Effects of nuclear spins on the transport properties of the edge of two-dimensional topological insulators
Chen-Hsuan Hsu Peter Stano Jelena Klinovaja Daniel Loss
cond-mat.mes-hall









^1RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
^2Department of Applied Physics, School of Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
^3Institute of Physics, Slovak Academy of Sciences, 845 11 Bratislava, Slovakia
^4Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland




The electrons in the edge channels of two-dimensional topological insulators can be described as a helical Tomonaga-Luttinger liquid. They couple to nuclear spins embedded in the host materials through the hyperfine interaction, and are therefore subject to elastic spin-flip backscattering on the nuclear spins. We investigate the nuclear-spin-induced edge resistance due to such backscattering by performing a renormalization-group analysis. Remarkably, the effect of this backscattering mechanism is stronger in a helical edge than in nonhelical channels, which are believed to be present in the trivial regime of InAs/GaSb quantum wells. In a system with sufficiently long edges, the disordered nuclear spins lead to an edge resistance which grows exponentially upon lowering the temperature. On the other hand, electrons from the edge states mediate an anisotropic Ruderman-Kittel-Kasuya-Yosida nuclear spin-spin interaction, which induces a spiral nuclear spin order below the transition temperature. We discuss the features of the spiral order, as well as its experimental signatures. In the ordered phase, we identify two backscattering mechanisms, due to charge impurities and magnons. The backscattering on charge impurities is allowed by the internally generated magnetic field, and leads to an Anderson-type localization of the edge states. The magnon-mediated backscattering results in a power-law resistance, which is suppressed at zero temperature. Overall, we find that in a sufficiently long edge the nuclear spins, whether ordered or not, suppress the edge conductance to zero as the temperature approaches zero. 

 
Effects of nuclear spins on the transport properties of the edge of two-dimensional topological insulators
    Daniel Loss^1,4
    December 30, 2023
==========================================================================================================



§ INTRODUCTION


The helical edge states are the essential feature of two-dimensional topological insulators (2DTIs), whose examples include HgTe/(Hg,Cd)Te quantum wells <cit.>
and InAs/GaSb quantum wells <cit.>.
Since elastic backscattering of the helical edge states must be accompanied by spin flips, the edge channel conductance is insensitive to perturbations that respect time-reversal symmetry. However, it remains to be clarified whether alternative backscattering mechanisms, arising from the broken time-reversal symmetry or inelastic processes, can still cause a substantial resistance to destroy the edge conductance quantization. Among other proposed mechanisms <cit.>, 
in Ref. <cit.> we pointed out that in general nuclear spins could be such a resistance source of 2DTIs.

Nuclear spins, typically present in usual 2DTI host materials, couple to the electrons in the edge channels via the hyperfine interaction, and thus result in elastic spin-flip backscattering. It is therefore necessary to examine whether the nuclear spins are detrimental to the edge states. Furthermore, because the edge states in InAs/GaSb heterostructure persist even in the regime where the energy band is not inverted <cit.>, the topological (helical) character of the edge states in the band-inverted (nominally topological) regime remains to be verified. 

This work extends results obtained in Ref. <cit.>, the main findings of which are summarized as follows. 
First, as in non-topological systems <cit.>, the electron-electron interaction strongly enhances the backscattering effects in one dimension. Therefore, in strongly interacting systems the effects of the nuclear spins on edge resistance become prominent in spite of the typically weak hyperfine couplings in 2DTI host materials <cit.>. 
Second, the resistance caused by randomly oriented nuclear spins is bigger for a longer edge, a lower temperature, and stronger electron-electron interactions. Third, a nuclear spin order is stabilized by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Finally, the transport properties of the edge states are influenced by the ordering of the nuclear spins. The impurities[To avoid possible confusion, we will use the term “impurity” when we refer to charge (nonmagnetic) impurity, and reserve the term “(dis-)order” for the (dis-)ordering of the nuclear spins.] and magnons enable additional backscattering processes in the ordered phase. The resistance due to the impurities ultimately dominates over the one from the magnons as the temperature approaches zero, leading to our conclusion that in general the nuclear spins suppress the edge conductance at very low temperatures.
 
The topic is, however, rather complex due to the intertwining of electron and nuclear subsystems. Therefore, a separate account is necessary. Specifically, here we additionally provide the details of our renormalization-group (RG) analysis on the nuclear-spin-induced backscattering in the bosonization framework, giving the reader a comprehensive understanding of various backscattering mechanisms in the disordered and ordered phases.  
We also compare the effects of the nuclear spins in various materials, including InAs/GaSb and HgTe/(Hg,Cd)Te quantum wells, and GaAs quantum wires. Remarkably, this provides signatures to reveal the helical nature of the edge states, since disordered nuclear spins lead to a stronger effect in a helical edge in InAs/GaSb than in nonhelical channels in a GaAs wire or in the trivial regime of InAs/GaSb, despite comparable values of material parameters. 
Further, we investigate the RKKY-induced nuclear spin order, whose elaborate description was not included in our previous work. Here we show how the helical edge states mediate an anisotropic RKKY interaction,[Whereas here we focus on the RKKY interaction mediated by the edge states of a 2DTI, we note that the behavior of the RKKY interaction on the surfaces of three-dimensional topological insulators can also be very rich <cit.>.] inducing a spiral nuclear spin order.[The term “spiral nuclear spin order” is used to distinguish the order in a 2DTI edge from the nuclear spin helix in a nonhelical channel, as will be explained in Sec. <ref>. Moreover, we adopt this term also in order to avoid possible confusion between the ordering of the nuclear spins and the helicity of the edge states.] We also explain in detail the differences between the order in a helical edge and the one in a spin-degenerate, nonhelical wire. Remarkably, the helical character of the edge states enhances the instability toward the nuclear spin ordering. In addition, we discuss experimental signatures of the spiral nuclear spin order in optical and charge transport measurements, which gives guidance for  possible experimental verifications of the formation of the spiral order predicted in our work.

The additional backscattering mechanisms due to the spiral order also deserves further elucidation. 
Since a macroscopic magnetic (Overhauser) field originating from the nuclear spin order admixes the edge states with the opposite spins, the edge states do not retain perfect helicity anymore, becoming susceptible to the backscattering on impurities <cit.>. Here we present the derivation of the Overhauser-field-assisted backscattering action, and show that the form can be understood from the spin rotational symmetry of electron-electron interaction. 
Finally, here we give the derivation of the effective action due to the magnon-mediated backscattering, including the emission and the absorption processes. Whereas the efficiency of these processes depends on the magnon energy and the magnon occupation, thus leading to rich temperature dependence of the resulting edge resistance, 
the derived effective action allows us to intuitively visualize the effects of the magnon-mediated backscattering on the resistance.
  
This paper is organized as follows. In Sec. <ref>, we introduce the Hamiltonian, including the edge-state electron Hamiltonian and the hyperfine interaction between the electron spins and the nuclear spins. 
In Sec. <ref>, we investigate the backscattering caused by disordered nuclear spins and compare the localization effect in various materials and experimental setups. Using an RG analysis we calculate the edge resistance in the short-edge, the high-temperature, and the strong-coupling regimes, as well as the differential resistance in the high-bias regime. 
In Sec. <ref>, we investigate the RKKY-induced spiral nuclear spin order. We first compare the spiral order in a 2DTI with the nuclear spin helix in a spin-degenerate wire in Sec. <ref> and then discuss its experimental signatures in Sec. <ref>. 
In addition, we examine the self-consistency condition of the RKKY approach in Sec. <ref>. 
In Sec. <ref>, we examine the effects of the magnons and the impurities on the 2DTI edge states in the ordered phase. In Sec. <ref> we analyze the resistance due to the Overhauser-field-assisted backscattering on impurities, and find that it gives rise to a resistance that is comparable to the one due to the disordered nuclear spins.
We then discuss the features to distinguish the two resistance sources.
In Sec. <ref>, we calculate the resistance due to the magnon-mediated backscattering, including both the emission and the absorption processes. 
Finally, in Sec. <ref> we summarize the resistance from all these backscattering mechanisms
and discuss the relevance of nuclear spins to the observed edge resistance in experiments.
The technical details are presented in the appendices.




§ HAMILTONIAN 
 

We begin by modeling the edge states and the nuclear spins with the Hamiltonian, 𝐻=𝐻_el+𝐻_hf. Throughout the paper we assume that the edge states consist of the right-moving down-spin R_↓ and the left-moving up-spin L_↑ electrons (see Fig. <ref>; the opposite edge of the 2DTI is assumed to be far away, and decoupled from the edge under consideration). In the bosonization framework, we describe these edge states as a helical Tomonaga-Luttinger liquid <cit.>,

    𝐻_el = ∫ħ dr/2π { uK [ ∂_rθ(r) ]^2 + u/K[ ∂_rϕ(r) ]^2 },

where the bosonic fields θ and ϕ relate to the original R_↓ and L_↑ fermionic fields through 



    R_↓ (r)    =   U_R/√(2π a) e^i k_Fr e^i[-ϕ(r) + θ(r)], 
    
    L_↑ (r)    =   U_L/√(2π a) e^-i k_Fr e^i[ϕ(r) + θ(r)],

 
with the Klein factors U_R and U_L, the Fermi wave number k_F, and the edge coordinate r. The electron-electron interaction is parametrized by the Luttinger liquid parameter K. Here u=v_F/K is the renormalized velocity with the Fermi velocity v_F. 
The short-distance cutoff a, required by the bosonization prescription, is taken as the transverse decay length of the wave function of the edge states, a=ħ v_F/Δ, with the 2DTI bulk gap Δ.[Since the energy bands of the edge states merge into the bulk energy bands above the 2DTI gap Δ, the helical Tomonaga-Luttinger liquid description of the edge states breaks down above Δ (here the energies are measured from the Dirac point). Therefore, for our bosonization procedure we choose Δ as the high-energy cutoff, below which the edge states retain their linear energy dispersion and helical nature. The corresponding short-distance cutoff is thus given by the evanescent decay length of the edge-state electron wave function into the bulk <cit.>.] 
The Fermi energy is given by ϵ_F≡ħ v_F k_F/2. The helical Tomonaga-Luttinger liquid action can be obtained by integrating out the θ field in Eq. (<ref>),   

    𝑆_el/ħ   ≡   ∫  dr dτ/2π u K{[ ∂_τϕ(r,τ) ]^2 + u^2 [ ∂_rϕ(r,τ) ]^2 },

with the imaginary time τ and the bosonic field 
ϕ(r,τ).
 
Before continuing, we comment on  the Luttinger liquid parameter K. In existing experiments, its value is largely unknown, and only few attempts have been made to extract it experimentally.  In Ref. <cit.>,  deduced values for InAs/GaSb 2DTIs are K = 0.21–0.22, indicating strong electron-electron interaction in that sample. [
On the other hand, we note that the value extracted for K depends crucially on the theory to which the experiments are fitted <cit.>; see also the discussion in Sec. <ref>.]
To be able to make quantitative predictions, we therefore use K=0.2 (see Table <ref>) throughout this paper, unless stated otherwise. It represents the nuclear-spin-induced effects on the edge transport in the presence of strong electron–electron interactions.





The hyperfine interaction is given by  

    𝐻_hf = A_0/ρ_nuc∑_n[ Ψ_el^†( x_n) σ/2Ψ_el( x_n) ] · I_n,
 
which describes the coupling of an electron spin σ /2 ≡ (σ^x, σ^y, σ^z)/2 to nuclear spins I_n (with magnitude I) with the coupling constant A_0 at positions x_n with the nuclear index n. Here σ^μ denotes the μ component of the Pauli matrix vector in spin space. We take the nuclear density to be ρ_nuc = 8/a_0^3 with the lattice constant a_0 <cit.>, and write the electron operator Ψ_el as the product of the transverse (x_⊥) and the longitudinal (r) parts,

    Ψ_el( x)  = ψ_⊥ ( x_⊥) ψ_(r).
 
In the above, the transverse part of the edge electron wave function ψ_⊥ is a complex scalar, while the longitudinal part is a two-component spinor, ψ_ = (L_↑, R_↓)^T.  
For simplicity, we assume that the system is homonuclear and take the average values for the nuclear spin I and the hyperfine coupling constant A_0. We also assume the electron wave function to be uniform in the transverse direction such that it is approximated as a constant | ψ_⊥|^2 = 1 / (W_effa), with W_eff denoting the effective width of the wave function perpendicular to the 2DTI plane.[Assuming that the electrons are confined by a square potential well along the z direction (perpendicular to the 2DTI plane), the inhomogeneous hyperfine coupling is then proportional to the electron density ∝sin^2 (π z/W_qw) with the lithographic quantum well thickness W_qw. In order to incorporate such inhomogeneity, we define the effective width W_eff by averaging the hyperfine coupling over z, such that 1/W_eff≡ (1/W_qw) ∫_0^W_qwdz  sin^2 (π z/W_qw) p(z), weighted by the probability distribution p (z) = (2/W_qw)sin^2 (π z/W_qw). As a result, we find that W_eff = 4W_qw/3, which is used to approximate the transverse electron wave function.] With these approximations, the hyperfine interaction can be written in a one-dimensional form,

    𝐻_hf≈A_0/N_⊥∑_jS(r_j) ·Ĩ_j,

with the number of nuclei per cross section N_⊥=W_effa a_0ρ_nuc. Here r_j denotes the common value of the edge coordinate of the nuclei belonging to the j-th cross section. In the above, we define the effective spin operator, 

    S(r) ≡ N_⊥/ρ_nuc| ψ_⊥|^2 ∑_α,βψ_,α^† (r) σ_αβ/2ψ_,β (r) ,

which interacts with a classical spin Ĩ_j≡∑_n ∈ j I_n composed of N_⊥ nuclear spins within the j-th cross section. In Fig. <ref>, the composite spins Ĩ are plotted as the green arrows along the edges, in which the cross sections are drawn as yellow blocks, demonstrating the three-dimensionality of the nuclear subsystem.
A summary of the adopted parameters is given in Table <ref>. We will examine various backscattering mechanisms arising from Eq. (<ref>), and their contributions to the edge resistance.

As a remark, the dipolar interaction between the nuclear spins is much weaker than the hyperfine interaction <cit.>, and is not explicitly included in the above. In the disordered phase, however, it leads to the dissipation of the accumulated nuclear spin polarization during the backscattering process due to the accompanied electron-nuclear flip-flops <cit.>. In order to incorporate the effect of the dipolar interaction, we assume that the nuclear spin polarization is destroyed by such a dissipation channel, and adopt unpolarized nuclear spin orientation for our analysis in the disordered phase [see Eq. (<ref>)]. On the other hand, the RKKY interaction between the nuclear spins dominates the dipolar interaction, so the latter is neglected in our analysis of the nuclear spin ordering. 
In addition, the Kondo temperature associated with a single nuclear spin is T_Kondo=Δ e^-ϵ_F/A_0 <cit.>, so the Kondo physics should not be relevant, as ϵ_F≫ A_0 under typical experimental conditions (see also Refs. <cit.> for discussions on the limitations of the RKKY description). 



