Parafermionic conformal field theory on the lattice

Parafermionic conformal field theory on the lattice

Roger S. K. Mong, David J. Clarke, Jason Alicea, Netanel H. Lindner, Paul Fendley
Department of Physics and Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA 91125, USA
Department of Physics, Technion, 32000 Haifa, Israel
Department of Physics, University of Virginia, Charlottesville, VA 22904-4714 USA
August 28, 2019

Finding the precise correspondence between lattice operators and the continuum fields that describe their long-distance properties is a largely open problem for strongly interacting critical points. Here we solve this problem essentially completely in the case of the three-state Potts model, which exhibits a phase transition described by a strongly interacting ‘parafermion’ conformal field theory. Using symmetry arguments, insights from integrability, and extensive simulations, we construct lattice analogues of nearly all the relevant and marginal physical fields governing this transition. This construction includes chiral fields such as the parafermion. Along the way we also clarify the structure of operator product expansions between order and disorder fields, which we confirm numerically. Our results both suggest a systematic methodology for attacking non-free field theories on the lattice and find broader applications in the pursuit of exotic topologically ordered phases of matter.

1 Introduction

A major triumph of theoretical physics was the precise understanding, via the renormalization group, of how lattice models of statistical mechanics are described by continuum field theories at and near critical points. In 1+1-dimensional quantum critical points with linear dispersion and two-dimensional classical systems with rotational invariance, all fields and their exact scaling dimensions often can be identified precisely by using conformal field theory (CFT) [1]. Once one identifies which particular CFT describes a lattice model of interest, it is then possible to utilize powerful non-perturbative techniques to do many exact computations. Moreover, certain models—called integrable—allow exact computations to be performed even away from criticality [2].

Despite the many successes of this approach, the connection between the original lattice formulation of a theory and its continuum counterpart is often difficult to make precise. One question that frequently arises is the following: Given some microscopic operator , what is its expansion in terms of continuum fields at criticality? When a CFT describes the continuum limit, we can turn the question on its head and ask: Which combination of lattice operators yields a particular continuum field as the lowest-scaling-dimension component in such an expansion? The former admits a unique answer, but the latter does not.

These questions are essential to understand for many applications involving critical phenomena—e.g., extracting correlation functions of physical quantities or perturbing critical points with specific microscopic operators. Yet the answers are surprisingly incomplete given the vast body of literature on critical systems. Resolving the connection between lattice operators and chiral continuum fields has proven especially difficult. Here the additional challenge derives from the fact that a chiral operator on the lattice is, by definition, non-local.

One reason chiral operators are particularly interesting is because of the many formal connections between chiral fields in CFT and anyons in topologically ordered systems. Although it may naively seem as if studying a lattice model is not necessary to analyze an elegant continuum theory, one central point of this paper is that this exercise not only results in a great deal of intuition into the problem, but also uncovers deep results. For example, the lattice analogues of certain local fields in minimal CFTs are given in terms of single-site operators using the modular matrix of topological field theory [3]. In fact, this analysis of Pasquier’s resulted in his discovery of a special case of the Verlinde formula [4] before Verlinde!

The Ising model comprises one of the few examples where the lattice/continuum correspondence is essentially completely understood. Here there are only two non-trivial relevant local operators, the spin and energy fields, which respectively possess scaling dimensions and . Symmetry alone allows one to relate their ‘ultraviolet’ and ‘infrared’ manifestations. For example, the spin field is the most relevant operator that is odd under the symmetry corresponding to a global spin flip. Thus the operator measuring the spin at a lattice site has the spin field as its leading contribution in the continuum limit. The lattice analogue of the energy or “thermal” field is found simply by noticing that perturbing the 2d classical Ising model off the critical point by changing the temperature preserves the spin-flip symmetry. The only symmetry-preserving relevant operator is the energy field, so the lattice expression can be extracted directly from the action (or Hamiltonian in the quantum spin-chain case). The most interesting chiral operators in the Ising model, the left- and right-moving components and of the free-fermion field, are also well understood. These non-local operators are obtained by the so-called Jordan-Wigner mapping [5], and can be elegantly understood as a product of spin and disorder operators [6]. A simple consistency check on these relations follows from the fact that the energy field is the product .

