Thermoelectric properties of Coulomb-blockaded fractional quantum Hall islands

We show that it is possible and rather efficient to compute at non-zero temperature the thermoelectric characteristics of Coulomb blockaded fractional quantum Hall islands, formed by two quantum point contacts inside of a Fabry-Perot interferometer, using the conformal field theory partition functions for the chiral edge excitations. The oscillations of the thermopower with the variation of the gate voltage as well as the corresponding figure-of-merit and power factors, provide finer spectroscopic tools which are sensitive to the neutral multiplicities in the partition functions and could be used to distinguish experimentally between different universality classes. We also propose a procedure for measuring the ratio r=v_n/v_c of the Fermi velocities of the neutral and charged edge modes for filling factor \nu=5/2 from the power-factor data in the low-temperature limit.

Introduction: Investigating the thermoelectric properties of strongly correlated two-dimensional electron systems is expected to reveal important information about the structure of the neutral excitations [1] and other specific characteristics of their universality classes. The Coulomb-blockade (CB) patterns of fractional quantum Hall (FQH) islands in states with different universality classes appear to be indistinguishable at low temperature [2], while their thermoelectric properties are different. The thermoelectric conductance of candidate FQH states at filling factors ν = 2/3 and ν = 5/2 in a CB island have been computed [1] from the conformal field theory (CFT) data of the underlying effective field theories for the edge excitations. The thermopower, known also as the Seebeck coefficient, has been previously computed for metallic quantum dots [3,4] indicating to be a better spectrometric tool than the transport coefficients, while showing the same periodicity as the CB conductance peaks. So far, the computations of thermopower for CB islands in quantum Hall states have been limited to the case of integer ν where it is similar to that of the metallic islands. Recently, the thermopower for the ν = 1/m Laughlin FQH states has been computed [5] showing that it is similar to the integer quantum Hall states, except that the oscillation period in the gate voltage is extended from 1 to m.
The chiral edge excitations determining the topological order of the FQH universality classes have been successfully described by CFTs [6][7][8][9]. In this Letter we will show how to use the CFT partition function for a general chiral FQH state, as a thermodynamic potential for the experimental setup of [1,10], in order to calculate the thermopower for a CB island, or quantum dot (QD), at non-zero temperature. Measuring the power factor computed from the thermopower could experimentally distinguish between the different ν = 5/2 states.
The thermopower is defined [3] as the potential difference V between the left and right leads of the single-electron transistor (SET), formed by the CB island and the gates [3], when their temperature differs by ∆T = T R − T L T L , under the condition that the electric current I is 0. It is usually computed [3] as the ratio S = G T /G of the thermal and electric conductances, however, it can be alternatively expressed in terms of the average energy ε of the electrons tunneling through the Coulomb-blockaded quantum dot [3] where e is the electron charge and T is the temperature of the CB island. The alternative approach is more suitable for disconnected systems such as the SET, because the conductances G T and G are both zero in the CB valleys [3], so it is not appropriate to put G in the denominator of (1), while the voltage V is non-zero and can be measured experimentally [11]. The knowledge of the thermopower and the conductances allow us to compute [12] the thermoelectric figure-ofmerit ZT = S 2 GT /G T for the CB island and the corresponding power factor P T , which is defined in terms of the electric power P generated by ∆T as where R = 1/G is the electric resistance of the CB island. The power factor P T seems to be measurable directly by applying an AC voltage of frequency f 0 /2, to the side gate, while measuring the thermoelectric current at frequency f 0 [13]. It is worth stressing that the sharp zeros of the power factor P T , at very low temperatures, mark precisely the positions of the maximum of the CB conductance peaks and could be used to determine experimentally the ratio r = v n /v c of the Fermi velocities v n and v c for the neutral and charged modes respectively, see the supplemental material. On the other hand the ratio of the two maxima of the power factor P T around each CB peak, just like the ratio of the two extrema of G T in [1], appears to be rather sensitive to the presence