Parafermionization, bosonization, and critical parafermionic theories

theories Yuan Yao 2, ∗ and Akira Furusaki 3 Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan Condensed Matter Theory Laboratory, RIKEN CPR, Wako, Saitama 351-0198, Japan Quantum Matter Theory Research Team, RIKEN CEMS, Wako, Saitama 351-0198, Japan Abstract We formulate a Zk-parafermionization/bosonization scheme for (1+1)-dimensional lattice models and field theories on a torus, starting from a generalized Jordan-Wigner transformation on lattices, which extends the Majorana-Ising duality when k = 2. The Zk-parafermionization enables us to investigate the critical theories of parafermionic chains whose fundamental degrees of freedom are parafermionic, and, surprisingly, we find that they cannot be described by any existing conformal field theory. The modular transformations of these parafermionic low-energy critical theories as general consistency conditions are found to be unconventional in that their partition functions on a torus transform differently from any conformal field theory when k > 2. Explicit forms of partition functions are obtained by the developed parafermionization for a large class of critical Zk-parafermionic chains, whose operator contents are distinct from their bosonic duals.

Parafermions naturally emerge in critical Z k -clock models [14,[20][21][22] which are made of generalized bosonic spins. The typical example is the well-known critical quantum transverse Ising model (k = 2), which is "equivalent" to a massless Majorana fermionic chain, i.e., a Z 2 parafermion [23]. However, the Majorana fermions and the Ising spins are significantly different in nature in that the Majorana fermions, which are local excitations in Majorana system and obey fermion statistics, are forbidden to exist in the local excitation content of the Ising chain, whose local excitations are bosonic. Thus, more precisely speaking, the massless Majorana fermion chain is actually equivalent to a proper stacking [24][25][26] of the critical Ising chain and a gapped Kitaev chain in its Z 2 -topologically nontrivial phase [27] providing the fermionic nature. Therefore, the critical theories of fermions (e.g., the critical theory of Majorana chains) are called "fermionic conformal field theories" to be distinguished from the critical bosonic theories, i.e., the so-called conformal field theories (CFTs). The classification [28] and the minimal models of fermionic CFTs [26] are of intense interest recently.
In fact, the construction of the general critical theory of the Z k -parafermionic systems still remains an open problem for k > 2, and the significant distinctions of parafermion statistics from boson and fermion statistics imply that these parafermionic critical theories with k > 2 may not be described by any existing CFT, e.g., bosonic or fermionic CFTs.
In this respect we note that there have been progress on the concept called "parafermionic CFTs" [21,22,29,30], which are field theories containing parafermionic operators, but still obeying the conventional modular invariance of bosonic CFTs. Indeed, they are found to describe the bosonic systems like Z k clock models [22,29,30], rather than genuinely parafermionic systems of our current interest whose fundamental degrees of freedom are parafermions. Therefore, the study on the (genuinely) parafermionic critical systems and the fundamental constraints on their low-energy effective theories is important and interesting in its own right and enriches the general framework and methodology of CFTs.
In this work, we investigate the fundamental properties of general critical theories of underlying parafermionic chains and their relation to parafermionic topological phases. As a main method of our study, we first develop a (1+1)-dimensional parafermionization together with a bosonization as its inverse to relate a parafermionic theory to a bosonic theory by a one-to-one correspondence. It can be regarded as an attachment construction using a nontrivial topological phase of a parafermionic chain [12,[14][15][16][17][18] generalizing the Kitaevchain attachment argument in k = 2 [25,26]. From this viewpoint, parafermionic chains and bosonic chains are expected to be indistinguishable locally since their differences result from this global topological factor. The parafermionization method also enables us to study the general properties of partition functions of critical parafermionic chains, which obey unconventional modular transformations, distinct from any existing bosonic or fermionic CFT. The source of this unconventional modular invariance is also interpreted from a lattice viewpoint, and we propose it as a general consistency condition on any critical theory of parafermionic systems. We also apply the parafermionization to explicitly calculate the partition function of a large class of critical parafermionic chains, from which their intrinsic fractional statistics can be read off.
