Exact logarithmic four-point functions in the critical two-dimensional Ising model

TL;DR

This paper derives an exact formula for boundary four-point connectivities in the critical 2D Ising model, revealing logarithmic singularities and validated by Monte Carlo simulations, advancing understanding of Logarithmic Conformal Field Theories.

Contribution

It provides the first exact formula for boundary four-point connectivities in the critical Ising model, highlighting logarithmic singularities and confirming predictions through simulations.

Findings

Exact boundary four-point connectivity formula derived

Logarithmic singularities identified in the solution

Monte Carlo simulations confirm the theoretical predictions

Abstract

Based on conformal symmetry we propose an exact formula for the four-point connectivities of FK clusters in the critical Ising model when the four points are anchored to the boundary. The explicit solution we found displays logarithmic singularities. We check our prediction using Monte Carlo simulations on a triangular lattice, showing excellent agreement. Our findings could shed further light on the formidable task of the characterization of Logarithmic Conformal Field Theories and on their relevance in physics.

Equations

$$Z=\sum _{\{s(x)\}}\prod _{⟨x,y⟩}[(1-p)+p{\delta }_{s(x),s(y)}],$$

$$h_{r,s}=\frac{[r(m+1)-sm{]}^2-1}{4m(m+1)},m\in \mathbb{R}.$$

$$⟨\prod _{i=1}^4{\varphi }_{1,3}(z_i){⟩}_{\mathbb{H}}={\left[\frac{z_{42}{z}_{31}}{z_{21}{z}_{43}{z}_{32}{z}_{14}}\right]}^{2h_{1,3}}F(\eta );$$

$$[2\eta (1-\eta ){]}^2{F}^{{}^{\prime \prime \prime }}-3(1-\eta +{\eta }^2)F^{\prime }+3(2\eta -1)F=0.$$

$$f(\eta )=\frac{p(\eta )E(\eta )+q(\eta )K(\eta )}{\sqrt{(1-\eta )\eta }},$$

$$R(\eta )=\frac{P_{(14)(23)}(\eta )}{P_t(\eta )}=A{\int }_0^{\eta }d{\eta }^{\prime }f({\eta }^{\prime })$$

$$R=1-{\epsilon }^{1/2}[{\mathrm{a}}_0+{\mathrm{a}}_1\epsilon +{\mathrm{a}}_2{\epsilon }^2(1+\mathrm{b}\log \epsilon )+\mathrm{O}({\epsilon }^3)].$$

Full text

Introduction –

Conformal symmetry in two dimensions BPZ has been of extraordinary usefulness to study classical statistical mechanics models at criticality since the 80’s. It has notably found also extensive applications in the quantum realm, spanning from gapless one-dimensional systems Haldane , quantum Hall effect MR and entanglement Entanglement . Two dimensional Conformal Invariant Quantum Field Theories (CFTs) and in particular Liouville theory are moreover the cornerstone of world-sheet geometry in string theory Polyakov . The simplest CFTs that capture the critical behaviour of lattice models and quantum spin chains are unitary. Moreover, when their Hilbert space splits into a finite number of representations of the conformal (or some larger) symmetry are usually termed rational. CFTs are classified according to their central charge $c$; for instance for a free massless boson $c=1$.

However unitary and rational theories are far from exhausting the physically relevant conformally invariant theories. In the beginning of the 90’s the groundbreaking discovery Cardy_perc of an exact formula for the crossing probability in critical percolation forced to analyze theories violating these two assumptions. Percolation is a simple stochastic process where bonds or sites on a lattice can be occupied independently with a certain probability. Since the partition function is not affected by finite-size effects, the central charge of a putative CFT describing critical percolation is zero BCM . The existence of a non-trivial formula for the crossing probability makes it a prominent example of a non-unitary CFT (the only unitary CFT with $c=0$ is trivial). For subsequent developments leading to the formulation of the Stochastic Lowener Evolution we refer to the reviews BB ; Cardy_SLE . At the same time, it was recognized that as far as the conditions of unitarity and rationality are relaxed, CFT correlation functions can display striking logarithmic singularities that are actually the signatures of intricate realizations of the conformal symmetry RSal ; G . The class of non-unitary and generally non-rational CFTs where these unconventional features show up, was christened Logarithmic CFTs. Such theories were promptly argued to play a fundamental role in the characterization of disordered systems in two dimensions Caux ; Cardy_99 ; GL , for instance by means of supersymmetry GLR ; RS .