of neutral degeneracies in the edge modes due to finite-temperature asymmetries in the conductance peaks [10]. Moreover, it has been shown that P T and S could be significantly enhanced in the single-electron-transistor setup due to the Coulomb blockade [14]. Therefore, P T and S could eventually be used to distinguish between different FQH universality classes having the same CB peak pattern [2]. For large CB islands, the total energy of E QD of the QD with N electrons is defined within the Constant Interaction model [15] as where N 0 is the number of electrons in the bulk of the QD and N − N 0 = N el is the number of electrons on the edge, E i (B), i = 1, . . . , N 0 , are the energies of the occupied single-electron states in the bulk of the QD. Since we intend to use the CFT partition function of the CB island as a Grand potential the average · · · β ,µ is taken within the Grand canonical ensemble for the FQH edge at inverse temperature β = (k B T ) −1 and chemical potential µ and H CFT β ,µ is the Grand canonical average of the Hamiltonian on the edge. At low temperature the number of electrons on the QD is quantized to be integer and the chemical potential of the QD with N electrons is denoted by µ N . We will be interested in the sequential tunneling regime [3,10], when the electrons in the SET tunnel through the QD one by one, which is the dominating mechanism at low temperature for small conductances between the CB island and the "leads", like in [3,10]. The leads are assumed to be large FQH liquids with energy spacing much smaller than the energy spacing ∆ε of the CB island. In this case, within the linear response approximation for low temperature differences ∆T , the average energy of the electrons tunneling through the CB island could be computed as the difference between the average total energy of the QD with N + 1 electrons and that for N electrons in presence of AB flux φ (or, equivalently, normalized gate voltage) The variation of the side-gate voltage V g induces (continuously varying) "external charge" eN g = C g V g on the edge [15,16], which is equivalent to the AB flux-induced variation of the particle number N φ = νφ , so that we can use instead the AB flux φ determined from C g V g /e ≡ νφ with φ = (e/h) (BA − B 0 A 0 ), where A 0 is the area of the CB island and B 0 is the magnetic field at V g = 0. Using the AB flux φ instead of the gate voltage is convenient because φ can be interpreted mathematically as a continuous twisting of the u(1) charge of the underlying chiral algebra [17,18], which is technically similar to the rational (orbifold) twisting of the u(1) current [19]. The Grand potential Ω(β , µ) = −β −1 ln Z(β , µ), for the FQH edge states, is defined as usual [20] in terms of the Grand canonical partition function Z(β , µ) = tr H e −β (H CFT −µN el ) , where H CFT = ∆ε(L 0 − c/24) is the Hamiltonian for the edge states expressed in terms of the zero mode L 0 of the Virassoro stress tensor [18] (with central charge c). The Luttinger liquid particle number operator N el = − √ νJ 0 is expressed in terms of the zero mode J 0 of the normalized u(1) current and ∆ε =h2πv c /L is the non-interacting energy spacing in the QD. The Hilbert space H of the FQH edge states, over which the trace is taken, depends on the type and number of the localized FQH quasiparticles in the bulk.
When the magnetic field B or the area A or the gate voltage V g are changed from their initial values, B 0 , or A 0 respectively, the partition function Z(β , µ) = Z(τ, ζ ) is modified by shifting the modular parameters, as proven in [17] (see Eq. (34)) where the modular parameters τ and ζ , used to construct (rational) CFT partition functions [18], are related to the temperature T and chemical potential µ by τ = iπT 0 /T , T 0 = hv c /πk B L, ζ = (µ/∆ε)τ. To understand Eq. (5) physically we recall the Aharonov-Bohm relation: the electron field operator ψ el (z), where z = e iϕ is the electron coordinate on the edge circle, is modified in presence of AB vector potential A as ψ A el (z) = z −φ ψ el (z), where φ is the dimensionless AB flux (see Eq. (26) in [17]). The AB flux changes the boundary conditions of all charged particles operators and the adiabatic variation of the flux changes the Hilbert space of the edge excitations by a well known procedure called twisting in the conformal field theory [17][18][19]. In particular, the partition function Z(β , µ) changes as [17] where H is the untwisted Hilbert space, corresponding to φ = 0, the thermodynamic parameters β and µ are independent of φ , and all flux dependence is moved to the twisted operators of energy H CFT (φ ) and charge imbalance [16] N imb (cf. Eqs. (32) and (33) in [17]) The ultimate effect of the AB flux on the partition function Z(β , µ) is shifting ζ as in (5) or, equivalently, µ → µ + φ ∆ε.