The paper is organized as follows. In Sec. II, we first introduce a generalized Jordan-Wigner transformation to obtain the fermionization. Next, it is re-interpreted as the attach- Let us consider a quantum Z k -generalization of Ising degrees of freedom or k-spin, σ j and τ j , at each site j in a one-dimension lattice: where ω ≡ exp(i2π/k), and σ's and τ 's are bosonic in the sense that they all commute at different sites and the local Hilbert space at the site j is (minimally) k-dimensional. They are local operators that can be represented as where 1 k is the k × k unit matrix and the matrices other than 1 k appear only in the kdimensional local Hilbert space at the site j.
Such a generalized k-spin picture in a finite chain is, roughly speaking, equivalent to a parafermionic system by the following Fradkin-Kadanoff transformation [20] generalizing the Jordan-Wigner transformation: where, for a finite chain j = 1, 2, · · · , L, the product terminates at the most "left" site 1: Note that the product for γ 2j includes τ j . As a generalization of Majorana fermions (k = 2), the parafermionic degrees of freedom satisfy which we will take as a defining feature of the parafermionic chains. The Hilbert space of the parafermionic chain is defined to be the same as that of the following local k-spins which turns out to be the inverse of the Fradkin-Kadanoff transformation (4). However, the parafermionic and the dual bosonic models are intrinsically different in the definition of locality of operators [31], while they still share the same Hilbert space. When k > 2, there is one additional significant aspect in a finite chain with sites j = 1, 2, · · · , L as follows. The last relation in Eq. (5) signifies the absence of a unitary translation symmetry U such that U γ j U † = γ j+1 if j = 1, 2, · · · , 2L−1 and U γ 2L U † = γ 1 , because U , if existing, acting on both sides of γ 2L γ 1 = ω * γ 1 γ 2L would be inconsistent with γ 1 γ 2 = ωγ 2 γ 1 when ω * = ω, i.e., k > 2.
Therefore, the branch cut connecting γ 2L and γ 1 is not equivalently oriented with the other links between γ j and γ j+1 when k > 2. Nevertheless, we can still define a different translation transformation V transl by the help of the auxiliary bosonic spins conversely defined by the parafermions as in Eq. (6), satisfying with (σ L+1 , τ L+1 ) ≡ (σ 1 , τ 1 ), which completely determines, by the relation (4), the action of V transl on the original parafermionic chain {γ j } with a finite length 2L as: for the parafermions at j = 1, 2, · · · , L − 1, while, for an infinitely long chain j = · · · , −2, −1, 0, 1, · · · , the finite-length corrections (ω (k−1)/2 γ † 2 γ 1 = τ † 1 ) coming from the leftmost site disappear. Additionally, the boundary parafermions transform as, where Q f is a generator of global Z k symmetry defined below. Such a translation transformation also keeps the algebra (5) since, if we defineγ j = V transl γ j V † transl , thenγ j 's satisfy the same algebra (5) by replacing γ j →γ j . Thus, V transl is the appropriate translation transformation for parafermionic chains of a finite or infinite length, and it will be used when we formulate the parafermions on a general space-time torus later in Sec. IV and Appendix A.
We will use the subscript "b" to label the bosonic spin system (1) and "f " for the parafermionic system (5). In either picture, there exist global Z k symmetries: and they are the same Q f = Q b by Eq. (4). In this paper, we will focus only on parafermionic systems with such Z k symmetries. Now we derive the exact correspondence between the parafermionic chains and k-spin bosonic chains of finite length L under twisted boundary conditions. Without loss of generality (see the discussion later), we consider the nearest-neighbor coupling and compare the edge-closing term with the corresponding terms in the bulk: where a 1 is a mod-k integer-valued parameter specifying a Z N -twisted boundary condition of the k-spin chain imposed by and we have restricted to the bosonic Hilbert subspace by the Z k symmetry: where q b is defined modulo k. From Eq. (11) and Q f = Q b , we obtain the following mapping between the Hamiltonians: where H f (s 1 ) denotes the Hamiltonian of the parafermionic chain twisted by Q s 1 f and H b (a 1 ) for the bosonic chain twisted by Q a 1 b . For general edge-closing terms H edge with a finite range l edge L, we can define the unit twisting as acting , we expect that the parafermionic chain and its bosonic dual obtained by Eq. (6) are locally indistinguishable since, on an infinitely long chain j = · · · , −1, 0, 1 · · · without boundaries, the Hilbert-space dependent boundary twistings are irrelevant. We will see that the global aspect of the difference can be understood by a topological-phase attachment in Sec. III.
