Particle-Hole Conjugation of Read-Rezayi states

Consequently, the electron operator has scaling dimension ³₂. The neutral sector does not enter the charge current, J = _2π¹ ∂ϕ, so the level-k RR state has a quantized Hall conductance σ_xy = _k+2^{k e}_h². If this fractional quantum Hall state occurs in the second Landau level and the lowest Landau level (of both spins) is filled and inert, then σ_xy = 2 + _k+2^k _e²

h. The energy momentum tensor is the sum of the two energy momentum tensors, T = T_c+T_Zk. Consequently, the thermal Hall conductivity of the level-k RR state is proportional to the sum of the two central charges [76]:

κ^RR_xy = 3k k+ 2

π²k²_B

3h T. (4.5)

If this this fractional quantum Hall state occurs in the second Landau level, then κxy = 2 + _k+2^3k π²k_B²

3h T.

replacing electrons, on top of a filled LL of electrons. Using this construction, the edge between the level-k RR state (ν = 1− _k+2^k = _k+2² ) and the vacuum (ν = 0) is mapped to the edge between the level k-RR state (ν = _k+2^k ) and a ν = 1 state.

Hence, the theory of this edge is described by a levelk-RR edge theory and a counter propagating bosonic charge mode which is the edge theory of the ν = 1 state. The low-energy effective Lagrangian is:

L_RR= 1

4π∂_xφ₁(i∂_τ+v₁∂_x)φ₁+

k+ 2 k

1

4π∂_xφ₂(−i∂_τ+v₂∂_x)φ₂ + L_Z

k

− 2

4πv₁₂∂_xφ₁∂_xφ₂+ξ(x)ψ₁e^{ik+2k φ2}e^−iφ1 +h.c., (4.6) where φ₁ is the ν = 1 edge charge mode, φ₂ and the parafermions Zk belong to the RR edge, and v₁₂ >0 is a repulsive density-density interaction along the edge.

The final term in Equation 4.6 is inter-mode electron tunneling which tunnels electrons from the outer ν = 1 edge to the inner edge with a random coefficient ξ which, for simplicity, we take to be of Gaussian white noise form: hξ(x)ξ^∗(x⁰)i = W δ(x−x⁰). In the absence of inter-mode tunneling, this theory will not realize the universal value of the two-terminal conductance we seek,σ_xy = 1−_k+2^k . Heuristically, if there are only two terminals, we cannot distinguish between forward and backward propagation, and the charge conductivities of the filled Landau level and the RR state of holes will add up in absolute value. The tunneling term allows the counter- propagating modes to equilibrate and achieve a universal two-terminal conductance, as is the case for the ν= 2/3 quantum Hall state [100].

To determine the relevance of the random inter-edge tunneling, we need to examine the scaling dimension of W. In general, the Renormalization Group flow of the strength of the tunneling, W, follows [101]:

dW

d` = (3−2∆)W (4.7)

where ∆ is the scaling dimension of the term coupled to the random ξ(x). In the absence of density-density interactions,v12= 0, the scaling dimension of the tunneling

operator ψ₁e^{ik+2k φ2}e^−iφ1 is ∆ = ³₂ + ¹₂ = 2. The inter-mode electron tunneling term is irrelavent in this case: dW /d` = −W, as may be seen by using the replica trick to integrate out ξ. However, for v₁₂ sufficiently large,W becomes relevant. To see this, we introduce a new set of fields defined by

φ_ρ=φ₁−φ₂, φ_σ =φ₁−φ₂(k+ 2)/k, (4.8) corresponding to charged and neutral bosonic modes, respectively. In these variables, the Lagrangian takes the form L_RR =L_ρ+L_σ+L_tun+L_ρσ, with:

L_ρ = 1 4π

k+ 2 2

∂_xφ_ρ(i∂_τ +v_ρ∂_x)φ_ρ, L_σ = 1

4π k

2∂_xφ_σ(−i∂_τ +v_σ∂_x)φ_σ +L_Zk(v_n), L_ρσ = 2v_ρσ∂_xφ_σ∂_xφ_ρ,

L_tun =ξ(x)ψ₁e^iφσ +ξ^∗(x)ψ₁^†e^−iφσ, (4.9) and v_σ, v_ρ,v_ρσ are functions of v₁,v₂ and v₁₂, e.g.,

4πv_ρσ = (k/2)²v₁+ (k+ 2/2)²v₂−(k(k+ 2)/4 + (k/2)²)v₁₂. (4.10) If v_ρσ = 0, then the electron tunneling operator has scaling dimension [ψ₁e^iφσ] = 1 and the inter-mode electron tunneling term is relevant: dW /d`=W.

