# Dense loops, supersymmetry, and Goldstone phases in two dimensions

Jesper-Lykke Jacobsen, Nicholas Read, Hubert Saleur

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Jesper-Lykke Jacobsen, Nicholas Read, Hubert Saleur. Dense loops, supersymmetry, and Goldstone phases in two dimensions. Physical Review Letters, 2003, 90, pp.090601. 10.1103/Phys-RevLett.90.090601. hal-00002307

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# Dense Loops, Supersymmetry, and Goldstone Phases in Two Dimensions

J. L. Jacobsen,¹ N. Read,² and H. Saleur³

¹Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, Bâtiment 100, F-91405 Orsay, France

²Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120

³Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089

(Received 2 May 2002; published 4 March 2003)

Loop models in two dimensions can be related to $O(N)$ models. The low-temperature dense-loops phase of such a model, or of its reformulation using a supergroup as symmetry, can have a Goldstone broken-symmetry phase for $N < 2$. We argue that this phase is generic for $-2 < N < 2$ when crossings of loops are allowed, and distinct from the model of noncrossing dense loops first studied by Nienhuis [Phys. Rev. Lett. 49, 1062 (1982)]. Our arguments are supported by our numerical results, and by a lattice model solved exactly by Martins et al. [Phys. Rev. Lett. 81, 504 (1998)].

DOI: 10.1103/PhysRevLett.90.090601

PACS numbers: 05.50.+q, 05.20.–y

It has long been understood that theories of an $N$-component scalar field $\phi$ with $O(N)$ symmetry in Euclidean space of dimension $d$ can be related to statistical-mechanics models in which configurations of loops are given Boltzmann weights depending on their lengths and intersection properties [1]. In the $N \to 0$ limit, unwanted closed loops vanish [1], and the usual self-avoiding walks (polymers) are related to the generic critical point in the $O(N \to 1)$ theory. The $N \to 0$ limit can be avoided by using a model with a Lie superalgebra as symmetry (“supersymmetry”) [2]. Other values of $N$ are also of interest. In $d=2$, exact results based on lattice models [3] show that there also exists a massless low-temperature (low-$T$) regime of the model, for $-2 < N \le 2$. The properties of this phase—in which the loops become dense—are well understood using Coulomb gas techniques [3].

In this Letter, we revisit the low-$T$ phase of the $O(N \le 2)$ models in $d=2$. The supersymmetry approach, which can be generalized to all integers $N$, makes it clear that for $N < 2$ there should be a massless phase described by the Goldstone modes of the broken $O(N)$ or supersymmetry, with very simple scaling dimensions. We argue that the usual dense-loops phase [3], in which loops never intersect, is not this Goldstone phase (in contrast to earlier expectations [4]): In fact, it is not generic, and an arbitrarily weak perturbation that allows loop intersections causes a crossover to the generic Goldstone phase for $-2 < N < 2$. We present numerical results that support our interpretation. There is also a soluble model that appears to be in the Goldstone phase [5].

The generic continuum $O(N)$-invariant action $S$ for the scalar field $\phi$ includes the interaction term $-\lambda(\phi \cdot \phi)^2$ (by convention the Boltzmann weight is $e^S$). In terms of loops, this is the model introduced by de Gennes [1], in which the loops can cross, but for $\lambda > 0$ these crossings are disfavored. It has a second-order transition for $d > 2$, but for $d=2$ a transition occurs only for $N \le 2$; for $N > 2$, there is only the high-$T$ or massive unbroken-symmetry phase. (For $N$ not a positive integer, the theory is supposed to be defined by analytic continuation of its perturbation expansion.)

For the $N$ integer, this can be made rigorous by the use of supersymmetry, as we describe below.) When $-2 \le N \le 2$, the phase transition is second order, and the associated critical exponents have been conjectured by Nienhuis [3] and others [6]. For $N < -2$, the transition is expected to be first order.