The partition-function correspondence with the inverse temperature β as the imaginary time can be obtained as by the Hamiltonian correspondence (14), the projection operator onto the Hilbert subspace the correspondence Q f = Q b , and where Z s 1 ,s 2 and Z a 1 ,a 2 are parafermionic and bosonic partition function under corresponding boundary-condition twistings, and we have inserted Q (1+s 2 ) f to twist the temporal direction as well. Here, the convention of "(1+s 2 )" is made so that (s 1 , s 2 ) reduces to the conventional Z 2 spin structure when k = 2, and we will call it a "paraspin" structure. Additionally, this convention is convenient in that s 1 and s 2 are on an equal footing in Eq. (15). In the following discussion, a 1,2 and s 1,2 are all defined mod k.

III. ATTACHMENT CONSTRUCTIONS AND BOSONIZATIONS AS INVERSE
To manifest the physical meaning of the parafermionization (15), we rewrite it as where the coefficient is defined as When k = 2, z s 1 ,s 2 a 1 ,a 2 reduces to the partition function of the nontrivial topological phase of the Kitaev chain as the Z 2 -Arf invariant [25,26], coupled with a background Z 2 -gauge field (a 1 , a 2 ).

A. Attachment of a gapped parafermionic chain
Here, we will argue that Eq. (19) has k-fold degenerate gapped ground states representing the dangling edge modes and having in H open is to normalize the ground-state energy density to be zero. The complicated polynomial summation will be useful for closing the chain as we will see later. In addition, the lattice model above is exactly solvable since the nearest-neighbor hoppings commute with each other and thus the energy of ground states can be saturated by which implies that, by Eq. (10), On the other hand, we can extract the charge sector with Q f = ω q f by the projection operator (16). Thus, in the ground-state sector, we have Since only the q f sector takes a nonzero positive value of P q f , we can gap out the other ground state(s) by the following Hamiltonian: With the last interedge coupling, the model is still exactly solvable and gapped. By a direct observation, Eq. (24) is exactly the Z k -twisted Hamiltonian by an s 1 = q f − 1 twisting, i.e., the Z k charge of the gapped unique ground state being q f = 1 + s 1 . Thus, we obtain the partition function as deeply into the gapped phase. After coupling it to a background Z k -gauge field (a 1 , a 2 ), the partition function is precisely that in Eq. (19).
Therefore, we can view the parafermionization (18)  thereby obtaining the corresponding properties of the critical parafermionic theories. As we will see in the next section, such an invertibility of the parafermionization will play an essential role when we investigate the requirement on modular-transformation condition of the effective field theories of critical parafermionic chains (5).

IV. MODULAR TRANSFORMATION OF CRITICAL PARAFERMIONS
In this section, we will consider general parafermionic chains at criticality, in which no relevant length scale exists except for the divergent correlation length in the thermodynamic limit. Conformal field theories are powerful tools to describe various universality classes of critical spin models, and their partition functions are modular invariant when the low-energy effective theories are formulated on a space-time torus [23]. We expect that the field theories of critical parafermionic chains also have a general modular transformation rule on a torus which is parametrized by a complex number, to be introduced below from the lattice viewpoint.