We now show that when the disorder is a relevant perturbation, the edge theory flows to a new fixed point described by a freely-propagating charged boson (responsi- ble for the universal quantized Hall conductance) and a backward propagating neutral sector that possesses an SU(2) symmetry. We will argue that due to the disordered tunneling the neutral modes will equilibrate and propagate at common average ve- locity ¯v and show that the velocity mismatch and the mixing termL_ρσ are irrelevant.

An SU(2) symmetry will thus emerge in the neutral sector. Note that for k= 2 this reduces to the result obtained for the anti-Pfaffian [60, 64].

Let us write the neutral sector action L_σ as L_SU(2)k +L_δv, with:

L_SU(2)k = 1 4π

k

2∂xφσ(−i∂τ + ¯v∂x)φσ +L_Zk(¯v), (4.11) L_δv = (L_Zk(v_n)− L_Zk(¯v)) + 1

4π k

2(v_σ −v¯)(∂_xφ_σ)².

The Lagrangian L_SU(2)k is, in fact, equivalent to (the opposite chirality version of) the chiral WZW action, Equation 4.2: the chiral boson φ_σ restores the U(1) which was gauged out in Equation 4.1. A simple way to see this is to note that the currents:

J⁺ =√

kψ₁e^iφσ, J⁻ =√

kψ^†₁e^−iφσ, J^z = k

2∂_xφ_σ, (4.12) obey the same SU(2)_k Kac-Moody commutation relations as the WZW currents:

J^a=−ik

2πtr T^ag⁻¹(i∂_τ −¯v∂_x)g

, (4.13)

where T^a, a=x, y, z are SU(2) generators and J^± =J^a±iJ^y.

We notice that the tunneling term L_tun can be written in terms of the currents:

L_tun=ξ(x)J⁺+ξ^∗(x)J⁻. (4.14)

It is convenient to use the WZW representation since the tunneling term can be eliminated from the action by the gauge transformation g →gU with

U =P e^{¯viRxdx0~ξ(x0)·T~} (4.15) where P denotes path ordering and ξ(x~ ⁰) = (2Re(ξ(x⁰)),−2Im(ξ(x⁰)),0). Under this gauge transformation

L_SU(2)k → L_SU(2)k −~ξ·J ,~ (4.16) thus with this gauge transformation we are able to eliminate the tunneling term L_tun

from the action of the edge..

We now turn to the effect of this gauge transformation on the velocity anisotropy terms. The velocity terms in L_σ can be written in the form:

vatr Sa∂xg⁻¹∂xg

, (4.17)

whereS_ais a matrix satisfying tr(S_aT_bT_c) =δ_abδ_acandv_a,a=x, y, zcan be expressed in terms of vσ, vn. The matrix vaSa picks out the current Ja, Equation 4.13, from the action, and gives it the velocity v_a. Let us separate the traceless part M of the matrix vaSa:

M =v_aS_a−tr(v_aS_a)×I/3. (4.18) Then L_δv takes the form L_δv = tr(M ∂_xg⁻¹∂_xg) Under the gauge transformation g → gU, Lδv → tr(M⁰∂xg⁻¹∂xg), where M⁰ = U M U^† is random since the gauge transformation is a function of ξ(x). The renormalization group flow of the mean square average ofM⁰, WM⁰, is dWM⁰/dl = (3−2∆)WM⁰ [101], where ∆ is the scaling dimension of the term to whichM⁰ couples. In this case,M⁰ couples to∂_xg⁻¹∂_xg ∝J² which has scaling dimension ∆ = 2 (i.e. M⁰is a velocity). HenceWM⁰ and the velocity anisotropy are irrelevant. The part of the velocity term which is invariant under the gauge transformation is the average velocity ¯v = tr(vaSa)/3.

The mixing term, L_ρσ is irrelevant. It can be written as L_ρσ = 2v_ρσ(²_kJ^z)·∂_xφ_ρ; under the gauge transformation g → gU the current J^z gets rotated with a ran- dom coefficient. Consequently, deviations from v_ρσ = 0 are irrelevant, much like the velocity anisotropy term above.