Unfortunately, analogous results are not so simple to obtain for more general models, even using very sophisticated techniques. Great progress was made in the ’70s using the Coulomb-gas approach pioneered by Kadanoff and others, where critical properties of a wide variety of classical lattice models were argued to be identical to those of a free boson, and then various heuristic arguments were used to identify the exact scaling dimensions of some operators [7]. With the advent of conformal field theory in the ’80s, these results were adapted and generalized. An example close to hand here is the antiferromagnetic three-state Potts model [8]. The ferromagnetic 3-state Potts model studied here also has been treated by this approach, but the Coulomb-gas results are even more heuristic, since the free-boson theory needs to be modified by including a charge at infinity [9, 10]. Thus while this approach has yielded many valuable results, it typically is somewhat ad hoc, and moreover rarely yields chiral operators.

Another reason for revisiting the lattice/continuum correspondence arises from recent work in the mathematical physics/probability community. In the context of two-dimensional classical lattice models, lattice chiral operators are known as discrete holomorphic operators. One reason for mathematicians’ interest is the potential that these can be used to rigorously prove that a given lattice model turns into a particular CFT in the continuum limit, a strategy successfully used in the Ising case [11]. Another reason is that demanding an operator be discrete holomorphic in many cases provides a simple way of finding integrable models [12]. In fact, using considerations from topological field theory it is possible to find a general method for constructing discrete holomorphic operators of this type [13].

It turns out, however, that such definitions are typically not sufficient to fix lattice operators precisely. Theories more complicated than Ising or a free boson admit multiple operators that behave similarly under discrete rotations; in CFT language such operators possess conformal spins differing by integers. A given lattice operator will then represent some linear combination of these continuum operators, and it is not a priori obvious how to separate the constituent pieces. In the Ising case, the fact that the chiral fermions are non-interacting makes it possible to find additional constraints sufficient to determine the precise connection between lattice and continuum. In interacting cases, it has not been so. For example, the equations usually solved for discrete holomorphicity in classical models typically amount to lattice versions of only half the Cauchy-Riemann equations.

One of our main results is showing how in a non-free field theory we can overcome this obstacle and precisely identify chiral lattice operators. The example is the famed three-state Potts model—which provides a very natural generalization of the Ising model in some respects, but is strongly interacting. The three-state Potts model generalizes the Ising model by replacing the variable on each site by a three-state “spin”. With ferromagnetic nearest-neighbor interactions, the model is ordered at low temperatures and remains disordered at high temperatures, as with Ising. A non-trivial critical point separates the two phases, and the “parafermion” CFT describing the continuum limit is well understood [14]. The naming arises because the CFT includes a chiral parafermion field – a analogue of the chiral fermion in Ising, with similar algebraic structure but without the feature that makes the model easily solvable. Likewise, the associated quantum lattice Hamiltonian, given in (1) below, can be rewritten in terms of lattice parafermion operators [15].

As the lattice parafermion operator and CFT parafermion field share common symmetry properties, it is natural to expect that taking the continuum limit of the former recovers the latter. We show, however, that the actual correspondence is more subtle. On symmetry grounds one can not exclude the possibility that another continuum operator will also appear in an expansion of the lattice parafermion [16]; see Sec. 5 for a detailed discussion. In fact, this “correction” field turns out to exhibit a smaller scaling dimension than the parafermion field and thus, surprisingly, provides the dominant contribution to the expansion! We construct a precise linear combination of lattice parafermion operators for which this contribution cancels, leaving the chiral parafermion field as the leading piece. Arguments invoking symmetry and integrability allow us to similarly infer the lattice analogues of many other continuum fields. We confirm these identifications by applying density-matrix renormalization group (DMRG) simulations to numerically compute two-point functions and hence the scaling dimensions of these lattice operators. The DMRG method is particularly well-suited to this problem since it enables essentially exact computations for ground-state properties of one-dimensional quantum systems; in all cases we obtain perfect agreement with expectations based on our CFT-field correspondence.