To investigate the properties under modular transformation, we first define a parafermionic chain on a discrete space-time torus and then take a proper continuum limit. The procedure is analogous to the bosonic case reviewed in Appendix A. The space-time torus is discretized by introducing a "lattice" spacing β 0 along the imaginary time and the (spatial) lattice spacing a 0 , which implies that the system length is La 0 ; see FIG. 1. The partition function twisted by (s 1 , s 2 ) in the space-time torus, similarly to Eq. (A5), is defined as where T is the time-ordering operator and V transl is defined by Eq. (8) with the additional "1" in the exponent (1+s 2 ) to reproduce the spin-structure convention when k = 2 as mentioned before. The imaginary-time evolution is induced by the time-dependent Hamiltonian during the time t ∈ [0, (β/β 0 )β 0 ]: Similarly, we have also defined a time-dependent Z k generator: Then, by the bosonization transformation (6) and its resultant Hamiltonian correspondence (14), we obtain the correspondence of partition functions where Z latt a 1 ,a 2 is the partition function of its bosonic dual formulated on the torus in Eq. (A5) rewritten as We consider the following standard continuum limit [32][33][34][35]: β 0 , a 0 → 0 with β 0 /a 0 , β and La 0 fixed, and various lattice couplings are scaled to keep the correlation length ξ fixed, where ξ has the same length dimension as a 0 and the (dimensionless) lattice correlation length is ξ/a 0 . Then we have the following two dimensionless parameters which are invariant in the continuum limit: defining the complex number τ in Eq. (28). Without changing the low-energy physics (e.g., long-distance correlations), we can take the critical parafermionic chain H f in the thermodynamic limit L → ∞ (ξ ∼ La 0 and ξ/a 0 → ∞), to be at the corresponding infrared renormalization-group (RG) fixed point. Its bosonic dual H b is also at the critical RG fixed point, since H f and H b are locally indistinguishable without boundaries, e.g., in the thermodynamic limit. Up to some non-universal factor [34], the critical bosonic partition function Z latt a 1 ,a 2 in Eq. (33) converges to that of CFTs in the continuum limit as where q ≡ exp(2πiτ ) andq ≡ q * with τ ≡ τ 1 + iτ 2 , and L b 0 (a 1 ) andL b 0 (a 1 ) are a 1 -twisted conformal-transformation generators of the bosonic CFT with a central charge c [23]. It follows from Eq. (32) that the parafermionic partition function Z latt s 1 ,s 2 in Eq. (29) converges to in the continuum limit. Indeed, the underlying lattice system with the partition function Z s 1 ,s 2 (τ ) is at a critical point since it depends only on the dimensionless ratio τ 2 = β/La 0 or τ 1 = (β/Lβ 0 ) rather than any length scale. In addition, we know that the partition function of a critical bosonic system as a CFT, in the absence of Z k or gravitational anomaly, obeys the modular-invariance condition due to the emergent large-diffeomorphism invariance in conformal invariant field theories [36]: Z a 1 ,a 2 +a 1 (τ + 1) = Z a 1 ,a 2 (τ ); (38a) Z −a 2 ,a 1 (−1/τ ) = Z a 1 ,a 2 (τ ).
By the help of the invertibility (26), we derive the modular transformation of the partition function of the critical parafermionic chain as where These two transformations reduce to Z s 1 ,s 2 +s 1 (τ + 1)| k=2 = Z s 1 ,s 2 (τ )| k=2 ; (41a) in the case of fermionization when k = 2, in agreement with the bosonic case (38). Thus the fermionic spin structure appears as a Z 2 -gauge field on the torus. However, for general k > 2, the paraspin structure no longer behaves as Eq. (41) under the modular transformation although it plays a similar role of Z k twisting on a torus.
To understand this unconventional transformation when k > 2, let us consider, for simplicity, τ ∈ iR as in FIG. 2. We start from a T transformation, τ → τ + 1. In addition to the Z k twisting, the solid arrows in FIG. 2 (a) also carry the algebraic information in that they connect the last site 2L and the "next" site 1, which, according to the algebra in Eq. (5), in a distinct way from the other links connecting two neighboring γ j and γ j+1 when ω = ω * . Such a piece of algebraic information is irrelevant or invisible when k = 2 because ω * | k=2 = ω| k=2 . When k > 2, the partition function on the T -transformed space-time torus cannot be identified with Z s 1 ,s 2 +s 1 (τ + 1) due to the additional appearance of the algebraic change on the link circled in FIG. 2 (a), crossed by the temporal arrow during the continuum limit a 0 , β 0 → 0, although the s 1 -twisting information on that link can be fused with s 2 to be (s 2 + s 1 ) by a temporal Z k -gauge transformation. For an S transformation τ → −1/τ , we assume that there are also solid arrows along the temporal direction to represent similar branch cuts so that the space and time are on the same footing in order to have a S-transformation property since S transformations partially have the effect of interchanging space and time. However, no matter how these temporal arrows are oriented, either the spatial or temporal orientation will be reversed by the S transformation as in FIG. 2 (b), for example. That the orientation is relevant when k > 2 makes the current S-transformation rule (40b) unconventional as well. On the other hand, the cases of k = 2 do not have these problems since the orientation of the closing link is irrelevant due to ω| k=2 = −1 = ω * | k=2 there. We will also see in the next section that no matter how we adjust the reference "periodic" point of s 1,2 = 0, the modular transformation (39) cannot be the same as the conventional form when k > 2.
