Open phase-diagram questions in the spin-anisotropic quantum spherical model: the $\lambda>1$ regime and a possible oscillating ordered sub-phase

The spin-anisotropic quantum spherical model (SAQSM) is an exactly solvable $d$-dimensional quantum rotor model in which an anisotropy parameter $\lambda$, motivated by the spin anisotropy of the quantum XY chain, acts both as a boson pair-creation rate and as an interaction between the canonically conjugate momenta. Wald and Henkel (J. Stat. Mech. 2015 P07006) diagonalised the model exactly and derived the zero-temperature quantum critical line $g_c(\lambda,d)$ for $|\lambda|\le 1$, finding it in the universality class of the classical spherical model in $d+1$ dimensions and, surprisingly, re-entrant for $1<d\lesssim 2.065$. Two parts of the analysis were left open: the critical behaviour for $\lambda>1$, where the Bessel-function asymptotics of the spherical constraint break down; and the existence (or exclusion) of an ordered sub-phase with spatially oscillating correlations analogous to the fermionic XY chain's $g<\sqrt{1-\lambda^2}$ phase. (The source's broader outlook remarks - on connections to long-ranged interactions and random matrix theory, and on general kinetic terms in finite-$n$ O($n$) quantum rotor models - are background context and are not part of the problem.) No published work addresses these questions, so completing the phase diagram and ordering characterisation of the SAQSM remains open.

Background

The spherical model of Berlin and Kac (1952) is a classical ferromagnet in which the Ising variables are replaced by real continuous spins $S_{\vec n}$ subject to a global constraint $\langle\sum_{\vec n} S_{\vec n}^2\rangle=N^d$ (the mean spherical constraint) on a hypercubic lattice of $N^d$ sites; the constraint is enforced by a spherical field $\mu$, and the model is exactly solvable in any dimension $d$. It is equivalent to the $n\to\infty$ limit of the classical O($n$) model and serves as the standard laboratory for non-mean-field criticality above two dimensions. Its quantum extension promotes the spins to operators $\hat S_{\vec n}$ with canonically conjugate momenta $\hat P_{\vec n}$ (a quantum rotor at each site), with a coupling $g$ multiplying the kinetic term that controls the strength of the zero-point fluctuations. At $T=0$ the quantum spherical model undergoes a quantum phase transition (a transition driven by a non-thermal coupling, here $g$) whose critical behaviour lies in the universality class of the classical spherical model in $d+1$ dimensions, since the imaginary-time direction adds one effective dimension.

A separate exactly solvable line is the quantum XY chain in a transverse field, built from Pauli matrices with a field $g$ and a spin anisotropy $\lambda$ mixing the $x$ and $y$ spin components. After a Jordan-Wigner transformation (a nonlocal map of spins to spinless fermions), $\lambda$ controls the rate of pair creation and annihilation of those fermions. In one dimension its ground-state phase diagram contains a disordered phase ($g>1$), an Ising-universality critical line ($g=1$), an ordered ferromagnetic phase, and, for $g<\sqrt{1-\lambda^2}$, an ordered sub-phase whose spin correlations oscillate at an incommensurate wavevector; that sub-phase is characterised by level crossings between the even and odd fermion-parity sectors, with Néel-type ground states on its boundary.

Wald and Henkel (2015) asked what happens when the analogous anisotropy is introduced into the bosonic quantum spherical model: writing the rotor model in bosonic ladder operators, they let the two-boson creation/annihilation rate be $\lambda$ instead of the fixed value 1. In terms of spins and momenta this defines the spin-anisotropic quantum spherical model (SAQSM),

$H=\sum_{\vec n}\left[\frac{g}{2}\hat P_{\vec n}^2+\frac{\mu}{2}\hat S_{\vec n}^2-B\hat S_{\vec n}-\frac{1}{4s}\sum_{j=1}^{d}\left((1+\lambda)\mu\,\hat S_{\vec n}\hat S_{\vec n+\vec e_j}+(1-\lambda)g\,\hat P_{\vec n}\hat P_{\vec n+\vec e_j}\right)\right],$

with $s=\mu/(2J)$, $J$ the exchange coupling, $B$ an external field, and the mean spherical constraint fixing $\mu$ (equivalently $s$). The anisotropy $\lambda$ therefore not only rescales the spin-spin coupling but also introduces an interaction $(1-\lambda)g$ between the momenta of neighbouring sites: at $\lambda=1$ the momentum interaction disappears and the standard quantum rotor spherical model is recovered, while $\lambda=0$ conserves the total boson number; a duality $\hat S\leftrightarrow\hat P$, $\mu\leftrightarrow g$, $\lambda\leftrightarrow-\lambda$ (at $B=0$) permits restricting to $\lambda\ge 0$.