 



§ ELASTIC BACKSCATTERING ON RANDOMLY ORIENTED NUCLEAR SPINS 


We first consider the disordered phase, where the nuclear spins are randomly oriented, and cause elastic electron backscattering via the hyperfine interaction Eq. (<ref>). 
Since the forward scattering [the S^z term in Eq. (<ref>)] has no influence on the transport properties <cit.>, it can be dropped. In the continuum limit, the remaining components of the electron spin operator can be written in terms of the right and left movers, 


    S^x(r)    =   1/2[ L_↑^† (r) R_↓ (r) + R_↓^†(r) L_↑(r) ], 
     
    S^y(r)    =   -i/2[ L_↑^† (r) R_↓ (r) - R_↓^†(r) L_↑(r) ],


and then bosonized using Eq. (<ref>). This gives rise to the backscattering Hamiltonian,

    𝐻_hf,b   =   ∫dr/2π a V_hf,2k_F (r) e^2i ϕ (r)  + H.c.,

where we keep only the slowly varying terms, and define the 2k_F component of the random potential caused by the nuclear spins as

    V_hf,2k_F (r)    ≡   A_0/2 N_⊥[ Ĩ^x (r) + i Ĩ^y (r) ] e^-2i k_F r.

Assuming that the nuclear spins are independent and unpolarized, we have 

    ⟨ V_hf,2k_F^† (r) V_hf,2k_F (r') ⟩_rs   =    M_hfδ(r-r'),

with the strength M_hf≡ a A_0^2 I(I+1)/(6 N_⊥). Here ⟨⋯⟩_rs denotes the expectation value with respect to the random nuclear spin state. Using the replica method <cit.> and the average from Eq. (<ref>), we obtain the contribution to the imaginary-time action from Eq. (<ref>), 

    δ𝑆_hf/ħ   =    - D_hf u^2/8π a^3 ∫_u|τ-τ'|>a  dr dτ dτ'  
          ×cos[  2 ϕ (r,τ)- 2ϕ (r,τ') ],

with the dimensionless coupling constant D_hf≡ 2a M_hf /(πħ^2 u^2). With the effective action composed of Eqs. (<ref>) and (<ref>), we perform the RG analysis to investigate how the disordered nuclear spins affect the transport properties of the edge states.

Following the procedure in Appendix <ref>, we build the RG flow equations by first evaluating the correlation function,

    < e^i [ϕ( r_1) - ϕ( r_2)] >_𝑆_el+δ𝑆_hf,

with respect to Eqs. (<ref>) and (<ref>). Then, upon changing the cutoff a→ a(l) = ae^l with the dimensionless scale l, we find the RG flow equations,



    d D_hf (l) /d l    =   [ 3-2K(l) ] D_hf (l), 
    d K (l) /d l    =    - K^2 (l) /2 D_hf(l), 
    d u (l) /d l    =    -u(l) K (l)/2 D_hf(l),


from which we see that the backscattering on disordered nuclear spins Eq. (<ref>) is RG relevant for K(l)<3/2. 
Assuming that the change in K can be neglected, we integrate the RG flow of the effective coupling to get D_hf (l)= D_hf (l=0) e^(3-2K)l, which allows us to find the localization length,

    ξ_hf = a D_hf^-1/(3-2K),

and the corresponding localization temperature T_hf≡ħ u / (k_Bξ_hf). For an edge longer than ξ_hf, the conductance gets exponentially suppressed below T_hf. For the parameters of InAs/GaSb 2DTIs (see Table <ref>), the estimated values of ξ_hf and T_hf suggest that the localization-delocalization transition is within an experimentally accessible regime. In contrast, the spinful nuclei in HgTe/(Hg,Cd)Te are naturally less  abundant, possess smaller spins, and have weaker hyperfine coupling <cit.>, leading to a much bigger localization length and much lower localization temperature. 
 
Since the Luttinger liquid parameter K and the number of nuclei per cross section N_⊥ vary with materials and setups, we also investigate the dependence of the localization temperature and length on these parameters. In Fig. <ref> we plot the localization temperature of InAs/GaSb and HgTe/(Hg,Cd)Te 2DTIs, in addition to nonhelical states in the trivial regime of InAs/GaSb and a spin-degenerate quasi-one-dimensional GaAs wire, as a function of the interaction parameter. The localization temperatures of these materials drastically increase with stronger interactions, a feature that can be tested through the sample preparation, e.g. by varying the quantum well width, or the distance between a screening metallic gate and the quasi-one-dimensional channel. 
In addition to the difference between the InAs/GaSb and the HgTe/(Hg,Cd)Te 2DTIs due to material parameters, there is a pronounced difference in the localization temperatures between a helical edge of InAs/GaSb, and nonhelical channels in the trivial regime of InAs/GaSb and a GaAs conductor, in spite of their comparable nuclear spins and hyperfine couplings. This difference arises from the fact that in a spin-degenerate wire the effective Luttinger liquid parameter is K_wire = (K_c + 1/K_s)/2, an average of values for the charge (K_c) and spin (K_s≈ 1) channels, and thus bounded by 1/2, whereas such averaging is absent in a helical edge. 
Consequently, in 2DTIs the interaction leads to a stronger effect on the nuclear-spin-induced localization, which may reveal the helical nature of the edge states in the band-inverted regime of InAs/GaSb. The dependence of ξ_hf on the interaction can be inferred from Fig. <ref>, using the fact that ξ_hf is inversely proportional to T_hf, and therefore not displayed here.
In the inset of Fig. <ref>, we plot the dependence of ξ_hf on N_⊥, which depends on the quantum well width and the transverse decay length, and can also vary for different samples. Since the dependence of ξ_hf (and therefore T_hf) on N_⊥ is a fractional power law, its value does not change much even if N_⊥ varies by an order of magnitude. From now on we shall adopt the parameters of InAs/GaSb 2DTIs, in which we expect the most significant effects from the nuclear spins. 
  

 

 
Before continuing, we note that other possible backscattering mechanisms may cause edge resistance, and therefore contribute to the total resistance R_total, in addition to the contact resistance from the leads and the nuclear-spin-induced resistance R. 
However, since here we want to find out whether the nuclear-spin-induced resistance is observable, i.e. whether R is comparable to the resistance quantum R_0≡ h/e^2, throughout the paper we discuss and plot R/R_0, instead of R_total. 





We now investigate the resistance R_hf caused by disordered nuclear spins, Eq.(<ref>). Using the effective coupling D_hf (l), we compute the edge conductivity as in Refs. <cit.>, and therefore the edge resistance,

    R_hf   ∝    R_0M_hf L /ħ^2 v_F^2 e^(2-2K)l^*.
 
Here the dimensionless scale l^* arises from the cutoff a(l^*) = a e^l^*, at which the RG flow stops, and thus depends on the experimental conditions. We identify four possible physical cutoffs, being the edge length L, the thermal length λ_T≡ħ u / (k_B T), the bias length λ_V≡ħ u / (e V), and the localization length ξ_hf, corresponding to the short-edge, the high-temperature, the high-bias, and the strong-coupling regimes, respectively. 
First, if the edge length is the shortest among all these scales, L< λ_T, λ_V, ξ_hf, we obtain

    R_hf (L)    ∝    R_0π D_hf/2K^2( L/a)^3-2K.

Second, if the temperature is so high that λ_T<L, λ_V, ξ_hf, we get

    R_hf (T)    ∝    R_0π D_hf L /2K^2 a( K k_BT/Δ)^2K-2.
 
Third, at high bias such that λ_V < L, λ_T, ξ_hf, the differential resistance depends on the bias voltage as 

    ( dV/dI)_hf   ∝    R_0π D_hf L /2K^2 a( K eV/Δ)^2K-2.
 
Finally, if ξ_hf < L, λ_T, λ_V, the RG flow reaches the strong-coupling regime, so the edge states are gapped, displaying a thermally activated resistance <cit.>,

    R_hf (T)    ∝    R_0π D_hfL /2K^2 a  e^Δ_hf /(k_BT),

with the gap Δ_hf = Δ( 2K D_hf)^1/(3-2K). The formulas Eqs. (<ref>)–(<ref>) were given in Ref. <cit.> (with slightly different notations), and are repeated here for reference. 
Further, here we additionally check that the edge resistance due to the disordered nuclear spins in the trivial regime of InAs/GaSb is much smaller than the one in the topological phase (by two orders of magnitudes, not shown), consistent with our conclusion from Fig. <ref>.
We also note that in the localized regime, the temperature dependence of the resistance may be affected by the tunneling between the instanton/kink states. For nonhelical systems, the conductivity/conductance due to such tunneling events has been investigated <cit.>. Since it is beyond the scope of this paper, here we only note that the variable-range-hopping behavior σ (T) ∝ Exp[-√(C/(k_BT))] with a constant C due to the tunneling between the kink states in the localized regime <cit.> may be relevant to the observation in Ref. <cit.>. 

Figure <ref> summarizes the dependence of the resistance on the most relevant and accessible parameters, as given, depending on the regime, by Eqs. (<ref>)–(<ref>). 
In panel (a) of Fig. <ref>, we show how the resistance scales with the edge length. The kink in the curve signifies the transition from the L^3-2K power law for a short edge [Eq. (<ref>)] to the linear L dependence for a long edge [Eq. (<ref>)]. 
The temperature dependence is shown in panels (b) and (c). The resistance initially increases as a T^2K-2 upon decreasing the temperature T. After that the resistance for a short edge L<ξ_hf saturates [panel (b)] because the RG flow stops at the edge length. In contrast, for a long edge L>ξ_hf it evolves into an exponential [panel (c)] due to the electronic gap in the spectrum. As a result, for an edge of the length longer than ξ_hf, the localization of the edge states caused by disordered nuclear spins is observable below the localization temperature T_hf. Finally, the differential resistance in the presence of a finite bias voltage V is plotted in the panel (d) for an edge shorter than ξ_hf. Starting from the high-bias regime, the differential resistance initially increases with a decreasing voltage as a power law [Eq. (<ref>)], and then saturates due to the cutoff given by the shorter of L and λ_T, Eqs. (<ref>) and (<ref>), respectively. 

We conclude here that the power law dependencies of the resistance are symptomatic for our theory.
Recently, such fractional power laws were reported <cit.> in short InAs/GaSb 2DTI samples as a function of the temperature and the bias voltage.
Similar measurements with longer samples can be therefore used to examine and distinguish various theories, including ours, proposed for the origin of the edge resistance. 
For such comparison, an independently extracted value of the parameter K for the edge states <cit.> would be highly desirable.
  
	



§ SPIRAL NUCLEAR SPIN ORDER 


In addition to the backscattering effects, the interplay between the nuclear spins and the strong electron-electron interaction leads to the formation of a spiral nuclear spin order, which will be discussed in this section.



 §.§ RKKY interaction and spiral nuclear spin order 


We now discuss the nuclear spin order stabilized by the edge electron-mediated RKKY interaction. Since the energy scale of the hyperfine coupling is much smaller than the Fermi energy, we can integrate out the electron degrees of freedom in the hyperfine interaction defined in Eq. (<ref>) to obtain the RKKY interaction, a pairwise coupling between the static nuclear spins <cit.>,   

    𝐻_RKKY   =   1/N_⊥^2∑_i,j,μ J^μ_ijĨ_i^μĨ_j^μ,

with μ=x,y,z in the spin space. Here the RKKY coupling J^μ_ij is proportional to the electronic spin susceptibility, and can be calculated along the line of Ref. <cit.>. Since the z component of the electron spin operator is marginally relevant in the helical Tomonaga-Luttinger liquid, the z component of the RKKY coupling J^z_ij decays as 1/(r_i-r_j)^2 <cit.>, negligible compared to the x and y components. This is a consequence of the broken SU(2) spin rotational symmetry of the edge states, and it leads to an anisotropic RKKY coupling |J^x_q|=|J^y_q| ≫ |J^z_q| in momentum space, where the x and y components of the RKKY coupling are given by

    J^x_q =    J^y_q
    
     =    - sin (π K) /8π^2K A_0^2/Δ(λ_T/2π a)^2-2K
       ×∑_κ=±| Γ( 1-K) Γ[  K / 2-i λ_T( q - 2 κ k_F)/ (4π) ]/Γ[ (2-K)/2 - i λ_T( q - 2 κ k_F)/ (4π) ]|^2,

with the Gamma function Γ(x). In addition to the anisotropy, the helicity of the electrons also leads to a stronger RKKY coupling, compared to the nonhelical case, because of the difference in the effective Luttinger liquid parameters (K_wire versus K), as explained in Sec. <ref>.
 
The RKKY coupling given by Eq. (<ref>) develops a dip at q= ± 2k_F, and therefore gives rise to an instability toward a nuclear spin order in a finite-size system.
Even though a similar RKKY-induced nuclear spin order also arises in nonhelical, spin-degenerate systems such as GaAs quantum wires and ^13C nanotubes <cit.>, we note four important differences regarding to the nuclear orders between a helical edge and a nonhelical wire. 

First, the ordered nuclear spins align ferromagnetically within each cross section. Along the edge (x axis), they rotate in the xy plane with a spatial period π/k_F. For illustration, the nuclear spin order is displayed in Fig. <ref>.
The plane within which the nuclear spins rotate is fixed by the 2DTI plane, due to the broken SU(2) symmetry in the edge states. This is different to the nuclear spin helix formed in a spin-degenerate system <cit.>, where the nuclear spins rotate in a plane which can have arbitrary orientation. 

Second, the remaining U(1) symmetry in a helical edge, corresponding to the rotation of nuclear spins around the spin quantization (z) axis, leads to one Goldstone mode in the magnon spectrum in an infinitely long system (cf. below). This is in contrast to nonhelical systems, where multiple Goldstone modes associated with the SU(2) symmetry emerge in the magnon spectrum <cit.>.
 
Third, the tendency toward the nuclear spin order is typically higher for a 2DTI edge, as a result of the stronger RKKY coupling. This is essentially due to, again, the difference in the effective Luttinger liquid parameters, and it leads to a higher transition temperature T_0 [see Eq. (<ref>) and Table <ref>]. In other words, the helical nature of the edge states promotes the formation of the nuclear spin order.

Finally, the nuclear spin ground state is related to the helicity of the electronic subsystem in a helical edge, unlike in a spin-degenerate wire. The expectation value of the nuclear spins in the ordered phase is given by

    ⟨Ĩ(r) ⟩_± = N_⊥ I m_2k_F [ cos (2k_F r) x̂±sin (2k_F r) ŷ],

with the sign ± labeling the anticlockwise/clockwise rotation of the nuclear spins, and m_2k_F denoting the order parameter such that m_2k_F(T=0)=1 for a complete order. In a nonhelical wire, both orders with the ± signs can be the ground state. As the temperature is lowered below T_0, the nuclear spins form an order with one of them (within a magnetic domain).
The ordered nuclear spins then generate a macroscopic Overhauser field, which acts back on the electron spins. Depending on the sign, either the conduction modes R_↓ and L_↑, or the other subbands are gapped out <cit.>. 
The nuclear spin helix in a nonhelical wire thus leads to a partial gap at the Fermi surface, halving the conductance <cit.>. 