Furthermore, since we consider general critical parafermions without reference to Hamiltonian, we conclude that the modular transformations (39) and (40) as the modular invariance requirement and consistency conditions for parafermionic systems (5) at criticality.
In this section, we will show that if the partition function z s 1 ,s 2 of an invertible topological field theory with a Z k -paraspin structure obeyed the traditional modular invariance by T and S, then it is always trivial (equals to 1) when k ∈ 2Z + 1, or behaves as an Z 2 -Arf invariant, in addition to the trivial phase on a torus when k ∈ 2Z, contradicting our result of Eq. (25).
Furthermore, the partition function Z s 1 ,s 2 would transform conventionally as Eq. (41) if the attached topological phase z s 1 ,s 2 on the bosonic theory Z a 1 ,a 2 obeys the modular invariance (42). In this sense, the traditional modular invariance is inapplicable to our parafermionic chains.
We start with a definition of modular-transformation orbits: in addition to the Z k property of the paraspin structure: We first prove that where "gcd" denotes the (non-negative) greatest common divisor. It can be shown as follows.
Then, by Dirichlet's theorem on arithmetic progressions, there exists an integer N > k such that N s 1 gcd(s 1 , s 2 , k) is a prime number thereby gcd N s 1 gcd(s 1 , s 2 , k) + s 2 gcd(s 1 , s 2 , k) , k gcd(s 1 , s 2 , k) = 1 and then gcd (N s 1 + s 2 , k) = gcd(s 1 , s 2 , k). In a short summary, the modular invariance requires the following form on the partition function of the topological field theory z s 1 ,s 2 : Since the partition function is defined up to a Z k phase, let us impose a normalization: which implies z 1,0 = z 0,−1 = 1. Therefore, the value of z 0,0 exactly gives the Z k charge of the untwisted sector.
When k ∈ 2Z, we can have two phases: z (s 1 ,s 2 ) = 1 and a nontrivial one satisfying which gives the result as reduced to the Arf invariant when treating s 1,2 as modulo 2, consistent with [37]. In conclusion, we could have one single Z 2 nontrivial phase only if k ∈ 2Z.
Therefore, when k > 2 especially k ∈ 2Z + 1, traditionally modular invariant Z k -paraspin topological field theories (42) cannot correctly describe the low-energy property of gapped Z k -parafermionic chains and modular invariance for parafermionic systems should be modified to be Eq. (39).

VI. FRACTIONAL STATISTICS
It is well known that the fermionization (k = 2) of the critical quantum transverse Ising model yields the massless Majorana fermions obeying fermion statistics rather than bosonic statistics, although the local operators of the Ising model are all bosonic. As we will see, when k > 2, the conformal spins of the fundamental field operators in the critical parafermionic theory can be neither integral (bosonic) nor half-integral (fermionic).

A. Critical parafermion dual to a critical three-state Potts model
As the simplest nontrivial bosonic example k = 3, let us consider the critical 3-state Potts model at its ferromagnetic self-dual point with a twisted boundary condition: Its (fixed-point) critical partition function under the periodic boundary condition is where χ h is the Rocha-Caridi character of the Virasoro representation with dimension h and its form under general twisting can be seen later in Eq. (65). Although the complete primary operator content of the Potts model contains parafermionic operators, they do not enter into the local-operator content determining the partition function Z 0,0 Potts , namely any character product χ h i χ * h j in the expansion of Z 0,0 Potts above satisfies the bosonic statistics: It implies the Potts model is bosonic in nature, as it should be.
