Criticality and phase diagram of quantum long-range O($N$) models

Several recent experiments in atomic, molecular, and optical systems motivated a huge interest in the study of quantum long-range systems. Our goal in this paper is to present a general description of their critical behavior and phases, devising a treatment valid in $d$ dimensions, with an exponent $d+\sigma$ for the power-law decay of the couplings in the presence of an $\text{O}\left(N\right)$ symmetry. By introducing a convenient ansatz for the effective action, we determine the phase diagram for the $N$-component quantum rotor model with long-range interactions, with $N=1$ corresponding to the Ising model. The phase diagram in the $\sigma \text{−}d$ plane shows a nontrivial dependence on $\sigma$. As a consequence of the fact that the model is quantum, the correlation functions are anisotropic in the spatial and time coordinates for $\sigma$ smaller than a critical value, and in this region the isotropy is not restored even at criticality. Results for the correlation length exponent $\nu$, the dynamical critical exponent $z$, and a comparison with numerical findings for them are presented.

Figure 1

Phase diagram of quantum LR rotor models for discrete (a) and continuous (b) symmetries. The cyan shaded area represents the mean field validity region, while the gray shaded area is the single-phase region. The mean field boundary ${\sigma }_{*}=2$ is represented by a gray dashed line, and renormalized boundaries are solid lines in red, blue, green, and purple for $N=1,2,3,4$ respectively.

Figure 2

The dynamical critical exponent $z$ as a function of $\sigma$ in $d=1,2$, panels (a) and (b) respectively. The red, blue, and green curves are for $N=1,2,3$ respectively; the mean field result is shown as a gray dashed line. The (blue and red) dots represent MC data for $d=1$ derived for a spin system coupled to a bosonic bath with spectral density proportional to ${\omega }^{\sigma }$ (data taken from [52]), with the red dots for the Ising model (in good agreement with our predictions) and the blue dots for the $\text{O}\left(2\right)$ model.

Figure 3

${\left(z\nu \right)}^{-1}$ as a function of $\sigma$ in $d=1,2$, panels (a) and (b) respectively. The red, blue, and green curves are for $N=1,2,3$ respectively, the mean field result is shown as a gray dashed line. Again dots represent MC data for the Ising (red) and the $\text{O}\left(2\right)$ models (blue) in $d=1$; data are from Ref. [52]. The red triangles represent the estimation obtained with the perturbative continuous unitary transformation taken from [54]. These data, however, cannot reproduce the expected values in the mean field region, and seem unable to correctly account for the expected value of the SR boundary ${\sigma }_{*}=2-{\eta }_{\text{SR}}$. Our results for the correlation length exponent $\nu$ have been obtained using the effective dimension relation (31) to map the values of $\nu$ for classical SR systems found in [39] to the quantum LR case; see the procedure outlined in [23]. It should be noted that for the SR Ising case it is necessary to employ two different definitions of anomalous dimensions in order to obtain reliable results; see [23, 38, 39]. In panel (a) leading order terms of $\varepsilon =(d_{uc}-d)$ and $\tilde{\varepsilon }=(d-d_{lc})$ expansions are shown as a gray dashed line and black crosses respectively.

Figure 4

Schematic representation of the relations between quantum and classical LR systems investigated in the paper. The green line indicates the exact mapping obtained via Trotter decomposition, while the red dotted lines indicate effective dimension relations, which, although not exact, produce very accurate results in the case of quantum LR models.