These findings further solidify the connection between the lattice model in Eq. (1) and the parafermion CFT. Numerous other implications, however, also follow. On a formal level, our analysis clarifies the structure of operator product expansions in the CFT—particularly those that involve the parafermion fields, where in Sec. 6 we identify an omission in previous results. Furthermore, just as the Ising model has been instrumental in constructing phases that support Ising non-Abelian anyons, so too has the three-state Potts model been central to accessing phases with more exotic anyonic content. Read-Rezayi quantum Hall phases [17] provide a classic example where parafermions play a key role. More recently, Teo and Kane introduced a coupled-chain construction of Read-Rezayi states by hybridizing counterpropagating parafermion fields from adjacent critical chains. Inspired by their work, the present authors and other collaborators [18] introduced a superconducting analogue of the () Read-Rezayi phase by weakly coupling critical Potts chains. Such a system can be ‘engineered’ in Abelian quantum Hall/superconductor hybrids that localize lattice parafermion zero modes [19, 20, 21, 22, 23, 24]. This phase is remarkable in that it supports Fibonacci anyons—which possess universal braid statistics—yet is built from well-understood Abelian phases of matter. Some of the relations derived here were first quoted in Ref. [18]; indeed, the expansion of the lattice parafermion operators that we elucidate below proved key to the entire analysis. The present manuscript thus puts these earlier results on firmer footing and greatly expands them. We expect the methodology pursued here to enable similar progress in other lattice systems, possibly paving the way to constructions of still more exotic two-dimensional phases of matter from critical chains.

2 The three-state Potts model and its symmetries

An obvious way to generalize the two-dimensional classical Ising model is to replace the variable on each site by a -state “spin”. When the interactions are nearest neighbor and only depend on whether the adjacent spins are the same or different, this is known as the -state Potts model. In an isotropic system, there is then just one coupling, which can be taken to be the temperature. With ferromagnetic interactions, the model orders at low temperatures and remains disordered at high temperatures, as with Ising. Also as with Ising, there is a duality symmetry exchanging high and low temperatures, and the phase transition occurs at the self-dual point. As opposed to Ising, however, this phase transition is first order for [25]. There does occur a self-dual critical point in a -state model when the symmetry permuting the spins is broken down to . This “parafermion” [15] critical point is integrable [26]; conserved charges have been computed explicitly [27]. Much of what we say in the following has an analogue for general , but there the fine tuning necessary to extract the physics of interest is considerable. We therefore will confine our analysis to the three-state Potts model.

2.1 The Hamiltonian and the spin operators

It is both intuitively and technically convenient to study the physics of the three-state Potts model by taking an anisotropic limit where the system can be described by a quantum Hamiltonian. Taking this approach also has the advantage of making direct contact with the physics discussed in Ref. [18]. The Hilbert space for an -site chain is , i.e., a three-state system at each lattice site. The symmetry permutes the three orthogonal basis states on each site. The Hamiltonian for the three-state Potts quantum chain is


Throughout we assume . The operator shifts the spin on site , while measures its value. Precisely, denoting the three states by , and , the operators on a particular site can be written as


where . These operators obey the algebra


For the ground state forms a ferromagnet that spontaneously breaks the symmetry, while for a disordered paramagnetic phase arises. At the phase transition point the system is critical; we defer discussion of its critical properties to the next section.

Equation (1) exhibits a number of symmetries that play an important role throughout this paper. On an infinite chain (or a chain with periodic boundary conditions), the Hamiltonian preserves simple translations that shift the operators by one site. The Hamiltonian’s full permutation symmetry can be usefully decomposed into a symmetry—which cyclically permutes , and —and a unitary ‘charge conjugation’ operation that swaps . The symmetry is generated by


and transforms operators according to . In particular, we have


One can therefore say that carries ‘charge’ under whereas is neutral. Here we take to be the eigenvalue under , so thus is defined modulo 3. Charge conjugation acts on operators via


where , and . Note that swaps the sign of the charge carried by —hence the term ‘charge conjugation’. The Hamiltonian is also invariant under both parity (spatial inversion) and time-reversal symmetry. Parity takes site to site ; that is, and . The time-reversal generator is anti-unitary and conjugates () but leaves invariant ().