It follows from (6) that ∂ Ω/∂ µ = − N imb β ,µ and taking into account (7) we find that the thermodynamic average of the electron number in presence of AB flux (setting µ = 0) is where Ω φ = −k B T ln Z φ . This general construction of the electron number operator average in presence of AB flux allows us to compute also the flux dependence of the conductance of the Coulomb island according to Eq. (10) in [10] and Eq. (8) at µ = 0, i.e., Next, we can compute the average quantum dot energies with N electrons on the edge at temperature T and chemical potential µ in presence of AB flux from the standard Grand canonical ensemble relation [20] Thermopower for general FQH states: The Grand canonical partition function Z(β , µ) for a general FQH disk can be written as [21][22][23][24] Z l,Λ (τ, ζ ) = where n H and d H are the numerator and denominator of the filling factor ν = n H /d H while ω is the neutral topological charge of the electron operator, which is always non-trivial when n H > 1. The u(1) partition function K l (τ, ζ ; m) for the charged part is completely determined by the filling factor ν and coincides with that for a chiral Luttinger liquid with a compactification radius [21] R c = 1/m, in the notation of [17,18] where q = e 2πiτ , η(τ) = q 1/24 ∏ ∞ n=1 (1 − q n ) is the Dedekind function [18] and CZ(τ, ζ ) = exp(−πν H (Im ζ ) 2 /Im τ) is the Cappelli-Zemba factor introduced to restore the invariance of K l (τ, ζ ; m) with respect to the Laughlin spectral flow [21]. The u(1) charge label l in Z l,Λ is determined by the total electric charge of the localized quasiparticles in the bulk Q el (bulk) = l/d H , while the weight Λ is determined by the total neutral topological charge of the quasiparticles localized in the bulk. The partition function ch Λ (τ ) represents the neutral edge modes corresponding to the total neutral topological charge Λ in the bulk. The modular parameter τ = rτ with r = v n /v c is modified in order to take into account the observation [1,25] that the Fermi velocity v n of the neutral edge modes might be smaller than v c . The * in Z l,Λ denotes the fusion product [18] of the topological charge Λ with the (smultiple) neutral topological charge ω of the electron [17,22]. The electric charge of the edge excitations, with quantum numbers n and s, encoded in Z l,Λ is Q l Λ (n, s) = l/d H +s+n H n. The neutral topological weight Λ and the electric charge l have to satisfy a general Z n H pairing rule, see Eq. (19) in [22].
Next we can introduce the AB flux into the partition function Z l,Λ by the shift (5) and then move the flux and chemical potential dependences into the charge index of the K function (9), due to the identity [17] In order to compute the average tunneling energy (4) we need to specify the parameters µ N and µ N+1 corresponding to QD with N and N + 1 electrons, respectively. To this end we emphasize that the parameter µ entering (6) is not the true chemical potential, except at zero gate voltage, because it is not coupled to the electron number operator but to the charge imbalance [16]. As can be seen in the supplemental material, when the bulk electron number is N 0 = n H n 0 , where n 0 is a positive integer, the partition function (10) is independent of the bulk component of µ and the edge components of µ N and µ N+1 can be chosen as µ N = −∆ε/2 and µ N+1 = ∆ε/2, being independent of the neutral contributions to the electron energy, or of the ratio r = v n /v c , or even of ν. Thermoelectric properties of a ν = 5/2 CB island: We will illustrate the general approach to thermopower using as an example a ν = 5/2 CB island, assuming that only the higher ν = 1/2 edge is strongly backscattering as in Ref. [1]. So far only the fractional electric charge e/4 of the fundamental quasiparticles has been confirmed experimentally [26][27][28] and this is consistent with several FQH candidates: the Pfaffian MR state [29], the anti-Pfaffian [30,31], the 331 Halperin state [32] and the u(1) × SU(2) 2 state [33]. In the rest of this Letter we will use the CFT partition functions to compute numerically the thermopower, conductance, thermal conductance and power factor for these FQH islands. The neutral functions in (10) for the Pfaffian state are ch 0, by plotting the thermopower for the Pfaffian state without bulk quasiparticles with r = 1 is given in FIG. 1, which is a central result in this work. Two important characteristics of the thermopower for fractional quantum Hall states