\beta_j)+(\beta_j-\frac{\pi}{4})\log(2\sin\beta_j)\big]$ for both the vertex and IRC forms (which they announce to coincide exactly), and $z_\infty^{\rm field}=\exp\big(\frac{4\eta^2}{\pi}\sum_j\Lambda(\beta_j)\big)$ for the vertex model in external fields; at the symmetric point $z_\infty=\exp(8\eta^2G/\pi)$ with $G\approx0.9159$ the Catalan constant. Crucially, the paper contains no proofs or derivations of any of these statements. The proof of the tetrahedron equations, the explanation of the vertex–IRC equivalence (via a three-dimensional extension of Baxter's 'propagation through the vertex' technique, a local transformation relating the partition functions of two different models), the derivations of the partition functions, and the discussion of the quasi-classical (low-temperature) limit $\mathsf{b}\to0$ and its announced relation to circular quadrilateral lattices — lattices in 3D whose faces are planar quadrilaterals inscribed in circles, a classical object of integrable discrete geometry — are all deferred to a companion paper listed in the bibliography as 'to be published, 2026'. That companion paper has not appeared, so the derivations exist nowhere in the public literature.

Problem Statement

Establish rigorously and completely the exact solvability of the quantum-dilogarithm 3D lattice models announced in arXiv:2512.23338. The problem comprises five linked deliverables serving one objective — converting the announced results into proved mathematics and completing the associated exact thermodynamics.

1. Integrability. Prove, using only the inversion relation and pentagon identity of the quantum dilogarithm (with the Fourier self-duality for the simplified vertex weights of its eq. (2.23)), that the universal vertex-type R-matrix of the source (its eqs. (2.19)/(2.23)) satisfies the vertex-type Zamolodchikov tetrahedron equation (its eq. (2.22)), and that the IRC weights (its eq. (2.28)) satisfy the IRC-form tetrahedron equation (its eq. (2.31)), uniformly for the spin sets $\mathcal{S}=\mathbb{R}$, $\mathcal{S}=\mathbb{R}\times\mathbb{Z}_N$ and $\mathcal{S}=\mathbb{T}\times\mathbb{Z}$.

2. Equivalence of formulations. Prove the announced exact coincidence of the vertex-model and IRC-model partition functions under periodic boundary conditions.

3. Exact thermodynamics. Derive the announced closed forms for the partition function per site of the Faddeev-dilogarithm models on the infinite cubic lattice — the Lobachevsky-function expressions of the source's eqs. (4.8) and (4.15), including the equality of the vertex and IRC partition functions (eq. (4.13); this is the infinite-lattice consequence of the general equivalence required in deliverable 2 and is therefore not a separate requirement) and the saddle-point evaluation of the $(6M-3)$-fold integral representation for the external-field model; the source announces obtaining these from the functional relations, symmetry and factorization properties of the layer-to-layer transfer matrix.

4. Remaining two dilogarithm families. Compute the partition function per site in the infinite-lattice limit for the Andersen–Kashaev model ($\mathcal{S}=\mathbb{R}\times\mathbb{Z}_N$) and the Woronowicz model ($\mathcal{S}=\mathbb{T}\times\mathbb{Z}$).

5. Quasi-classical geometry. Establish the announced correspondence between the quasi-classical (low-temperature) limit $\mathsf{b}\to0$ of the Faddeev-dilogarithm models and the construction of circular quadrilateral lattices in 3D, in the specific sense that in this limit the leading contribution to the Faddeev-dilogarithm partition function is given by the stationary configuration whose geometry is that of a 3D circular quadrilateral lattice, governed by the classical discrete equations of ref. 9. A rigorous demonstration of this leading-order correspondence resolves the deliverable, as does a rigorous disproof of the $\mathsf{b}\to0$ reduction to that classical geometry; a stronger identification of the full quantum and classical structures is not required.

