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Exact Logarithmic Four-Point Functions in the Critical Two-Dimensional Ising Model
Giacomo Gori1 and Jacopo Viti2,*
1SISSA & CNR-IOM, Via Bonomea 265, 34136 Trieste, Italy
2ECT & Instituto Internacional de Física, UFRN, Lagoa Nova 59078-970 Natal, Brazil
(Received 17 April 2017; revised manuscript received 4 August 2017; published 6 November 2017)
Based on conformal symmetry we propose an exact formula for the four-point connectivities of Fortuin-
Kasteleyn clusters in the critical Ising model when the four points are anchored to the boundary. The explicit
solutionwe found displays logarithmic singularities.We check our prediction usingMonteCarlo 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.
DOI: 10.1103/PhysRevLett.119.191601
Introduction.—Conformal symmetry in two dimensions characterization of disordered systems in two dimensions
[1] has been of extraordinary usefulness to study classical [12–14], for instance, by means of supersymmetry [15,16].
statistical mechanics models at criticality since the 1980s. It With these motivations, logarithmic CFTs were inves-
has notably found also extensive applications in the tigated in greater detail in the last decade, either from a
quantum realm, spanning from gapless one-dimensional purely algebraic point of view [17], either constructing
systems [2], the quantum Hall effect [3], and entanglement lattice regularizations [16,18–20] or generalizing the study
[4]. Two dimensional conformal invariant quantum field of crossing formulas in critical percolation [21–23].
theories (CFTs) and, in particular, Liouville theory are Recently [24], it has also been suggested that an analytic
moreover the cornerstone of world-sheet geometry in string nonunitary extension ofLiouville theory [25]might describe
theory [5]. The simplest CFTs that capture the critical the connectivity properties of critical bulk percolation and,
behavior of lattice models and quantum spin chains are more generally, the Q-state Potts model. The domain of
unitary. Moreover, when their Hilbert space splits into a applicability of Liouville theory in statistical mechanics is
finite number of representations of the conformal (or some currently an important open problem; see Refs. [26,27] and
larger) symmetry are usually termed rational. CFTs are also Ref. [28].
classified according to their central charge c; for instance, Despite these huge advances, a satisfactory understanding
for a free massless boson c ¼ 1. of logarithmic CFTs is still a long way off. Moreover, it is
However unitary and rational theories are far from fair to say that few examples of explicit logarithmic
exhausting the physically relevant conformally invariant singularities have been found in familiar statistical models:
theories. In the beginning of the 1990s the groundbreaking notably only in percolation [29,30] (c ¼ 0), dense polymers
discovery [6] of an exact formula for the crossing prob- [31–33]; see alsoRefs. [34,35] for applications to disordered
ability in critical percolation forced to analyze theories systems. An exact Coulomb gas approach to CFT correla-
violating these two assumptions. Percolation is a simple tion functions [36–38] closely related to those considered in
stochastic process where bonds or sites on a lattice can be this Letter reveals an infinite number of logarithmic cases.
occupied independently with a certain probability. Since These arise from operator mixing as is the case here. These
the partition function is not affected by finite-size effects, results suggest the possibility of logarithmic behavior in
the central charge of a putative CFT describing critical multiple self-avoiding random walks (SAWs) [38].
percolation is zero [7]. The existence of a nontrivial In this Letter we show how a logarithmic singularity due
formula for the crossing probability makes it a prominent to operator mixing also arises in the context of arguably the
example of a nonunitary CFT (the only unitary CFT with best-known model of statistical mechanics: the two-dimen-
c ¼ 0 is trivial). For subsequent developments leading to sional Ising model; see Refs. [39,40] for a pedagogical
the formulation of the stochastic Lowener evolution we introduction to the richness of the model. This is remark-
refer to the reviews [8,9]. At the same time, it was able since it shows unambiguously that critical properties
recognized that as far as the conditions of unitarity and of Ising clusters are ruled by a logarithmic CFT. Moreover
rationality are relaxed, CFT correlation functions can the four-point function we consider here, a four point
display striking logarithmic singularities that are actually connectivity in the Fortuin-Kasteleyn (FK) representation
the signatures of intricate realizations of the conformal of the Ising model, is a natural observable that can be easily
symmetry [10,11]. The class of nonunitary and generally simulated with Monte Carlo (MC) methods. Logarithmic
nonrational CFTs where these unconventional features singularities in Ising connectivities were also argued to
show up, was christened logarithmic CFTs. Such theories exist in Refs. [41,42], in this Letter we demonstrate it
were promptly argued to play a fundamental role in the explicitly using CFT.