In a helical edge, however, the position of the electronic gap opened in the edge state spectrum depends on the ± sign in Eq. (<ref>). Provided that the edge states consist of R_↓ and L_↑ electrons and the Fermi level is placed above the Dirac point at q=0, the order with the positive sign, ⟨Ĩ⟩_+, would mix R_↓(q+2k_F) and L_↑(q), and therefore gap out the electrons at the Fermi surface [panel (a) of Fig. <ref>], reducing the RKKY coupling. On the other hand, ⟨Ĩ⟩_- would mix R_↓(q) and L_↑(q+2k_F). In this case, a gap Δ_m is induced below the Fermi surface, as shown in panel (b) of Fig. <ref>.
By establishing the nuclear spin order, the entire system of the nuclei and the electrons may acquire the magnetic energy gain, in addition to the Peierls energy (from opening an electronic gap at the Fermi surface) and the Knight energy (from the electron spin polarization) <cit.>.  Hence, the orders with the opposite signs lead to distinct energy gains due to the different gap positions. 
We examine the two scenarios [positive versus minus signs in Eq. (<ref>)], and find that the total energy gain of ⟨Ĩ⟩_- is higher, due to the stronger RKKY coupling and therefore the magnetic energy gain is larger (typically, the magnetic energy dominates the Peierls and Knight energies).
As a consequence, it is energetically favorable for the nuclear spins to order without opening a gap at the Fermi surface. 
To distinguish from the order in a nonhelical wire, which respects distinct symmetries, the order in a 2DTI edge predicted in this work is thus dubbed a spiral nuclear spin order.   





To proceed, we take ⟨Ĩ⟩_- [see Eq. (<ref>), with the minus sign] as the ground state for our spin-wave analysis (see Appendix <ref>), from which we obtain the magnon spectrum in an infinitely long edge,

    E_mag(q) 
    =I m_2k_F/N_⊥√(2J^x_2k_F( 2J^x_2k_F - J^x_q-2k_F -  J^x_q+2k_F)),

which we plot in Fig. <ref>. There is a zero-energy Goldstone mode at q=0 as a consequence of the U(1) rotational symmetry, as discussed. Importantly, however, in a finite-size system the Goldstone mode is gapped out, and the remaining magnon spectrum is basically dispersionless, as the RKKY resonance dip is very narrow. We thus approximate the magnon energy as E_mag(q) ≈ħω_mag≡ 2I|J^x_2k_F|m_2k_F/N_⊥, 
allowing us to analytically compute the temperature dependence of the order parameter (see Appendix <ref>),

    m_2k_F(T) = 1- 1/2(T/T_0)^3-2K,  as T  T_0,

shown in the left inset of Fig. <ref>. Here the transition temperature T_0 is defined such that m_2k_F(T_0)=1/2, leading to

    k_BT_0   =   [ A_0^2 I^2 /3N_⊥( Δ/2π K)^1-2K C(K) ]^1/(3-2K), 
    
    
    C(K)   ≡   sin (π K)/16π^3| Γ( 1-K) Γ( K/2 )/Γ[ (2-K)/2 ]|^2,
 
which depends crucially on the Luttinger liquid parameter K, as demonstrated in Fig. <ref>. As discussed above, in nonhelical wires the effective Luttinger liquid parameter K_wire is bounded by 1/2, so the feedback effect from the Overhauser field is essential to enhance T_0 to millikelvin range <cit.>. In contrast, there is no such lower bound for K for a helical edge, and we obtain T_0 in the order of tens of mK. The helical character of the edge states thus substitutes for the role of the feedback effect on boosting T_0. Since the fractional power-law dependence of T_0 on N_⊥ gives T_0∝ N_⊥^-0.38 (assuming K=0.2), increasing N_⊥ by a factor of 10 only decreases T_0 by a factor of 2.4, suggesting that a moderate change in N_⊥ does not lead to a significant change in the estimated value of T_0. 
 

 
 
Due to the temperature dependencies of the RKKY coupling strength [see Eq. (<ref>)] and of the order parameter [see Eq. (<ref>)], the magnon excitation energy also depends on the temperature. In the right inset of Fig. <ref>, we plot the temperature dependence of the magnon energy, which grows upon decreasing the temperature as a power law, 

    ħω_mag (T)   =   2I/N_⊥ |J^x_2k_F(T)|m_2k_F (T).
 
This temperature dependence affects the efficiency of the magnon-mediated backscattering, and therefore enters the magnon-induced resistance [see Eq. (<ref>) below]. In addition, the gap Δ_m opened in the electron spectrum also depends on the temperature,

    Δ_m (T)    =   Δ[ 2K A_0I m_2k_F(T)/Δ]^1/(2-K).

We obtained this formula using a self-consistent variational approach <cit.>. This gap is, however, below the Fermi surface, and thus not directly observable in transport experiments.
Since the spiral nuclear spin order has no influence on the electron subsystem at the Fermi surface, the previously considered detection methods <cit.> are not directly applicable. In a clean and short system the edge states remain gapless despite of the formation of the spiral nuclear spin order. Nevertheless, the gap below the Fermi surface provides an alternative to detect the spiral nuclear spin order, which we discuss in the following subsection.




 §.§ Experimental signatures of the nuclear spin order 

 
The gap Δ_m below the Fermi surface results in experimental signatures which can reveal the spiral order.
Since the nuclear spin dynamics  is much slower than the one of electrons, one may change the gate voltage quickly to shift the electron Fermi energy in the gap, while the pitch of the nuclear spin order, and therefore the position of the gap, remains fixed. Then, indirect evidences for the spiral order can be searched for by measuring the dc conductance, which reduces to zero if the Fermi energy is placed inside the gap. In addition to the gap position, this measurement can also determine the position of the charge neutral point, which would otherwise be difficult to locate due to the constant density of states for the edge states. 
Alternatively, one can reach states away from the Fermi surface with finite-frequency measurements. For instance, the Drude peak in the ac conductivity shifts from zero frequency to a finite frequency associated with the gap, when the scanned Fermi energy is inside the gap.

With these intuitions, we now proceed to explicit formulas. We first consider the case when the Fermi energy is outside of the gap. 
In optical experiments, the ac conductivity can be measured without the influence of the leads. Following Ref. <cit.>, we compute the ac conductivity (see Appendix <ref> for the details),

    σ(ω)    =   e^2/ħ (uK) [ δ(ω) + i/π𝒫(1/ω) ],

with the Dirac delta function δ (x) and the principal value 𝒫(x). The real part of the ac conductivity shows a Drude peak at zero frequency with the weight uK (e^2/ħ), and the imaginary part is connected to the real part through the Kramers-Kronig relations.





When measuring the charge transport through edge states over the finite length L, however, the effects of the Fermi liquid leads must be incorporated. To this end, we apply the Maslov-Stone approach <cit.> to compute the nonlocal conductivity σ_nl and the dc conductance G_dc by modeling the leads as a Tomonaga-Luttinger liquid with a different parameter K_L. The nonlocal conductivity σ_nl relates the charge current I_c to the external electric field E_ext through

    I_c(r,t)    =   ∫_-L/2^L/2 dr' ∫dω/2π  e^-iω tσ_nl(r,r',ω) E_ext(r',ω).
 
In general, the nonlocal conductivity depends on both r and r', and here we give only its expression at the origin [(r,r')=(0,0)], which is related to the dc conductance by G_dc = lim_ω→ 0Re[σ_nl(0,0,ω)]. With the details of derivation presented in Appendix <ref>, the real and imaginary parts of the nonlocal conductivity at (r,r')=(0,0) are given by



    Re[σ_nl(0,0,ω)] = e^2/hK^2/K_L1/sin^2 ( ω L/2u) + ( K/K_L)^2 cos^2 ( ω L/2u) , 
       Im[σ_nl(0,0,ω)] = e^2 K /htan( ω L/2u) ( K/K_L)^2 - 1/( K/K_L)^2 + tan^2 ( ω L/2u) .


As shown in Fig. <ref>, the real part Re[σ_nl(0,0,ω)] oscillates between the maximal value K_L (e^2/h) at the angular frequencies ω = 2n π u/L and the minimal value (K^2/K_L) (e^2/h) at the angular frequencies ω = (2n+1) π u/L with an integer n, similar to a fractional helical Tomonaga-Luttinger liquid <cit.>. 
Using Eq. (<ref>), we get the following expression for the dc conductance,

    G_dc   =   e^2/h K_L.

Importantly, G_dc is independent of the Luttinger liquid parameter K of the edge states. We note that Eq. (<ref>) is valid in a short edge L≪ξ_hf, where the resistance caused by the nuclear spins [Eqs. (<ref>)–(<ref>)] is insignificant.
Even though Eq. (<ref>) suggests that measuring G_dc in the gapless regime does not reveal any feature of the nuclear spin order or the Tomonaga-Luttinger liquid, it can be used to contrast the measurement when the Fermi energy is in the gap, as we discuss below.





We now consider the case when the Fermi energy is quickly tuned into the gap Δ_m, where the action acquires a sine-Gordon term [see Eq. (<ref>)]. The dc conductance in this case is absent, G_dc=0, instead of being given by Eq. (<ref>). Therefore, the zero dc conductance in the range [-2ħ v_F k_F - Δ_m/2, -2ħ v_F k_F + Δ_m/2] when scanning the Fermi energy by a back gate can serve as an experimental signature for the spiral order, as well as a method to determine the gap value. 

An alternative is provided by the ac conductivity probed optically with the Fermi energy inside the gap,

    σ(ω)    =   e^2/ħ (uK) [ δ(ω-Δ_m^2/ħ^2ω) . 
           + . i/π𝒫(1/ω- Δ_m^2/(ħ^2ω)) ].

In this case, the Drude peak is shifted to a finite frequency corresponding to Δ_m. As shown in Fig. <ref>, the gap Δ_m [see Eq. (<ref>)] depends on the temperature, so does the position of the Drude peak. Therefore, tracking the evolution of the Drude peak position with the temperature can then verify the temperature dependence of the gap Δ_m, and thus the generalized Bloch law given by Eq. (<ref>). 
We find that to access the Drude peak, say, at T=54mK, it requires a microwave source with the frequency, Δ_m(T=54mK)/h ≈ 12GHz, which is experimentally accessible. 

We conclude this section with some remarks on the proposed microwave measurements. First, the maximal frequency that a microwave source can reach sets a practical constraint for observing the Drude peaks. Thus, in Fig. <ref>, we mark the temperature region in which the corresponding gap value can be reached by assuming this maximal frequency to be 20 GHz. 
Second, the temperature fluctuations near the transition temperature lead to the fluctuation of the gap as indicated in Fig. <ref>, so a precise temperature control would be required for a clear peak in the measurements.
Third, it is necessary for the edge length to be longer than the Fabry-Perot length defined as L_FP≡ h v_F/Δ_m such that the effect of the leads is negligible. For our parameters, we find L_FP to be in the order of μm. 




 §.§ Self-consistency of the RKKY approach 


In this subsection we comment on the self-consistency condition of the RKKY approach, which allows us to derive the RKKY interaction Eq. (<ref>) from the hyperfine interaction Eq. (<ref>) <cit.>.
Even though similar discussions are also given in Refs. <cit.> for nonhelical systems, here we point out the importance of the finite-size effect.

For the self-consistency of the RKKY approach, we examine the following conditions. First, the RKKY approach requires the energy scale of the electron subsystem to be larger than the coupling between the electron and nuclear subsystems, such that the higher-order terms after performing the Schrieffer-Wolff transformation can be dropped. As mentioned in Sec. <ref>, this is justified by the weak hyperfine coupling compared to the electron Fermi energy. Second, the separation of the time scales of the electron and nuclear spin dynamics can be examined by comparing the Fermi velocity v_F and the magnon velocity <cit.>. We check that, around T_0, v_F is larger than the maximal magnon velocity, computed from the slope of the magnon spectrum in the vicinity of zero momentum [see Fig. <ref>]. This verifies that the dynamics of the nuclear spins is slower than the electrons. Finally, we check that the energy scale of the RKKY term Eq. (<ref>) is bounded by the original hyperfine interaction Eq. (<ref>). 
Since E_ RKKY∝ |Ĩ_q=2k_F|^2 while E_ hf∝Ĩ_2k_F, the ratio E_ RKKY/E_ hf∝Ĩ_2k_F∝ m_2k_F(T) decreases with the fraction of the ordered nuclear spins m_2k_F. Therefore, E_ RKKY is bounded by E_ hf for T  T_0.
  
At lower temperatures, however, there arise some subtleties when examining the ratio E_ RKKY/E_ hf.
First, the magnitudes of the RKKY coupling |J_2k_F^x| and thus E_ RKKY diverge at zero temperature, whereas E_ hf does not. As a result, upon decreasing the temperature, the energy scale of E_ RKKY inevitably exceeds E_ hf at some point. This issue arises because Eq. (<ref>) was derived assuming an infinite system <cit.>, leading to an unphysical divergence at zero temperature. For a realistic system, the finite-size effect has to be taken into account. Since the thermal length λ_T below T_0 is comparable with a typical edge length of O(10 μ m), the finite-size effect becomes relevant at such low temperatures, and the edge length emerges as a cutoff for the divergence in Eq. (<ref>). Second, in the ordered phase, the hyperfine coupling in E_ hf is renormalized by the electron-electron interaction <cit.>. Third, the electron subsystem can also be affected by the ordering of the nuclear spins. The direct comparison between E_ RKKY and E_ hf then incorrectly neglects the different contributions from the electron subsystem before and after applying the RKKY approach. 

To reflect these issues and make a sensible check on the self-consistency, 
we introduce the edge length L as a cutoff by replacing q → q + iπ/L in the zero-temperature expression of the RKKY coupling, which is given by Eq. (30) in Ref. <cit.>. This gives E_ RKKY / E_ hf∼ O(10) for L = 10 μm and K=0.2. 
This value is the upper bound for the ratio, and it will be reduced by finite temperature and therefore we conclude that the bound is fulfilled up to a numerical factor of order unity. Using the finite-size expression of the RKKY interaction, we now reestimate the value for T_0, which is reduced. Importantly, this reduction is modest, since T_0 depends on the magnitude of the RKKY coupling only weakly, as indicated by Eq. (<ref>). More precisely, it is given by a fractional power law (T_0 ∝ |J_2k_F^x|^0.38 for K=0.2), and for L=10 μm the reduction is a factor of ∼ 0.4.
We therefore conclude that at typical parameters that we use, our approach is self-consistent. Due to analytical inconveniences accompanying the finite-size regularization, we use the RKKY interaction of an infinite system, Eq. (<ref>), elsewhere in the article. This somewhat (by a factor of order unity) overestimates the transition temperature, an error which is of little importance here.




§ RESISTANCE IN THE ORDERED PHASE 


After discussing the RKKY-induced spiral order, we consider how it modifies the transport properties of the edge states. We find that there are two additional backscattering mechanisms in the ordered phase. First, the spin-flip backscattering can arise as a combination of the Overhauser field induced by the nuclear spin order and the impurities. The former provides the spin flip, whereas the 4k_F component of the random potential of the latter provides the necessary momentum. This Overhauser-field-assisted backscattering process is sketched in Fig. <ref>. 
Second, magnons, the excitations of the nuclear spin ground state, can also cause electron backscattering. 
It differs from the electron-nuclear spin coupling in the disordered phase, since now it takes a finite exchange energy, set by the magnon energy, for the electron and nuclear spins to flip-flop. Therefore, in the ordered phase it costs energy ħω_ mag for the electrons to backscatter.
In addition, thermally excited magnons can be absorbed by electrons and cause additional backscattering. In the following, we show how these backscattering mechanisms arise from the hyperfine interaction 𝐻_hf defined in Eq. (<ref>).