have to be emphasized: when the gate voltage approaches a position of a CB peak the thermopower vanishes at the maximum of the peak, just like it does for metallic islands [3]; second, at the centers of the CB valleys thermopower decreases rapidly (jumping discontinuously at T = 0) crossing the x-axis exactly at the center of the valley like in metallic islands [3]. This modified saw-tooth shape of the thermopower is similar to that in superconducting SET [34]. The neutral partition functions for the 331 state are expressed in terms of (9) as ch 0 (q) = K 0 (τ, 0; 4) and ch ω (q) = K 2 (τ, 0; 4), while for the SU(2) 2 they are expressed in terms of the functions from (14.183) in [18]  The oscillations of the electric and thermal conductances G and G T = G.S for all paired FQH states with odd number of quasiparticles in the bulk are given in the supplemental material. The partition functions in this case are reduced to K l+1/2 (τ, ζ ; 2) ≡ K 2l+1 (τ, 2ζ ; 8) + K 2l−3 (τ, 2ζ ; 8) because the neutral partition functions are degenerate ch q.p. ≡ ch el * q.p. , see the supplemental material. Therefore, the neutral degrees of freedom are completely decoupled from the charged ones and the thermoelectric properties are independent of r and basically the same as for the g = 1/2 Luttinger liquid. This includes the anti-Pfaffian state, with odd number of bulk quasiparticles, as well, cf. [1]. The conductances's peaks are equally spaced, with period ∆φ = 2, and completely symmet-ric for all temperatures, showing similar results as in [1].  2), together with the peaks of the conductance (right-Y scale). The power factor shows sharp dips corresponding precisely to the maxima of the conductance peaks. Notice that this figure is qualitatively similar to Fig. 3c in Ref. [13], which suggests that the method used there for measuring the thermoelectric current might be convenient for measuring the power factors of for the ν = 5/2 FQH state as well.
Finally in FIG. 4 we plotted the power factors for those states with even number of bulk quasiparticles. The power factor for the (even bulk q.p.s) Anti-Pfaffian state is reduced by an order of magnitude, see the supp. material. Again the sharp dips of P T mark precisely the maxima of the conductance peaks and can be used to determine precisely their positions. Because the distances between these dips depend on r = v n /v c they could be used to measure r for the ν = 5/2 island with even number of bulk quasiparticles as follows: if we denote the shorter distance in gate voltage between neighboring dips of P T by ∆V 1 and the longer one by ∆V 2 and we set x = ∆V 2 /∆V 1 then 2(x − 1)/(x + 1) → r when T → 0.
Furthermore, the apparent asymmetries in P T might allow to distinguish between the different states by measuring the ratio P max 1 /P max 2 between the maxima of P T surrounding the first CB conductance peak with φ > 0. The plot of these ratios as functions of T are shown in FIG. 5. Measuring P max 1 /P max 2 , as in [13], at three different temperatures, would be sufficient to choose one paired state among the others, see the supp. material.
Conclusion: We demonstrated that the CFT partition functions of Coulomb blockaded FQH islands can be efficiently used to calculate the thermoelectric characteristics of the islands which could eventually distinguish between inequivalent FQH universality classes with similar CB peaks patterns, at finite temperature even when v n /v c < 1. Introducing AB flux φ = e(BA − B 0 A 0 )/h through the AB relation modifies the electron filed by ψ el (z) → z −φ ψ el (z), where z = e iϕ is the electron coordinate on the edge circle, see Sect. 2.8 in [17]. This twisting of the electron operator can be implemented by a conjugation [17] with a flux-changing operator U β (here β ∈ R is the twist parameter and should not be confused with the inverse temperature) defined by its commutation relations with the Laurent modes of the normalized [S13] charge density J(z) = ∑ n∈Z J n z −n−1 , namely [J n ,U β ] = βU β δ n,0 . It is not difficult to see that U β acts on the electron as ψ el (z) → U β ψ el (z)U −β = z −φ ψ el (z) when the twist is β = − √ νφ [17]. Then, the twisted electric charge is obtained by the same action [17] J el 0 → U β J el 0 U −β = J el 0 + νφ and the twisted Hamiltonian, which is defined as [17] H CFT → U β H CFT U −β = H CFT +∆εφ J el 0 +∆ενφ 2 /2, reproduces Eq. (7) in the main text, while the twisted partition function is expressed as in (6).