0031-9007=17=119(19)=191601(6) 191601-1 © 2017 American Physical Society
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Finally, our findings could shed further light on the variables. The fundamental observables in the random
extremely challenging problem of the characterization of cluster model are the connectivities and they offer a purely
logarithmic CFTs and on their applications to physics. geometrical interpretation of the magnetic Potts model
Four points connectivities in the Ising model.—It is phase transition. Connectivities represent the different
convenient to introduce the Ising FK clusters, starting from probabilities with which n points of the plane can be
the ferromagneticQ-state Potts model [43]; the Ising case is partitioned into FK clusters. If the points are on the
recovered by setting Q ¼ 2. The model is defined on a boundary of the domain D, the total number of n-point
finite simply connected domain D of the plane, see Fig. 1, connectivities is clearly equal to the number of noncrossing
and the choice of the underlying lattice is irrelevant at the partitions of a set of n elements, i.e., the catalan numberCn;
critical point. To introduce the notion of cluster connectiv- for example, if n ¼ 4 there are C4 ¼ 14 of them. These
ities, we should first recall the FK representation [44]. The functions are, however, not linearly independent, since they
Q-state Potts model is defined in terms of spin variables satisfy sum rules: for instance, the sum over all n-point
sðxÞ taking 1;…; Q different values; its partition function connectivities has to be 1. Following Ref. [45], it is possible
can be expressed as a product over the lattice edges as to show that a valid choice of n-point linearly independent
X Y connectivities is given by all the probabilities associated
¼ ½ð Þ þ  ð Þ with configurations where no point is disconnected fromZ 1 − p pδsðxÞ;sðyÞ ; 1 all the others (non-singleton partitions). In the specific
fsðxÞg hx;yi example of n ¼ 4 and x1, x2, x3, x4 on the boundary of D,
see again Fig. 1, a possible choice of linearly independent
where p is a parameter related to the temperature and the connectivities is Pð1234Þ, Pð12Þð34Þ, and Pð14Þð23Þ. The func-
product in Eq. (1) extends only to next-neighboring sites. tion P denotes the probability that all the four points
Suppose then to expand such a product: Each term in the ð1234Þx , x , x , and x are on the same FK cluster; P is
expansion can be represented graphically by drawing a 1 2 3 4 ð12Þð34Þ
bond between x and y if the factor pδ is selected, and instead the probability that x and xð Þ ð Þ 1 2
are in the same cluster,
s x s y x3 and x4 are in the same cluster but these two are nowleaving empty the bond if such a factor is absent. The set of different and analogously for P . We also omitted for
nonempty bonds in each term of the expansion then defines ð14Þð23Þ
a graph G on the underlying lattice that is called the FK simplicity the explicit spacial dependence. Notice that
graph. Such a graph might contain N different connected when the points x1, x2, x3, and x4 are anchored to thec boundary the function P
components (including isolated points), dubbed FK clus- ð13Þð24Þ does not appear since two
ters. Moreover, on each cluster the spin values are con- clusters cannot cross.