To this end, we perform the Holstein-Primakoff transformation [see Appendix <ref> for details] on the hyperfine interaction [see Eq. (<ref>)], which is then written as 𝐻_hf = ⟨𝐻_hf⟩_gs + 𝐻_e-mag, where ⟨⋯⟩_gs denotes the expectation value with respect to the nuclear spin ground state <cit.>. The first term arises from the ground state of the spiral order,

    ⟨𝐻_hf⟩_gs   =   B_Ov/2π a∫ dr  cos[ 2ϕ(r) - 4k_F r ],
 
with the Overhauser field given by B_Ov≡ A_0I m_2k_F.
The second term describes the coupling between the electrons and the magnons, 

    𝐻_e-mag   ≈    A_0/2  L^2√(I m_2k_F/2 N_⊥)∑_q,q' 1/i( b_q'^† + b_- q')  
          × L_↑^†(q) R_↓(q+q'-2k_F) + H.c.,

where we keep the lowest-order backscattering terms in magnon operators.
In the above, b_q^† (b_q) creates (annihilates) a magnon with momentum q. 
In the following subsections, we then investigate the edge resistance caused by these additional backscattering processes defined by Eqs. (<ref>) and (<ref>).




 §.§ Impurity-induced resistance in the ordered phase 


The Overhauser field [see Eq. (<ref>)] contains an oscillating integrand except for the special case 4k_Fa = integer×π, which we do not consider. Therefore, Eq. (<ref>) is irrelevant in the RG sense, and does not cause any electron backscattering at the Fermi surface on its own. Nonetheless, it causes a mixing of the right- and left-moving electrons with opposite spins, lifting the topological protection of the helical edge states against impurities. To proceed, we model the impurity Hamiltonian as 

    𝐻_imp   =   ∫ dr   V_imp(r) [R_↓^†(r) R_↓(r) + L_↑^†(r) L_↑(r) ] ,
 
where the Gaussian random potential V_imp(r) satisfies V_imp(r) V_imp(r') = M_impδ (r-r'), with ⋯ denoting the average over the random potential. We estimate the impurity strength M_imp=ħ^2 v_F^2/(2πλ_mfp) with the mean free path of the 2DTI bulk of λ_mfp∼ 0.1–1 μm <cit.>, as listed in Table <ref>.
To derive the effective action for the nuclear-order-assisted backscattering on impurities, we perform a Schrieffer-Wolff transformation <cit.> and average over impurities <cit.>. We defer the details of calculations in Appendix <ref>. Here, we present the result,

    δ𝑆_hx/ħ   =    - M_hx/(2πħ a)^2  ∫_u|τ-τ'|>a  dr dτ dτ'  
          ×cos[ 2 ϕ (r,τ)- 2ϕ (r,τ') ],

which is identical to Eq. (<ref>) upon replacing the coupling 
M_hf→ M_hx≡ M_impB_Ov^2/(64 ħ^2 v_F^2 k_F^2).
Therefore, the RG flow equations can be derived as in Appendix <ref>, leading to a set of RG flow equations identical to Eq. (<ref>) with the replacement D_hf→ D_hx≡ 2a M_hx /(πħ^2 u^2).
We then find that Eq. (<ref>) is RG relevant for K(l)<3/2, which leads to the Anderson-type localization in an edge longer than the associated localization length,

    ξ_hx = a D_hx^-1/(3-2K),

depending on the temperature through m_2k_F(T) [see Eq. (<ref>)].
Since for the above values of the mean free path this backscattering strength is comparable to the strength of backscattering on disordered nuclear spins, the  localization length at zero temperature ξ_hx(T=0) is also comparable to ξ_hf. We define the characteristic temperature T_hx≡ħ u / [k_Bξ_hx(T=0)] through the zero-temperature localization length, and find that typically T_hx >T_0. This means for a sufficiently long edge L > ξ_hf≈ξ_hx(T=0), the electrons get localized by the impurities once the nuclear spins start to order at T_0. 


 


In Fig. <ref>, we plot T_hx and ξ_hx(T=0) as functions of the Luttinger liquid parameter for various materials. Again, a common property shared by all the curves is that the backscattering effects are enhanced by the electron-electron interaction. In contrast to the disordered phase (see Fig. <ref>), however, here the estimated quantities for the helical and nonhelical states are comparable in the strongly interacting regime, indicating that the localization effects in the ordered phase are not as markedly different for a helical and a nonhelical channel as in the disordered phase. The reason behind this is that the Overhauser field in a spin-degenerate wire provides a synthetic spin-orbit interaction in the ordered phase, making the remaining gapless electrons helical <cit.>. After ordering, the effective Luttinger liquid parameter of the remaining gapless modes is not bounded by 1/2 anymore. As a consequence, the estimated values of T_hx and ξ_hx(T=0) in the helical and nonhelical systems become similar in the presence of strong electron-electron interaction.

Since typically T_hx > T_0, the impurities induce an exponentially growing resistance below T_0, 

    R_hx (T)    ∝    R_0π D_hxL /2K^2 a  e^Δ_hx /(k_BT),

with a gap Δ_hx = Δ( 2 K D_hx)^1/(3-2K). Despite its similarity to Eq. (<ref>), R_hx contains the temperature-dependent D_hx and Δ_hx while their counterparts D_hf and Δ_hf are independent of T. As a result, the resistances arising from the two scenarios [Eq. (<ref>) versus Eq.(<ref>)] are distinct due to the different temperature dependencies of the gaps and prefactors, as well as the dependence of D_hx and Δ_hx on ϵ_F and M_imp.

Before moving to the magnon-mediated backscattering, let us comment on two complications not considered in this work. First, we remark that an applied voltage may lead to the nuclear spin polarization along the spin quantization (z) axis <cit.>, and thus modify the nuclear spin order. While the z component of the nuclear spin polarization does not directly cause the spin-flip backscattering, it reduces the xy components of the Overhauser field from B_Ov to B_Ov√(1-P_N^2(T)), with a temperature-dependent factor, P_N(T). However, unless the nuclear spins are nearly full-polarized, which requires a very high applied voltage at very low temperatures, the residual xy components of the Overhauser field can still cause the spin-flip backscattering on impurities. Therefore, the voltage-induced dynamic nuclear polarization would not alter our conclusion qualitatively.

Second, in the ordered phase the gap Δ_hx reduces the RKKY coupling, and therefore the strength of the nuclear-order-assisted backscattering. However, since the effective range of the RKKY coupling is related to the electron Fermi wavelength λ_F≡ 2π / k_F, the gapped electrons can still mediate the RKKY interaction within the scale of λ_F, provided that it is much shorter than the length scale associated with the gap, ħ v_F / Δ_hx, as discussed in Refs. <cit.>. We thus expect our results to remain qualitatively valid if the condition λ_F≪ħ v_F /Δ_hx holds.
We have checked that for our case it is fulfilled, so that the RKKY interaction remains effective, even though the coupling strength is reduced by the gap Δ_hx. 
 



 §.§ Resistance due to the magnon-mediated backscattering 


We now turn to the electron-magnon interaction described by Eq. (<ref>) with the magnon dispersion given by Eq. (<ref>). With the approximated magnon energy [see Eq. (<ref>)], we are able to reformulate the electron-magnon backscattering as an electron-phonon backscattering problem. In particular, it is analogous to a Tomonaga-Luttinger liquid consisting of spinless fermions coupled to dispersionless phonons <cit.>.
We then proceed by integrating out the magnons, and obtain the contribution to the effective action from the magnon-mediated backscattering. The details of calculations are relegated to Appendix <ref>, in which we get the following expressions,



    δ𝑆_mag= δ𝑆_mag^em + δ𝑆_mag^abs,  
       δ𝑆_mag^em/ħ = - M_mag/(2πħ a )^2  ∫_u|τ-τ'|>a dr dτ dτ'   e^-ω_mag |τ-τ'| 
       ×[  1 + n_B(ħω_mag) ] cos[  2 ϕ (r,τ)- 2ϕ (r,τ') ] ,
     
       δ𝑆_mag^abs/ħ = - M_mag/(2πħ a )^2  ∫_u|τ-τ'|>a dr dτ dτ'   e^ω_mag |τ-τ'| 
       ×
     n_B(ħω_mag) cos[  2 ϕ (r,τ)- 2ϕ (r,τ') ],


with M_mag≡ A_0^2 a I/(4N_⊥) being the backscattering strength and the Bose-Einstein distribution given by

    n_B(E)     =   1/e^E / (k_BT) - 1.

In comparison with Eqs. (<ref>) and (<ref>), the magnon-mediated backscattering acquires extra exponential factors e^±ω_mag |τ-τ'| in the integrand, corresponding to the process where a magnon is absorbed (+) or emitted (-) because of a finite energy exchange due to the electron spin flip through a magnon. In addition, the efficiency of the magnon-mediated backscattering depends on the magnon occupation n_B(ħω_mag), and therefore on the temperature.

One can visualize the effects of the magnon-mediated backscattering on the resistance by examining Eq. (<ref>): if the magnon energy is much larger than the temperature, the backscattering is suppressed exponentially by either the exponential factor in Eq. (<ref>) (for magnon emission), or by the Boltzmann factor of the magnon occupation in Eq. (<ref>) (for magnon absorption). We then expect that the magnon-induced resistance to be suppressed in the T → 0 limit. On the other hand, if the magnon energy is comparable to the temperature (that is, when T is near T_0), the order parameter is small and there are many thermally excited magnons. Then, the electron-magnon backscattering events become efficient and give rise to a resistance similar to the one caused by disordered nuclear spins [Eqs. (<ref>)-(<ref>)], which can be considered as the ordered nuclear spins with zero-energy magnons. 

We confirm these observations by performing the RG analysis. In the low-temperature limit, where the contribution from magnon emission dominates, we have δ𝑆_mag≈δ𝑆_mag^em with n_B(ħω_mag) → 0. When the short-distance cutoff a increases under the RG flow, the exponential factor in Eq. (<ref>) decreases, and δ𝑆_mag^em becomes vanishingly small as a>u/ω_mag. Therefore, we derive the RG flow equations up to the scale l_mag^*≡ln [u/(a ω_mag)] as in Refs. <cit.> (similar to the procedure given in Appendix <ref>), which are given by 



    d Y_mag (l) /d l  =[3-2K(l) ] Y_mag (l), 
       d K (l) /d l  = - K^2 (l) /2 Y_mag(l) ω_mag a /u e^- ω_mag a(l)/u, 
       du(l)/dl = -u(l)K(l)/2 Y_mag(l) ω_mag a /u e^-ω_mag a(l)/u,
 

with Y_mag≡ 2 M_mag / (πħ^2 u ω_mag).
Using Eq. (<ref>), we calculate the resistance caused by the magnon emission at low temperatures T<T_x. The temperature T_x, defined by ω_mag (T_x) = D_mag^1/(4-2K) u/a, gives the limit at which the RG flow reaches the strong-coupling regime.
For T<T_x, we integrate the RG flow up to l_mag^* to obtain 

    R_mag^em (T)    ∝    R_0π D_magL/2 K^2 a[ K ħω_mag(T) /Δ]^2K-3,
 
with D_mag≡ 2a M_mag /(πħ^2 u^2). Upon decreasing the temperature, the resistance due to magnon emission decreases as a power law of the magnon energy, whose temperature dependence is given by Eq. (<ref>). 
 
On the other hand, in the range T_x T  T_0, the backscattering due to the magnon absorption process Eq. (<ref>) becomes efficient. In this case, the magnon energy is so low compared to the temperature that it can be approximated by zero. We therefore expect that the associated resistance R_mag^abs(T) takes the form of R_hf [Eqs. (<ref>)–(<ref>)], with the strength weighted by the fraction of the disordered nuclear spins. Since typically T_0<T_hf, the regime corresponding to Eq. (<ref>), which is valid only for T>T_hf, is never reached in the ordered phase.  In addition, we restrict ourselves in the low-bias regime, in which the high-bias resistance Eq. (<ref>) is not relevant.
Both of the remaining equations [see Eqs. (<ref>) and (<ref>)] then give the same temperature dependence,

    R_mag^abs (T)    ∝    R_0π D_hfL/2 K^2 a[ 1-m_2k_F (T) ] ,
 
decaying as a T^3-2K. 
In addition to the limit set by T_x, we note another limit described by R_mag (T) ≡ R_mag^em (T) + R_mag^abs (T) ≤ R_hf (T), with R_hf(T) determined by Eqs. (<ref>) and (<ref>). This arises from the self-consistency check: 
the resistance from the backscattering that requires an energy [see Eq. (<ref>)] should be bounded by the resistance when such an energy cost is absent [see Eq. (<ref>)]. This allows us to define T_b through R_mag (T_b) = R_hf(T_b), and numerically find T_b≈ T_x (see Fig. <ref>), consistent with the limit set by T_x. Overall, we find the magnon-induced resistance to be dominated by Eq. (<ref>) for T  T_x≈ T_b, whereas Eq. (<ref>) also contributes in the range T_x≈ T_b T  T_0. 

 





§ DISCUSSION 


We now summarize our results on the resistance discussed in Secs. <ref> and <ref>. For the demonstration, we plot the temperature dependence of the edge resistance of the 2DTI of length L<ξ_hf in Fig. <ref>. The opposite regime, L>ξ_hf, has been presented in Ref. <cit.>. Well above T_0, the resistance R_hf (the black solid curve) initially increases as a power law, which becomes a plateau in the range T_0<T<ħ u / (k_BL) [Eq. (<ref>)]. 
Below T_0, at which the nuclear spins form an order, the resistance is initially dominated by the magnon-mediated backscattering. The resistance R_mag^abs from the magnon absorption drops as a power law (the black dashed curve). Below T_b ≈ T_x, the resistance due to the magnon emission is given by R_mag^em, which decays as a power law (the blue curve) different from R_mag^abs. Both R_mag^em and R_mag^abs decay to zero as T → 0, as expected. 
At very low temperatures, the established Overhauser field dominates the resistance by allowing backscattering on impurities, leading to an exponential form of the resistance R_hx (the red curve).
In addition, in the inset of Fig. <ref>, we plot the temperature dependence of the gap Δ_hx, which, as opposed to the constant Δ_hf, distinguishes the two exponential regions due to disordered and ordered nuclear spins as T>T_0 and T<T_0, respectively. The temperature dependent gap Δ_hx can therefore serve as an experimental signature of the spiral nuclear spin order, in addition to those discussed in Sec. <ref>.
We conclude that the nuclear spins, whether ordered or not, suppress the edge conductance of a sufficiently long 2DTI sample as the temperature approaches zero.

Finally, we remark that our estimation with realistic material parameters allows us to discuss the relevance of nuclear spins to the edge resistances of HgTe/(Hg,Cd)Te and InAs/GaSb 2DTIs observed in experiments. As mentioned in Sec. <ref>, the effects of nuclear spins in HgTe/(Hg,Cd)Te are insignificant even for strong interactions, suggesting that the observed finite edge resistance in HgTe/(Hg,Cd)Te 2DTIs <cit.> is unlikely due to nuclear spins.