In order to make connection with the notation of Ref. [17], whose results for the AB transformation will be used below, we denote q = e 2πiτ = e −β ∆ε and e 2πiζ = e β µ .
The form of the twisted Hamiltonian H CFT (φ ) defined in Eq. (7) in the main text is typical (see, e.g. Eq. (17) in Ref. [S1]) for two-dimensional interacting electron systems in which single-particle energies depend quadratically on the orbital momentum n and hence depend quadratically on the AB flux after the shift [S1] n → n − φ . Also we emphasize that while N el denotes the electron number operator, in case when the AB flux is non-zero, the operator N imb , defined in (7), is equal to the charge imbalance operator [16], i.e. it is the difference between the true electron number operator, which changes only by integers, and the externally induced charge νφ , which varies continuously (either by external gate voltage or AB flux variation). To understand their relation physically we note that for φ = 0 the derivative of the Grand potential However, the latter is equal to K l+sd H +n H φ (τ, n H ζ ; n H d H ) due to the K-function identity shown before Eq. (10), i.e., the effect of adding AB flux is simply to shift the K-function index l → l + n H φ . Then shifting l in (S.1) yields in the sum an extra term proportional to φ which after the average gives rise to which explains Eqs. (8) and (7) in the main text. It is interesting to note that the operator Q imb ≡ −N imb coincides with the zero mode of the twisted u(1) current π φ (J el 0 ) defined in [17], which is precisely the operator that appears in the twisted partition function (6)  Now we continue with the explanation why we choose µ N = −∆ε/2 and µ N+1 = ∆ε/2. First, let us see why the partition function (10) is independent of the bulk chemical potential µ 0 but depends only on the edge part µ = µ tot − µ 0 as stated in the text. The electron number average derived in general from (S.2) with a total chemical potential µ tot = µ 0 + µ contains a bulk term ν µ 0 /∆ε and an edge term ν µ/∆ε − ∂ Ω φ /∂ µ. Because the bulk term, corresponding to φ = 0, must be equal to N 0 we find µ 0 = d H ∆εN 0 /n H . Next we assume, as in the main text, that the number of electrons in the bulk is N 0 = n H n 0 , where n 0 is a positive integer, so that the bulk chemical potential becomes µ 0 /∆ε = n 0 d H . Now, if we substitute µ tot into the partition function (10) we see that the latter is independent of the bulk chemical potential µ 0 because the sum over n is invariant with respect to the shift n → n + n 0 , i.e., Z l,Λ φ (τ, µ 0 + µ) = Z l,Λ φ (τ, µ). Therefore the edge partition function (10), as well as all thermodynamic averages, depend only on the edge part µ of the total chemical potential µ tot = µ 0 + µ.
