Correlation length of the quantum transverse-field Ising model on the Bethe lattice: the bound $\mu \le 1/(q-1)$ and the conjectured sharp critical value $\xi_c = 1/\ln(q-1)$

The ferromagnetic transverse-field Ising model on the Bethe lattice (the infinite regular tree) undergoes a continuous quantum phase transition at which, in sharp contrast with Euclidean lattices, the tree-distance correlation length of the ground state appears to remain finite and bounded by $1/\ln(q-1)$. Nagaj, Farhi, Goldstone, Shor and Sylvester (2008) conjectured, from tree tensor network simulations, that at the critical point of the $q=3$ model the transfer-matrix second eigenvalue saturates the sharp value $\mu_2 = 1/2$, i.e. the correlation length equals exactly $1/\ln 2$. Subsequent work confirmed the bound numerically at larger bond dimension and supplied two nonrigorous explanations (a Trotterized stability argument and a susceptibility-threshold argument; the classical analogue of the latter is an exact theorem), but no proof, refutation, or decisive larger-bond-dimension convergence study exists. The problem posed is to establish or refute (i) the bound $\mu \le 1/(q-1)$ on the correlation decay rate of Bethe-lattice transverse-field Ising ground states, rigorously or by controlled numerics, and (ii) the exact saturation $\mu_c = 1/(q-1)$ at the quantum critical point, rigorously (numerics alone cannot settle the exact equality). Both statements are part of the deliverable: the general-$q$ form is the extension the source does not pose, with $q=3$ (the case conjectured in the source) as its primary instance; the general-$q$ statement is motivated by the exact classical result $\xi_c = 1/\ln(q-1)$.

Background

The transverse-field Ising model (TFIM) is the paradigmatic model of a continuous quantum phase transition: for a lattice of spin-1/2 sites with Pauli matrices $\sigma^z, \sigma^x$ it reads $H = -J\sum_{\langle ij\rangle}\sigma^z_i\sigma^z_j - h\sum_i\sigma^x_i$, with a ferromagnetically ordered ground state at weak transverse field $h$, a paramagnet at strong $h$, and a diverging correlation length at the critical ratio $h_c/J$ on Euclidean lattices. The Bethe lattice is the infinite regular tree of coordination $q$: every site has $q$ neighbors and the graph contains no closed loops. It is the geometry on which the cavity method (a recursive message-passing technique that becomes exact on trees) is founded, and it is locally identical to large random regular graphs, which is why it serves as the reference system for analyses of quantum spin models on sparse graphs, including studies motivated by the quantum adiabatic algorithm (an optimization algorithm that slowly deforms a simple quantum Hamiltonian into a cost Hamiltonian). A finite Cayley tree, a finite tree grown outward from a center, has a boundary comparable in size to its bulk, so the infinite translation-invariant Bethe lattice must be distinguished from its finite approximants.

Nagaj, Farhi, Goldstone, Shor and Sylvester (2008) generalized the infinite time-evolving block decimation method (iTEBD, an imaginary-time evolution algorithm for translationally invariant matrix product states on the infinite line) to the Bethe lattice with $q = 3$. A matrix product state (MPS) is a variational wavefunction built from local tensors contracted through auxiliary bonds whose dimension $\chi$ bounds the entanglement the state can carry; on a tree geometry the analogous ansatz is the tree tensor network state. For a translationally invariant such ansatz, correlation functions along a branch of the tree are governed by a transfer matrix $B$ acting on the auxiliary bond space: the connected correlation function decays as $C(r) \propto \mu_2^r$ in the tree distance $r$, where $\mu_2$ is the second-largest eigenvalue of $B$ in modulus, and the correlation length is $\xi = -1/\ln\mu_2$. On the infinite line at criticality $\mu_2 \to 1$ as $\chi \to \infty$ and $\xi$ diverges. On the $q = 3$ Bethe lattice, Nagaj et al. instead observed that $\mu_2$ never exceeds a value close to $1/2$: the correlation length stays finite at the quantum critical point (located at $x_T \approx 0.451$ in their parametrization $x = J/h$, i.e. $h_c/J \approx 2.2$), and they conjectured that the exact limiting value is $\mu_2 = 1/2$, that is, $\xi_{\lim} = 1/\ln 2 \approx 1.44$ lattice spacings — a finite correlation length at a continuous quantum transition.

The number $1/2$ is not arbitrary. For the classical Ising model on the $q$-regular tree, the two-point function in the paramagnetic phase is exactly $C(d) = \tanh(\beta J)^d$ (with $\beta$ the inverse temperature) and the transition occurs at $(q-1)\tanh(\beta_c J) = 1$, so at criticality the decay rate is exactly $1/(q-1)$ and $\xi_c = 1/\ln(q-1)$ exactly (Semerjian, Tarzia, Zamponi 2009). The structural reason is geometric: the number of sites at tree distance $d$ from a given site grows as $(q-1)^d$, so a susceptibility $\chi_{\mathrm{susc}} = \sum_d N_d C(d)$ diverges as soon as the decay rate reaches $(q-1)^{-d}$; a diverging susceptibility, the hallmark of the transition, therefore does not require a diverging tree-distance correlation length. The open quantum question is whether the same sharp saturation holds for the TFIM ground state. Two heuristic arguments support it: first, the stability argument of Nagaj et al., which Trotterizes the quantum model (the Suzuki–Trotter mapping rewrites the quantum partition function as a classical model on a lattice whose layers are Cayley trees, each quantum site becoming a column of classical spins indexed by discrete imaginary time) and argues that the linearized cavity recursion $T^\perp$ around its attractive fixed point has all eigenvalues $\le 1/2$, so that correlations decay at least as fast as $2^{-d}$; second, the susceptibility-threshold argument of Li, von Delft and Xiang (2012), that the longitudinal susceptibility of the Bethe-lattice ground state diverges exactly when $\xi$ reaches $1/\ln(q-1)$, so that at a continuous, susceptibility-driven transition $\xi_c = 1/\ln(q-1)$. Both arguments are acknowledged as nonrigorous, and the quantum model has no exact solution: even the critical field is known only numerically, from tree tensor network simulations and from the quantum cavity method (the extension of the classical message-passing technique to quantum spins via imaginary-time path integrals) combined with quantum Monte Carlo. Numerically, $\mu_2$ approaches $1/2$ from below with growing bond dimension, but published simulations stop at $\chi \approx 30$ (the original implementation was limited to $\chi \le 8$ by an $O(\chi^8)$ update cost), which is insufficient to separate $1/2$ from nearby values.