Deliverables 1–3 supply the unpublished content of the announced companion paper and are currently available nowhere in the literature; deliverables 4–5 complete the research program that the source announces but does not carry out. All five serve the single objective of placing the new family of exactly solvable 3D lattice models on a rigorous foundation: deliverables 1–2 establish integrability, which underwrites the transfer-matrix functional relations used in deliverable 3; deliverable 4 extends the thermodynamics to the remaining two models of the family; deliverable 5 interprets the same models in the classical limit. A rigorous disproof of any announced identity or closed formula (for example an explicit counterexample within the source's stated parameter domains, or a detected inconsistency) resolves the corresponding deliverable in the negative and is equally acceptable.

References

1. V. V. Bazhanov, R. M. Kashaev, V. V. Mangazeev, S. M. Sergeev, Quantum Dilogarithms and New Integrable Lattice Models in Three Dimensions, arXiv:2512.23338 (2025, v2 June 2026), https://arxiv.org/abs/2512.23338
2. A. B. Zamolodchikov, Tetrahedron equations and the relativistic S-matrix of straight-strings in 2+1-dimensions, Communications in Mathematical Physics 79 (1981) 489–505, doi:10.1007/BF01209309, https://link.springer.com/article/10.1007/BF01209309
3. R. J. Baxter, On Zamolodchikov's solution of the tetrahedron equations, Communications in Mathematical Physics 88 (1983) 185–205, doi:10.1007/BF01209476, https://link.springer.com/article/10.1007/BF01209476
4. V. V. Bazhanov, R. J. Baxter, New solvable lattice models in three dimensions, Journal of Statistical Physics 69 (1992) 453–485, doi:10.1007/BF01050423, https://link.springer.com/article/10.1007/BF01050423
5. V. V. Bazhanov, R. J. Baxter, Star-triangle relation for a three-dimensional model, Journal of Statistical Physics 71 (1993) 839–864, doi:10.1007/BF01049952, https://arxiv.org/abs/hep-th/9212050
6. L. D. Faddeev, R. M. Kashaev, Quantum Dilogarithm, Modern Physics Letters A 9 (1994) 427–434, doi:10.1142/S0217732394000447, https://arxiv.org/abs/hep-th/9310070
7. S. L. Woronowicz, Operator equalities related to the quantum E(2) group, Communications in Mathematical Physics 144 (1992) 417–428, doi:10.1007/BF02101100, https://link.springer.com/article/10.1007/BF02101100
8. J. E. Andersen, R. Kashaev, Complex Quantum Chern-Simons, arXiv:1409.1208 (2014), https://arxiv.org/abs/1409.1208
9. V. V. Bazhanov, V. V. Mangazeev, S. M. Sergeev, Quantum geometry of three-dimensional lattices, Journal of Statistical Mechanics: Theory and Experiment (2008) P07004, doi:10.1088/1742-5468/2008/07/P07004, arXiv:0801.0129, https://arxiv.org/abs/0801.0129
10. J. Yagi, State integral models and the tetrahedron equation, Journal of Physics A: Mathematical and Theoretical 59 (2026) 285203, doi:10.1088/1751-8121/ae848f, arXiv:2603.09230, https://arxiv.org/abs/2603.09230
11. M. Shim, X. Sun, H. E. Wang, J. Yagi, Solving the tetrahedron equation by Teichmüller TQFT, JHEP 06 (2026) 141, doi:10.1007/JHEP06(2026)141, arXiv:2602.22060, https://arxiv.org/abs/2602.22060
12. S. Iwao, K. Motegi, R. Ohkawa, Tetrahedral L-operators, tensor Schur polynomials and q-deformed loop elementary symmetric functions, arXiv:2604.22141 (2026), https://arxiv.org/abs/2604.22141
13. S. M. Sergeev, On spectral equations for an evolution operator of a q-oscillator lattice, arXiv:2512.24043 (2025), https://arxiv.org/abs/2512.24043