In order to determine the values of µ N and µ N+1 , which are needed for the computation of the thermodynamic averages of the energy H CFT (φ ) and the electron number N el (φ ), we argue that the difference µ N+1 − µ N corresponds to the difference between the energy of the last occupied single-particle state in the CB island and the first available unoccupied single-particle state. This difference is proportional to the flux difference between the two states, i.e. µ N+1 − µ N = φ ∆ε. The first unoccupied single-particle state in the QD can be obtained from the last occupied one by the Laughlin spectral flow [21]: we apply the Laughlin argument [S2] to the last occupied single particle state by changing adiabatically the AB flux threading the electron disk from 0 to 1; when the flux becomes 1 the discrete spectrum of the Hamiltonian H CFT (φ ) becomes the same as the spectrum for φ = 0 while the single-electron states are mapped onto themselves, i.e., if at φ = 0 an electron is described by the (unnormalized) wave function z l e −|z| 2 /4 , then at φ = 1 it would have the wave function z l+1 e −|z| 2 /4 . Therefore the difference between the two chemical potentials corresponding to the last occupied and the first unoccupied single-particle states is exactly µ N+1 − µ N = ∆ε. Put another way, the Laughlin spectral flow, which transforms the last occupied single-particle level to the first unoccupied one, is expressed by the (modular) transformation [21] ζ → ζ + τ and it can be implemented by Eq. (5) with φ = 1. Taking into account that ζ = (µ/∆ε)τ, as defined in the main text before Eq. (5), we conclude that the Laughlin spectral flow is equivalent to (µ/∆ε) → (µ/∆ε) + 1, or µ → µ + ∆ε so that ∆µ = ∆ε. Next, assuming that µ N + µ N+1 = 0, which is equivalent to fixing the QD in the center of a CB valley for φ = 0, we finally obtain µ N = −∆ε/2 and µ N+1 = ∆ε/2. It is worth stressing that µ N+1 − µ N is independent of the neutral degrees of freedom of the electrons in the QD, hence it is independent of the ratio r = v n /v c of the Fermi velocities of the charged and neutral edge modes. It is also independent of the filling factor ν. Note however that, except for φ = 0, µ N is not the true chemical potential, because it is coupled in the partition function (6) to the charge imbalance operator N imb defined in (7) instead of the particle number N el . That is why, µ N+1 − µ N is not equal to the addition energy ∆µ phys = µ phys N+1 − µ phys N , where µ phys N is the true (physical) chemical potential. The addition energy ∆µ phys N corresponds to the energy spacing between CB peaks and it can be interpreted as the difference between the energies of the ground states with N + 1 and N electrons and depends on the neutral degrees of freedom and on the filling factor ν.
In this work we have considered only chiral FQH states. The reason is that for such FQH states all edge modes move in the same direction, which is determined by the direction of the magnetic field perpendicular to the FQH sample, and there is certainly a unitary rational CFT describing the edge states [6][7][8][9]24]. For the non-chiral FQH states, such as the ν = 2/3 [13, S3] and probably ν = 5/2 as well [30, 31, S4], there might be counter-propagating neutral modes, or upstream modes, and it is not completely clear if conformal symmetry exists in the limit v n /v c → 1, so that the effective field theory partition function is unknown. For example, in a recently proposed Abelian candidate for ν = 5/2, known as the 113 Halperin state [S5], the standard partition function is divergent because the K-matrix is not positive definite There are also open problems, such as equilibration of counter-propagating modes in disorder-dominated phases, non-universality of the Hall conductance without equilibration, edge reconstruction, etc. However, as soon as the partition function for any FQH state is fixed the method described here would allow to compute the thermopower, figure-of-merit, power factor and conductances of a Coulomb-blockaded island inside this state.
The only exception is the Anti-Pfaffian state for which we have used the partition function given in Refs. [1,24].

FULL CFT PARTITION FUNCTIONS FOR ν = 5/2 PAIRED FQH STATES
Before we continue with the explicit listing of all partition functions used in the main text we would like to make an important comment. The modular parameter ζ in Eq. (10) carries an additional multiplicative factor n H , the numerator fo the filling factor ν = n H /d H , which can be understood as follows [22]: the K functions (9) entering the full partition functions (10) are actually the partition functions for the Luttinger liquid which can be described by a chiral boson with a compactification radius [21] R c = 1/m, where m = n H d H . This is because the u(1) component of the electron field e iφ (z)/ √ ν , which is fixed by the requirement to have electric charge 1, has a statistical angle θ /2π = d H /n H that is not integer for n H > 1. Therefore, for n H > 1 we need to consider a smaller subalgebra containing only clusters of n H electrons [22], which can be generated by e in H φ (z)/ √ ν e i √ n H d H φ (z) . Theses fields have integer statistics and are local, however their corresponding u(1) compactification radius n H d H is bigger [22]. The electric charge operator Q can be expressed in terms of the normalized [S13] chiral u(1) charge J 0 by Q = √ νJ 0 . On the other hand the normalized [S13] charge J 0 is related to the Luttinger liquid number operator N = J 0 / √ m, whose spectrum appears in the Luttinger liquid partition function (9) as a conjugate of ζ , so that combining both we can express Q by N as follows This implies that the modular parameter ζ in the partition function (10) must be multiplied by n H . This detail has one important consequence: the charge index l + sd H , which is divided by n H d H in (10), is deformed by adding n H φ in presence of extra AB flux. Therefore the AB flux enters (S.2) as φ n H /d H which is necessary for the correct implementation of the charge-flux relation in general FQH states [22, S6, S7].