strained to be the same because of the Kronecker delta in Exact solution.—We turn now to the exact determination
Eq. (1). Summing over their possible Q values leads to the of these three functions in the critical Ising model, using
following rewriting of the Q-state P
¼ P
otts model partition arguments inspired by the seminal work of Ref. [6]. At the
function as a sum over graphs: Z ð − pÞn̄ pn QN , critical point, conformal invariance allows one to map any1 b b cG simply connected domain D of the plane by the Riemann
where nb and n̄b are the number of occupied and empty mapping theorem into the unit disk. Moreover, the points
bonds in the graph G. For arbitrary non-negative Q, the x , x , x , x are mapped to points w , w , w , w lying at
graph representation for Z is a generalized percolation 1 2 3 4 1 2 3 4the boundary (circumference) of such a disk. The three
problem known as random cluster model, where bonds connectivities Pð1234Þ; Pð12Þð34Þ, and Pð13Þð24Þ can be singledoccupied with probability p are not independent random
out by computing Potts partition functions with specific
boundary conditions for the dual Potts spins [45]. 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 ≥ 4, and it would be nonphysical
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 → 2. Configurations of dual FK
clusters with such a particular choice of boundary con-
FIG. 1. TheQ-state Potts model is defined on a finite domainD ditions cannot contain clusters that cross from regions with
of the plane. FK graphs G on the underlying square lattice are boundary conditions α to regions with boundary conditions
drawn in blue. The figure shows a particular configuration where β if α ≠ β. Dual FK clusters are represented schematically
the four boundary points x1, x2, x3, and x4 are all connected by an by blue dashed curves in Fig. 2. Applying a duality
FK cluster thus contributing to Pð1234Þ. transformation to the Potts model partition function [43],
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where only a finite numbers of Virasoro algebra represen-
tations should be considered and furthermore allows us to
classify all the possible conformal boundary conditions
[46]. However, when analyzing the connectivity properties
of the Ising FK clusters, the identification of ϕ1;3 with ϕ2;1
is no longer possible. According to the general theory [1],
see also Ref. [47], the four-point function of ϕ1;3 satisfies a
linear differential equation of degree 3. If we map the unit
disk to the upper half plane H and call z1;…; z4 the images
on the real axis of the boundary points w1;…; w4 we have
FIG. 2. On the left, schematic representation of allowed dual Y4   2h
FK clusters (dashed blue curves) in the Potts model when ϕ1;3ðziÞ ¼
z z 1;342 31 FðηÞ; ð3Þ
boundary conditions that fix the values of the dual boundary ¼ H z21zi 1 43z32z14
spins to 1,2,3, and 4 are chosen. On the right, the Kac table,
obtained from the scaling dimension hr;s in Eq. (2) for Q ¼ 2 where zij ¼ zi − zj and η ¼ ðz21z43=z42z31Þ is the har-
corresponding to the Ising model. monic ratio (0 < η < 1). For the Ising model, the function
FðηÞ is the solution of the differential equation [48]
these configurations are in one-to-one correspondence with
2 000 2 0
configurations where a single FK cluster, the continuous ½2ηð1− ηÞ F − 3ð1− ηþ η ÞF þ 3ð2η− 1ÞF¼ 0: ð4Þ
red curve in Fig. 2, connects the four boundary points. The
reasoning above allows us to compute the connectivities as Equation (4) has three linearly independent solutions
Potts partition functions with insertion of local operators F1;1ðηÞ, F1;3ðηÞ, and F1;5ðηÞ. The behavior for small η
ϕð j Þ that switch the values of the dual spins at the of each of the functions Fr;s is of the form η
hr;s and the
α β
boundary from α → β, α; β ¼ 1;…; Q. In the jargon of exponent hr;s coincides with the scaling dimension of the
