High to low temperature: $O(N)$ model at large $N$

TL;DR

The paper analyzes the large-$N$ $O(N)$ vector model on $S^1\\times S^2$ without singlet constraints, uncovering a non-trivial fixed point described by a thermal mass solving a gap equation. It develops both high- and low-temperature expansions for the free energy and energy density, and validates them against each other using a Borel-Padé resummation, demonstrating that they correspond to the same fixed-point solution. A key result is the temperature-dependent ratio of the interacting fixed-point free energy to that of the Gaussian theory, which starts at $4/5$ at high temperature, dips to $0.760753$, and returns toward unity as temperature decreases. The work also provides explicit expressions for the thermal mass and exponential low-temperature corrections, and discusses the limitations and range of validity of the resummation, highlighting the method’s potential for finite-temperature CFT and holographic contexts on curved geometries.

Abstract

We study the $O(N)$ vector model for scalars with quartic interaction at large $N$ on $S^1\times S^2$ without the singlet constraint. The non-trivial fixed point of the model is described by a thermal mass satisfying the gap equation at large $N$. We obtain the free energy and the energy density for the model as a series at low temperature in units of the radius of the sphere. We show these results agree with the Borel-Padé extrapolations of the high temperature expansions of the free energy and energy density obtained in our previous work. This agreement validates both the expansions and demonstrates that low temperature expansions obtained here correspond to the same solution of the gap equation studied earlier at high temperature. We obtain the ratio of the free energy of the theory at the non-trivial fixed point to that of the Gaussian theory at all values of temperature. This ratio begins at $4/5$ when the temperature is infinity, decreases to a minimum value of $0.760753$, then increases and approaches unity as the temperature is decreased.

Figures (12)

• Figure 1: Plots for the ratio of free energies at the non-trivial fixed point to the Gaussian fixed point. Here $Z_{\rm crit}$ and $Z_{\rm free}$ denote the partition functions at the non-trivial fixed point and the Gaussian fixed point respectively. Figure \ref{['intro fig a']} plots this ratio computed from the low temperature expansions of $\log Z$, truncated till $O(e^{-\frac{6\beta}{r}})$ for both the fixed points. Figure \ref{['intro fig b']} is obtained using the Borel-Padé re-summation of the high temperature expansions of $\log Z$ at both the fixed points, using the Padé of order $[14,14]$. Figures \ref{['intro fig a']} and \ref{['intro fig b']} plot the same function. The low temperature expansion, Figure \ref{['intro fig a']} describes the function correctly when $\beta/r$ is large, as the effect of truncation of the higher order terms is negligible. In contrast, the Borel-Padé extrapolations of the high temperature expansions, Figure \ref{['intro fig b']} remain sufficiently accurate for small values of $\beta/r$. Both the expansions consistently exhibit the minimum and the value at the minimum.
• Figure 2: The figures compare the Padé approximant of the Borel transform of $f(\frac{\beta}{r})$ given in \ref{['log Z free+1/12..']} described by the orange curve with the Borel transform ${\cal B}f(\frac{\beta}{r})$ itself given by the blue curve, for different orders $p$ of the Padé approximation. The agreement between the two curves occurs till the Padé approximant of the Borel transform $[p,p]{\cal B}f(\frac{\beta}{r})$ admits its first pole on the positive real axis. The agreement provides an internal consistency check to the numerical implementation of the Borel-Padé re-summation technique for the free theory. Note the occurrence of closely located poles and zeros for higher order Padé approximations. In general, such poles usually turn out to be spurious poles arising from Padé approximation and may not be present in the actual Borel transform.
• Figure 3: $\log Z_{\rm free}$ in the free theory; the orange curve denotes the Borel-Padé re-sum of the high temperature expansion \ref{['free pat fn using Cardy']} for $\log Z_{\rm free}$ and the blue curve stands for the low temperature expansion \ref{['free small b/r']}(truncated till $l=n=10$). These two curves overlap on each other for a finite range of $\frac{\beta}{r}$ using Padé approximants of different orders. The figures inside the boxes focus on small values of $\frac{\beta}{r}$ corresponding to each plots. For very large values of $\frac{\beta}{r}$ the Borel-Padé sum fails to approximate the function correctly. From the plots within the boxes, for very small values of $\frac{\beta}{r}$ two curves diverge slowly from each other. It is due to the fact that the low temperature expansion is plotted till finite orders in $e^{-\frac{\beta}{2r}}$, but as $\frac{\beta}{r}$ decreases, an increasing number of subleading contributions become significant.
• Figure 4: Free theory; here we plot the absolute value of the difference between the low temperature expansion \ref{['free small b/r']} and the Borel-Padé re-sum of high temp expansion for the $\log Z_{\rm free}$ divided by its low temperature expansion against $\frac{\beta}{r}$. This demonstrates the numerical accuracy of the agreement between the Borel-Padé resum of high temp expansion and low temperature expansion \ref{['free small b/r']} plotted in figure \ref{['fig 2']}.
• Figure 5: thermal expectation of energy for free theory in units of radius i.e., $r\langle E\rangle$; The Borel-Padé re-sum \ref{['P B rsum E']} of the high temperature expansion of the thermal expectation of energy, plotted as the orange curve, is compared against the low temperature expansion \ref{['E free']} (truncated till $l=n=10$) given by the blue curve. The figures inside the boxes magnify the small $\frac{\beta}{r}$ regions for the corresponding graphs. The agreement between the orange and blue curves is observed over a finite range of $\frac{\beta}{r}$. The limited domain of validity of the Borel-Padé resummation at large $\frac{\beta}{r}$ causes the two curves to deviate significantly from each other. At small $\frac{\beta}{r}$(see the figure inside boxes), the truncated low temperature expansion starts accumulating error, as the higher order terms become significant, though Borel-Padé resum for the high temperature expansion works very well in this regime.