2.2 Dual variables

While the preceding section enumerated the complete set of symmetries manifest in the three-state Potts Hamiltonian expressed in terms of the spin operators and , additional symmetries are revealed upon recasting the model in dual variables. Namely, disorder operators [6] can be defined here using a generalization of the Kramers-Wannier duality of the Ising model. In the quantum Hamiltonian limit they are defined as


which live on bonds between sites of the original lattice. We use conventions where these bonds are labeled by half-integers—e.g., the bond between sites and is denoted by . The disorder operator in effect adds a domain wall between sites and by cycling all spins to the left of bond . Conjugating the Hamiltonian by the disorder operator, , leaves all terms invariant except for the ferromagnetic term that couples sites . This suggests that the operator conjugate to ought to be


Indeed, the and operators defined as above obey exactly the same algebra as and :


In a very precise sense these operators are the duals of the original spin operators.

Remarkably, the Hamiltonian written in terms of , retains the same form as Eq. (1),


with the roles of and swapped. Hence we can define the duality transformation


which is an additional symmetry of the Hamiltonian at the critical point . Since the transformation takes and , two applications of the duality transformation yield a simple lattice translation by one site: and . In this representation one can also identify a symmetry (present even for ) generated by . This symmetry acts on the dual operators according to


Hence and respectively possess charge and under . For an infinite chain, one can interpret this transformation as a phase applied to (since here ). For a periodic chain, the transformation can instead be thought of as acting on the space of boundary conditions.

Importantly, the duality mapping exploited above is not unique. Instead of defining the disorder operator in terms of a string of ’s emanating to , we can of course also choose the opposite convention with


This ‘primed’ disorder operator creates a domain wall by cycling the spins to the right of bond , and relates to the disorder operator defined earlier via . Despite the reversed string orientation in , the operator that winds the (dual) spin measured by is again defined as in Eq. (9), and the relations (10) hold true with replaced by . Consequently there is an alternative duality transformation which satisfies


This alternative duality is simply a mirror version of our earlier definition: . Two applications thus again yields a simple lattice translation, i.e., and .

Table 1 summarizes the symmetry properties of the original Potts operators, their duals, and the generators , of , transformations. (Lattice translations are suppressed since they act trivially on all operators.) Note that parity swaps the two types of disorder operators and . This fact will become important in Sec. 5.

Table 1: Symmetry properties of various operators in the three-state Potts model. In the second and third columns and denote the and charges carried by each operator. (Recall that the eigenvalues of the generators and are given by and , respectively.)

2.3 Lattice parafermions

The easiest and most powerful way of analyzing the Ising model is to rewrite the transfer matrix/quantum Hamiltonian in terms of Majorana fermion operators. One elegant way of defining these lattice fermions is by taking the product of adjacent order and disorder operators [6]. The resulting fermionic Hamiltonian is of considerable interest in its own right (particularly when the fermions themselves comprise the physical degrees of freedom), providing a simple but profound example of a topologically nontrivial phase [28].

Parafermions in the three-state Potts model naturally generalize the Majorana fermions in Ising. These operators are obtained by performing a Fradkin-Kadanoff transformation [15, 29] analogous to the Jordan-Wigner transformation of the Ising model. Although the resulting lattice parafermions are similarly formed by products of order and disorder operators, a crucial difference arises: As discussed in Sec. 2.2, the Potts model allows for two distinct duality transformations and hence two types of disorder operators ( and ). Consequently, one can define two classes of lattice parafermions as follows:111We employ slightly different conventions for the lattice parafermions here compared to Ref. [18], to adhere more closely to conventions in the CFT literature. The parafermions denoted by in [18] are related to those above via , .




