The spin-anisotropic quantum spherical model (SAQSM) of Wald and Henkel (2015) is a quantum spherical model on a dd-dimensional hypercubic lattice whose Hamiltonian couples nearest-neighbour rotor spins and, unusually, also their conjugate momenta. In the exactly solved spherical limit (n→∞n\to\infty, nearest-neighbour couplings) it displays a re-entrant zero-temperature quantum phase transition for spatial dimensions 1<d≲2.0651<d\lesssim2.065 at small anisotropy: the critical coupling gc(λ,d)g_c(\lambda,d) passes through a non-zero minimum at intermediate λ\lambda, an effect with no known classical analogue that arises because interactions between momenta cannot be absorbed into a change of variables. Whether this re-entrant quantum criticality is an artifact of the exactly solved limit or a robust physical mechanism is open. The problem is to determine the quantum phase diagram and critical line in extensions beyond that limit — O(n)O(n)-symmetric quantum rotor models with finite nn and general momentum-space kinetic couplings, and SAQSM-type models with long-range or other non-nearest-neighbour interactions — to decide whether the re-entrant structure and its universality class survive, and to identify the microscopic conditions under which momentum-mediated interactions produce re-entrant quantum criticality.

Background

The spherical model of Berlin and Kac replaces the fixed-length spins of the Ising model by real-valued spin variables subject to a single global constraint ∑n⃗Sn⃗2=N\sum_{\vec n}S_{\vec n}^2=N; this simplification keeps a genuine phase transition while making the model exactly solvable, and its critical behaviour coincides with that of the O(n)O(n) vector model at n→∞n\to\infty. In the quantum spherical model (equivalently, a lattice of quantum rotors), each site carries a coordinate operator Sn⃗S_{\vec n} and a conjugate momentum Pn⃗P_{\vec n} with [Sn⃗,Pm⃗]=iℏ δn⃗m⃗[S_{\vec n},P_{\vec m}]=i\hbar\,\delta_{\vec n\vec m}, and the coupling gg in the kinetic term g2Pn⃗2\frac{g}{2}P_{\vec n}^2 tunes a zero-temperature quantum phase transition between an ordered and a disordered phase; its criticality is in the universality class of the classical spherical model in d+1d+1 dimensions (Vojta 1996).

Wald and Henkel (2015) introduced a spin-anisotropic extension motivated by the spin anisotropy λ\lambda of the quantum XY chain. In that fermionic chain, the anisotropy controls pair creation and annihilation of Jordan-Wigner fermions (obtained from the spins by a non-local transformation mapping spin chains to free fermions) and produces, besides disordered and uniformly ordered phases, an ordered phase with spatially oscillating correlations at small anisotropy. Introducing the analogous pair-creation rate 0≤λ≤10\le\lambda\le1 into the bosonic quantum spherical model gives the SAQSM,

H=∑n⃗[g2Pn⃗2+μ2Sn⃗2−BSn⃗−14s∑j=1d((1+λ)μ Sn⃗Sn⃗+e⃗j+(1−λ)g Pn⃗Pn⃗+e⃗j)]H=\sum_{\vec n}\left[\frac{g}{2}P_{\vec n}^2+\frac{\mu}{2}S_{\vec n}^2-BS_{\vec n}-\frac{1}{4s}\sum_{j=1}^d\left((1+\lambda)\mu\,S_{\vec n}S_{\vec n+\vec e_j}+(1-\lambda)g\,P_{\vec n}P_{\vec n+\vec e_j}\right)\right]

with s=μ/(2J)s=\mu/(2J). In the spin-momentum representation the pair terms become an interaction between the conjugate momenta of neighbouring sites, so the SAQSM is an exactly solvable bosonic lattice model in which interactions between momenta shape the phase diagram. At λ=1\lambda=1 the momentum interaction vanishes and the usual quantum rotor spherical model is recovered; at λ=0\lambda=0 the total boson number is conserved and the critical behaviour changes.

The exact solution shows a line of quantum critical points g=gc(λ,d)g=g_c(\lambda,d) at T=0T=0 whose critical behaviour — exponents, universal amplitude combinations of finite-size scaling theory (the dimensionless combinations describing how critical observables scale with system size), and full scaling forms — is universal along the line and coincides with the thermal transition of the classical spherical model in d+1d+1 dimensions. The surprise is the shape of this line: for 1<d≲2.0651<d\lesssim2.065 and sufficiently small λ\lambda, the slope ∂gc/∂λ∣λ=0\partial g_c/\partial\lambda|_{\lambda=0} is negative (behaving as −λd/2-\lambda^{d/2} for 1<d<21<d<2) while the slope at λ=1\lambda=1 is positive, so by continuity gc(λ,d)g_c(\lambda,d) has a non-zero minimum at an intermediate λ\lambda. At fixed gg in a suitable range, increasing λ\lambda then drives the system from ordered to disordered and back to ordered — a re-entrant quantum phase transition with no known classical analogue, which persists at small nonzero temperature. The authors stress that this demonstrates that interactions between conjugate momenta cannot always be absorbed into a change of variables. Re-entrant second-order transitions had been found earlier by simulation in a quantum anisotropic planar rotor model (Hetényi, Müser and Berne 1999), so re-entrance in rotor-type models is not unique to the spherical limit; however, that anisotropy acts in orientation space and shares no microscopic mechanism with the SAQSM's momentum coupling.

