Finite-temperature conductance of strongly interacting quantum wire with a nuclear spin order

Finite-temperature conductance of strongly interacting quantum wire with a nuclear spin order

Pavel P. Aseev    Jelena Klinovaja    Daniel Loss Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
Abstract

We study the temperature dependence of the electrical conductance of a clean strongly interacting quantum wire in the presence of a helical nuclear spin order. The nuclear spin helix opens a temperature-dependent partial gap in the electron spectrum. Using a bosonization framework we describe the gapped electron modes by sine-Gordon-like kinks. We predict an internal resistivity caused by an Ohmic-like friction these kinks experience via interacting with gapless excitations. As a result, the conductance rises from at temperatures below the critical temperature when nuclear spins are fully polarized to at higher temperatures when the order is destroyed, featuring a relatively wide plateau in the intermediate regime. The theoretical results are compared with the experimental data for GaAs quantum wires obtained recently by Scheller et al. [Phys. Rev. Lett. 112, 066801 (2014)].

pacs:
73.23.-b, 71.10.Pm, 75.30.-m

I Introduction

Basic electronic properties of three-dimensional (3D) interacting electron systems are usually well described within the Landau Fermi-liquid picture where low-energy excitations are single-electron quasiparticles. This is not the case in 1D systems where interaction cannot be considered as a small perturbation. Rather than electronic quasiparticles, the low energy excitations are collective density waves (bosons), and the system can be described as a Luttinger liquid (LL) GiamarchiBook (); VoitRepProg1995 ().

In recent years, helical and quasi-helical LLs, special classes of LLs exhibiting spin-filtered transport, have received much attention. The helical LL describes, for example, edges of two-dimensional topological insulators KonigJPhysSocJapan2008 (); HasanKaneRevModPhys2010 (). Quasi-helical LLs can, for example, emerge if a magnetic field is applied to a quantum wire with Rashba spin-orbit interaction (SOI) StredaPRL2003 (). (Quasi-)helical LLs have applications as Cooper pair splitters SatoPRL2010 () or spin filters StredaPRL2003 (), and are an essential ingredient for topological quantum wires with Majorana bound states AliceaRepProgPhys2012 ().

Quasi-helical LLs can also be generated by a helical magnetic field which is equivalent to the combination of a homogeneous magnetic field and Rashba SOI BrauneckerPRB2010 (). An intrinsic helical magnetic field arises as a result of hyperfine coupling between interacting electrons and nuclear spins: the Ruderman-Kittel-Kasuya-Yosida (RKKY) FrohlichProcSocLondon1940 (); RudermanKittelPhysRev1954 (); KasuyaProgTheorPhys1956 (); YosidaPhysRev1957 () interaction diverges at momentum due to electron backscattering inducing a helical order of nuclear spins  BrauneckerPRL2009 (); BrauneckerPRB2009 (); MengPRB2013 (); MengEPLJB2014 (); HsuPRB2015 () (see Fig. 1). This helical order reveals itself as a spatially rotating Overhauser field acting on electrons. As a result, a partial gap strongly enhanced by electron-electron interactions opens around the Fermi level. While in infinite systems this order would be suppressed by long-wavelength magnons LossPRL2011 (), the helical order still can exist in a finite-length wire BrauneckerPRB2009 (); MengEPLJB2014 ().

The possible experimental evidence for a nuclear spin order has been observed by Scheller et al. SchellerPRL2014 () by measuring temperature dependence of conductance in a cleaved edge overgrowth GaAs quantum wire. Remarkably, at low temperatures, the conductance is instead of the expected for a single channel in a spin-degenerate quantum wire. At higher temperatures the conductance becomes SchellerPRL2014 (). This can be explained by the lifting of electron spin degeneracy at low temperatures in the presence of a helical nuclear spin order BrauneckerPRL2009 (); BrauneckerPRB2009 (). Further ways to confirm the presence of the nuclear spin helix were suggested theoretically, for example, by means of nuclear magnetic resonance StanoLossPRB2014 (), nuclear spin relaxation ZyuzinPRB2014 (), and quantum Hall effect anisotropies MengEPLJB2014 ().