(To obtain the right-hand sides we employed the identity .) By examining the expressions on the right we see that these classes differ in the orientation of the strings: the “left” parafermion operators have a string going off to , while the “right” parafermions have a string emanating to . We call these operators semi-local because they involve strings that are related to symmetry generators and hence commute with Hamiltonian terms far from the string termination. These operators are not independent, as the ’s may be written in terms of the ’s and ; nevertheless both representations are very useful to retain since they transform into one another under certain symmetries as discussed below. In lattice parafermion language, the Hamiltonian (1) reads


Using properties of the original Potts-model operators and their duals, it is straightforward to derive the following relations,


Hence comprise generalizations of Majorana fermion operators. It is particularly noteworthy that the ’s do not anticommute off-site, but rather swapping their order acquires a phase factor of (or ). One can also use the symmetry properties of , to back out the transformations of under the two symmetries, as well as , , and . Table 1 summarizes the results. Notice that both parity and time-reversal swap the right and left parafermion representations, hinting that such operators are related to chiral fields in the continuum limit.

Understanding the properties of the parafermion operators under duality is essential to the following analysis. Deducing these properties requires some care because, for example, even though the two factors in act on separate sites and commute, their duals under overlap on site and fail to commute. One therefore must define how duality acts on products of operators. We require that

  • the duality operator be linear;

  • ;

  • if and are local operators, then duality is distributive over multiplication, i.e., ;

  • amounts to translation by one Potts-model site.

Using these conditions, we find that222For instance, our requirements on duality imply the following,

Hence . Setting yields , so that


which may be summarized succinctly as333Applying duality twice translates the lattice parafermion operators by two sites of the parafermion chain, i.e., , but this corresponds to translation by a single Potts site as required. . Applying similar logic for the left parafermions yields


Both transformations are obvious in hindsight, as swapping even and odd parafermion sites interchanges the and terms in the Hamiltonian (18), precisely as duality should. Determining the action of on left parafermion operators or on right parafermion operators is more complicated, but the full results appear in Table 1.

3 The parafermion conformal field theory

When the Hamiltonian (1) is tuned to the ferromagnetic critical point with —which we assume hereafter—its long-distance physics is described by a well-studied conformal field theory [1] known as the three-state Potts or parafermion CFT [10]. For brevity we often call this the CFT (although the theory exhibits a full permutation symmetry) [14]. Conformal symmetry is infinite-dimensional in 1+1 dimensions, and so the resulting constraints allow many properties to be understood exactly. In this section we review some properties of the CFT. We also describe how one can deduce the behavior of the many interesting fields under the Hamiltonian’s discrete symmetries.

3.1 Primaries

The CFT has central charge [10] and is a rational conformal field theory. The fundamental characteristic of a rational conformal field theory is that all the operators/states of the theory can be expressed in terms of a finite set of operators dubbed primary fields. That is, every state in the Hilbert state may be constructed by acting with a primary field and the generators of the (possibly extended) conformal algebra.

With appropriate boundary conditions, the left- and right-moving conformal symmetries are independent. When space-time is written in terms of complex coordinates, the corresponding generators are the holomorphic and antiholomorphic parts of the energy-momentum tensor, respectively. Thus one can decompose any field into representations of these independent symmetries. A given field therefore can be characterized by left and right scaling dimensions (, ), so that its total scaling dimension is while its conformal spin is . Local fields possess integer conformal spin and exhibit correlators that remain invariant under rotations; parafermions (and fermions for that matter) do not represent local fields in this sense.

The CFT supports additional spin currents denoted and . It is then useful to extend the usual conformal (Virasoro) algebra by these generators to obtain what is known as the “ algebra” [30]. This is the simplest non-trivial CFT with this symmetry algebra. Fortunately, for our purposes here the intricacies of the extended algebra are largely unimportant. All we need to know is the list of primary fields and that the field content can be generated by operator product expansions (OPEs) of the primaries with the left- and right-moving stress-energy tensors , , and with , . The “descendant fields” obtained in this fashion yield all the operators/states in the theory.