On the other hand, whether nuclear spins in InAs/GaSb 2DTIs can lead to an appreciable edge resistance depends on the experimental conditions. To be explicit, we expect our mechanism to be relevant for strong interactions, long edge lengths, and low temperatures. Since, however, the interaction parameter K is typically unknown in real samples, it is difficult to draw conclusions on the relevance of the nuclear spins. Recently, a value of K=0.21–0.22 was extracted in InAs/GaSb 2DTIs <cit.>. [5] 
However, based on our estimation in Secs. <ref> and <ref>, the corresponding localization length is much longer than the edge length L∼ 1 μm of the samples in Ref. <cit.>. We therefore believe that the observed edge resistance in their experiment is probably dominated by other sources than nuclear spins. Nevertheless, we expect the nuclear spins in InAs/GaSb to become relevant for longer samples with small K at low temperatures.




We thank P. P. Aseev, Y.-Z. Chou, R. S. Deacon, M. R. Delbecq, S. Hoffman, P. Marra, and K. Muraki for helpful discussions. This work was supported financially by the the JSPS Kakenhi Grant No. 16H02204, the Swiss National Science Foundation (Switzerland), and the NCCR QSIT.






§ DERIVATION OF THE RG FLOW EQUATIONS 


In this appendix, we sketch the derivation of the RG flow equations for the backscattering action 𝑆_el+δ𝑆_hf in the disordered phase. The other backscattering processes [Eqs. (<ref>) and (<ref>)] can be treated similarly. To this end, we compute the correlation function as in Refs. <cit.>,

    < e^i [ϕ( r_1) - ϕ( r_2)] >_𝑆_el+δ𝑆_hf
       ≡𝑍^-1∫ Dϕ  e^-𝑆_el/ħ e^-δ𝑆_hf/ħ  e^i [ϕ( r_1) - ϕ( r_2)] ,

with the bold font denoting the two-dimensional vectors r_j≡ (r_j, y_j) = (r_j, u τ_j) and the partition function,

    𝑍≡∫ Dϕ  e^-(𝑆_el+ δ𝑆_hf)/ħ.
 
We expand the correlation function up to the first-order terms in D_hf, corresponding to the second-order terms in the random potential V_hf,2k_F. The zeroth-order term gives

    < e^i [ϕ( r_1) - ϕ( r_2)]>_𝑆_el   =    
    e^- K F( r_1 -  r_2) / 2,

where we define the function,

    F( r_1 -  r_2)
       ≡1/2ln[ (r_1-r_2)^2 + u^2 (τ_1 - τ_2)^2 /a^2] + t_⊥/Kcos (2 θ_ r_1 -  r_2 ),

where θ_ r is the angle between the vector r=(r,y)=(r,uτ) and the spatial coordinate axis r, and the t_⊥ term generated by the RG flow gives the anisotropy between the spatial and the temporal coordinates. The first-order term in D_hf is given by 

    D_hf/ 8π a^3 ∫_|y-y'|>a dr dy dy' 
     
          ×{< e^i [ϕ( r_1) - ϕ( r_2)]cos[ 2ϕ (r,τ) - 2ϕ (r,τ')] >_𝑆_el. 
          . - < e^i [ϕ( r_1) - ϕ( r_2)]>_𝑆_el< cos[ 2ϕ (r,τ) - 2ϕ (r,τ')] >_𝑆_el}.

The correlation function can then be computed along the line of Refs. <cit.>, giving Exp[-K_eff F_eff( r_1 -  r_2) /2], where F_eff( r_1 -  r_2) takes the form of Eq. (<ref>) with the effective parameters,


    K_eff   =    K - K^2 D_hf/2∫_a^∞dz/a( z/a)^2-2K  , 
    
    t_⊥,eff   =     t_⊥ + K^2 D_hf/4∫_a^∞dz/a( z/a)^2-2K.


The RG flow equations can be obtained by increasing the cutoff a → a e^d l = a + da while keeping the correlation function the same. Finally, we obtain a set of three equations,


    d D_hf (l) /d l    =   [3-2K(l)] D_hf (l), 
    d K (l) /d l    =    - K^2 (l) /2 D_hf(l), 
    d t_⊥ (l)/d l    =   K^2 (l) /4 D_hf(l).


In addition, from Eq. (<ref>) we see that the renormalization of t_⊥ is equivalent to that of u, leading to 

    du(l)/dl   =    -2 u(l)/K(l)d t_⊥ (l)/d l .

The RG flow equations are then given in Eq. (<ref>) in Sec. <ref>. Since the RG flow equations are obtained with the perturbation in D_hf(l), they are valid only below the length scale l_hf^*, at which D_hf(l_hf^*) ∼ 1.




§ SPIN-WAVE ANALYSIS 


In this Appendix, we provide the details of the spin-wave analysis. For the sake of convenience, we start by locally rotating the spin axes, (Ĩ_j^x,Ĩ_j^y,Ĩ_j^z) → (Ĩ_j^1,Ĩ_j^2,Ĩ_j^3) such that in the new basis (ê_j^1,ê_j^2,ê_j^3) the ground state of the nuclear spins is described as a uniform ferromagnet, i.e. < Ĩ(r_j) > = N_⊥ I m_2k_F ê_j^1 <cit.>. In addition, the electron spin operators are also rotated accordingly. To proceed, we perform the Holstein-Primakoff transformation <cit.>, in which the nuclear spin operators are written in terms of the ground state of the spiral order in addition to the deviation caused by the magnon excitation, 



    Ĩ^1_j   =    N_⊥I  - a_j^† a_j, 
    Ĩ^2_j   ≈   √(N_⊥I /2)( a_j + a_j^†), 
    Ĩ^3_j   ≈   √(N_⊥I /2)1/i( a_j - a_j^†),


where a_j(a_j^†) is the annihilation (creation) operator of a magnon at site j. Note that in this Appendix, we assume T=0 for the ease of notation, so that m_2k_F=1. For finite temperatures, where the nuclear spins are partially ordered, the formulas are valid with the replacement, I → I m_2k_F. Using Eq. (<ref>), we derive the magnon Hamiltonian 𝐻_mag in momentum space from the RKKY interaction given by Eq. (<ref>),

    𝐻_mag   =   1/L∑_q( a^†_q, a_q)
    ( 
    [ h_11(q) h_12(q); h_12(q) h_11(q) ])
    (
    [   a_q; a^†_q ]),

where we have dropped the constant and the higher-order terms in the magnon operators. The functions h_11(q) and h_12(q) are defined as


    h_11(q)    ≡   I/4N_⊥( 2 J^z_q + J^x_q-2k_F +  J^x_q+2k_F - 4J^x_2k_F) 
       ≈   I/4N_⊥( J^x_q-2k_F +  J^x_q+2k_F - 4J^x_2k_F),
    
    h_12(q)    ≡   I/4N_⊥( J^x_q-2k_F +  J^x_q+2k_F -2 J^z_q) 
       ≈   I/4N_⊥( J^x_q-2k_F +  J^x_q+2k_F),


where we have used the fact that the RKKY coupling is highly anisotropic |J^x_q|, |J^y_q| ≫ |J^z_q|. The bilinear bosonic Hamiltonian Eq. (<ref>) can be diagonalized by the Bogoliubov transformation,  

    (
    [ a^†_q;  a_-q ])
       =   ( 
    [  u_BT(q) -v_BT(q); -v_BT(q)  u_BT(q) ])
    (
    [ b^†_q;  b_-q ]),

with the coefficients,



    [u_BT(q)]^2    =   1/2[ 1 + h_11(q)/√(h_11^2(q) - h_12^2(q))], 
    [v_BT(q)]^2    =   1/2[ -1 + h_11(q)/√(h_11^2(q) - h_12^2(q))].


Using Eqs. (<ref>)–(<ref>), the magnon Hamiltonian Eq. (<ref>) can be diagonalized as 

    𝐻_mag = 1/L∑_q E_mag(q) b_q^† b_q,
 
with the magnon excitation energy E_mag(q) given by Eq. (<ref>) and shown in Fig. <ref>. 
Since the magnon spectrum is almost dispersionless, we approximate E_mag(q) ≈ 2I|J^x_2k_F|/N_⊥, leading to u_BT(q) ≈ 1 and v_BT(q) ≈ 0, and therefore b_q≈ a_q. The transition temperature and the temperature dependence of the order parameter can be calculated by evaluating the magnon occupation number <cit.>, 

    N_⊥∑_q ≠ 01/e^E_mag(q)/k_BT-1,

and the results are given in Eqs. (<ref>)–(<ref>) in Sec. <ref>. 
 



§ TRANSPORT PROPERTIES OF A HELICAL TOMONAGA-LUTTINGER LIQUID 


In this Appendix we sketch the calculation of the conductance and the nonlocal conductivity of the helical edge states. When the Fermi energy is away from the gap, the action is given by Eq. (<ref>), from which we obtain the Green's function in the momentum-Matsubara frequency domain,

    G_ϕϕ(q,ω_n)    =   π u K/ω_n^2 + u^2q^2 ,

with the momentum q and Matsubara frequency ω_n. In optical measurements, the ac conductivity can be computed as in Refs. <cit.>, leading to 

    σ(ω)
       =   . e^2/π^2 ħω_n G_ϕϕ(q=0,ω_n)|_iω_n→ω+i0^+
       =   e^2/ħ (uK) [ δ( ω) + i/π𝒫( 1/ω) ],

as given in Eq. (<ref>) in Sec. <ref>. 

On the other hand, the nonlocal conductivity and the dc conductance of a finite-size system (in the presence of the leads) can be computed by using the Maslov-Stone approach <cit.>, in which the velocity and the Luttinger liquid parameter are taken to be spatially dependent, and change abruptly at the interfaces between the leads and the helical Tomonaga-Luttinger liquid. The action now takes the form,

    𝑆_el/ħ   =   ∫dτ dr/2π{u (r)/K(r)[ ∂_rϕ(r,τ) ]^2 . 
          . + 1/u(r)K(r) [ ∂_τϕ(r,τ) ]^2 },

where the spatial dependent velocity is defined as u(r)=v_F/K(r) with the Luttinger liquid parameter,

    K(r)    =   {[ K,   as  -L/2 ≤ r ≤ L/2,;        K_L,   otherwise. ].

The charge current is related to the external electric field by Eq. (<ref>), where the nonlocal conductivity is given by <cit.>

    σ_nl(r,r',ω)    =   . e^2/π^2 ħω_n  G_nl(r,r',ω_n)|_i ω_n→ω + i 0^+,

with the nonlocal propagator G_nl(r,r',ω_n). 
It satisfies

    {- ∂_r[ u(r)/K(r)∂_r] + ω_n^2/u(r)K(r)} G_nl(r,r',ω_n)    =   πδ (r-r').
 
and the boundary conditions,


    (i)         G_nl(r →±∞, r',ω_n) = 0 ; 
    (ii)         G_nl(r,r',ω_n)  is continuous at  r=r', ± L/2 ;
    (iii)        u(r)/K(r)∂_r G_nl(r,r',ω_n)  is continuous at  r=± L/2 ;
    
    (iv)         - .u(r)/K(r)∂_r G_nl(r,r',ω_n)|_r'-0^+^r'+0^+ = π.


We take the ansatz for the nonlocal propagator, 

    G_nl(r,r',ω_n) 
       =   {[                   A e^|ω_n|r/u_L,; B_+ e^|ω_n|r/u + B_- e^-|ω_n|r/u,; C_+ e^|ω_n|r/u + C_- e^-|ω_n|r/u,;                  D e^-|ω_n|r/u_L, ].
    [      as  r ≤ -L/2,; as  -L/2 ≤ r ≤ r',;  as  r' ≤ r ≤ L/2,;       as  r ≥ L/2, ]

which satisfies the condition (i). Here we define u_L≡ v_F / K_L. The unknowns A, B_±, C_±, and D are functions of r' and ω_n, and can be solved for by applying the boundary conditions (ii)–(iv) <cit.>. 
The solutions for these unknowns are



    A    =   π K/2 |ω_n| e^|ω_n|L/2u_L[ cosh( |ω_n|r'/u) /sinh( |ω_n|L/2u)
    +K/K_Lcosh( |ω_n|L/2u)  
    - sinh( |ω_n|r'/u)  /cosh( |ω_n|L/2u) +K/K_Lsinh( |ω_n|L/2u) ], 
    
    B_±   =   π K/4 |ω_n| e^±|ω_n|L/2u( 1 ±K/K_L) [ cosh( |ω_n|r'/u) /sinh( |ω_n|L/2u)
    +K/K_Lcosh( |ω_n|L/2u)  
    - sinh( |ω_n|r'/u)  /cosh( |ω_n|L/2u) +K/K_Lsinh( |ω_n|L/2u) ], 
    
    
    C_±   =   π K/4 |ω_n| e^∓|ω_n|L/2u( 1 ∓K/K_L) [ cosh( |ω_n|r'/u) /sinh( |ω_n|L/2u)
    +K/K_Lcosh( |ω_n|L/2u)  
    + sinh( |ω_n|r'/u)  /cosh( |ω_n|L/2u) +K/K_Lsinh( |ω_n|L/2u) ], 
    
    
    D    =   π K/2 |ω_n| e^|ω_n|L/2u_L[ cosh( |ω_n|r'/u) /sinh( |ω_n|L/2u)
    +K/K_Lcosh( |ω_n|L/2u)  
    + sinh( |ω_n|r'/u)  /cosh( |ω_n|L/2u) +K/K_Lsinh( |ω_n|L/2u) ].


 
For the dc signals the r and r' dependence in the nonlocal conductivity will eventually vanish, allowing us to focus on the origin [(r,r')=(0,0)]. We then get the propagator,

    G_nl(0,0,ω_n)    =   π K/2 |ω_n| 1+ K/K_Ltanh( |ω_n|L/2u) /K/K_L + tanh( |ω_n|L/2u),

and the nonlocal conductivity σ_nl(0,0,ω), as given in Eq. (<ref>). The dc conductance is then G_dc = lim_ω→ 0Re[σ_nl(0,0,ω)] = K_L e^2/h, as given in Eq. (<ref>).

When the Fermi energy is quickly tuned into the gap, the action acquires an RG-relevant sine-Gordon term,

    δ S_m   =   B_Ov/2π a∫_0^βħ dτ∫ dr  cos[ 2 ϕ (r,τ) ],

leading to the propagator, 

    G_ϕϕ(q,ω_n)    =   π u K/ω_n^2 + u^2q^2 + Δ_m^2/ħ^2.

The dc conductance and the ac conductivity can then be computed following the same procedure, and the results are given in Sec. <ref>. 
 
 


§ SCHRIEFFER-WOLFF TRANSFORMATION 


In this Appendix we perform the Schrieffer-Wolff transformation to obtain the effective Hamiltonian for the Overhauser-field-assisted backscattering on impurities. In the absence of the Overhauser field, (nonmagnetic) impurities cannot cause the spin-flip backscattering, so the helical edge states cannot be localized by the impurities. The Overhauser field, however, acts on electron spins as a spatially rotating Zeeman field, which breaks the time-reversal symmetry. Assuming that the nuclear spin order is given by ⟨Ĩ(r) ⟩_- [see Eq. (<ref>)], the Overhauser field then causes a mixing of the R_↓(q) and L_↑(q+2k_F) particles, inducing a gap Δ_m below the Fermi surface [panel (b) of Fig. <ref>]. Whereas Eq. (<ref>) itself does not lead to any backscattering at the Fermi surface, here we show that a second-order spin-flip backscattering at the Fermi surface can still arise as a combination of the Overhauser field and the impurities, as sketched in Fig. <ref>. Here `second-order' means that the effective backscattering potential is determined by the product of the Overhauser field and the impurity potential.