The partition functions of the paired FQH states with ν = 5/2 with quasiparticles in the bulk depend on their number modulo 2 (modulo n H in general) [1, 23, S8]. Therefore we consider only two cases: even (no quasiparticles in the bulk) and odd (one quasiparticle in the bulk). The partition functions for a CB island in all paired FQH states with ν = 5/2 (we consider only the highest Landau level with ν = 1/2 and n H = 2, d H = 4) with even number of quasiparticles in the bulk can be written as a sum of two products, e.g. for zero quasiparticles, Z e (τ, ζ ) = K 0 (τ, 2ζ ; 8)ch 0 (τ ) + K 4 (τ, 2ζ ; 8)ch ω (τ ) (S. 3) where τ = rτ with r = v n /v c , the K functions are defined in (9) and ch 0 (τ ) is the neutral partition function of the vacuum sector, while ch ω (τ ) is the neutral partition function of the one-electron sector. For the Pfaffian and the 331 state without bulk quasiparticles the neutral characters ch 0 (τ ) and ch ω (τ ) are given explicitly in the main text, before FIG. 1 and FIG. 2. For the SU(2) 2 state we have essentially given these characters in the main text, omitting a common multiplicative µindependent factor which has no contribution to the average tunneling energy (4). Here we give the complete formulas for the SU(2) 2 characters taken from Eq. (14.183) in [18] ch 0 (τ) = χ The partition functions for the paired ν = 1/2 FQH states with odd number of quasiparticles in the bulk can be given in a similar way, e.g. with one quasiparticle in the bulk it is where the K functions are defined in (9), ch σ (τ ) is the neutral partition function for the one-quasiparticle sector labeled by the topological charge σ and ch ω * σ is the neutral partition function of the topological sector labeled by ω * σ . In all three cases of the Pfaffian, SU(2) 2 and Anti-Pfaffian the neutral characters satisfy ch ω * σ (τ ) ≡ ch σ (τ ) as a consequence of the fusion rules ψ el × σ σ , where σ is the lowest-CFT dimension quasiparticle field and ψ el is the electron field corresponding to the neutral topological weight ω. For the 331 state the sectors with σ and ω * σ are represented by different topological charges having opposite fermion parity, however the neutral partition functions of the two sectors coincide because K −1 (τ, 0; 4) ≡ K 1 (τ, 0; 4). Therefore in all four cases of paired FQH states including the Anti-Pfaffian we have ch ω * σ (τ ) ≡ ch σ (τ ) which explains the second line of (S.5). Next we can use the identity mentioned before FIG. 3 in the main text to find that K 1 (τ, 2ζ ; 8) + K −3 (τ, 2ζ ; 8) = K 1/2 (τ, ζ ; 2) and the partition function (S.5) can be written as a product of a charged part K 1/2 (τ, ζ ; 2) and a neutral part ch σ (τ ). It is now obvious that the neutral part of the CFT has no contribution to the average tunneling energy (4) for odd number of quasiparticles in the bulk and the distance between the centers of the consecutive peaks of the electric and thermal conductances is always ∆φ = 2, as shown in FIG. S2, cf.
Ref. [1]. For very low temperature and away from the transition points and discontinuities it would be sufficient to take ch ω s * Λ (τ ) = D ω,Λ q r∆(ω,Λ) , like in Ref. [1], where ∆(ω, Λ) is the scaling dimension of the leading term and D ω,Λ is the neutral multiplicity. However, in general more terms are necessary in order to obtain smooth and realistic plots. For the numerical calculation of the thermoelectric properties of the ν = 5/2 paired states presented here, we have used 1000 terms for the charged and 200 terms for the neutral partition functions and accuracy up to 36 digits after the decimal point. The oscillation of the thermopower, for a CB island formed in a ν = 1/3 Laughlin quantum Hall state with l = 0 and the CB conductance, can be seen in [5]. The thermopower has a saw-tooth behavior with period 3 flux quanta and the positions For comparison with FIG. 3 and FIG. 4 in the main text, we also plot in FIG. S5 the power factor P T computed from Eq. (2) for the Pfaffian state without bulk quasiparticles at T = T 0 and r = 1. Again the sharp dips in the power factor correspond to the maxima of the conductance peaks but for v n = v c the ratio of the two maxima of P T around a conductance peak is obviously 1.