The chiral building blocks of the fields are known as primary chiral vertex operators [31]; we call these chiral primaries for short. All primary fields, both local and non-local, can be built from linear combinations of products of chiral and anti-chiral primaries. It is important to note that in any conformal field theory other than that of a free boson, this decomposition is non-trivial. Some of the fields are not simply the product of holomorphic and antiholomorphic fields; they are the sum of such products.

The six local primary fields of the CFT have long been known [32]. A set of local fields has the property that all their correlators remain unchanged under rotations of the system; i.e., their conformal spin is an integer. For a given CFT, there is not a unique such set. As with the Ising model [1], in parafermion theories one can form a set of local fields containing either the spin or the disorder field, but not both: the OPE of the two contains fractional powers of [cf. Eq. (25)]. By convention we view the spin field as local. This choice uniquely determines the set of local primaries, which we denote by 1, , , , , and .

There is of course the identity field, labeled . The spin fields and each have dimensions , and correspond to the scaling limit of the spin operators , described above. Charge conjugation interchanges them, so they form a doublet under the symmetry [33]. The energy field possesses dimensions . Perturbing the critical theory by this field describes the scaling limit of the three-state Potts model away from criticality with [10]. We denote the chiral primaries comprising as ; a full labeling includes the fusion channels [31], but we will not need this information. We likewise label the chiral primaries that are part of and by and , respectively, with the antichiral primaries labeled as , , and .

As opposed to the spin and energy fields, the remaining two primaries of conformal spin zero split into a simple product of holomorphic and antiholomorphic fields. It is thus convenient to denote them in terms of this product as and . The chiral components are the holomorphic “parafermion” fields , each with dimensions , and their antiholomorphic cousins which have dimensions [14]. While the parafermion fields are closely related to the scaling limit of the lattice parafermion operators described in the previous section, we will show later that this relationship is more subtle than one might naively anticipate. Identifying the precise connection between such lattice and continuum operators is the central goal of this work.

We stress that , , and are not physical fields, in the sense they cannot be realized separately by local or semi-local lattice operators in the three-state quantum Potts chain. For this reason expressions like are deceptive (though sometimes used in the literature). Instead we write , , , etc. On the other hand, and are physical fields arising from the operator product expansion of the spin and disorder fields, each of which can be realized on the lattice. Moreover, by taking appropriately twisted boundary conditions in the parafermion conformal field theory, states corresponding to the chiral parafermion fields do occur [32, 34]. This is why we can safely express the remaining local primaries as , and .

It is worth noting that there is still a finite number of primary fields here even if the symmetry algebra is not extended. These primary fields occur in the CFT with the “diagonal” modular invariant [32]. This CFT describes the continuum limit of another lattice model sometimes known as the tetracritical Ising model, or the model in the nomenclature of Ref. [35] (the three-state Potts model corresponds to ). The two CFT’s are related by an orbifold [36], which on the lattice amounts to a generalization of Kramers-Wannier duality [37]. A nice illustration of the relation between these CFT’s is given by constructing a field using the OPE of the energy field with the spin-3 current . Precisely,


where and are (known) constants [30]. The field carries dimensions (7/5,2/5); the notation indicates that its chiral parts are comprised of chiral vertex operators in a sector labeled by . Taking the OPE of with yields another field with dimensions (2/5,7/5), while the OPE with both and gives a field of dimensions (7/5,7/5). Since these fields cannot be constructed from with only the stress-energy tensor , they are primary fields in the tetracritical Ising CFT. In other words, is a descendant of when considering the entire algebra but is not a descendant under the Virasoro algebra. Indeed, it appears in the Kac table of the minimal conformal field theory [1]. Similarly, the field itself is a descendant, but not a Virasoro descendant of the identity field—which is to say one can clearly construct from combinations of and ’s, but not with ’s alone. Likewise, it also appears in the Kac table of primary fields.