The existence of the transition for $N \le 2$ implies the existence of a low-$T$ phase. From the point of view of the generic $O(N)$ $\phi^4$ theory, at low $T$ the symmetry is broken to $O(N-1)$ according to Landau theory, and the low-$T$ phase would be expected to be a Goldstone phase with massless excitations, described by a nonlinear sigma model with target manifold $O(N)/O(N-1) \cong S^{N-1}$, a sphere. For $N=2$, the $O(2)$ symmetry is not really broken, but there is a power-law phase with continuously varying exponents, the Berezinskii-Kosterlitz-Thouless (BKT) phase [7]. For $N < 2$, use of the perturbative beta function in the $S^{N-1}$ sigma model, defined by analytic continuation in $N$, implies that the coupling between the Goldstone modes in the sigma model becomes weak at large length scales [8], such as Goldstone phases in higher $d$. This yields a low-$T$ phase with simple forms for the scaling dimensions. This is possible in $d=2$ for $N < 2$ because the Mermin-Wagner theorem no longer applies when the continuous $SO(N)$ symmetry cannot be realized as unitary operations on a vector field. We discuss this theory more fully below, using supersymmetry.

In contrast, Refs. [3,9] defined a particular lattice model of strictly noncrossing loops. This was done by truncating the high-$T$ expansion of a lattice version of the $\phi^4$ theory as above [9], in order to simplify calculations of critical properties while hopefully remaining in the same universality class. The model has Boltzmann weight

$$e^S \equiv \prod_{\langle ij \rangle} (1 + K \vec{S}_i \cdot \vec{S}_j), \quad (1)$$

where $i, j$ are vertices on the honeycomb graph, $\langle ij \rangle$ denotes its edges, and the variables $\vec{S}_i$ are $N$-component fixed-length spins, $\vec{S}_i^2 = N$. The partition function can be evaluated as $Z = \sum N^L K^E$, where the sum is over graphs
consisting of self-avoiding mutually avoiding loops, $L$ is the number of loops, $E$ the number of edges they occupy, and $K \sim 1/T$. The critical $K$ for this model is known exactly, $K_c = (2 + \sqrt{2-N})^{-1/2}$ for $N \le 2$ [3]. The transition is second order for $-2 \le N \le 2$ only, and the critical exponents at this critical point (known as *dilute loops*, or as self-avoiding walks for $N \to 0$) are known by a variety of techniques [3,6]. The low-$T$ region for each $-2 < N < 2$ is attracted to a unique massless phase (that varies with $N$) for all $K_c < K < \infty$. This phase has nontrivial scaling dimensions, which are again known [3,10]. It is known as *dense loops* (or as dense polymers for $N \to 0$). In the limit $N \to 2$, this phase coincides with the critical endpoint of the BKT phase (which appears in the $N=2$ model for $K \ge K_c$). For $N < -2$, we expect that the dense-loops phase is massive.

Both the critical and dense-loops phases are believed to be universal as continuum phases, in the sense that they are independent of the choice of lattice used in defining them. Recent work [11] on the $T=0$ limits of certain lattice models identifies other massless phases of nonintersecting *fully packed* loops, that depend on the lattice used. We are not concerned with these here, nor with the transition for $N > 2$ in the region $|K| > 1/N$, where the Boltzmann weight ($1$) can be negative [12].

From the critical exponents for dilute loops, it is natural to hope [3], that they are in the same universality class as the transition in the generic $O(N) \phi^4$ theory. The idea is that any weak repulsion of the lines leads to a crossover to the infinitely repulsive limit in which the loops never cross, so that the latter fixed point governs the transition for all $\lambda > 0$. We know of no reason to doubt this. As a check, we can reintroduce intersections of the lines in the dilute critical theory. The graphs of the expansion of the generic $O(N)$ model that are not present in the truncated model would involve, in particular, crossings of the loops, or multiple occupancies of edges. In the continuum description, the most relevant (and most generic) of these is where two lines cross. This may be included as a perturbation of the dilute critical theory by adding to the action the integral of a multiple of the corresponding scalar operator, the so-called four-leg operator, which has conformal weight [6] $h_4 = g - (g-1)^2/(4g)$, where we parameterized $N = -2 \cos \pi g$, $g \in [1, 2]$. We see that this operator is marginal at $N=2$ and irrelevant (i.e., $h_4 > 1$) for $-2 \le N < 2$. This strongly suggests that allowing more general interactions or configurations of loops in the lattice model would not change the universality class, at least when the perturbations are small, and, hence, that the *dilute* critical point [3] is indeed generic.