As mentioned in the main text, the sharp zeros of the power factor can be used to determine experimentally the ratio r = v n /v c because the CB peak pattern for the all paired FQH states proposed for ν = 5/2 with even number of bulk quasiparticles [1,23] consists of a longer flux period ∆φ 2 = 2 + r and a shorter one ∆φ 1 = 2 − r, as shown in FIG. S5, while that for the states with odd number of bulk quasiparticles is equidistant, i.e., ∆φ 1 = ∆φ 2 = 2. This equidistant pattern of CB peaks could be used as a reference [1,2,10,23]. Since, according to Eq. (5), the gate voltage V g is simply proportional to the AB flux φ we have that the ratio of the gate voltage FIG. S5: The Power factor P T of a CB island for the Pfaffian state for ν = 5/2, without quasiparticles in the bulk, with r = 1, at temperature T /T 0 = 1. periods is the same, i.e., x = ∆V 2 /∆V 1 = ∆φ 2 /∆φ 1 ≥ 1 and therefore For experimental purposes the ratio 2(x − 1)/(x + 1) at temperatures T ≤ T 0 /2 is very close to its zero-temperature value. We also plot in FIG. S6 the power factor for the Anti-Pfaffian state with r = 1/6 at T = T 0 computed from the partition function of Ref. [1,24]. We considered the partition function for the disorder-dominated phase of the Anti-Pfaffian state [30,31] with even number of bulk quasiparticles, in which the charged and neutral modes have already equilibrated [S3, S10] and consequently the Hall conductance is universal, as that of Eq. (S.3) with ch 0 = [χ (2) 0 ] −1 and ch ω = [χ (2) 2 ] −1 , where χ The power factor for the Anti-Pfaffian state with even number of bulk quasiparticles is similar to that of the SU(2) 2 state though the places of higher and lower peaks, respectively the short and long periods in the gate voltage, are exchanged. This leads to different behavior of the ratio P max 1 /P max 2 , as shown in FIG. 5 in the main text, which might also be a clear signature of the Anti-Pfaffian or SU(2) 2 state.
The ratios of the maxima of the power factor around a conductance peak can be recalculated if the measured r is different from 1/6.

PARTITION FUNCTIONS FOR THE Z 3 PARAFERMION FQH STATES
As another illustration of the approach we consider a CB island in which the FQH state is in the Z 3 Read-Rezayi FIG. S6: The Power factor P T and the conductance G of a CB island for the Anti-Pfaffian state for ν = 5/2, without quasiparticles in the bulk, with r = 1/6, at temperature T /T 0 = 1.
Notice that the sum ∑ (l) is restricted by the condition n 1 + 2n 2 = l mod 3. Introducing AB flux as in Eq. (5) and we calculate numerically the thermopower and the conductance of the CB island at temperature T /T 0 = 1 and r = 1, see Fig. S7. Although the plot of the thermopower for r = 1 might be unrealistic, as experiments and numerical calculations suggest that v n < v c , it is instructive for the analysis of the characteristics of the thermopower in general which are similar to those for metallic islands: the thermopower grows linearly with the gate voltage V g and the edges are smoothened at finite temperature; it is always 0 at the maximum of a conductance peak expressing the fact that the tunneling energy is zero at the conductance peaks; there are sharp jumps of thermopower (discontinuous at T = 0) in the middle of the CB valleys between neighboring conductance peaks, expressing particle-hole symmetry [3]. The difference with the metallic islands can be seen in the ratio between the maxima of the thermopower and power factor at the neighboring conductance peaks which depends on r = v n /v c and this observable dependence is rather sensitive to the number of quasiparticles in the bulk of the CB islands.