Table 2: The chiral (holomorphic) primaries under the Virasoro algebra, together with their scaling dimensions . Note that under the extended algebra the fields and are descendants of and , respectively.

Table 2 enumerates the set of holomorphic chiral primaries and their scaling dimensions.

3.2 Parafermions

This subsection discusses the parafermion fields introduced above in greater depth. First, however, it is worth briefly digressing on the nature of physical fields. Not all physical fields need to be local; for example, the disorder field introduces a branch cut in space-time but proves to be very useful. The complete set of physical operators in the CFT includes semi-local fields that—like —contain a string that is invisible to the stress-energy tensor far away. In operator language, such strings represent ‘half’ of a symmetry generator, as exemplified by the lattice operators and (which together yield ). For identifying semi-local field combinations, it is useful to separate the holomorphic primary fields into two groups: contains , , and , while contains , , and . Likewise, we divide the antiholomorphic fields into analogous groups denoted and . The field (along with its descendants) is then semi-local if and , or if and . With the inclusion of semi-local fields the set of permissible primary fields expands beyond the six local primaries discussed in Sec. 3.1. Thus, combinations such as , , are all acceptable, but it appears that neither or are physical as they involve fields from different sets.444We note that if one builds a string from the charge-conjugation operator , there can be additional fields from the tetracritical Ising model, which are beyond the 6 chiral primaries of the parafermion CFT [33]. They will not be considered in this paper.

With this in mind we turn now to the parafermion operators, which are particularly important since, for example, they provide a simple way of understanding the appearance of topological properties [19]. To begin identifying the link between their lattice and continuum realizations, recall from Sec. 2.3 that the lattice parafermion operators were defined as products of order and disorder operators, following Ref. [15]. Likewise, the holomorphic and antiholomorphic parafermion fields are naturally defined in conformal field theory by taking the operator product of order and disorder fields [14],


If only descendants of parafermion fields comprised the ‘other terms’ above, then the identification of the lattice analogues of the parafermions would be obvious. A main message of our paper is that this is not so. Rather, both of these operator products contain another field coming from a different sector,


where , are constants and the ellipses denote subleading terms. The operators and are not discussed in Ref. [14], but there is no obvious reason why they should be absent. For instance carries the same and charge as the parafermion field . Moreover, they do indeed appear in the partition function with twisted boundary conditions (Eq. (B3) in Ref. [34]). Section 6 confirms the presence of these operators. This is particularly important given that has a smaller scaling dimension than and thus constitutes the most singular term in the OPE’s for the spin and disorder fields. Consequently the identification of the lattice analogues of the parafermions is subtler than is might first appear and requires a careful analysis of the discrete symmetries.

3.3 Symmetry properties of CFT fields

In Sec. 2 we reviewed the symmetries of the three-state quantum Potts chain, and the corresponding transformation properties of lattice operators in the theory (recall Table 1). Here we sketch how one can leverage those results to deduce the symmetry properties of CFT fields at criticality. This exercise, the outcome of which appears in Table 3, will prove instrumental in allowing us to complete our infrared/ultraviolet correspondence below. It is simplest to begin with the local spin and disorder fields and , which by definition are the continuum limits of the spin and disorder operators and (up to subleading corrections). All symmetry properties of those fields can therefore be immediately read off from those of the lattice operators. It turns out that this is the only link between the original Potts model and the CFT that we will need—the transformation properties of the remaining CFT fields can be inferred from consistency with OPE’s. Consider, for instance, Eqs. (25). It is obvious from these OPE’s that carries the same (known) and charges as , and with some care their transformations under , , , and the dualities follow as well. We also note that all the neutral fields () arise from OPEs between and and thus their symmetry properties can be inferred from those of the spin fields. The charge of also follows as it arises from the OPE of with the field. In total this procedure allows one to fill in all rows of Table 3.

Table 3: Partial list of physical fields in the CFT, group by and charges. The second column indicates which fields are primary while the third and fourth list the scaling dimension and spin of each. (A spin of means that the corresponding fields by themselves do not have well-defined spin; they can, however, be combined to yield fields with a spin of either or , e.g., and ) Remaining columns specify their symmetry properties. (We note that factors of , , and have been suppressed in the table.)