The model of nonintersecting loops was obtained by truncating the high-$T$ expansion, and this would be expected not to change the behavior for sufficiently high $T$ (small $K$); apparently, this remains true down to $T = T_c$. However, this leaves open the possibility that the low-$T$ behavior of the models is different, which we will now examine. The dense-loops phase is described by a theory

similar to that of the dilute case, with the same parametrization of $N$ and $h_4$ as above, but with $g \in [0, 1]$. One sees that the four-leg operator is now relevant in the whole region $-2 < N < 2$. As $N \to -2$, $h_4 \to -\infty$. Since the four-leg perturbation is relevant in the low-$T$ phase, it is *dangerously* irrelevant in the critical theory.

It is thus clear that the *dense-loops* phase is nongeneric from an $O(N)$-model point of view. In $d > 2$, de Gennes' model appears generic, as the topological effect of strict nonintersection, present in $d=2$, is absent (though in $d=3$ there remains the topology of knots and linking instead). Even a model in $d > 2$ of strict nonintersection, when reduced to $d=2$ by confining $d-2$ dimensions to finite intervals, gives an effective model in which intersections still occur. There is also a symmetry interpretation for the nongenericity: The dense phase of strictly nonintersecting loops has a larger symmetry, $U(N)$ instead of $O(N)$, which is based on the possibility of consistently orienting all the loops in the partition sum [3,8]. Though this symmetry may be broken by the boundary conditions, the massless dense-loops phase appears to be most easily understood in terms of a nonlinear sigma model with this larger symmetry [8]. The four-leg operator is a symmetry-breaking perturbation, that reduces the symmetry to the generic $O(N)$. We note that the importance of the self-crossings has been observed in the related context of Lorentz lattice gases [13].

We now argue that the flow induced by the four-leg perturbation leads to the generic Goldstone phase of the $\phi^4$ theory, and begin by examining the structure of this phase, before giving numerical evidence that such a flow does in fact occur.

The loop (high-$T$) expansion of the $O(N)$ model can be reproduced by considering instead a model with $OSp(m|2n)$ symmetry, with $N = m - 2n$; in particular, each closed loop incurs a factor of $N$ [2,8]. The model then makes complete sense even for $N$ nonpositive (but $N$ must be an integer). The phase diagram, beta functions, and scaling dimensions are the same for all $m$ at fixed $N$ (though multiplicities of operators may vanish for $m$ small [8]). The Goldstone phase should thus be described by a sigma model with target space $OSp(m|2n)/OSp(m-1|2n) \cong S^{m-1|2n}$, a supersphere, and a single coupling constant $g_\sigma$ [8,14]. The perturbative beta function is, to leading order, $\beta \equiv dg_\sigma/d\ln L \propto (N-2)g_\sigma^2$ so, for $N < 2$, the model flows to weak coupling for $g_\sigma \ge 0$ (which is the expected physical sign). Again, the Mermin-Wagner theorem does not apply here, and symmetry breaking is allowed for $N < 2$. The weak-coupling fixed point is a theory of free massless scalars, with $m-1$ bosonic and $2n$ fermionic components, so is conformal with central charge $c = m-1-2n = N-1$, as it would be for $S^{N-1}$. There will be logarithmic corrections due to the marginally irrelevant coupling $g_\sigma$. We emphasize that, in the sigma model, $OSp(m|2n)$ symmetry does not allow any other marginal or relevant couplings to be added to the action,
so that this fixed point is robust, unlike the dense-loops
theory above.

Explicitly, the target manifold can be parametrized by
commuting coordinates $x_{i=1,...,m}$, and anticommuting co-
ordinates $\eta_{j=1,...,2n}$, subject to the constraint $\sum_{i=1}^{m} x_i^2 + \sum_{j=1}^{n} \eta_{2j-1} \eta_{2j} = 1$. The action of the sigma model is

$$
S = - \frac{1}{g_{\sigma}} \int d^2 r \left[ \sum_{i=1}^{m} (\partial_{\mu} x_i)^2 + \sum_{j=1}^{n} \partial_{\mu} \eta_{2j-1} \partial_{\mu} \eta_{2j} \right]. \quad (2)
$$

