Corrections to finite-size scaling in two-dimensional O(N) sigma-models

TL;DR

This paper analyzes finite-size scaling corrections in two-dimensional O(N) sigma-models at N=∞, revealing how these corrections depend on the type of action and discussing complex behaviors in models with four-spin interactions.

Contribution

It provides a detailed calculation of leading finite-size correction terms for a broad class of O(N) sigma-models, highlighting differences based on action improvements.

Findings

Corrections behave as (f(z) log L + g(z))/L^2, with f(z) vanishing for Symanzik improved actions.

On-shell improved actions do not exhibit the same correction behavior as Symanzik improved actions.

Models with four-spin interactions display more complex finite-size correction behaviors.

Abstract

We have considered the corrections to the finite-size-scaling functions for a general class of $O(N)$ $\sigma$-models with two-spin interactions in two dimensions for $N=\infty$. We have computed the leading corrections finding that they generically behave as $(f(z) \log L + g(z))/L^2$ where $z = m(L) L$ and $m(L)$ is a mass scale; $f(z)$ vanishes for Symanzik improved actions for which the inverse propagator behaves as $q^2 + O(q^6)$ for small $q$, but not for on-shell improved ones. We also discuss a model with four-spin interactions which shows a much more complicated behaviour.

Equations

$$H\,=\,N\beta\sum_{xy}J(x-y)\,\mbox{\boldmath$\sigma$}_{x}\cdot\mbox{\boldmath$\sigma$}_{y}\;\;.$$

$$\beta =\frac{1}{LT}\sum _{p_x,p_y}\frac{1}{w(p)+m_{L,T}^2}\equiv {\mathcal{I}}_{L,T}(m_{L,T}^2)$$

$${\mathcal{I}}_{L,T}(m_{L,T}^2)=\frac{1}{2\pi }\log L+F_0(z;\rho )+{\Lambda }_0$$

$$+\frac{z^2}{4\pi L^2}(3{\delta }_1+4{\delta }_2)\log L+\frac{1}{L^2}{F}_1(z;\rho )$$

$${\Lambda }_0=\int \frac{d^{2}p}{(2\pi {)}^2}\left(\frac{1}{w(p)}-\frac{1}{{\hat{p}}^2}\right)$$

$$w(q)=q^2+{\delta }_1\sum _{\mu }{q}_{\mu }^4+{\delta }_2(q^2{)}^2+O(q^6)$$

$${\left(\frac{\mu (\infty )}{\mu (L)}\right)}^2=\frac{32}{x^2}{e}^{-4\pi F_0(x;\infty )}\times$$

$$\left(1+\frac{{\Delta }_1(x)}{L^2}\log L+\frac{{\Delta }_2(x)}{L^2}+O(L^{-4}\log L)\right)$$

$${\Delta }_1(x)=(3{\delta }_1+4{\delta }_2)(32e^{-4\pi F_0(x;\infty )}-x^2)$$

$${\Delta }_2(x)=\frac{2{\pi }^2}{3}{\delta }_1-4\pi ({\delta }_1+{\delta }_2)x+O(x^2)$$

$$\frac{\mathcal{O}(L,T)}{\mathcal{O}(\infty )}=1+O(z^p{e}^{-z})+O({\rho }^p{z}^p{e}^{-\rho z})$$

$${\xi }^{(2)}(L)={\xi }^{(2)}(\infty )+O(1/L^2)$$

$${\left({\tilde{\xi }}^{(2)}(L,T)\right)}^2=$$

$$\frac{1}{4\hat{G}(0)}\sum _{i=1-L/2}^{L/2}\sum _{j=1-T/2}^{T/2}(i^2+j^2)G(i,j)$$

$$H\,=\,N\beta\sum_{\langle xy\rangle}\left[(1-r)\mbox{\boldmath$\sigma$}_{x}\cdot\mbox{\boldmath$\sigma$}_{y}+{r\over 2}\left(\mbox{\boldmath$\sigma$}_{x}\cdot\mbox{\boldmath$\sigma$}_{y}\right)^{2}\right]$$

$${\left(\frac{\mu (\infty )}{\mu (L)}\right)}^2=\frac{32}{x^2}{e}^{-4\pi F_0(x;\infty )}\times$$

$$\left(1+\frac{{\Delta }_1(x)}{L^2}\log L+\frac{{\Delta }_2(x)}{L^2}+r\frac{{\Delta }_3(x,L)}{L^2}\right)$$

$${\Delta }_3(x,L)={\overline{\Delta }}_3(x)\frac{(\log L+2\pi F_0(x;\infty ){)}^2}{\log L+2\pi F_0(x;\infty )-2r}$$