With these motivations, Logarithmic CFTs were investigated in greater detail in the last decade, either from a purely algebraic point of view Ridout_rev , either constructing lattice regularizations RS ; Logmin ; RS1 ; DJS or generalizing the study of crossing formulas in critical percolation Watts ; Simmons ; Flores . Recently DV , it has also been suggested that an analytic non-unitary extension of Liouville theory Zam might describe the connectivity properties of critical bulk percolation and more generally the $Q$-state Potts model. The domain of applicability of Liouville theory in statistical mechanics is currently an important open problem see IJS ; PRS and also RS15 .

Despite these huge advances, a satisfactory understanding of Logarithmic CFTs is still a long way off. Moreover, it is fair to say that few examples of explicit logarithmic singularities have been found in familiar statistical models: notably only in percolation VJS ; FSK2 ($c=0$), dense polymers Sal_pol ; GK ; K00 ; see also Santa2 ; Vasseur_2016 for applications to disordered systems. An exact Coulomb gas approach to CFT correlation functions KF ; KF_1 ; KF_2 closely related to those considered in this letter reveals an infinite number of logarithmic cases. These arise from operator mixing as is the case here. These results suggest the possibility of logarithmic behavior in multiple Self Avoiding Random Walks (SAWs) KF .

In this paper we show how a logarithmic singularity due to operator mixing also arises in the context of arguably the best-known model of statistical mechanics: the two-dimensional Ising model; see Aldo_rev ; Giuseppe_book for a pedagogical introduction to the richness of the model. This is remarkable since it shows unambiguously that critical properties of Ising clusters are ruled by a Logarithmic CFT. Moreover the four-point function we consider here, a four point connectivity in the Fortuin and Kasteleyn (FK) representation of the Ising model, is a natural observable that can be easily simulated with Monte Carlo (MC) methods. Logarithmic singularities in Ising connectivities were also argued to exist in Cardy_revLog ; VJ_log , in this letter we demonstrate it explicitly using CFT.

Finally, our findings could shed further light on the extremely challenging problem of the characterization of Logarithmic CFTs and on their applications to physics.

Four points connectivities in the Ising model –

It is convenient to introduce the Ising FK clusters, starting from the ferromagnetic $Q$-state Potts model Wu ; The Ising case is recovered by setting $Q=2$. The model is defined on a finite simply-connected domain $\mathcal{D}$ of the plane, see Fig. 1, and the choice of the underlying lattice is irrelevant at the critical point. To introduce the notion of cluster connectivities, we should first recall the Fortuin-Kastelein (FK) representation FK . The $Q$-state Potts model is defined in terms of spin variables $s(x)$ taking $1,\dots,Q$ different values; its partition function can be expressed as a product over the lattice edges as

[TABLE]

where $p$ is a parameter related to the temperature and the product in (1) extends only to next-neighboring sites. Suppose then to expand such a product: Each term in the expansion can be represented graphically by drawing a bond between $x$ and $y$ if the factor $p\delta_{s(x)s(y)}$ is selected and leaving empty the bond if such a factor is absent. The set of non-empty bonds in each term of the expansion then defines a graph $G$ on the underlying lattice that is called FK graph. Such a graph might contain $N_{c}$ different connected components (including isolated points), dubbed FK clusters. Moreover, on each cluster the spin values are constrained to be the same because of the Kronecker delta in (1). Summing over their possible $Q$ values leads to the following rewriting of the $Q$-state Potts model partition function as a sum over graphs: $Z=\sum_{G}(1-p)^{\bar{n}_{b}}p^{n_{b}}Q^{N_{c}},$ where $n_{b}$ and $\bar{n}_{b}$ are the number of occupied and empty bonds in the graph $G$.