Interestingly, all the operators above have spins that are multiples of 1/3, while the characters like χ 2/3 χ * 7/5 with anomalous spins are not allowed to occur in the partition function.
It is consistent with the fact that the fundamental degrees of freedom are Z 3 parafermions.
The partition functions under other paraspin structures are given in Appendix C. Besides the bosonic operators like the identity operator Φ 0,0 and the energy operator Φ 3,3 , the operator content also contains the fundamental parafermionic field Φ 2/3,−2/3 , which is genuinely parafermionic with fractional statistics. Furthermore, this parafermion operator Φ 2/3,−2/3 is even not mutually local with itself due to its multivalued two-point correlator calculated in traditional CFTs [21,22]. In addition, the field of Φ 2/3,2/3 which is mutually local (by the calculations in traditional CFTs) with the existing Φ 2/3,−2/3 does not show up above. However, the state-operator correspondence (see Appendix B) of traditional bosonic/fermionic CFTs implies that all the operators in Eq. (64) are local/mutually local [36]. Therefore, the concept of (mutual) locality of the critical parafermionic field theories is sharply distinct from the bosonic/fermionic CFTs.

B. Critical parafermion dual to Z k -clock models
General critical Z k -clock models are expected to realize the following partition function [21,22]: where η(τ ) is the Dedekind's η function and the integral multiplicity L l,l (0 ≤ l,l ≤ k) is given by the modular invariant solutions L l,l χ h l χ * hl of the SU(2) current algebra system [23]. Here, c l m (τ )'s are string functions derivable from the cosetŝu(2) k /û(1) construction satisfying c l m = c l m+2k = c l −m = c k−l k+m with c l m =l mod 2 ≡ 0 [23], and |η(τ )| 2 c l m (τ )cl * m−2a 1 (τ ) can be seen as the partition function of the primary field with the Z k ×Ẑ k charge of (m − a 1 , 0).
The self-duality can be proven by the invariance under Z k gauging, which generalizes the Kramers-Wannier duality [33,38,39] in that gauging Z k effectively neutralizes the Z k charge or eliminates the nontrivially Z k -charged primary fields, e.g., spin order parameters, from the local-operator spectrum which thus only contains purelyẐ k -charged fields, i.e., various disorder parameters. The parafermionization of Eq. (65) by Eq. (37) yields the partition function in the universality class of a large class of critical Z k -parafermionic chains, where h l m is the conformal dimension related to c l m (τ ) [21,22]: in which m is set to stay in the interval [−l, 2k − l) by its Z 2k cyclicity given before.

VII. CONCLUSIONS
In this work, we propose a one-dimensional Z k -parafermionization/bosonization scheme on critical parafermionic chains starting from a generalized Jordan-Wigner transformation.
It is shown to be equivalent to an attachment construction of attaching a nontrivial topological phase of a gapped parafermionic chain, generalizing the conventional fermionization/bosonization. Such a parafermionization enables us to study the critical parafermionic system whose fundamental degrees of freedom are fractionally statistical fields beyond bosons and fermions. We find that the critical theories of the parafermions generally obey uncon- where T is the time-ordering operator. Such an evolution and the corresponding partition function Z latt r,s can be visualized on the discrete space-time lattice as in FIG. 3. If the lattice model H b (σ 1 , σ 2 , · · · ) on an infinite chain is at criticality, i.e., if the correlation length diverges ξ/a 0 → ∞ in the unit of the lattice constant a 0 as the system length L → ∞, then its universal properties are described by its lattice RG fixed point. Then, without changing the low-energy physics, e.g., long-distance correlations, we take our lattice Hamiltonian H b (σ 1 , σ 2 , · · · ) to be at the corresponding infrared RG fixed point, which is described by a CFT after the following continuum limit is taken: a 0 → 0, while β 0 /a 0 , β and La 0 fixed, and various coupling constants in the lattice Hamiltonian are scaled to keep the correlation length ξ ∼ La 0 fixed [40]. We define the following ratios: which are invariant during the continuum limit. Then, up to some non-universal factor, the partition function Z latt r,s converges to the partition function of the corresponding CFT where q ≡ exp(2πiτ ) andq ≡ q * with L b 0 (r) andL b 0 (r) the r-twisted conformal-transformation generators of the corresponding bosonic CFT with a central charge c [23]. The partition function Z r,s (τ ) has no relevant length scale and it only depends on the dimensionless number τ , which reflects the criticality of the lattice model.