$$\frac{1}{L^2}[b_{00}+x(b_{11}\log L+b_{10})$$

$$+x^2(b_{22}{\log }^{2}L+b_{21}\log L+b_{20})+O(x^3)]$$

$$P_n(\log L)+Q_n(\log L)/L^2+O(L^{-4}{\log }^{n}L)$$

Full text

We have studied the finite-size-scaling (FSS) behaviour of $O(N)$ models for large $N$. The purpose of the study was twofold: first of all we wanted to understand how mean values computed on a finite lattice of size $L$ converge to their infinite-volume values; moreover we wanted to understand the functional form of the correction to FSS. For this purpose we have considered a generic hamiltonian of the form

[TABLE]

The coupling is required to be local so that the Fourier transform $\hat{J}(q)$ is continuous. Moreover, to have the correct continuum limit we require $\hat{J}(q)-\hat{J}(0)$ to have a unique zero in the Brillouin zone at $q=0$ with $\hat{J}(q)-\hat{J}(0)\sim q^{2}$ for $q\to 0$.

We have considered the $N\to\infty$ limit of the model defined by (1) at fixed $\beta$ in a finite box of size $L\times T$ and in a strip of width $L$, in both cases using periodic boundary conditions in the finite direction(s), and we have studied the FSS limit $L\to\infty$, $T\to\infty$ with $T/L\equiv\rho$ fixed ($\rho=\infty$ for the strip) and $m(L,T)L$ fixed where $m(L,T)$ is (some) mass scale.

On a finite lattice the theory is parameterized by a mass $m_{L,T}$ related to $\beta$ by

[TABLE]

where $w(p)=2(\hat{J}(q)-\hat{J}(0))/\hat{J}^{\prime\prime}(0)$ and the sum extends over the points $(p_{x},p_{y})=2\pi(n_{x}/L,n_{y}/T)$, $0\leq n_{x}\leq L-1$ and $0\leq n_{y}\leq T-1$. When the lattice is infinite in one (or both) direction(s) the sum is substituted by the corresponding integral.

The basic tool for the computation of the FSS functions is the expansion in the FSS limit of the r.h.s of (2). We find a general result of the form

[TABLE]

where $z\equiv m_{L,T}L$ and the neglected terms are of order $O(\log L/L^{4})$. We want to make a few general remarks on this result:

The leading logarithm is universal and it is indeed connected to the leading term of the $\beta$-function. 2. 2.

$F_{0}(z;\rho)$ is a universal function (i.e. independent on the specific $J(x)$) which depends only on the modular parameter $\rho$. 3. 3.

In the leading (for $L\to\infty$) term of the expansion, the only dependence on $J(x)$ is due to $\Lambda_{0}$ given by

[TABLE]

where $\hat{p}^{2}=4\sum_{i}\sin^{2}(p_{i}/2)$. 4. 4.

The $\log L/L^{2}$ term is not universal but depends only on the small-$q^{2}$ behaviour of $w(q)$; indeed it depends on $\delta_{1}$ and $\delta_{2}$ defined by

[TABLE]

Given the expansion (3) it is easy to obtain the FSS functions (including the corrections of order $O(1/L^{2})$) of various quantities. If one considers the true mass gap $\mu(L)$ on a strip of width $L$ (this quantity cannot obviously be defined on a $L\times T$ lattice) one obtains in terms of $x=\mu(L)L$

[TABLE]

where $\Delta_{1}(x)$ and $\Delta_{2}(x)$ have been explicitly computed. The leading contribution is in agreement with the result by Lüscher [1]. The function $\Delta_{1}(x)$ is very simple: explicitly

[TABLE]

and thus this term vanishes for Symanzik actions for which $\delta_{1}=\delta_{2}=0$, but not for on-shell improved actions for which $\delta_{1}=0$ but $\delta_{2}$ is arbitrary. In particular it does not vanish for the so-called “perfect” laplacian [2, 3] unless $\kappa=4$. Notice moreover that in the perturbative (PT) regime this term is proportional to $x^{2}$ and thus appears only in two-loop PT computations. For the function $\Delta_{2}(x)$ we obtain in the PT limit $x\to 0$

[TABLE]

Thus the tree-level correction vanishes for $\delta_{1}=0$ (the so-called on-shell improved actions) independently of $\delta_{2}$. The vanishing of the second (one-loop) term requires instead $\delta_{1}+\delta_{2}=0$: this is verified by Symanzik actions for which $\delta_{1}=\delta_{2}=0$ (and in this case the one-loop corrections behave as $1/L^{4}$) but not by generic on-shell improved actions. Notice however that this cancellation does not happen for the complete function and thus the corrections to FSS always behave as $1/L^{2}$ (it is for instance easy to see that terms of order $x^{2}$ do not cancel in $\Delta_{2}(x)$ even for $\delta_{1}=\delta_{2}=0$).