A canonical transformation diagonalises $H$ into free quasi-bosons with spectrum $\Lambda_{\vec k}=\sqrt{s-\frac{1+\lambda}{2}\sum_j\cos k_j}\,\sqrt{s-\frac{1-\lambda}{2}\sum_j\cos k_j}$, and the spherical constraint becomes an integral equation which for $|\lambda|\le 1$ was analysed asymptotically through modified Bessel functions $I_n(\rho)$ of an argument $\rho$ depending on $(s,\lambda,d)$. This yields the exact quantum critical line $g_c(\lambda,d)$ for $0<\lambda\le 1$, along which the critical behaviour is universal and in the universality class of the classical spherical model in $d+1$ dimensions, with $g_c(1,d)>0$ only for $d>1$. Two features are specific to the anisotropy. First, for dimensions $1<d\lesssim 2.065$ and small enough $\lambda$, the critical line $g_c(\lambda,d)$ passes through a non-zero minimum, so that at fixed $g$ the ordered phase is left and then re-entered as $\lambda$ grows - a re-entrant quantum phase transition with no known classical-analogue (the one-dimensional fermionic XY chain has exactly $g_c(\lambda)=1$, and the scarce numerical data in two dimensions suggest monotonicity). Second, although the equal-time spin-spin correlator was derived exactly, no analogue of the fermionic oscillating sub-phase could be detected. The authors also state precisely what their methods could not do: the analysis is restricted to $|\lambda|\le 1$ because for $\lambda>1$ the Bessel argument $\rho$ can vanish, so the constraint analysis has to be redone and the critical behaviour in that regime is unknown (though numerics show no singularity at $\lambda=1$ and straightforward generalisation is expected); and only the leading small-temperature corrections to $g_c$ were computed (they show the re-entrant transition survives for small $T>0$), the paper's own analysis being concentrated on the ground-state transition at $T=0$. The re-entrant shape of $g_c(\lambda,d)$ was itself analysed in the paper's appendix E, which splits the spherical constraint into a non-analytic contribution $G^{(1)}$ (the leading behaviour in $\lambda$ for $1<d<2$) and an analytic contribution $G^{(2)}$.

Problem Statement

For the spin-anisotropic quantum spherical model (SAQSM) on a $d$-dimensional hypercubic lattice of $N^d$ sites, defined by the Hamiltonian

$H=\sum_{\vec n}\left[\frac{g}{2}\hat P_{\vec n}^2+\frac{\mu}{2}\hat S_{\vec n}^2-B\hat S_{\vec n}-\frac{1}{4s}\sum_{j=1}^{d}\left((1+\lambda)\mu\,\hat S_{\vec n}\hat S_{\vec n+\vec e_j}+(1-\lambda)g\,\hat P_{\vec n}\hat P_{\vec n+\vec e_j}\right)\right],$

where $\hat S_{\vec n}$ are the spin operators, $\hat P_{\vec n}$ their canonically conjugate momenta, $s=\mu/(2J)$ with $J$ the exchange coupling, $B$ an external field, and $\vec e_j$ the unit vector along lattice direction $j$, together with the mean spherical constraint $\langle\sum_{\vec n}\hat S_{\vec n}^2\rangle=N^d$ which fixes $\mu$ (equivalently $s$), the exact solution of Wald and Henkel established the zero-temperature quantum critical line $g_c(\lambda,d)$ and its universality class for $|\lambda|\le 1$. The problem is to complete the phase diagram and ordering characterisation of this model. Its two components are the parts of this phase diagram that the source itself leaves unresolved:

(1) The $\lambda>1$ regime. Extend the exact analysis of the spherical constraint to $\lambda>1$, including the contributions from the regions where the argument $\rho$ of the modified Bessel functions $I_n(\rho)$ in the integral-equation form of the spherical constraint vanishes - contributions that the $|\lambda|\le 1$ asymptotics excluded. Determine the critical line $g_c(\lambda,d)$ and the critical behaviour for $\lambda>1$, thereby confirming or refuting the authors' expectation that the $\lambda\le 1$ results generalise straightforwardly across $\lambda=1$.

(2) The structure of the ordered phase. Determine whether the ordered phase of the SAQSM contains a sub-phase with spatially oscillating (incommensurate) spin-spin correlations, analogous to the $g<\sqrt{1-\lambda^2}$ sub-phase of the fermionic quantum XY chain; if such a sub-phase exists, locate its boundary in the $(g,\lambda,d)$ parameter space, and if it does not, give an argument within the exact solution that excludes it.

Completing both components completes the phase diagram and ordering characterisation of the SAQSM.

References

1. S. Wald and M. Henkel, "Quantum phase transition in the spin-anisotropic quantum spherical model", J. Stat. Mech. (2015) P07006, DOI 10.1088/1742-5468/2015/07/p07006, arXiv:1503.06713, https://arxiv.org/abs/1503.06713
2. T. H. Berlin and M. Kac, "The Spherical Model of a Ferromagnet", Physical Review 86, 821-835 (1952), DOI 10.1103/PhysRev.86.821, https://doi.org/10.1103/PhysRev.86.821
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