4 Identifying local fields with lattice operators

We are now ready to begin addressing the most important issue of this paper—finding lattice operators in the critical three-state Potts chain that in the continuum limit yield particular fields in the corresponding CFT. Here we find lattice realizations of local fields, which according to our previous definition are those that are realizable in terms of local lattice operators (without strings). This case is therefore simpler than that of the non-local operators undertaken in the next section. In what follows we construct lattice realizations of each local primary CFT field as well as the energy-momentum tensor and the dimension (2/5,7/5) and (7/5,2/5) operators and . We utilize the symmetries reviewed in Secs. 2 and 3 (together with integrability in one case) as a guiding principle and present density-matrix-renormalization-group (DMRG) simulations that verify our results; for some details on the method see the Appendix.

Section 3 discussed at length the local fields in the three-state Potts model that are both relevant and Lorentz invariant (rotationally invariant if the two dimensions are interpreted classically). They consist of the identity 1, the spin field and its conjugate , the energy field , and the parafermion bilinears and . In CFT language, these form the primary fields of the extended symmetry algebra with conformal spin zero (i.e., the scaling dimensions for the constituent right- and left-moving components match). As we will detail shortly, however, this list does not exhaust the local, relevant fields here; there are others with conformal spin 1.

We already asserted that the spin field represents the continuum counterpart of the lattice operator . Nevertheless it is worth making some additional remarks regarding the inevitability of this identification. Because is both the most relevant operator and breaks symmetry, essentially any lattice operator with the latter property will have as its leading component the spin field in the continuum limit (except in special cases where other symmetries preclude this field from appearing). As the nomenclature suggests, the simplest such operator is indeed , which when diagonalized measures which of the three states is present on site . The identity is also consistent with the operator product expansion of . Finally, one can verify using DMRG that at criticality for large as shown in Fig. 1, in agreement with the expansion .

Symmetry also allows one to identify the lattice analogue of the energy field —so named because when added to the action it changes the temperature in the two-dimensional classical Potts model. In the quantum chain, this simply corresponds to making . The energy field is neutral under both and ; even under , , and ; but odd under both dualities and . An appropriate combination of lattice operators sharing these symmetries is


Note that adding such a term uniformly to the critical Hamiltonian indeed shifts the system off of criticality, and into a either the ferromagnetic or paramagnetic phase depending on the sign of the coupling constant.

The preceding identification of the spin and energy fields with lattice operators is fairly obvious and has long been known [10, 38]. The same is not true for the parafermion bilinears. For reference, the analogous fermion bilinear in the Ising model is precisely the energy field—they are not independent perturbations unlike in the present context. This is a succinct reason why the Ising model can be solved for any temperature; Hamiltonians composed of fermion bilinears are typically easily solvable, and hence their correlators can be readily computed. In the three-state Potts and other parafermion models, a parafermion bilinear is a much more complicated object distinct from the energy field. In fact, it does not even preserve the symmetry (or more generally the symmetry in the -state Potts case with ). Moreover, it shares the the same symmetry properties as the spin field yet exhibits a larger scaling dimension. When constructing a lattice analogue one therefore must effectively subtract off the bits that would scale onto the spin field. Doing this by brute force seems prohibitively difficult.

Luckily, in all parafermion models integrability provides a means of finding a lattice analogue of the parafermion bilinears. Consider a perturbed parafermion CFT described by the continuum Hamiltonian


Here is the Hamiltonian at criticality while the term breaks the symmetry (except in the Ising case). This field theory is integrable and is analyzed in depth in Ref. [39]. Remarkably, there also exists an integrable -breaking deformation [40] of the corresponding critical self-dual lattice models [26]. These lattice Hamiltonians are known explicitly, and the quantum Hamiltonians can be extracted by taking a particular anisotropic limit. For the three-state Potts model corresponding to , taking the limit very close to criticality yields