To proceed, we consider the total Hamiltonian, which consists of two parts, 𝐻_tot = 𝐻_el + δ𝑉, with the perturbation δ𝑉=𝐻_imp + 𝐻_Ov. 
For convenience, we use the fermionic expression for the electron part, 𝐻_el=𝐻_0+𝐻_2+𝐻_4, with the three terms corresponding to the kinetic energy, g_2, and g_4 processes, respectively. To be explicit, we have



    𝐻_0 = -i ħ v_F∫ dr [
    R_↓^†(r) ∂_r R_↓(r) - L_↑^†(r) ∂_r L_↑(r) ], 
       𝐻_2 = g_2∫ dr   R_↓^†(r)  R_↓(r)  L_↑^†(r)  L_↑(r),  
       𝐻_4 = g_4/2∫ dr [
    (R_↓^†(r)  R_↓(r) )^2 + ( L_↑^†(r)  L_↑(r) )^2 ].


The impurity Hamiltonian 𝐻_imp is given by Eq. (<ref>), and the Overhauser field felt by the electrons is described by

    𝐻_Ov≡⟨𝐻_hf⟩_gs   =   B_Ov/2∑_j  e^2ik_Fr_j L_↑^†(r_j) R_↓(r_j) + H.c.,

which is the fermionic form of ⟨𝐻_hf⟩_gs in Eq. (<ref>).

We then perform the canonical Schrieffer-Wolff transformation <cit.>  such that

    𝐻_SW   ≡    e^S_SW𝐻_tot e^-S_SW
       =   𝐻_el + δ𝑉 + [S_SW,𝐻_el + δ𝑉] 
           + 1/2 [S_SW,[S_SW, 𝐻_el + δ𝑉]] + ⋯,

where we keep terms up to the second order in δ𝑉 and S_SW, and choose [S_SW, 𝐻_el] + δ𝑉=0 to eliminate the first-order term in δ𝑉. This gives S_SW = ℒ_el^-1δ𝑉 with the Liouvillian superoperator ℒ_el𝑂≡ [𝐻_el, 𝑂] <cit.>. 
Using the integral representation, ℒ_el^-1 = -i ∫_0^∞ dt  .e^-η t + i t ℒ_el|_η→ 0, we arrive at the effective Hamiltonian 𝐻_SW≈𝐻_el + 𝐻_hx, where the backscattering term is given by 

    𝐻_hx≡   1/2 [S_SW,δ𝑉] 
    = -i/2∫_0^∞ dt   e^-η t
    [δ𝑉̃(t),δ𝑉]_η→ 0,

with the tilde defining an operator in the interaction picture, δ𝑉̃(t) ≡𝐻̃_imp(t) + 𝐻̃_Ov (t). The commutators can be computed straightforwardly, and the results can be simplified by approximating g_2≈ g_4. This approximation can be understood through the form of Eq. (<ref>). Namely, for the Hamiltonian involving only the density-density interaction and therefore insensitive to the spins and the velocities of the electrons, the g_2 and g_4 processes are indistinguishable.  

After performing the integral over time, we finally arrive at

    𝐻_hx   ≈   1/L^2∑_q,q'  V_hx(q') R_↓^†(q+q'+2k_F) L_↑(q) + H.c.,

with the effective coupling for the second-order backscattering process V_hx (q) ≡ B_Ov V_imp (4k_F+q)/ (8 ħ v_F k_F). 
After performing inverse Fourier transform and bosonizing 𝐻_hx, the q' ≠ 0 components give oscillating integrand, and therefore vanish upon integration. Consequently, the effective backscattering potential is given by the product of B_Ov/2 [the strength of 𝐻_Ov in Eq. (<ref>)], and V_imp (4k_F) [the 4k_F component of the random potential in Eq. (<ref>)], divided by the energy difference between the initial and the intermediate states, 4ħ v_F k_F, as expected from Fig. <ref>. Finally, utilizing the replica method <cit.>, we can average the random potential V_imp in 𝐻_hx, leading to the effective backscattering action Eq. (<ref>).




§ MAGNON-MEDIATED BACKSCATTERING 


In this Appendix we provide the derivation of the effective action for the magnon-mediated backscattering process. In the bosonized form and the continuum limit, the electron-magnon interaction, described by Eq. (<ref>), can be written as 

    𝐻_e-mag   =   ∫dr /2π a [ g_e-magφ (r) e^-2iϕ (r) + H.c.],

where we introduce the effective electron-magnon coupling,

    g_e-mag≡ -i A_0/2√(ω_mag I/ħ N_⊥),

and the bosonic field,

    φ (r) ≡√(ħ/2 ω_mag)1/L∑_ q b_q e^i q r + H.c..

Here φ is analogous to the displacement field in the electron-phonon problem <cit.>. 
With these definitions, the magnon Hamiltonian [see Eq. (<ref>)] can be written, up to a constant term, as

    𝐻_mag  
    =1/2∫ dr  [ Π^2 (r) + ω_mag^2 φ^2 (r) ],

with Π being the canonically conjugate momentum to φ. In the above we have used the fact that both φ and Π are Hermitian.
Therefore, the terms involving the magnons, 𝐻_mag+𝐻_e-mag, lead to the contribution to the imaginary-time action δ𝑆_mag≡δ𝑆_mag^(0) + δ𝑆_mag^(1), where 

    δ𝑆_mag^(0)/ħ   =   1/2∫ dr dτ [ -2i Π(r,τ) ∂_τφ (r,τ) + Π^2 (r,τ) . 
          . + ω_mag^2 φ^2 (r,τ) ], 
    δ𝑆_mag^(1)/ħ   =   ∫dr dτ/2π aħ [ g_e-magφ (r,τ) e^-2iϕ (r,τ) + H.c.].

We first integrate out the Π field in the term δ𝑆_mag^(0). In the momentum space and Matsubara frequency domain, it is given by

    δ𝑆_mag^(0)/ħ   =   1/2β L∑_q,ω_n [ δ G_mag (q,ω_n) ]^-1| φ (q,ω_n) |^2 ,

where the magnon propagator is defined as

    δ G_mag (q,ω_n)    =    - 1/(iω_n)^2 - ω_mag^2,

which is independent of the momentum, as we are considering the magnons with dispersionless energy band. Finally, by integrating out the remaining φ field in the action δ𝑆_mag, we get 

    δ𝑆_mag/ħ   =    -| g_e-mag|^2 /(2π)^2 aħ∫ dr dr' dτ dτ'  δ G_mag (r-r',τ - τ') 
          ×[ e^-2i[ϕ(r,τ) - ϕ(r',τ') ] + H.c.],

with the magnon propagator,

    δ G_mag (r,τ)    =   1/β L∑_q,ω_n δ G_mag (q,ω_n) e^i(qr-ω_nτ) .

The summation over the momentum and the Matsubara frequency can be done straightforwardly <cit.>, which leads to

    δ G_mag (r,τ) =  1/2 ω_magδ(r)  
       ×[ e^-ω_mag|τ| + 2 n_B(ħω_mag) cosh( ω_magτ) ].

The resulting effective magnon-mediated backscattering action is then given in Eq. (<ref>) in Sec. <ref>. 

Since the magnon energy ħω_mag, see Eq. (<ref>), depends on the temperature, the behavior and the validity of Eq. (<ref>) also depend on the temperature. First, in the low-temperature regime where k_BT ≪ħω_mag, we may approximate n_B(ħω_mag) ≈ 0 to obtain the effective action dominated by magnon emission, from which we derive Eq. (<ref>). Second, in the T  T_0 regime, the magnon energy is so low that the procedure of integrating out the magnon fields is no longer valid. In this case we compute Eq. (<ref>) using the resistance in the disordered phase, R_hf, which can be considered as the resistance due to the ordered nuclear spins, but with zero-energy magnons. 