Completely analogous formulae can be derived for other observables on the strip. In particular one can consider the inverse second-moment correlation length: the result is similar to that of the true mass gap $\mu$, the only difference being the function $\Delta_{2}(x)$. However, up to $O(x^{2})$ its expansion is still given by (8).

Similar results are valid on a torus. In this last case we take as variable $z\equiv m_{L,T}L$, $m_{L,T}$ being the mass appearing in the gap equation (2) which is the inverse of the second-moment correlation length $\xi^{(2)}(L,T)$.

It is interesting to consider the limit $z\to\infty$ (infinite-volume limit): in this case we find generically, as expected,

[TABLE]

i.e. the infinite-volume limit is essentially reached exponentially ($p$ is an exponent which depend on the specific observable ${\cal O}$). A notable exception is the second-moment correlation length which has corrections of order $O(1/z^{2})$. The reason is essentially due to the fact that the standard definitions of $\xi^{(2)}$ on a finite volume are such that, for $L\to\infty$,

[TABLE]

These $1/L^{2}$ terms are the cause of the $1/z^{2}$ corrections in the FSS functions. It is however possible to define a correlation length which does not suffer from this problem: define $\tilde{\xi}^{(2)}(L,T)$ by (assume for simplicity $L$ and $T$ even)

[TABLE]

for any two-point function $G(x)$ (here $\hat{G}$ denotes the Fourier transform of $G$). It is easy to see that $\tilde{\xi}^{(2)}(L,T)\to\xi^{(2)}(\infty)$ exponentially and that, for $z\to\infty$, the corrections are of order $O(e^{-\rho z/2},e^{-z/2})$.

We have also considered a more general class of models of the form

[TABLE]

with nearest-neighbour interactions only (this is the only case which can be easily solved in the large-$N$ limit [4]). Here $r$ is a free parameter which interpolates between the standard $N$-vector hamiltonian ($r=0$) and the $RP^{N-1}$ standard hamiltonian ($r=1$). We wanted indeed to understand if the functional form of the corrections to FSS we have found for generic two-spin interactions is general or not: one reason to believe that two-spin interactions can give rise to a simpler behaviour is the fact, for instance, that the $\beta$-function for models defined by (1) has only the leading term, all others vanishing. Instead, for $r\not=0$, the hamiltonian (12) has a $\beta$-function which is non vanishing to all orders in $1/\beta$. And indeed for the models defined by (12) we have found a more complicated behaviour. Considering for instance the mass gap in a strip we have found

[TABLE]

with corrections of order $O(L^{-4}\log L)$. Here

[TABLE]

As expected the FSS function is universal ($r$-independent). The new feature is the appearance of the term (14): in the limit $L\to\infty$ at $z$ fixed, this new term behaves as $\log L/L^{2}$ and thus the leading correction has the same functional form of the case we considered before; however in this case beside $1/L^{2}$ corrections there are also $1/(L^{2}\log L)$ terms and so on. The origin of these terms can be understood if we expand them in the PT limit, $z\to 0$ with $L$ fixed (notice that here we are making an illicit exchange of limits). Since $F_{0}(x;\infty)=1/(2x)(1+O(x))$, we obtain an expansion on the form

[TABLE]

which is indeed the pattern one expects from multiloop sums. An $n$-loop sum will generically behave as

[TABLE]

where $P_{n}(y)$ and $Q_{n}(y)$ are $n$-degree polynomials. Of course to obtain the correct corrections to FSS one must resum the PT expansion in order to be able to exchange the limits (the PT expansion is obtained for $z\to 0$ at $L$ fixed and we want to obtain the behaviour for $L\to\infty$ for small, but fixed $z$). Our calculation shows that these infinite series of logarithms resum giving rise to corrections which still behave as $\log L/L^{2}$. Of course, it would be interesting to know if this is still true for generic models ($N=\infty$ models usually have a behaviour which is simpler than that of generic ones): indeed one could be worried by the possibility that the terms in $\log L/L^{2}$ resum non trivially to give corrections with a non trivial power, i.e. corrections to FSS behaving as $L^{p}/L^{2}$, $p>0$. Unfortunately we do not have any answer to this question.

A detailed presentation will appear in a forthcoming paper [5].