The simplest example is $m = n = 1$ ($N = -1$), which corresponds to the $S^{0|2}$ supersphere. The constraint can be solved as $x_1 = 1 - \frac{1}{2}\eta_1\eta_2$. After elimination of $x_1$ and some rescaling, one finds

$$
S = - \int d^2 r \left[ \partial_{\mu} \eta_1 \partial_{\mu} \eta_2 - \frac{g_{\sigma}}{2} \eta_1 \eta_2 \partial_{\mu} \eta_1 \partial_{\mu} \eta_2 \right]. \quad (3)
$$

At long wavelengths, the coupling $g_σ$ renormalizes to
wards zero, and the theory becomes free symplectic fer-
mions, with central charge $c = -2$, as claimed. In
general, for $N < 1$, the partition function of the fixed-
point theory vanishes, $Z = 0$. Similar behavior is well
known in the dense-polymer ($N = 0$) phase.

The supersymmetry analysis is valid for *N* integer. For
general *N*, we are forced again to interpolate formally.
The availability of the supersymmetry results greatly
strengthens our confidence in this procedure and in the
existence of the Goldstone phase for all *N* < 2.

Both at the dilute critical point and in the dense-loops
phase, $c = 1 - 6(g-1)^2/g$ in the parametrization above.
For dense loops, $g \in [0, 1]$, this coincides with $c = N-1$
for $N=1, 2$. The $N=2$ case is the BKT phase with $c=1$
throughout, and not of interest here. We claim that, for all
$-2<N<2$, the perturbed dense-loops phase flows to
the Goldstone phase, which is a distinct massless phase
even when it has the same $c$. For $1<N<2$, $c$ decreases
during the flow, but for $-2<N<1$ it increases. This
latter behavior would not violate Zamolodchikov's $c$
theorem, as the theories involved are nonunitary. In par-
ticular, the $N=0$ ($c=-2$) dense-polymer theory should
flow to a $c=-1$ free theory. Note that for $N=1$ the
partition function is $Z=1$ (modulo nonuniversal con-
stants) in both the dense-loops and Goldstone phases, a
consequence of $c=0$.

Some particularly interesting operators in loop models
in general are the *k*-leg operators (also known as fuseaux
or watermelon operators), insertions of which give the
probability that *k* lines terminate at the same point. In the
general *O(N)* point of view, the leading such operator
would be represented by *k* insertions of the field *φ* at the
same point (with no derivatives), and must necessarily be
in a totally symmetric rank-*k* tensor representation of
*O(N)*, which can be assumed to be traceless on all pairs
of indices. For the *OSp(m|2n)* formulation, this becomes
a supersymmetric tensor, that vanishes when contracted
using the same invariant bilinear form as in the constraint
in the sigma model. It is useful to choose the index values

of the set of tensors in a correlation function to occur in
distinct pairs, to force the lines in the loop model to
connect the positions of the operators in some specified
way. This is possible if *m* is sufficiently large, a typical
situation in applications of supersymmetry.

At the dilute critical point and in the dense-loops
phase, there are nontrivial scaling dimensions $X_k$ (or con-
formal weights $h_k = X_k/2$) for the k-leg operators, which
are known exactly [6,10]; $h_4$ was given earlier. (In these
phases, the symmetric-tensor k-leg operators are in gen-
eral degenerate with others [8], and in the dense-loops
phase, we assume there is a singlet four-leg operator,
which we used as the perturbation above; it is the most
relevant invariant scalar operator.) In the Goldstone
phases, on the other hand, the symmetric tensors corre-
spond to functions such as spherical harmonics on the tar-
get space, and their scaling dimensions vanish as $g_σ → 0$.
Since the symmetry is broken for $N < 2$, the precise form
of each operator depends on which components are
chosen. If the expectation of the field is $x_1 = 1$, all other
$x_i$ and $η_j = 0$, then any component of the k-leg operator
can be rewritten in terms of the (rescaled, free) $m-1$
$x_i$'s and the $2n$ $η_j$'s, and the correlation functions will
contain various logarithmic factors depending on which
components are chosen. Clearly, the scaling dimensions
of the leading k-leg operators are all zero, and there will
be subleading operators with integer conformal weights
(the precise form of correlation functions may require
more attention to the limit $g_σ → 0$). When the dense-
loops phase is perturbed by the four-leg operator, it
should be possible to see the effective scaling dimensions
of the k-leg operators (say, in two-point functions) cross
over to zero, for all $k$.

