The partition function of interfaces from the Nambu-Goto effective string theory

TL;DR

The paper addresses describing interface fluctuations with a noncritical Nambu-Goto string. It derives an exact toroidal partition function via covariant quantization and modulus integration, and demonstrates that it reproduces known two-loop results from functional integrals. It then confirms, by comparison with high-precision Monte Carlo data for the 3D Ising model, that NG theory provides an accurate description for large interfaces, with deviations at small areas pointing to additional degrees of freedom or alternative string models. The work strengthens Nambu-Goto as a practical effective description for interfaces and informs the ongoing search for the correct string dual of gauge theories.

Abstract

We consider the Nambu-Goto bosonic string model as a description of the physics of interfaces. By using the standard covariant quantization of the bosonic string, we derive an exact expression for the partition function in dependence of the geometry of the interface. Our expression, obtained by operatorial methods, resums the loop expansion of the NG model in the "physical gauge" computed perturbatively by functional integral methods in the literature. Recently, very accurate Monte Carlo data for the interface free energy in the 3d Ising model became avaliable. Our proposed expression compares very well to the data for values of the area sufficiently large in terms of the inverse string tension. This pattern is expected on theoretical grounds and agrees with previous analyses of other observables in the Ising model.

Figures (4)

• Figure 1: The mapping of the toroidal string world-sheet, of modular parameter $\tau$, into the target space is organized in many distinct sectors, labeled by the integers $w_i$ and $m_i$ (see the text). By selecting the sector with (say) $w_1=1$ and $m_2=1$ we are considering the fluctuations of an extended interface, which is a torus because of the target space periodicity.
• Figure 2: Consider an embedding of the world-sheet torus into the target space $T^2$ aligned along the directions $x^1,x^2$, characterized by the wrapping numbers $(m^1,w^1)$ and $(m^2,w^2)$. It corresponds, in the covering space of this $T^2$, to a parallelogram defined by the vectors $\vec{w} = (w^1 L_1,w^2 L_2)$ and $\vec{m} = (m^1 L_1,m^2 L_2)$, whose area is $\vec{w}\wedge \vec{m} = L_1 L_2(w^1 m^2 - w^2 m^1)$.
• Figure 3: On the left, (some of) the coverings corresponding to the chosen sector of the partition function are depicted. Beside the one which exactly corresponds to the fundamental cell of the target space $T^2$, i.e. the case with $(m^1,w^1)=(0,1)$ and $(m^2,w^2)=(1,0)$ there are "slanted" ones corresponding to generic values of $m^1$. The generators $S$ and $T$ of the world-sheet modular group, which act by $\mathrm{SL}(2,\mathbb{Z})$ matrices as indicated in eq. (\ref{['bos8']}), map these coverings to different ones with the same area. For instance, on the right, we draw the $S$- and $T$-transform of the "fundamental" covering discussed above (the solid one in the leftmost drawing).
• Figure 4: Two sets of Monte Carlo data for the interface free energy data provided in nuovomc are compared to our theoretical predictions following from eq.s (\ref{['Fc0']},\ref{['bos21']}), represented by the solid red line for the data set 1 and by the dashed blue line for data set 2. The only free parameter is the additive constant $\mathcal{N}$, corresponding to an overall normalization of the NG partition function, fitted to the data using the points to the right of the vertical grey dashed line (see the text for more details). The error bars in the MC data are visible only for the rightmost points, but they were kept into account in the fit. The sum over the level $m$ in eq. (\ref{['bos21']}) was truncated at $N=100$.