Direct evaluation of the dynamical critical exponent from ordered-phase imaginary-time correlations in the dissipative quantum spherical model

The transverse-field Ising model with every spin coupled to its own Ohmic heat bath is the paradigmatic model of noisy quantum spins, and its $n\to\infty$ limit, the dissipative quantum spherical model, was solved exactly by Takada and Nishimori [J. Phys. A: Math. Theor. 49, 435001 (2016)]: the static critical exponents equal those of the classical spherical model in $d+2$ spatial dimensions, and the dynamical exponent $z=2$ follows only indirectly from the effective-dimensionality argument $d_{\mathrm{eff}}=d+z$. In the paramagnetic phase the imaginary-time correlations decay algebraically, $\sim 1/\rho^{2}$, so no temporal correlation time exists there, and the authors explicitly left open the direct dynamical characterization: derive the connected imaginary-time correlation function in the ordered (ferromagnetic) phase, establish whether it decays exponentially so that a correlation time $\xi_{\tau}$ can be defined, and extract $z$ directly from $\xi_{\tau}\sim\xi^{z}$. No published work is known to compute this ordered-phase correlator or to evaluate $z$ directly from it; the problem remains open.

Background

A transverse-field Ising model (TFIM) is a lattice of spin-1/2 degrees of freedom whose Hamiltonian couples a classical Ising interaction $-J_{0}\sum_{\langle ij\rangle}\sigma_i^z\sigma_j^z$ to a non-commuting field $-A\sum_i\sigma_i^x$; varying $A/J_{0}$ at zero temperature drives a quantum phase transition. In realistic devices each spin also couples to an environment, modeled after Caldeira and Leggett as an independent bath of harmonic oscillators attached to every site. When the bath spectral density $J(\omega)=\sum_k g_{ik}^{2}\delta(\omega-\omega_{ik})$ grows linearly, $J(\omega)=\alpha\omega$ (Ohmic dissipation), integrating out the oscillators and applying the Suzuki-Trotter decomposition (a path-integral discretization of the quantum partition function in imaginary time) maps the partition function to a $(d+1)$-dimensional classical spin model whose interactions along the extra imaginary-time axis are long-ranged, decaying as $1/|\tau-\sigma|^{2}$ (Chakravarty and Rudnick, 1995). Because this extra direction is anisotropic and long-ranged, the effective dimensionality of the quantum system is $d+z$, where $z$ is the dynamical critical exponent (the exponent relating the correlation time to the correlation length, $\xi_{\tau}\sim\xi^{z}$) and is itself set by the dissipation. For Ohmic damping, quantum Monte Carlo (QMC) simulations (Werner et al., 2005; Sperstad et al., 2010) and renormalization-group (a framework that integrates out fluctuations scale by scale) and large-$N$ (the limit of $n$-component spins as $n\to\infty$) analyses (Pankov et al., 2004; Sachdev, Werner and Troyer, 2004) find $z\approx 2$, so a $d$-dimensional Ohmically damped system shares its static critical behavior with the $(d+2)$-dimensional classical short-range model. Exact benchmarks are valuable here, both because the naive quantum-to-classical mapping is known to fail in related sub-Ohmic problems (Vojta, Tong and Bulla, 2005) and because environmental noise is central to the performance of quantum annealing (an optimization method that exploits quantum fluctuations, whose working Hamiltonian is precisely a TFIM).

The spherical model of Berlin and Kac (1952) replaces Ising spins by real variables under a global constraint $\sum S^{2}=N$; it is exactly solvable in any dimension, including with long-range interactions (Joyce, 1966), and is the $n\to\infty$ limit of the $O(n)$ model (spins with $n$ continuous components and rotationally invariant interactions). Its exponents differ quantitatively from the Ising ones, but it reproduces the upper and lower critical dimensions and other qualitative features, making it a standard analytically controlled reference point.