Problem Statement

Prove or refute the sharp correlation-length conjecture of Nagaj et al. (2008) — formulated in the source for $q = 3$ — together with its natural general-$q$ extension, for the spin-1/2 ferromagnetic transverse-field Ising model on the infinite Bethe lattice.

Consider $H = -J\sum_{\langle ij\rangle}\sigma^z_i\sigma^z_j - h\sum_i\sigma^x_i$ on the infinite $q$-regular tree, $q \ge 3$ (all $q \ge 3$ are required; $q = 3$ is the case of primary interest). Let $C(r) = \langle\sigma^z_0\sigma^z_r\rangle - \langle\sigma^z\rangle^2$ be the connected equal-time ground-state correlation function between a site and its $r$-th descendant along a branch, and let $\mu = \limsup_{r\to\infty} |C(r)|^{1/r}$ denote its exponential decay rate, so that $C(r) \sim \mu^r$ up to subexponential factors and $\xi = -1/\ln\mu$ when the decay is purely exponential.

(i) Bound. Prove, or refute by explicit counterexample, that the decay rate satisfies $\mu \le 1/(q-1)$ (for $q = 3$: $\mu \le 1/2$, i.e. $\xi \le 1/\ln 2$) for the symmetric translation-invariant ground state throughout the disordered phase $h \ge h_c$, and the same bound on the connected correlations of the symmetry-broken (extremal) ground states in the ordered phase (where the source's numerics used an infinitesimal longitudinal field $B_z = 10^{-8}$).

(ii) Sharp critical value. Prove, or refute, that at the quantum critical point $h = h_c$ (known only numerically; $h_c/J \approx 2.2$ for $q = 3$ in the normalization above) the decay rate equals exactly $\mu = 1/(q-1)$ (for $q = 3$: $\mu_2 = 1/2$ and $\xi_c = 1/\ln 2$), i.e. that critical correlations decay as $(q-1)^{-r}$ up to subexponential factors — the quantum counterpart of the exact classical result $\tanh(\beta_c J) = 1/(q-1)$.

Acceptable resolutions include a rigorous proof, an exact solution (for example a quantum-cavity fixed-point analysis yielding the correlation function or $\xi(h)$ in closed form), or, for the bound (i), a controlled numerical convergence study at substantially larger bond dimension than the published $\chi \approx 30$, whose error bars separate $\mu = 1/(q-1)$ from nearby values; a finite-precision numerical study can bound $\mu$ and locate it near $1/(q-1)$ but cannot establish the exact equality in (ii), which requires a rigorous proof or exact solution.

References

1. D. Nagaj, E. Farhi, J. Goldstone, P. Shor, and I. Sylvester, "Quantum transverse-field Ising model on an infinite tree from matrix product states", Phys. Rev. B 77, 214431 (2008). DOI: 10.1103/PhysRevB.77.214431; arXiv:0712.1806; https://arxiv.org/abs/0712.1806
2. F. Krzakala, A. Rosso, G. Semerjian, and F. Zamponi, "Path-integral representation for quantum spin models: Application to the quantum cavity method and Monte Carlo simulations", Phys. Rev. B 78, 134428 (2008). DOI: 10.1103/PhysRevB.78.134428; arXiv:0807.2553; https://arxiv.org/abs/0807.2553
3. G. Semerjian, M. Tarzia, and F. Zamponi, "Exact solution of the Bose-Hubbard model on the Bethe lattice", Phys. Rev. B 80, 014524 (2009). DOI: 10.1103/PhysRevB.80.014524; arXiv:0904.3075; https://arxiv.org/abs/0904.3075
4. Á. Nagy, "Simulating quantum systems on the Bethe lattice by translationally invariant infinite-tree tensor network", Annals of Physics 327, 542–552 (2012). DOI: 10.1016/j.aop.2011.11.012; arXiv:1106.3033; https://arxiv.org/abs/1106.3033
5. W. Li, J. von Delft, and T. Xiang, "Efficient simulation of infinite tree tensor network states on the Bethe lattice", Phys. Rev. B 86, 195137 (2012). DOI: 10.1103/PhysRevB.86.195137; arXiv:1209.2387; https://arxiv.org/abs/1209.2387
6. S. Sahu and B. Swingle, "Efficient tensor network simulation of quantum many-body physics on sparse graphs", arXiv:2206.04701 (2022). DOI: 10.48550/arXiv:2206.04701; https://arxiv.org/abs/2206.04701
7. T. Barthel, "Tree tensor network states represent low-energy states faithfully", arXiv:2512.20215 (2025). DOI: 10.48550/arXiv:2512.20215; https://arxiv.org/abs/2512.20215