For arbitrary non-negative $Q$, the graph representation for $Z$ is a generalized percolation problem known as Random Cluster Model, where bonds occupied with probability $p$ are not independent random variables. The fundamental observables in the Random Cluster Model are the connectivities and they offer a purely geometrical interpretation of the magnetic Potts model phase transition. Connectivities represent the different probabilities with which $n$ points of the plane can be partitioned into FK clusters. If the points are on the boundary of the domain $\mathcal{D}$, the total number of $n$-point connectivities is clearly equal to the number of non-crossing partitions of a set of $n$ elements, i.e. the Catalan number $C_{n}$; for example if $n=4$ there are $C_{4}=14$ of them. These functions are however not linearly independent, since they satisfy sum rules: for instance the sum over all $n$-point connectivities has to be one. Following DVPotts , it is possible to show that a valid choice of $n$-point linearly independent connectivities is given by all the probabilities associated to configurations where no point is disconnected from all the others (non-singleton partitions). In the specific example of $n=4$ and $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$ on the boundary of $\mathcal{D}$, see again Fig. 1, a possible choice of linearly independent connectivities is: $P_{(1234)}$, $P_{(12)(34)}$ and $P_{(14)(23)}$. The function $P_{(1234)}$ denotes the probability that all the four points $x_{1}$, $x_{2}$, $x_{3}$ and $x_{4}$ are on the same FK cluster; $P_{(12)(34)}$ is instead the probability that $x_{1}$ and $x_{2}$ are in the same cluster, $x_{3}$ and $x_{4}$ in the same cluster but these two are now different and analogously for $P_{(14)(23)}$. We also omitted for simplicity the explicit spacial dependence. Notice that when the points $x_{1},~{}x_{2},~{}x_{3}$ and $x_{4}$ are anchored to the boundary the function $P_{(13)(24)}$ does not appear since two clusters cannot cross.

Exact solution–

We turn now to the exact determination of these three functions in the critical Ising model, using arguments inspired by the seminal work Cardy_perc . At the critical point, conformal invariance allows one to map any simply connected domain $\mathcal{D}$ of the plane by the Riemann mapping theorem into the unit disk. Moreover, the points $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$ are mapped to points $w_{1}$, $w_{2}$, $w_{3}$, $w_{4}$ lying at the boundary (circumference) of such a disk. The three connectivities $P_{(1234)},~{}P_{(12)(34)}$ and $P_{(13)(24)}$ can be singled out by computing Potts partition functions with specific boundary conditions for the dual Potts spins DVPotts . As an example let us suppose to fix the values of the dual Potts spins at the boundary of the disk to be $1,2,3$ and $4$ as in Fig. 2 left and to compute the Potts partition function in this case. Notice that this assignment will require at least four available colors, i.e. $Q\geq 4$, and it would be non-physical for the Ising model. It has however certainly sense if we assume $Q$ real and imagine to compute connectivities in the Random Cluster Model at any values of $Q$ and take eventually the limit $Q\rightarrow 2$. Configurations of dual FK clusters with such a particular choice of boundary conditions cannot contain clusters that cross from regions with boundary conditions $\alpha$ to regions with boundary conditions $\beta$ if $\alpha\not=\beta$. Dual FK clusters are represented schematically by blue dashed curves in Fig. 2. Applying a duality transformation to the Potts model partition function Wu , these configurations are in one-to-one correspondence with configurations where a single FK cluster, continuous red curve in Fig. 2, connects the four boundary points. The reasoning above allows to compute then connectivities as Potts partition functions with insertion of local operators $\phi_{(\alpha|\beta)}$ that switch the values of the dual spins at the boundary from $\alpha\rightarrow\beta$, $\alpha,\beta=1,\dots,Q$. In the jargon of CFT, the fields $\phi_{(\alpha|\beta)}$ are called boundary-condition-changing operators. In this way, we can argue for example that $P_{(1234)}$ has to be proportional to the correlation function of $\langle\phi_{(4|1)}(w_{1})\phi_{(1|2)}(w_{2})\phi_{(2|3)}(w_{3})\phi_{(3|4)}(w_{4})\rangle$.

Let us briefly recall that in the simplest case, the scaling fields $\phi_{r,s}$ of any CFT can be classified by two positive integers $r,s$ such that their scaling dimensions are

[TABLE]

The parameter $m$ is related to the central charge $c$ of the CFT through $c(m)=1-\frac{6}{m(m+1)}$, and in turns for the Potts model $Q=4\cos^{2}[\pi/(m+1)]$. The values $h_{r,s}$ can be represented into a lattice, dubbed the Kac table; for a CFT with $c=1/2$ as the Ising model, the Kac table is represented in Fig. 2 on the right.