CFT, the fields ϕðαjβÞ are called boundary-condition- field ϕr;s that is produced in the operator product algebra
changing operators. In this way, we can argue, for example, [1]: ϕ1;3×ϕ1;3¼ϕ1;1þϕ1;3þϕ1;5. Although there is not a
that P has to be proportional to the correlation general procedure to solve the differential equation (4), weð1234Þ can proceed as follows. First, we observe that function
function of hϕð4j1Þðw1Þϕð1j2Þðw2Þϕð2j3Þðw3Þϕð3j4Þðw4Þi. F1;1ðηÞ has to coincide apart from the prefactor in Eq. (3)Let us briefly recall that in the simplest case, the scaling with the four point function of the boundary spin σ and
fields ϕr;s of any CFT can be classified by two positive such a function [1] is the simple monodromy invariant [49]
integers r, s such that their scaling dimensions are polynomial 1 − ηþ η2. It is also easy to understand what
F1;1 should be in terms of connectivities. Since the four-
¼ ½rðmþ 1Þ − sm
2 − 1
h ; m ∈ R: ð2Þ point function of the boundary spin operator can be fullyr;s
4mðmþ 1Þ defined in the minimal Ising model, it has to correspond to
the unique partition function that requires only two colors
The parameter m is related to the central charge c of the to be constructed, namely, hϕð2j1Þðw1Þϕð1j2Þðw2Þϕð2j1Þðw3Þ×
CFT through cðmÞ ¼ 1 − ½6=mðmþ 1Þ, and in turn for the ϕð1j2Þðw4Þi. This, in turn, is proportional to the sum Pt ¼
Potts modelQ ¼ 4 cos2½π=ðmþ 1Þ. The values hr;s can be Pð1234Þ þ Pð14Þð23Þ þ Pð12Þð34Þ of the three linearly indepen-
represented into a lattice, dubbed the Kac table; for a CFT dent connectivities. Using the known solution F1;1ðηÞ
with c ¼ 1=2 as the Ising model, the Kac table is we can reduce the degree of thRe differential equation (4)
represented in Fig. 2 on the right. by substituting FðηÞ ¼ F ðηÞ η dη01;1 0 Gðη0Þ. The function
The boundary condition changing operator ϕðαjβÞ was GðηÞ is finally obtained through a rational pull back of the
identified in Ref. [6] for any values of Q as the field ϕ1;3. Gauss hypergeometric function [50]; see also Ref. [48].
Notice that at c ¼ 1=2, the dimension of ϕ1;3 is h1;3 ¼ 1=2 One gets two linearly independent solutions G1;2 for GðηÞ,
and coincides with the one of the Ising order parameter σ, related by the transformation η→ 1− η: G1ðηÞ ¼ fðηÞ and
when inserted at the boundary [46]; in this case the spin G2ðηÞ ¼ fð1− ηÞ. The function fðηÞ is
operator σ transforms as the field ϕ2;1. In the construction
of the simplest conformal field theory describing the Z pðηÞEðηÞ þ2 fðηÞ ¼ pðffiffiffiffiffiffiffiffiffi
qffiffiffiðÞffi
ηffiffiffiÞKðηÞ ; ð5Þ
universality class these two fields can be actually identified 1 − η η
and, consequently, the operator product algebra of fϕ1;1;
ϕ2;1;ϕ1;2g closes. The self-consistent closure of the oper- with KðηÞ and EðηÞ the elliptic integrals of the first and
ator product algebra was used as a criterion in Ref. [1] to second kind, respectively, and pðηÞ and qðηÞ rational
build the whole family of minimal conformal models, functions of η [48]. The behavior for small η finally fixes,
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R
up to an overall constant, F η 0 01;5ðηÞ ¼ F1;1ðηÞ 0 dη fðη Þ.
The third linear independent solution to Eq. (4) can be
chosen to be F1;5ð1 − ηÞ, which is actually a linear
combination [51] of all the F’s. Coming back to the
connectivities we observe that in the limit w1 → w2,
Pð14Þð23Þ contains configurations where two FK clusters
are separated by a dual line. These configurations are
realized by the insertion of the operator ϕ1;5 [52] at the
boundary and it was argued in Ref. [41] that in this case
logarithmic singularities should arise. We conjecture then FIG. 3. (Left) Triangular lattice (L ¼ 9) with the four points z0 ,
the following identification for the universal probability 0 0 1
ð Þ z2, z3, and z
0
4 highlighted. (Right) A realization of FK clustersratio, which we denote with R η
ð Þ Z
contributing to the probability Pð12Þð34Þ.