105
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Bernevig et al.(2006)Bernevig, Hughes, and Zhang]Bernevig:2006
  author author B. A. Bernevig, author T. L. Hughes,  and author S.-C. Zhang, @noop  journal journal
  Science volume 314, pages 1757
  (year 2006)NoStop
[König et al.(2007)König, Wiedmann, Brüne,
  Roth, Buhmann, Molenkamp,
  Qi, and Zhang]Konig:2007
  author author M. König, author S. Wiedmann,
  author C. Brüne, author A. Roth, author
  H. Buhmann, author L. W. Molenkamp, author X.-L.  Qi,  and author S.-C.  Zhang, 10.1126/science.1148047 journal journal Science volume
  318, pages 766 (year 2007)NoStop
[Roth et al.(2009)Roth,
  Brüne, Buhmann, Molenkamp, Maciejko, Qi, and  Zhang]Roth:2009
  author author A. Roth, author C. Brüne,
  author H. Buhmann, author L. W. Molenkamp, author
  J. Maciejko, author
  X.-L. Qi,  and author
  S.-C. Zhang, 10.1126/science.1174736 journal journal
  Science volume 325, pages 294
  (year 2009)NoStop
[Gusev et al.(2013)Gusev,
  Olshanetsky, Kvon, Raichev,
  Mikhailov, and Dvoretsky]Gusev:2013
  author author G. M. Gusev, author E. B. Olshanetsky, author Z. D. Kvon, author O. E. Raichev,
  author N. N. Mikhailov,  and  author S. A. Dvoretsky, 10.1103/PhysRevB.88.195305 journal journal Phys. Rev. B volume 88, pages 195305 (year 2013)NoStop
[Gusev et al.(2014)Gusev,
  Kvon, Olshanetsky, Levin,
  Krupko, Portal, Mikhailov,  and Dvoretsky]Gusev:2014
  author author G. M. Gusev, author Z. D. Kvon,
  author E. B. Olshanetsky,
  author A. D. Levin, author Y. Krupko, author
  J. C. Portal, author
  N. N. Mikhailov,  and author
  S. A. Dvoretsky, 10.1103/PhysRevB.89.125305 journal journal Phys.
  Rev. B volume 89, pages 125305
  (year 2014)NoStop
[Ma et al.(2015)Ma,
  Calvo, Wang, Lian,
  Mühlbauer, Brüne, Cui,
  Lai, Kundhikanjana, Yang,
  Baenninger, König, Ames,
  Buhmann, Leubner, Molenkamp,
  Zhang, Goldhaber-Gordon, Kelly, and Shen]Ma:2015
  author author E. Y. Ma, author M. R. Calvo,
  author J. Wang, author
  B. Lian, author M. Mühlbauer, author C. Brüne, author Y.-T.  Cui, author K. Lai, author W. Kundhikanjana,
  author Y. Yang, author
  M. Baenninger, author
  M. König, author C. Ames, author H. Buhmann, author P. Leubner,
  author L. W. Molenkamp, author S.-C. Zhang, author
  D. Goldhaber-Gordon, author
  M. A. Kelly,  and author
  Z.-X. Shen, doi:10.1038/ncomms8252 journal journal Nat.
  Commun. volume 6, pages 7252
  (year 2015)NoStop
[Olshanetsky et al.(2015)Olshanetsky, Kvon, Gusev, Levin, Raichev, Mikhailov, and  Dvoretsky]Olshanetsky:2015
  author author E. B. Olshanetsky, author Z. D. Kvon, author G. M. Gusev,
  author A. D. Levin, author O. E. Raichev, author
  N. N. Mikhailov,  and author
  S. A. Dvoretsky, 10.1103/PhysRevLett.114.126802 journal journal
  Phys. Rev. Lett. volume 114, pages
  126802 (year 2015)NoStop
[Deacon et al.(2017)Deacon,
  Wiedenmann, Bocquillon, Domínguez, Klapwijk, Leubner,
  Brüne, Hankiewicz, Tarucha,
  Ishibashi, Buhmann, and Molenkamp]Deacon:2017
  author author R. S. Deacon, author J. Wiedenmann,
  author E. Bocquillon, author F. Domínguez, author T. M. Klapwijk, author
  P. Leubner, author C. Brüne, author E. M.  Hankiewicz, author S. Tarucha, author K. Ishibashi, author H. Buhmann,  and author L. W. Molenkamp, 10.1103/PhysRevX.7.021011 journal journal Phys.
  Rev. X volume 7, pages 021011
  (year 2017)NoStop
[Liu et al.(2008)Liu,
  Hughes, Qi, Wang, and  Zhang]Liu:2008
  author author C. Liu, author T. L. Hughes,
  author X.-L. Qi, author K. Wang,  and author
  S.-C. Zhang, 10.1103/PhysRevLett.100.236601 journal journal
  Phys. Rev. Lett. volume 100, pages
  236601 (year 2008)NoStop
[Knez et al.(2011)Knez,
  Du, and Sullivan]Knez:2011
  author author I. Knez, author R.-R. Du,  and  author G. Sullivan, 10.1103/PhysRevLett.107.136603 journal journal Phys. Rev. Lett. volume 107, pages 136603 (year 2011)NoStop
[Suzuki et al.(2013)Suzuki,
  Harada, Onomitsu, and Muraki]Suzuki:2013
  author author K. Suzuki, author Y. Harada,
  author K. Onomitsu,  and  author K. Muraki, 10.1103/PhysRevB.87.235311 journal journal Phys. Rev. B volume 87, pages 235311 (year 2013)NoStop
[Charpentier et al.(2013)Charpentier, Fält, Reichl, Nichele, Pal, Pietsch, Ihn,
  Ensslin, and Wegscheider]Charpentier:2013
  author author C. Charpentier, author S. Fält,
  author C. Reichl, author F. Nichele, author
  A. N. Pal, author P. Pietsch, author T. Ihn, author K. Ensslin,  and author W. Wegscheider, 10.1063/1.4821037 journal
  journal App. Phys. Lett. volume 103,  pages 112102 (year 2013)NoStop
[Knez et al.(2014)Knez,
  Rettner, Yang, Parkin,
  Du, Du, and Sullivan]Knez:2014
  author author I. Knez, author C. T. Rettner,
  author S.-H. Yang, author S. S. P. Parkin, author
  L. Du, author R.-R. Du,  and author G. Sullivan, 10.1103/PhysRevLett.112.026602 journal journal
  Phys. Rev. Lett. volume 112, pages
  026602 (year 2014)NoStop
[Suzuki et al.(2015)Suzuki,
  Harada, Onomitsu, and Muraki]Suzuki:2015
  author author K. Suzuki, author Y. Harada,
  author K. Onomitsu,  and  author K. Muraki, 10.1103/PhysRevB.91.245309 journal journal Phys. Rev. B volume 91, pages 245309 (year 2015)NoStop
[Mueller et al.(2015)Mueller, Pal, Karalic, Tschirky, Charpentier, Wegscheider,
  Ensslin, and Ihn]Mueller:2015
  author author S. Mueller, author A. N. Pal,
  author M. Karalic, author T. Tschirky, author
  C. Charpentier, author
  W. Wegscheider, author
  K. Ensslin,  and author
  T. Ihn, 10.1103/PhysRevB.92.081303 journal journal Phys.
  Rev. B volume 92, pages 081303
  (year 2015)NoStop
[Qu et al.(2015)Qu,
  Beukman, Nadj-Perge, Wimmer,
  Nguyen, Yi, Thorp,
  Sokolich, Kiselev, Manfra,
  Marcus, and Kouwenhoven]Qu:2015
  author author F. Qu, author A. J. A. Beukman,
  author S. Nadj-Perge, author M. Wimmer, author
  B.-M. Nguyen, author
  W. Yi, author J. Thorp, author M. Sokolich, author A. A.  Kiselev, author M. J.  Manfra, author C. M.  Marcus,  and author L. P.  Kouwenhoven, 10.1103/PhysRevLett.115.036803 journal journal
  Phys. Rev. Lett. volume 115, pages
  036803 (year 2015)NoStop
[Li et al.(2015)Li,
  Wang, Fu, Du, Schreiber, Mu, Liu, Sullivan, Csáthy, Lin, and  Du]Li:2015
  author author T. Li, author P. Wang, author H. Fu, author
  L. Du, author K. A. Schreiber, author X. Mu, author X. Liu, author G. Sullivan,
  author G. A. Csáthy, author X. Lin,  and author
  R.-R. Du, 10.1103/PhysRevLett.115.136804 journal journal
  Phys. Rev. Lett. volume 115, pages
  136804 (year 2015)NoStop
[Du et al.(2015)Du,
  Knez, Sullivan, and Du]Du:2015
  author author L. Du, author I. Knez, author G. Sullivan,  and author R.-R. Du, 10.1103/PhysRevLett.114.096802 journal journal
  Phys. Rev. Lett. volume 114, pages
  096802 (year 2015)NoStop
[Nichele et al.(2016)Nichele, Suominen, Kjaergaard,
  Marcus, Sajadi, Folk,
  Qu, Beukman, de Vries,
  van Veen, Nadj-Perge, Kouwenhoven, Nguyen, Kiselev, Yi, Sokolich, Manfra, Spanton, and Moler]Nichele:2016
  author author F. Nichele, author H. J. Suominen, author M. Kjaergaard, author C. M. Marcus, author E. Sajadi,
  author J. A. Folk, author F. Qu, author
  A. J. A. Beukman, author
  F. K. de Vries, author
  J. van Veen, author
  S. Nadj-Perge, author
  L. P. Kouwenhoven, author
  B.-M. Nguyen, author
  A. A. Kiselev, author
  W. Yi, author M. Sokolich, author M. J.  Manfra, author E. M.  Spanton,  and author K. A.  Moler, http://stacks.iop.org/1367-2630/18/i=8/a=083005 journal
  journal New J. Phys. volume 18,  pages 083005 (year 2016)NoStop
[Couëdo et al.(2016)Couëdo, Irie, Suzuki, Onomitsu, and Muraki]Couedo:2016
  author author F. Couëdo, author H. Irie,
  author K. Suzuki, author K. Onomitsu,  and author K. Muraki, 10.1103/PhysRevB.94.035301 journal journal Phys.
  Rev. B volume 94, pages 035301
  (year 2016)NoStop
[Akiho et al.(2016)Akiho,
  Couëdo, Irie, Suzuki,
  Onomitsu, and Muraki]Akiho:2016
  author author T. Akiho, author F. Couëdo,
  author H. Irie, author
  K. Suzuki, author K. Onomitsu,  and author K. Muraki, 10.1063/1.4967471 journal journal Appl. Phys. Lett. volume 109, pages 192105 (year 2016),  http://arxiv.org/abs/https://doi.org/10.1063/1.4967471
  https://doi.org/10.1063/1.4967471 NoStop
[Nguyen et al.(2016)Nguyen,
  Kiselev, Noah, Yi,
  Qu, Beukman, de Vries,
  van Veen, Nadj-Perge, Kouwenhoven, Kjaergaard, Suominen,
  Nichele, Marcus, Manfra,  and Sokolich]Nguyen:2016
  author author B.-M. Nguyen, author A. A. Kiselev,
  author R. Noah, author
  W. Yi, author F. Qu, author A. J. A.  Beukman, author F. K.  de Vries, author J. van Veen, author S. Nadj-Perge, author L. P. Kouwenhoven, author M. Kjaergaard, author H. J. Suominen, author F. Nichele,
  author C. M. Marcus, author M. J. Manfra,  and author M. Sokolich, 10.1103/PhysRevLett.117.077701 journal journal
  Phys. Rev. Lett. volume 117, pages
  077701 (year 2016)NoStop
[Mueller et al.(2017)Mueller, Mittag, Tschirky, Charpentier, Wegscheider, Ensslin, and  Ihn]Mueller:2017
  author author S. Mueller, author C. Mittag,
  author T. Tschirky, author C. Charpentier, author
  W. Wegscheider, author
  K. Ensslin,  and author
  T. Ihn, 10.1103/PhysRevB.96.075406 journal journal Phys.
  Rev. B volume 96, pages 075406
  (year 2017)NoStop
[Wu et al.(2006)Wu,
  Bernevig, and Zhang]Wu:2006
  author author C. Wu, author B. A. Bernevig,   and author S.-C. Zhang, 10.1103/PhysRevLett.96.106401 journal journal Phys. Rev. Lett. volume 96, pages 106401 (year 2006)NoStop
[Xu and Moore(2006)]Xu:2006
  author author C. Xu and author J. E. Moore,  10.1103/PhysRevB.73.045322 journal journal Phys. Rev. B volume 73, pages 045322 (year 2006)NoStop
[Maciejko et al.(2009)Maciejko, Liu, Oreg, Qi,
  Wu, and Zhang]Maciejko:2009
  author author J. Maciejko, author C. Liu,
  author Y. Oreg, author
  X.-L. Qi, author C. Wu,  and author S.-C.  Zhang, 10.1103/PhysRevLett.102.256803
  journal journal Phys. Rev. Lett. volume 102, pages 256803 (year
  2009)NoStop
[Jiang et al.(2009)Jiang,
  Cheng, Sun, and Xie]Jiang:2009
  author author H. Jiang, author S. Cheng,
  author Q.-f. Sun,  and  author X. C. Xie, 10.1103/PhysRevLett.103.036803 journal journal Phys. Rev. Lett. volume 103, pages 036803 (year 2009)NoStop
[Ström et al.(2010)Ström, Johannesson, and Japaridze]Strom:2010
  author author A. Ström, author H. Johannesson,  and author G. I. Japaridze, 10.1103/PhysRevLett.104.256804 journal journal Phys. Rev. Lett. volume 104, pages 256804 (year
  2010)NoStop
[Tanaka et al.(2011)Tanaka,
  Furusaki, and Matveev]Tanaka:2011
  author author Y. Tanaka, author A. Furusaki,   and author K. A. Matveev,  10.1103/PhysRevLett.106.236402 journal
  journal Phys. Rev. Lett. volume 106,  pages 236402 (year 2011)NoStop
[Hattori(2011)]Hattori:2011
  author author K. Hattori, 10.1143/JPSJ.80.124712 journal
  journal J. Phys. Soc. Jpn. volume
  80, pages 124712 (year 2011)NoStop
[Lunde and Platero(2012)]Lunde:2012
  author author A. M. Lunde and author G. Platero, 10.1103/PhysRevB.86.035112 journal journal Phys. Rev. B volume
  86, pages 035112 (year 2012)NoStop
[Budich et al.(2012)Budich,
  Dolcini, Recher, and Trauzettel]Budich:2012
  author author J. C. Budich, author F. Dolcini,
  author P. Recher,  and author B. Trauzettel, 10.1103/PhysRevLett.108.086602 journal journal
  Phys. Rev. Lett. volume 108, pages
  086602 (year 2012)NoStop
[Lezmy et al.(2012)Lezmy,
  Oreg, and Berkooz]Lezmy:2012
  author author N. Lezmy, author Y. Oreg,  and  author M. Berkooz, 10.1103/PhysRevB.85.235304 journal journal Phys. Rev. B volume 85, pages 235304 (year 2012)NoStop
[Schmidt et al.(2012)Schmidt, Rachel, von Oppen, and  Glazman]Schmidt:2012
  author author T. L. Schmidt, author S. Rachel,
  author F. von Oppen,  and  author L. I. Glazman, 10.1103/PhysRevLett.108.156402 journal journal Phys. Rev. Lett. volume 108, pages 156402 (year 2012)NoStop
[Delplace et al.(2012)Delplace, Li, and Büttiker]Delplace:2012
  author author P. Delplace, author J. Li,  and  author M. Büttiker, 10.1103/PhysRevLett.109.246803 journal journal Phys. Rev. Lett. volume 109, pages 246803 (year 2012)NoStop
[Crépin et al.(2012)Crépin, Budich, Dolcini, Recher, and Trauzettel]Crepin:2012
  author author F. Crépin, author J. C. Budich, author F. Dolcini,
  author P. Recher,  and author B. Trauzettel, 10.1103/PhysRevB.86.121106 journal journal Phys.
  Rev. B volume 86, pages 121106
  (year 2012)NoStop
[Del Maestro et al.(2013)Del Maestro, Hyart, and Rosenow]DelMaestro:2013
  author author A. Del Maestro, author T. Hyart,
   and author B. Rosenow, 10.1103/PhysRevB.87.165440 journal journal Phys. Rev. B volume 87, pages 165440 (year 2013)NoStop
[Altshuler et al.(2013)Altshuler, Aleiner, and Yudson]Altshuler:2013
  author author B. L. Altshuler, author I. L. Aleiner,  and author V. I. Yudson, 10.1103/PhysRevLett.111.086401 journal journal Phys. Rev. Lett. volume 111, pages 086401 (year
  2013)NoStop
[Väyrynen et al.(2013)Väyrynen, Goldstein, and Glazman]Vayrynen:2013
  author author J. I. Väyrynen, author M. Goldstein,  and author L. I. Glazman, 10.1103/PhysRevLett.110.216402 journal journal Phys. Rev. Lett. volume 110, pages 216402 (year
  2013)NoStop
[Geissler et al.(2014)Geissler, Crépin, and Trauzettel]Geissler:2014
  author author F. Geissler, author F. Crépin,
   and author B. Trauzettel,  10.1103/PhysRevB.89.235136 journal journal Phys. Rev. B volume 89, pages 235136 (year 2014)NoStop
[Kainaris et al.(2014)Kainaris, Gornyi, Carr, and Mirlin]Kainaris:2014
  author author N. Kainaris, author I. V. Gornyi, author S. T. Carr,   and author A. D. Mirlin,  10.1103/PhysRevB.90.075118 journal journal Phys. Rev. B volume 90, pages 075118 (year 2014)NoStop
[Väyrynen et al.(2014)Väyrynen, Goldstein, Gefen, and  Glazman]Vayrynen:2014
  author author J. I. Väyrynen, author M. Goldstein, author Y. Gefen,   and author L. I. Glazman,  10.1103/PhysRevB.90.115309 journal journal Phys. Rev. B volume 90, pages 115309 (year 2014)NoStop
[Chou et al.(2015)Chou,
  Levchenko, and Foster]Chou:2015
  author author Y.-Z. Chou, author A. Levchenko,   and author M. S. Foster,  10.1103/PhysRevLett.115.186404 journal
  journal Phys. Rev. Lett. volume 115,  pages 186404 (year 2015)NoStop
[Yevtushenko et al.(2015)Yevtushenko, Wugalter, Yudson, and  Altshuler]Yevtushenko:2015
  author author O. M. Yevtushenko, author A. Wugalter, author V. I. Yudson,  and author B. L. Altshuler, http://stacks.iop.org/0295-5075/112/i=5/a=57003
  journal journal EPL (Europhysics Letters)  volume 112, pages 57003 (year 2015)NoStop
[Väyrynen et al.(2016)Väyrynen, Geissler, and Glazman]Vayrynen:2016
  author author J. I. Väyrynen, author F. Geissler,  and author L. I. Glazman, 10.1103/PhysRevB.93.241301 journal journal Phys. Rev. B volume
  93, pages 241301 (year 2016)NoStop
[Xie et al.(2016)Xie,
  Li, Chou, and Foster]Xie:2016