Additionally, in the path-integral formalism of Z latt r,s at the fixed point, the local degrees of freedom {σ j } are coarse-grained to be φ(t, x) with x = ja 0 , e.g., φ(t, x) being the configuration of local spin density in the Ising model. The symmetry operation r mod k acting on φ(t, x) is denoted as r φ(t, x) ≡ ω r φ(t, x), and the translation transformation V transl acts on φ as |φ(t, x) → |φ(t, x − a 0 ) on the wave functional, where the minus sign should be noted. The r-twisted Hamiltonian and the operators ( (A5) correspond to the following boundary condition in the path integral: which is translated to, if we define z = (x + it)/(La 0 ), as the boundary condition in the path-integral functional integration: reproducing Eq. (A8) for CFTs.

Appendix B: Relation between local operators and partition functions on a torus
In this part, we will explain how to obtain the conformal dimensions of local operators in a CFT from its partition function on a torus by a state-operator correspondence as follows.
Let us consider a CFT with a fundamental field operator ϕ(z,z) and start from the complex plane {z ∈ C}. We insert a local operator Φ(z,z) made of ϕ(z,z) at the origin of the complx plane and denote the conformal spin of ϕ(z,z) as S ϕ = h ϕ −h ϕ , which is integral or half-integral if the CFT is bosonic or fermionic, respectively. By a path integral, this insertion defines a quantum state |Ψ on a unit circle: S 1 ≡ {z = exp(iθ) : θ ∈ [0, 2π)} with the wave functional as: Dϕ exp{−S[ϕ]}Φ(0, 0)| ϕ=ϕ 0 on S 1 , where S[ϕ] is the action and the path integral is performed on the disk D 2 ≡ {|z| ≤ 1} with the boundary condition ϕ = ϕ 0 on S 1 . Since Φ(0, 0) is a local operator, it does not introduce any branch cut for ϕ fields, i.e., PBC still held: ϕ exp(2πi)z, exp(−2πi)z = ϕ(z,z).
To evaluate the conformal dimensions (h Φ ,h Φ ), we simply act L 0 orL 0 on |Ψ , for example: Then we apply the following conformal coordinate transformation: to transform the theory from D 2 to a half-infinitely long cylinder parametrized by {w = t + ix|x ∼ x + 2π, t ∈ (−∞, 0]}. Then, both sides of Eq. (B3) become: It implies that the state |Ψ is obtained by a path integral on a half-infinitely long cylinder, but with its spatial boundary condition twisted by exp(i2πS ϕ ).
On the other hand, the partition function on a torus twisted by a 1 in the spatial direction, but without twistings along the temporal direction is Z a 1 ≡ Tr exp 2πτ L cyl 0 (a 1 ) +L cyl 0 (a 1 ) , for a purely imaginary τ ∈ iR. Then, we take a 1 = kS ϕ when exp(i2πS ϕ ) twisting is realizable by a Z k symmetry, as is the case with our paper. By the definition of the characters χ h 's, we obtain that (h Φ ,h Φ ) must be the conformal dimensions of one of the operators in the conformal family associated with the highest weight (h,h) appearing in (A h,h = 0) Since Φ is an arbitrary local operator, the local-operator content of the theory is composed by the operators in the conformal families with the heighest weight (h,h) for nonzero A h,h .
For the massless Majorana fermion, the fundamental degree of freedom is the real fermion ψ(z) with S ψ = −1/2. Therefore, we need to choose s 1 = 2S ψ = −1 and 1 + s 2 = 0, in the fermionic partition function, where the additional "1" in the "1+s 2 " above is due to our convention (Q f ) 1+s 2 of the operator insertion along the time direction for the parafermionic partition function (15).