ð Þ ¼ Pð14Þð23Þ η
η
R η ¼ A dη0fðη0Þ; ð6Þ 0 pffiffiffi
P ðηÞ reads z ¼ ½2zΓð5Þ2F1ð1 ; 2 ; 3 ; z2Þ= πΓð1Þ,t R 0 6 2 3 2 3 2
F1 being the
Gauss hypergeometric function. In the simulations the three
where the constant A ¼ ½ 10 dηfðηÞ−1 is chosen to ensure points z0 , z0 , and z01 3 4 have been fixed in the midpoint of each
that Rð1Þ ¼ 1. The conjecture (6) can be easily tested on an side, while the point z02 takes any position on the boundary
arbitrary geometry by applying a conformal mapping z0ðzÞ. between z01 and z03. Since the problem is symmetric under
Since all the dimensionful parameters in Eq. (3) cancel rotation of 2π=3 and 4π=3 around the center of the triangle,
when computing Eq. (6), one has only to express η in the
0 also the configurations obtained with these rotations havenew coordinates z . Finally, we observe that denoting
¼ been measured to enhance the statistics. An example of the1 − η ε one obtains [48] the small ε expansion for the simulated system together with a realization of FK clusters
ratio in Eq. (6): is presented in Fig. 3. The ratios Pð12Þð34Þ=Pt; Pð14Þð23Þ=Pt;
R¼ 1− ε1=2½a0 þ a1εþ a2ε2ð1þ b log εÞ þOðε3Þ: ð7Þ Pð1234Þ=Pt, because of the symmetry Pð12Þð34ÞðηÞ ¼
Pð14Þð23Þð1 − ηÞ are not independent and only one function
The logarithmic singularity arises from the mixing of the suffices to specify all of them, that is RðηÞ as defined in
level two descendants of ϕ1;3 with the field ϕ1;5 that have at Eq. (6). In Fig. 4 we show the simulation results together
c ¼ 1=2 the same conformal dimension h1;5 ¼ 5=2. This is with the CFT prediction for R for the four largest lattices.
the first example where a logarithmic singularity is explic- In the inset of Fig. 4 we show the deviations from the
itly calculated in the context of the critical Ising model. The exact result.
logarithmic behavior in Eq. (7) has a completely different
origin with respect to the well-known logarithmic diver-
gence of the specific heat at the critical temperature [39,40]. 1.0
−1
It shows that the phenomenon of mixing of scaling fields 10
and nondiagonalizability of the conformal dilation operator
0.8
could arise potentially at any rational value of the central
10−3
charge—a circumstance that was already recognized in
Refs. [41,42] and Refs. [38,53–55], but for which in the 0.6
Ising model no explicit result was available. In 10−5
Refs. [56,57], a possible source of logarithmic behavior
0.00 0.25 0.50 0.75 1.00
was also identified but appeared to be ruled out by 0.4 η CFT
numerical data. L = 33
Numerical results.—Simulations have been carried out 0.2 L = 65
on the Ising model at the exactly known critical temperature L = 129
on a triangular lattice in triangles of sides of lengths L ¼ 9, L = 257
17, 33, 65, 129, and 257 with open boundary conditions 0.00.00 0.25 0.50 0.75 1.00
collecting a number of samples up 1010. The random η
number generator employed is given in Ref. [58]. We
implemented the efficient Swendsen-Wang algorithm [59] FIG. 4. Universal ratio RðηÞ (6) for the lattice sizes L ¼ 33, 65,
that provides direct access to the FK clusters [44]. 129, and 257 denoted by triangles, diamonds, squares, and
circles, respectively. Errors are smaller than the symbol size.
In order to use our results for the upper half plane (6) in The CFT prediction is plotted with the continuous line. In the
the triangle geometry a Schwarz-Christoffel is in order.
0 inset deviations of MC data from the theory are shown with theGiven a z in H and a z belonging to tpheffiffiffi interior of an same symbols used in the main figure, lines are just guides to
equilateral triangle with vertices ð−1; 1; i 3Þ the mapping the eyes.
191601-4
R(η)
|RMC(η)−RCFT (η)|
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