  author author H.-Y. Xie, author H. Li, author Y.-Z. Chou,  and author M. S. Foster, 10.1103/PhysRevLett.116.086603 journal journal
  Phys. Rev. Lett. volume 116, pages
  086603 (year 2016)NoStop
[Kharitonov et al.(2017)Kharitonov, Geissler, and Trauzettel]Kharitonov:2017
  author author M. Kharitonov, author F. Geissler,  and author B. Trauzettel, 10.1103/PhysRevB.96.155134 journal journal Phys. Rev. B volume
  96, pages 155134 (year 2017)NoStop
[Wang et al.(2017)Wang,
  Meir, and Gefen]Wang:2017
  author author J. Wang, author Y. Meir,  and  author Y. Gefen, 10.1103/PhysRevLett.118.046801 journal journal Phys. Rev. Lett. volume 118, pages 046801 (year 2017)NoStop
[Chou et al.(2017)Chou,
  Nandkishore, and Radzihovsky]Chou:2017
  author author Y.-Z. Chou, author R. M. Nandkishore,  and author L. Radzihovsky, @noop  journal journal
  ArXiv e-prints  (year 2017), http://arxiv.org/abs/1710.04232 arXiv:1710.04232 [cond-mat.str-el]
  NoStop
[Groenendijk et al.(2017)Groenendijk, Dolcetto, and Schmidt]Groenendijk:2017
  author author S. Groenendijk, author G. Dolcetto,  and author T. L. Schmidt, @noop  journal journal
  ArXiv e-prints  (year 2017), http://arxiv.org/abs/1711.07375 arXiv:1711.07375 [cond-mat.mes-hall]
  NoStop
[Väyrynen et al.(2018)Väyrynen, Pikulin, and Alicea]Vayrynen:2018
  author author J. I. Väyrynen, author D. I. Pikulin,  and author J. Alicea, @noop  journal journal
  ArXiv e-prints  (year 2018), http://arxiv.org/abs/1803.01021 arXiv:1803.01021 [cond-mat.mes-hall]
  NoStop
[Hsu et al.(2017)Hsu,
  Stano, Klinovaja, and Loss]Hsu:2017
  author author C.-H. Hsu, author P. Stano, author J. Klinovaja,  and author D. Loss, 10.1103/PhysRevB.96.081405 journal journal Phys.
  Rev. B volume 96, pages 081405
  (year 2017)NoStop
[Giamarchi and Schulz(1988)]Giamarchi:1988
  author author T. Giamarchi and author H. J. Schulz, 10.1103/PhysRevB.37.325 journal
  journal Phys. Rev. B volume 37,  pages 325 (year 1988)NoStop
[Giamarchi(2003)]Giamarchi:2003
  author author T. Giamarchi, @noop  title Quantum Physics in
  One Dimension (publisher Oxford University Press, New
  York, year 2003)NoStop
[Gueron(1964)]Gueron:1964
  author author M. Gueron, 10.1103/PhysRev.135.A200 journal
  journal Phys. Rev. volume 135,  pages A200 (year 1964)NoStop
[Schliemann et al.(2003)Schliemann, Khaetskii, and Loss]Schliemann:2003
  author author J. Schliemann, author A. Khaetskii,  and author D. Loss, http://stacks.iop.org/0953-8984/15/i=50/a=R01 journal journal J. Phys. Condens. Matter volume 15, pages R1809 (year
  2003)NoStop
[Braun et al.(2006)Braun,
  Urbaszek, Amand, Marie,
  Krebs, Eble, Lemaitre, and  Voisin]Braun:2006
  author author P.-F. Braun, author B. Urbaszek,
  author T. Amand, author X. Marie, author
  O. Krebs, author B. Eble, author A. Lemaitre,  and author P. Voisin, 10.1103/PhysRevB.74.245306 journal journal Phys. Rev. B volume
  74, pages 245306 (year 2006)NoStop
[Lunde and Platero(2013)]Lunde:2013
  author author A. M. Lunde and author G. Platero, 10.1103/PhysRevB.88.115411 journal journal Phys. Rev. B volume
  88, pages 115411 (year 2013)NoStop
[Zyuzin and Loss(2014)]Zyuzin:2014b
  author author A. A. Zyuzin and author D. Loss,  10.1103/PhysRevB.90.125443 journal journal Phys. Rev. B volume 90, pages 125443 (year 2014)NoStop
[Klier et al.(2017)Klier, Sharma, Pankratov, and  Shallcross]Klier:2017
  author author N. Klier, author S. Sharma,
  author O. Pankratov,  and  author S. Shallcross,  @noop  journal journal ArXiv e-prints 
  (year 2017), http://arxiv.org/abs/1711.10760
  arXiv:1711.10760 [cond-mat.mes-hall] NoStop
[Meng et al.(2014a)Meng, Stano,
  Klinovaja, and Loss]Meng:2014a
  author author T. Meng, author P. Stano,
  author J. Klinovaja,  and  author D. Loss, 10.1140/epjb/e2014-50445-1 journal journal Eur.
  Phys. J. B volume 87, pages 203
  (year 2014a)NoStop
[Kornich et al.(2015)Kornich, Stano, Zyuzin, and Loss]Kornich:2015
  author author V. Kornich, author P. Stano,
  author A. A. Zyuzin,  and  author D. Loss, 10.1103/PhysRevB.91.195423 journal journal Phys.
  Rev. B volume 91, pages 195423
  (year 2015)NoStop
[Paget et al.(1977)Paget,
  Lampel, Sapoval, and Safarov]Paget:1977
  author author D. Paget, author G. Lampel,
  author B. Sapoval,  and  author V. I. Safarov, 10.1103/PhysRevB.15.5780 journal journal Phys. Rev. B volume 15, pages 5780 (year 1977)NoStop
[Russo et al.(2017)Russo, Barnes, and Economou]Russo:2017
  author author A. Russo, author E. Barnes,   and author S. E. Economou,  @noop  journal journal ArXiv e-prints 
  (year 2017), http://arxiv.org/abs/1711.01712
  arXiv:1711.01712 [cond-mat.mes-hall] NoStop
[Simon and Loss(2007)]Simon:2007
  author author P. Simon and author D. Loss,  10.1103/PhysRevLett.98.156401 journal
  journal Phys. Rev. Lett. volume 98,  pages 156401 (year 2007)NoStop
[Maciejko(2012)]Maciejko:2012
  author author J. Maciejko, 10.1103/PhysRevB.85.245108 journal journal Phys. Rev. B volume
  85, pages 245108 (year 2012)NoStop
[Schecter et al.(2015)Schecter, Rudner, and Flensberg]Schecter:2015
  author author M. Schecter, author M. S. Rudner,  and author K. Flensberg, 10.1103/PhysRevLett.114.247205 journal journal Phys. Rev. Lett. volume 114, pages 247205 (year
  2015)NoStop
[Yevtushenko and Yudson(2017)]Yevtushenko:2017
  author author O. M. Yevtushenko and author V. I. Yudson, @noop  journal journal
  ArXiv e-prints  (year 2017), http://arxiv.org/abs/1709.00325 arXiv:1709.00325 [cond-mat.str-el]
  NoStop
[Pribiag et al.(2015)Pribiag, Beukman, Qu, Cassidy, Charpentier, Wegscheider, and  Kouwenhoven]Pribiag:2015
  author author V. S. Pribiag, author A. J. A. Beukman, author F. Qu,
  author M. C. Cassidy, author C. Charpentier, author
  W. Wegscheider,  and author
  L. P. Kouwenhoven, @noop 
  journal journal Nat. Nanotechnol. volume 10, pages 593 (year
  2015)NoStop
[Schrade et al.(2015)Schrade, Zyuzin, Klinovaja, and  Loss]Schrade:2015
  author author C. Schrade, author A. A. Zyuzin,
  author J. Klinovaja,  and  author D. Loss, 10.1103/PhysRevLett.115.237001 journal journal
  Phys. Rev. Lett. volume 115, pages
  237001 (year 2015)NoStop
[Hou et al.(2009)Hou,
  Kim, and Chamon]Hou:2009
  author author C.-Y. Hou, author E.-A. Kim,  and  author C. Chamon, 10.1103/PhysRevLett.102.076602 journal journal Phys. Rev. Lett. volume 102, pages 076602 (year 2009)NoStop
[Ström and Johannesson(2009)]Strom:2009
  author author A. Ström and author H. Johannesson, 10.1103/PhysRevLett.102.096806 journal journal Phys. Rev. Lett. volume 102, pages 096806 (year
  2009)NoStop
[Teo and Kane(2009)]Teo:2009
  author author J. C. Y.  Teo and author C. L.  Kane, 10.1103/PhysRevB.79.235321
  journal journal Phys. Rev. B volume 79, pages 235321 (year
  2009)NoStop
[Egger et al.(2010)Egger,
  Zazunov, and Yeyati]Egger:2010
  author author R. Egger, author A. Zazunov,   and author A. L. Yeyati,  10.1103/PhysRevLett.105.136403 journal
  journal Phys. Rev. Lett. volume 105,  pages 136403 (year 2010)NoStop
[Pfeiffer et al.(1997)Pfeiffer, Yacoby, Stormer, Baldwin, Hasen, Pinczuk, Wegscheider, and West]Pfeiffer:1997
  author author L. N. Pfeiffer, author A. Yacoby,
  author H. L. Stormer, author K. L. Baldwin, author
  J. Hasen, author A. Pinczuk, author W. Wegscheider,  and author K. W. West, @noop  journal
  journal Microelectron. J. volume 28,  pages 817 (year 1997)NoStop
[Yacoby et al.(1997)Yacoby,
  Stormer, Baldwin, Pfeiffer,  and West]Yacoby:1997
  author author A. Yacoby, author H. L. Stormer,
  author K. W. Baldwin, author L. N. Pfeiffer,  and author K. W. West, @noop 
  journal journal Solid State Commun. volume 101, pages 77 (year
  1997)NoStop
[Auslaender et al.(2002)Auslaender, Yacoby, de Picciotto,
  Baldwin, Pfeiffer, and West]Auslaender:2002
  author author O. M. Auslaender, author A. Yacoby,
  author R. de Picciotto, author K. W. Baldwin, author
  L. N. Pfeiffer,  and author
  K. W. West, 10.1126/science.1066266 journal journal
  Science volume 295, pages 825
  (year 2002)NoStop
[Auslaender et al.(2005)Auslaender, Steinberg, Yacoby,
  Tserkovnyak, Halperin, Baldwin, Pfeiffer, and West]Auslaender:2005
  author author O. M. Auslaender, author H. Steinberg, author A. Yacoby,
  author Y. Tserkovnyak, author B. I. Halperin, author
  K. W. Baldwin, author
  L. N. Pfeiffer,  and author
  K. W. West, @noop  journal journal Science volume
  308, pages 88 (year 2005)NoStop
[Steinberg et al.(2008)Steinberg, Barak, Yacoby, Pfeiffer, West, Halperin, and  Le Hur]Steinberg:2008
  author author H. Steinberg, author G. Barak,
  author A. Yacoby, author L. N. Pfeiffer, author
  K. W. West, author
  B. I. Halperin,  and author
  K. Le Hur, @noop  journal journal Nat. Phys. volume
  4, pages 116 (year 2008)NoStop
[Braunecker et al.(2009a)Braunecker, Simon, and Loss]Braunecker:2009a
  author author B. Braunecker, author P. Simon,   and author D. Loss, 10.1103/PhysRevLett.102.116403 journal journal Phys. Rev. Lett. volume 102, pages 116403 (year 2009a)NoStop
[Braunecker et al.(2009b)Braunecker, Simon, and Loss]Braunecker:2009b
  author author B. Braunecker, author P. Simon,   and author D. Loss, 10.1103/PhysRevB.80.165119 journal journal Phys. Rev. B volume 80, pages 165119 (year 2009b)NoStop
[Nattermann et al.(2003)Nattermann, Giamarchi, and Le Doussal]Nattermann:2003
  author author T. Nattermann, author T. Giamarchi,  and author P. Le Doussal, 10.1103/PhysRevLett.91.056603 journal journal Phys. Rev. Lett. volume 91, pages 056603 (year
  2003)NoStop
[Aseev et al.(2017)Aseev,
  Klinovaja, and Loss]Aseev:2017
  author author P. P. Aseev, author J. Klinovaja,   and author D. Loss, 10.1103/PhysRevB.95.125440 journal journal Phys. Rev. B volume 95, pages 125440 (year 2017)NoStop
[Ilan et al.(2012)Ilan,
  Cayssol, Bardarson, and Moore]Ilan:2012
  author author R. Ilan, author J. Cayssol,
  author J. H. Bardarson,  and  author J. E. Moore, 10.1103/PhysRevLett.109.216602 journal journal Phys. Rev. Lett. volume 109, pages 216602 (year 2012)NoStop
[Müller et al.(2017)Müller, Thomale, Trauzettel,
  Bocquillon, and Kashuba]Muller:2017
  author author T. Müller, author R. Thomale,
  author B. Trauzettel, author E. Bocquillon,  and author O. Kashuba, 10.1103/PhysRevB.95.245114 journal journal Phys.
  Rev. B volume 95, pages 245114
  (year 2017)NoStop
[Simon et al.(2008)Simon,
  Braunecker, and Loss]Simon:2008
  author author P. Simon, author B. Braunecker,   and author D. Loss, 10.1103/PhysRevB.77.045108 journal journal Phys. Rev. B volume 77, pages 045108 (year 2008)NoStop
[Klinovaja and Loss(2013)]Klinovaja:2013a
  author author J. Klinovaja and author D. Loss, 10.1103/PhysRevB.87.045422 journal
  journal Phys. Rev. B volume 87,  pages 045422 (year 2013)NoStop
[Klinovaja et al.(2013)Klinovaja, Stano, Yazdani, and  Loss]Klinovaja:2013b
  author author J. Klinovaja, author P. Stano,
  author A. Yazdani,  and  author D. Loss, 10.1103/PhysRevLett.111.186805 journal journal
  Phys. Rev. Lett. volume 111, pages
  186805 (year 2013)NoStop
[Hsu et al.(2015)Hsu,
  Stano, Klinovaja, and Loss]Hsu:2015
  author author C.-H. Hsu, author P. Stano, author J. Klinovaja,  and author D. Loss, 10.1103/PhysRevB.92.235435 journal journal Phys.
  Rev. B volume 92, pages 235435
  (year 2015)NoStop
[Yang et al.(2016)Yang,
  Hsu, Stano, Klinovaja, and  Loss]Yang:2016
  author author G. Yang, author C.-H. Hsu,
  author P. Stano, author J. Klinovaja,  and author D. Loss, 10.1103/PhysRevB.93.075301 journal journal Phys.
  Rev. B volume 93, pages 075301
  (year 2016)NoStop
[Meng and Loss(2013)]Meng:2013
  author author T. Meng and author D. Loss,  10.1103/PhysRevB.87.235427 journal journal Phys. Rev. B volume 87, pages 235427 (year 2013)NoStop
[Scheller et al.(2014)Scheller, Liu, Barak, Yacoby, Pfeiffer, West, and Zumbühl]Scheller:2014
  author author C. P. Scheller, author T.-M. Liu,
  author G. Barak, author A. Yacoby, author
  L. N. Pfeiffer, author
  K. W. West,  and author
  D. M. Zumbühl, 10.1103/PhysRevLett.112.066801 journal journal
  Phys. Rev. Lett. volume 112, pages
  066801 (year 2014)NoStop
[Stano and Loss(2014)]Stano:2014
  author author P. Stano and author D. Loss,  10.1103/PhysRevB.90.195312 journal journal Phys. Rev. B volume 90, pages 195312 (year 2014)NoStop
[Maslov and Stone(1995)]Maslov:1995
  author author D. L. Maslov and author M. Stone,  10.1103/PhysRevB.52.R5539 journal journal Phys. Rev. B volume 52, pages R5539 (year 1995)NoStop
[Ponomarenko(1995)]Ponomarenko:1995
  author author V. V. Ponomarenko, 10.1103/PhysRevB.52.R8666 journal journal Phys. Rev. B volume
  52, pages R8666 (year 1995)NoStop
[Safi and Schulz(1995)]Safi:1995
  author author I. Safi and author H. J. Schulz, 10.1103/PhysRevB.52.R17040 journal journal Phys. Rev. B volume
  52, pages R17040 (year 1995)NoStop
[Meng et al.(2014b)Meng, Fritz,
  Schuricht, and Loss]Meng:2014b
  author author T. Meng, author L. Fritz,
  author D. Schuricht,  and  author D. Loss, 10.1103/PhysRevB.89.045111 journal journal Phys.
  Rev. B volume 89, pages 045111
  (year 2014b)NoStop
[Holstein and Primakoff(1940)]Holstein:1940
  author author T. Holstein and author H. Primakoff, 10.1103/PhysRev.58.1098 journal journal Phys. Rev. volume
  58, pages 1098 (year 1940)NoStop
[Schrieffer and Wolff(1966)]Schrieffer:1966
  author author J. Schrieffer and author P. Wolff, 10.1103/PhysRev.149.491 journal
  journal Phys. Rev. volume 149,  pages 491 (year 1966)NoStop
[Bravyi et al.(2011)Bravyi,
  DiVincenzo, and Loss]Bravyi:2011
  author author S. Bravyi, author D. P. DiVincenzo,  and author D. Loss, https://doi.org/10.1016/j.aop.2011.06.004
  journal journal Annals of Physics volume 326, pages 2793  (year
  2011)NoStop
[Braunecker et al.(2010)Braunecker, Japaridze, Klinovaja, and  Loss]Braunecker:2010
  author author B. Braunecker, author G. I. Japaridze, author J. Klinovaja,  and author D. Loss, 10.1103/PhysRevB.82.045127 journal
  journal Phys. Rev. B volume 82,  pages 045127 (year 2010)NoStop
[Braunecker and Simon(2013)]Braunecker:2013
  author author B. Braunecker and author P. Simon, 10.1103/PhysRevLett.111.147202 journal journal Phys. Rev. Lett. volume 111, pages 147202 (year
  2013)NoStop
[Voit and Schulz(1987)]Voit:1987
  author author J. Voit and author H. J. Schulz, 10.1103/PhysRevB.36.968 journal
  journal Phys. Rev. B volume 36,  pages 968 (year 1987)NoStop
[Voit(1995)]Voit:1995
  author author J. Voit, http://stacks.iop.org/0034-4885/58/i=9/a=002 journal journal Rep. Prog. Phys. volume 58, pages 977 (year 1995)NoStop
[Bruus and Flensberg(2004)]Bruus:2004
  author author H. Bruus and author K. Flensberg, @noop  title Many-Body Quantum
  Theory in Condensed Matter Physics - An Introduction (publisher Oxford University Press, New York, year
  2004)NoStop