Takada and Nishimori (2016) applied this to dissipative quantum spin systems: generalizing the Ohmically damped TFIM to $O(n)$ spins and taking $n\to\infty$, they obtained the exact solution of the $d$-dimensional dissipative quantum spherical model, that is, a $(d+1)$-dimensional classical spherical model with nearest-neighbor couplings in the spatial directions, nearest-neighbor couplings along imaginary time, and the Ohmic $1/|\tau-\sigma|^{2}$ temporal kernel. They found a second-order transition for every $d>0$ with phase boundary $G^{-1}\sim\exp(-C_d\alpha)$ (where $G=(KK_\perp)^{-1}$ combines the spatial and temporal nearest-neighbor couplings), static exponents identical to those of the classical spherical model in $d+2$ dimensions (for example $\nu=1/d$, $\eta=0$, with upper critical dimension $d=2$, above which exponents take their mean-field values), and hence $z=2$ through the $d_{\mathrm{eff}}=d+z$ argument. The value $\nu=1$ at $d=1$ differs from the QMC value $\nu=0.638(3)$ of the $d=1$ dissipative TFIM exactly in the way the spherical and Ising universality classes differ, and the $n\to\infty$ limit of the $\epsilon$-expansion of the dissipative $O(n)$-rotor exponents (Sachdev, Werner and Troyer, 2004; Werner, Troyer and Sachdev, 2005) is reproduced exactly.

What is missing is the direct dynamical characterization. In the paramagnetic phase no temporal correlation time can be defined at all: the Ohmic tail makes the imaginary-time correlation function decay algebraically, $G(0,\rho)\approx\alpha/(u\rho)^{2}$, where $u=2z_{s}-\tilde K$ (with $z_{s}$ the saddle-point parameter) measures the distance from the phase boundary and the spatial correlation length is $\xi=\sqrt{K/u}$; this is the spherical analog of the corresponding statement for the dissipative Ising chain. The authors therefore locate the missing dynamical information in the ordered phase: there the connected imaginary-time correlation function is, in their words, in principle computable but in practice difficult; they expected it to decay exponentially, which would define a correlation time $\xi_{\tau}$ and allow $z$ to be extracted directly from $\xi_{\tau}\sim\xi^{z}$, and they explicitly left this direct evaluation of $z$ open. One structural subtlety should be borne in mind: throughout the ordered phase the spherical constraint pins the saddle point at $u=0$, so the fluctuation propagator is massless; in the classical spherical model below its critical point (equivalently, the ideal Bose gas below condensation, to which it is exactly related) connected correlations are known to decay algebraically rather than exponentially. Whether the anticipated exponential decay and a finite $\xi_{\tau}$ actually hold at $n=\infty$ is therefore itself part of what the open computation must establish.

Problem Statement

Within the exactly solved dissipative quantum spherical model — the $n\to\infty$ limit of the $O(n)$ generalization of the $d$-dimensional transverse-field Ising model in which every site is coupled to its own bath of harmonic oscillators with Ohmic spectral density $J(\omega)=\alpha\omega$ — derive the connected imaginary-time correlation function in the ordered (ferromagnetic) phase, where the magnetization is nonzero and the spherical constraint pins the saddle-point parameter (denoted $z$ in the source paper; here written $z_{s}$ so that it is not confused with the dynamical exponent $z$ below) at $z_{s}=\tilde K/2$, so that the mass parameter $u=2z_{s}-\tilde K$ vanishes; here $\tilde K$ is the zero-momentum value of the interaction Fourier transform. Establish its asymptotic decay along the imaginary-time axis: the source work anticipates exponential decay, from which a finite correlation time $\xi_{\tau}$ would be defined and the dynamical critical exponent $z$ extracted from $\xi_{\tau}\sim\xi^{z}$ as the phase boundary $G^{-1}\sim\exp(-C_d\alpha)$ is approached, where $G=(KK_{\perp})^{-1}$ combines the spatial and temporal nearest-neighbor couplings, $C_d$ depends only on $d$, and $\xi$ is the spatial correlation length. The requested results are: (i) the explicit connected ordered-phase correlator, in integral representation and asymptotic form, for general spatial dimension $d>0$; (ii) the identification, from that asymptotic form, of whether the anticipated exponential decay and a finite $\xi_{\tau}$ actually hold at $n\to\infty$ or whether the decay is instead algebraic, given that the constraint renders the fluctuation propagator massless throughout the ordered phase — this identification is a corollary of (i) and is complete once stated, so it need not be developed as a separate computation; and (iii) the direct evaluation of $z$ from these ordered-phase temporal correlations, to be compared with the value $z=2$ already inferred indirectly from the equality of the static exponents with those of the classical spherical model in $d+2$ dimensions. The evaluation (iii) is complete once it fixes $z$ on whichever branch (ii) selects: if the decay is exponential, through the correlation time $\xi_{\tau}$ and the relation $\xi_{\tau}\sim\xi^{z}$; if the decay is algebraic, through the exponent governing the decay with the temporal separation $\rho$ and the exponent governing the decay with the spatial separation $r$, whose comparison (the anisotropy of the decay) determines $z$. This is the problem the authors explicitly left open ('We leave it an open problem to evaluate z directly along this line of analysis').

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