The boundary condition changing operator $\phi_{(\alpha|\beta)}$ was identified in Cardy_perc for any values of $Q$ as the field $\phi_{1,3}$. Notice that at $c=1/2$, the dimension of $\phi_{1,3}$ is $h_{1,3}=1/2$ and coincides with the one of the Ising order parameter $\sigma$, when inserted at the boundary Cardy_verlinde ; in this case the spin operator $\sigma$ transforms as the field $\phi_{2,1}$. In the construction of the simplest conformal field theory describing the $\mathbb{Z}_{2}$ universality class these two fields can be actually identified and consequently the operator product algebra of $\{\phi_{1,1},\phi_{2,1},{\phi_{1,2}}\}$ closes. The self-consistent closure of the operator product algebra was used as a criterion in BPZ to build the whole family of minimal conformal models, where only a finite numbers of Virasoro algebra representations should be considered and furthermore allows to classify all the possible conformal boundary conditions Cardy_verlinde . However when analyzing the connectivity properties of the Ising FK clusters, the identification of $\phi_{1,3}$ with $\phi_{2,1}$ is no longer possible. According to the general theory BPZ , see also DMS , the four-point function of $\phi_{1,3}$ satisfies a linear differential equation of degree $3$. If we map the unit disk to the upper half plane $\mathbb{H}$ and call $z_{1},\dots,z_{4}$ the images on the real axis of the boundary points $w_{1},\dots,w_{4}$ we have

[TABLE]

where $z_{ij}=z_{i}-z_{j}$ and $\eta=\frac{z_{21}z_{43}}{z_{42}z_{31}}$ is the harmonic ratio ($0<\eta<1$). For the Ising model, the function $F(\eta)$ is the solution of the differential equation Suppl

[TABLE]

The equation (4) has three linearly independent solutions $F_{1,1}(\eta)$, $F_{1,3}(\eta)$, $F_{1,5}(\eta)$. The behaviour for small $\eta$ of each of the functions $F_{r,s}$ is of the form $\eta^{h_{r,s}}$ and the exponent $h_{r,s}$ coincides with the scaling dimension of the field $\phi_{r,s}$ that is produced in the operator product algebra BPZ : $\phi_{1,3}\times\phi_{1,3}=\phi_{1,1}+\phi_{1,3}+\phi_{1,5}$. Although there is not a general procedure to solve the differential equation (4), we can proceed as follows. Firstly, we observe that function $F_{1,1}(\eta)$ has to coincide apart from the prefactor in (3) with the four point function of the boundary spin $\sigma$ and such a function BPZ is the simple monodromy invariant Foot polynomial $1-\eta+\eta^{2}$. It is also easy to understand what $F_{1,1}$ should be in terms of connectivities. Since the four-point function of the boundary spin operator can be fully defined in the minimal Ising model, it has to correspond to the unique partition function that requires only two colors to be construct, namely $\langle\phi_{(2|1)}(w_{1})\phi_{(1|2)}(w_{2})\phi_{(2|1)}(w_{3})\phi_{(1|2)}(w_{4})\rangle$. This in turn, is proportional to the sum $P_{t}=P_{(1234)}+P_{(14)(23)}+P_{(12)(34)}$ of the three linearly independent connectivities. Using the known solution $F_{1,1}(\eta)$ we can reduce the degree of the differential equation (4) by substituting $F(\eta)=F_{1,1}(\eta)\int^{\eta}_{0}d\eta^{\prime}G(\eta^{\prime})$. The function $G(\eta)$ is finally obtained through a rational pull-back of the Gauss hypergeometric function key , see also Suppl . One gets two linearly independent solutions $G_{1,2}$ for $G(\eta)$, related by the transformation $\eta\rightarrow 1-\eta$: $G_{1}(\eta)=f(\eta)$ and $G_{2}(\eta)=f(1-\eta)$. The function $f(\eta)$ is

[TABLE]

with $K(\eta)$ and $E(\eta)$ the elliptic integrals of first and second kind respectively, and $p(\eta)$ and $q(\eta)$ rational functions of $\eta$ Suppl . The behaviour for small $\eta$ finally fixes, up to an overall constant, $F_{1,5}(\eta)=F_{1,1}(\eta)\int^{\eta}_{0}d\eta^{\prime}f(\eta^{\prime})$. The third linear independent solution to (4) can be chosen to be $F_{1,5}(1-\eta)$ that is actually a linear combination note of all the $F$’s. Coming back to the connectivities we observe that in the limit $w_{1}\rightarrow w_{2}$, $P_{(14)(23)}$ contains configurations where two FK clusters are separated by a dual line. These configurations are realized by the insertion of the operator $\phi_{1,5}$ Cardy_inc at the boundary and it was argued in Cardy_revLog that in this case logarithmic singularities should arise. We conjecture then the following identification for the universal probability ratio which we denote with $R(\eta)$

