Low-temperature behavior of the $O\left(N\right)$ models below two dimensions

We investigate the critical behavior and the nature of the low-temperature phase of the $O\left(N\right)$ models treating the number of field components $N$ and the dimension $d$ as continuous variables with a focus on the $d\le 2$ and $N\le 2$ quadrant of the $(d,N)$ plane. We precisely chart a region of the $(d,N)$ plane where the low-temperature phase is characterized by an algebraic correlation function decay similar to that of the Kosterlitz-Thouless phase but with a temperature-independent anomalous dimension $\eta$. We revisit the Cardy-Hamber analysis leading to a prediction concerning the nonanalytic behavior of the $O\left(N\right)$ models' critical exponents and emphasize the previously not broadly appreciated consequences of this approach in $d<2$. In particular, we discuss how this framework leads to destabilization of the long-range order in favor of the quasi-long-range order in systems with $d<2$ and $N<2$. Subsequently, within a scheme of the nonperturbative renormalization group we identify the low-temperature fixed points controlling the quasi-long-range ordered phase and demonstrate a collision between the critical and the low-temperature fixed points upon approaching the lower critical dimension. We evaluate the critical exponents $\eta (d,N)$ and ${\nu }^{-1}(d,N)$ and demonstrate a very good agreement between the predictions of the Cardy-Hamber type analysis and the nonperturbative renormalization group in $d<2$.

Figure 1

Schematic representation of the $(d,N)$ plane in the vicinity of the Kosterlitz-Thouless point (2,2). The Mermin-Wagner line $[d=2$, $N>1]$ separates the systems that support the LRO in nonzero temperatures from those that do not. Below two dimensions, for sufficiently small values of $N\left[N<N_c\left(d\right)\right]$ the low-temperature phase is characterized by QLRO, while for $N>N_c\left(d\right)$ the system remains disordered for any nonzero temperature. For $d<2$, the Cardy-Hamber line coincides with $N_c\left(d\right)$, while for $d>2$ it is a locus of the hypothetical nonanalyticity of the critical exponents predicted within the framework of Ref. [16], but contrary to the findings of Ref. [17].

Figure 2

Illustration of the Cardy-Hamber scenario for the fixed point collision. The figure presents the evolution of different fixed point solutions $g^{*}(\epsilon ,N)$ upon varying $\epsilon$ for $N=1.5$ and 2.5. The six-pointed star denotes a point of FP collision for $N=1.5$; the five-pointed star denotes a point for $N=2.5$. The horizontal lines mark the overlapping critical fixed points $g^{*}(\epsilon ,2.5)=g^{*}(\epsilon ,1.5)=g_{\text{KT}}=\pi /2$ and were slightly separated for the purpose of transparency.

Figure 3

Dependence of the anomalous dimension of both the critical and the QLRO fixed points on $d$ for a series of values of $N$ within the simplified scheme. Points denote the anomalous dimension of the QLRO FP, and lines denote the critical FP. The stars mark the collision of the FPs at the lower critical dimension as obtained by the simplified calculation.

Figure 4

Dependence of the first RG eigenvalue on $\epsilon$ for a series of values of $N$. Points denote our results from the functional calculation, and lines denote the predictions from the Cardy-Hamber type analysis. The red line corresponding to $N=2.5$ features a discontinuity of the first derivative predicted by the CH type analysis [16], notably absent in the corresponding fRG calculations [17].

Figure 5

Dependence of the anomalous dimension of the critical FP ${\eta }_c$ on $\epsilon$ for a series of values of $N$. Points denote the results from the simplified scheme, and lines denote the data from the functional calculation. The values at the $N$-dependent lower critical dimensions ${\eta }_c\left[d_c\left(N\right)\right]$ are marked by the stars: five-pointed for the functional calculations and six-pointed for the simplified calculations.

Figure 6

Flow of the dimensionless potential minimum ${\tilde{\rho }}_0$ (left axis, blue circles) and the anomalous dimension $\eta$ (right axis, orange squares) for $(d,N)=(1.75,1.3)$ slightly below the critical temperature calculated within the functional scheme. Red vertical lines roughly demarcate the scales at which the flow is controlled by the critical and QLRO FPs.

Figure 7

Fixed point functions for $(d,N)=(1.75,1.3)$. The red horizontal line marks $-\alpha$ corresponding to the propagator pole. Upon decreasing $N$ towards 1, the fixed point potential builds up singularity as the range of values of $\rho$ where $U^{\prime }\left(\rho \right)$ is close to $-\alpha$ increases. (a) Critical FP and (b) QLRO FP.

Figure 8

Dependence of the correlation function exponent of the QLRO FP on $\epsilon$ for a series of values of $N$. Filled points denote the results from the functional scheme, empty points denote the results from the simplified scheme, and lines denote the predictions of the $2+\epsilon$ expansion [25]. The stars mark the lower critical dimensions: five-pointed as obtained in the functional scheme and six-pointed from the CH scenario. The lines are extended below the point of the FP collision (where the BZJ FP is no longer infrared stable) along the prediction $2+\epsilon$ expansion.

Figure 9

The correlation function exponent $d-2+\eta$ of the critical and QLRO FPs as functions of $\epsilon$ for a series of values of $N$. Points denote the exponent of the QLRO FP: filled points denote the results from the functional scheme, empty points denote the results from the simplified scheme, and lines denote the exponent of the critical FP. Stars mark the lower critical dimensions from the analysis of the critical FPs in the functional scheme.

Figure 10

Comparison of different estimates of the shape of the line of the lower critical dimension. Lines denote two different ways of charting the CH line based on the perturbative RG calculations. Points denote results from the present paper and from Ref. [15]. The NPRG points for $d>2$ mark the position of the CH line estimated within the functional RG scheme in Ref. [17], where it was obtained in the form of a crossover.