The exact solution rests on two restrictions: the spherical limit, i.e. O(n)O(n) symmetry with n→∞n\to\infty, which makes the model gaussian and the constraint exact, and nearest-neighbour couplings. The authors explicitly raise as open questions the quantitative importance of more general kinetic terms, for instance in O(n)O(n)-symmetric quantum rotor models with finite nn, and whether their conclusions — above all the re-entrant quantum criticality — have qualitative analogues under long-range interactions, noting that effectively non-integer dimensions can also be produced by long-ranged couplings. For integer physical dimensions, d=2d=2 lies inside the re-entrant window 1<d≲2.0651<d\lesssim2.065. Long-range quantum spherical models without momentum coupling have been analysed after 2015 (notably for entanglement properties), but the momentum-coupled case has not been touched.

Problem Statement

Determine whether the re-entrant quantum phase transition induced by interactions between conjugate momenta in the spin-anisotropic quantum spherical model (SAQSM) is robust beyond the exactly solved case — the spherical limit n→∞n\to\infty with nearest-neighbour couplings — and identify the microscopic conditions under which momentum-mediated interactions produce re-entrant quantum criticality. Two concrete instantiations of this question, raised explicitly by Wald and Henkel (2015):

(a) Finite nn and general kinetic terms: for O(n)O(n)-symmetric quantum rotor models with finite nn (e.g. n=2n=2 or n=3n=3) on dd-dimensional lattices (d=2d=2 lies inside the re-entrant window 1<d≲2.0651<d\lesssim2.065), with momentum–momentum couplings of the SAQSM type or more general momentum-space kinetic interactions, determine the zero-temperature quantum critical line gc(λ,d)g_c(\lambda,d), whether its re-entrant structure (the non-zero minimum at intermediate anisotropy λ\lambda) survives 1/n1/n corrections, and whether the universality class — that of the classical spherical model in d+1d+1 dimensions at n=∞n=\infty — changes.

(b) Interaction range: for SAQSM-type models with long-range spin and/or momentum couplings, or with other non-nearest-neighbour lattice interactions, determine the phase diagram and whether the re-entrant quantum critical line persists, noting that long-range couplings can mimic non-integer effective dimensions.

Whether momentum–momentum interactions generate a non-monotonic quantum critical line depends on coupling structure, interaction range, dimensionality, and nn.

References

  1. Sascha Wald and Malte Henkel, 'Quantum phase transition in the spin-anisotropic quantum spherical model', J. Stat. Mech. Theory Exp. P07006 (2015), arXiv:1503.06713, DOI: 10.1088/1742-5468/2015/07/P07006, https://arxiv.org/abs/1503.06713
  2. T. H. Berlin and M. Kac, 'The Spherical Model of a Ferromagnet', Phys. Rev. 86, 821 (1952), DOI: 10.1103/PhysRev.86.821, https://journals.aps.org/pr/abstract/10.1103/PhysRev.86.821
  3. Thomas Vojta, 'Quantum version of a spherical model: Crossover from quantum to classical critical behavior', Phys. Rev. B 53, 710 (1996), DOI: 10.1103/PhysRevB.53.710, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.53.710
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  8. André M. Timpanaro, Sascha Wald, Fernando Semião and Gabriel T. Landi, 'Dynamical chaotic phases and constrained quantum dynamics', Phys. Rev. A 100, 012117 (2019), arXiv:1807.04374, DOI: 10.1103/PhysRevA.100.012117, https://arxiv.org/abs/1807.04374
  9. L. V. T. Tavares, L. G. dos Santos, G. T. Landi, Pedro R. S. Gomes and P. F. Bienzobaz, 'Supersymmetric quantum spherical spins with short-range interactions', J. Stat. Mech. (2020) 023104, arXiv:1910.04007, DOI: 10.1088/1742-5468/ab6a06, https://arxiv.org/abs/1910.04007
  10. Sascha Wald, Malte Henkel and Andrea Gambassi, 'Non-equilibrium dynamics of the open quantum O(n)-model with non-Markovian noise: exact results', J. Stat. Mech. (2021) 103105, arXiv:2106.08237, DOI: 10.1088/1742-5468/ac25f6, https://arxiv.org/abs/2106.08237
  11. Sascha Wald, Saulo V. Moreira and Fernando L. Semião, 'In- and out-of-equilibrium quantum metrology with mean-field quantum criticality', Phys. Rev. E 101, 052107 (2020), arXiv:2001.03189, DOI: 10.1103/PhysRevE.101.052107, https://arxiv.org/abs/2001.03189
  12. Malte Henkel, 'Quantum Dynamics Far from Equilibrium: A Case Study in the Spherical Model', in: Lie Theory and Its Applications in Physics, Springer Proceedings in Mathematics and Statistics 396, pp. 111-128, Springer (2022), arXiv:2201.06448, DOI: 10.1007/978-981-19-4751-3_8, https://arxiv.org/abs/2201.06448