[TABLE]

where the constant $A=[\int_{0}^{1}d\eta f(\eta)]^{-1}$ is chosen to ensure that $R(1)=1$. The conjecture (6) can be easily tested on an arbitrary geometry by applying a conformal mapping $z^{\prime}(z)$. Since all the dimensionful parameters in (3) cancel when computing (6), one has only to express $\eta$ in the new coordinates $z^{\prime}$. Finally we observe that denoting $1-\eta=\varepsilon$ one obtains Suppl the small $\varepsilon$ expansion for the ratio in (6)

[TABLE]

The logarithmic singularity arises from the mixing of the level two descendants of $\phi_{1,3}$ with the field $\phi_{1,5}$ that have at $c=1/2$ the same conformal dimension $h_{1,5}=5/2$. This is the first example where a logarithmic singularity is explicitly calculated in the context of the critical Ising model. The logarithmic behaviour in (7) has a completely different origin respect to the well-known logarithmic divergence of the specific heat at the critical temperature Aldo_rev ; Giuseppe_book . It shows that the phenomenon of mixing of scaling fields and non-diagonalizability of the conformal dilation operator could arise potentially at any rational value of the central charge. A circumstance that was already recognized in Cardy_revLog ; VJ_log and VSJ2 ; GV ; SV ; KF but for which in the Ising model no explicit result was available. In SA_1 ; SA_2 , a possible source of logarithmic behaviour was also identified but appeared to be ruled out by numerical data.

Numerical results–

Simulations have been carried out on the Ising model at the exactly known critical temperature on a triangular lattice in triangles of sides of lengths $L=9,17,33,65,129,257$ with open boundary conditions collecting a number of samples up $10^{10}$. The random number generator employed is given in matsumoto_1998 . We implemented the efficient Swendsen-Wang algorithm swendsen_1987 that provides direct access to the FK clusters FK .

In order to use our results for the upper half plane (6) in the triangle geometry a Schwarz-Christoffel is in order. Given a $z$ in $\mathbb{H}$ and a $z^{\prime}$ belonging to the interior of an equilateral triangle with vertices $\left(-1,1,i\sqrt{3}\right)$ the mapping reads $z^{\prime}=\frac{2z\Gamma\left(\frac{5}{6}\right){}_{2}F_{1}\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};z^{2}\right)}{\sqrt{\pi}\Gamma\left(\frac{1}{3}\right)}$, ${}_{2}F_{1}$ being the Gauss hypergeometric function. In the simulations the three points $z_{1}^{\prime}$, $z_{3}^{\prime}$ and $z_{4}^{\prime}$ have been fixed in the midpoint of each side while the point $z_{2}^{\prime}$ takes any position on the boundary between $z^{\prime}_{1}$ and $z_{3}^{\prime}$. Since the problem is symmetric under rotation of $2\pi/3$ and $4\pi/3$ around the center of the triangle also the configurations obtained with these rotations have been measured to enhance the statistics. An example of the simulated system together with a realization of FK clusters is presented in Figure 3. The ratios $P_{(12)(34)}/P_{t},P_{(14)(23)}/P_{t},P_{(1234)}/P_{t}$, because of the symmetry $P_{(12)(34)}(\eta)=P_{(14)(23)}(1-\eta)$ are not independent and only one function suffices to specify all of them, that is $R(\eta)$ as defined in (6). In Figure 4 we show the simulation results together with the CFT prediction for $R$ for the four largest lattices. In the inset of Figure 4 we show the deviations from the exact result.

Conclusion–

In this letter we have calculated the four point connectivities of FK clusters in the critical Ising model and show that they can display logarithmic branch cuts. This is a first explicit example where such type of singularities are determined exactly for a theory that also has a non-trivial sector belonging to the series of unitary minimal models. Previous exact CFT studies focused on percolation and SAW FSK2 ; KF_1 ; KF_2 ; KF . Our findings are fully corroborated by numerical simulations, showing excellent agreement. Similar structures, expected in many other important two-dimensional models including critical percolation, fully deserve the attention of future investigations. It would be also of clear interest to analyze whether logarithmic singularities could be found in higher dimensions Bootstrap ; ABCD , for instance in the three dimensional critical Ising model.

Acknowledgements.

GG thanks the International Institute of Physics of Natal for the kind hospitality. JV is grateful to J. Dubail, G. Mussardo and R. Santachiara for a critical reading of the manuscript and many important and interesting observations. We thank also D. Bernard and A. Cappelli for correspondence and in particular J. Cardy for pointing out his result in Cardy_revLog about logarithmic observables in the Ising model.