Numerical transfer matrix calculations were done on a
square-lattice cylinder of circumference *L*. The edges of
the lattice are packed with one line per edge; two lines
can meet at each vertex in three ways, the weight being 1
for a noncrossing and *w* for a crossing. The natural
*O(N)*-invariant boundary condition for a periodic system
is that the fields *φ* are periodic, both for Bose and Fermi
components. This implies that a loop that winds the
periodic direction is given the same factor *N* as a

<figure><img src="image_2.png" /><figcaption>FIG. 1. Measures of the central charge vs *w* for *N* = 0. The
conjectured value for *L* → ∞ is *c* = −1 for all *w* > 0.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>FIG. 2. The k-leg exponents $X_k = 2h_k$ vs $w$ for $N = 0$.</figcaption></figure>

topologically trivial loop. In a conformally invariant system, $c$ can be extracted from the scaling of the free energy per vertex, $f(L) = f(\infty) - \pi c/(6L^2)$. We therefore use this formula to extract the effective value of $c$ in a crossover, using $f(L)$ at pairs of values of $L$. Results for $N=0$ (see Fig. 1) show a clear trend away from $c=-2$, and are consistent with a slow approach to $c=-1$ for $L \to \infty$.

For the k-leg scaling dimensions, we use the lowest transfer matrix eigenvalues $E_k$ in each sector with $k$ lines propagating along the cylinder. From conformal invariance [15], we define effective values of $X_k$ at finite $L$ from $E_k - E_0 = 2\pi X_k/L$; see Fig. 2 ($k=2$ is not included as, for $N=0$ in our model, $E_2 = E_0$ for all $L$ and $w$). The results are consistent with $X_k = 0$ in the fixed-point theory. In a situation with a marginally irrelevant operator, one expects [15] corrections to scaling to be $O(1/L \ln L)$, which presumably explains the slow convergence.

A useful check of our analysis is obtained by modifying the boundary conditions, in a similar way as in Ref. [8]. Making the Fermi components of the $\phi$ fields obey antiperiodic boundary conditions gives a noncontractible (winding) loop the usual weight $w_e = m - 2n = N$ if its winding number is even, and $w_o = m + 2n = N + 4n$ if it is odd. In the Goldstone phase, this will give the $n$ pairs of symplectic fermions the antiperiodic boundary condition, while leaving the bosonic scalar fields periodic. In the formula for $f(L)$, $c$ will be replaced by $c - 24h_{tw}$, where $h_{tw}$ is the lowest conformal weight in the sector with boundary conditions twisted as described. We expect $c - 24h_{tw} = m + n - 1$, which can be understood as the contribution of the twist operator of weight $h = -1/8$ for each symplectic fermion pair. Thus, we expect

$$c - 24h_{\text{tw}} = N - 1 + 3n = \frac{3w_o + w_e}{4} - 1, \quad (4)$$

for general $N, n$. This appears to be confirmed numerically for $N = w_e = 0$ and several values of $w_o$ in Fig. 3. Notice how this differs from the usual loop gas behavior ($w=0$), where $c - 24h_{tw} = 1 - 6 \arccos^2(w_o)/(\pi^2 g)$ [16].

For $N < -2$, we believe that the dense-loops phase is massive, and hence flows to the generic Goldstone

<figure><img src="image_4.png" /><figcaption>FIG. 3. Measures of $c - 24h_{tw}$ vs $w$ for $N = 0$. Dashed lines show the prediction of Eq. (4).</figcaption></figure>

phase only when $w$ is greater than a positive critical value.

Finally, it is important that a version of the OSp($m|2n$)-invariant model on the square lattice is integrable at $w = (2-N)/4$ and can be solved by the Bethe ansatz [5]. The model is massless for $N < 2$, and there is strong evidence that the central charge is $c = N - 1$, and that the lowest scaling dimension, other than that of the identity operator, is zero; the authors conjecture that these are the exact values, but do not identify the phase with the Goldstone phase.

In conclusion, we have argued that crossing of loops is a relevant perturbation in the dense-loops low-$T$ phase for $-2 < N < 2$, and that the long-distance behavior is governed by the Goldstone phase instead, by analytical, numerical, and exact approaches.

This work was supported by the NSF under Grant No. DMR-98-18259 (N. R.), and by the DOE (H. S.).

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