In this paper we study the temperature-dependence of the conductance in an interacting quantum wire with a helical nuclear spin order. Although the conductance at finite temperatures in quasi-helical one-dimensional (1D) electron systems (namely, in wires with Rashba SOI) has been previously studied in Ref. SchmidtPRB2014, , the main attention has been paid to weakly-interacting electrons. For zero-temperature and finite frequency conductances in strongly interacting Rashba wires see Ref. Meng_FritzPRB2014, . However, an essential ingredient for formation of the helical nuclear spin order is a strong electron backscattering, and, thus, our aim is to investigate how interactions affect the conductance of a quantum wire with a nuclear spin order at finite temperatures. Using a bosonization framework we describe the gapped electron modes by sine-Gordon-like solitons or kinks. These kinks are coupled to gapless excitations which leads to an Ohmic-like friction for these kinks and thus to a temperature-dependent resistivity. As a result, the conductance rises from at temperatures below the critical temperature when nuclear spins are fully polarized to at higher temperatures when the order is destroyed, featuring a relatively wide plateau in the intermediate regime, in qualitative agreement with the experimental observation by Scheller et al. SchellerPRL2014 (). Allowing in addition for different temperatures in the nuclear spin and electron system, the data can be fitted by our expression for the conductance over the entire temperature regime of the experiment.

The outline of the paper is as follows. In Sec. II, we introduce the model of a 1D quantum wire with a helical nuclear spin order. In Sec. III we discuss the electron transport in the wire disregarding electron-electron interactions both in the fermionic and bosonization frameworks. In Sec. IV we study how interactions affect the finite-temperature conductance using one-soliton and dilute soliton gas approximations. In Sec. V we revise the temperature dependence of the partial gap. Finally, in Sec. VI we compare the theoretical results with the available experimental data. The appendices contain technical details.

Ii The model

We consider a 1D semiconductor quantum wire of length aligned along the -axis with itinerant electrons coupled to localized nuclear spins via hyperfine coupling (see Fig. 1). The wire is adiabatically connected to normal Fermi liquid leads. Typically, the number of nuclear spins in the cross-section of the wire is large, , and the hyperfine coupling constant of the material is much smaller than the Fermi energy , . We adopt then the Born-Oppenheimer approximation: since the dynamics of nuclear spins is much slower than that of electrons, the effect of nuclear spin polarization can be described as a static Overhauser field. A helical order triggered by RKKY interaction appears below a critical temperature BrauneckerPRB2009 (), the Overhauser field is spatially rotating: , with a period determined by the Fermi momentum . Here, for definiteness, we assume that is rotating in the -plane, which need not be the case in general. The amplitude is assumed to be constant inside the wire and vanisihing in the leads (see below). The Hamiltonian in second quantization form for the electron subsystem and the associated Hamiltonian density are given by BrauneckerPRB2009 ()

 H=∫dxH(x), (1) (2)

where , is the Bohr magneton and the electron -factor, is a vector of Pauli matrices acting on the electron spin space, is a field operator annihilating an electron at position with spin (along the spin quantization axis ). Here and in the following we set .

The Overhauser field is weak compared to the Fermi energy and can be treated as a small perturbation. In order to describe interacting electrons with the LL model, we linearize the electron spectrum in the vicinity of Fermi points , and represent the fermionic fields in terms of slowly-varying left () and right () mover fields, The Hamiltonian density given by Eq. (2) can be now rewritten as

 H(x)=vF∑s[R†s(−i∂x)Rs+L†s(i∂x)Ls]+b(x)[R†↑L↓+L†↓R↑], (3)

where is the Fermi velocity.

Iii Non-interacting electrons

iii.1 Fermionic representation

First, we disregard electron-electron interactions, assuming that their only role is the formation of a nuclear spin order. In this case the dynamics of the electrons can be completely described in fermionic representation.

The Hamiltonian is block-diagonalized in the basis of the fields , describing two gapless modes with spectrum and fields describing gapped modes with spectrum (for the moment we ignore the leads), where we assumed a constant amplitude of the Overhauser field inside the wire. In the following the plus sign will denote the gapless modes, and the minus sign will denote the gapped modes. The gapless modes yield a temperature-independent contribution to the conductance . In the presence of the partial gap a contribution of gapped branches to the two-terminal conductance is temperature-dependent and is given by the generalized Landauer formula ButtikerPRB1985 (); Datta1997 (),

 G−(V=0)=G0+∞∫−∞dεT(ε)14Tcosh2(ε/2T), (4)

where is the applied voltage difference and is a transmission coefficient for the gapped modes. Here and in the following we set

If the wire is long enough compared to the magnetic length , the tunneling current (i.e., the contribution from energies below the gap ) can be neglected, and the integration can be performed only for the energies above the gap (see details in the Appendix A). The transmission coefficient and, hence, the temperature-dependent conductance itself are not universal in the sense that they depend on how the Overhauser field varies close to the leads. First, we assume that the Overhauser field vanishes in the leads, i.e., at , , and then abruptly turns on to some constant finite value in the wire at , , where is the Heaviside step-function. For energies above the gap, , we obtain (see Appendix A)

 T(ε)=ε2−b2ε2−b2cos2(√ε2−b2L/vF). (5)

For long enough wires , the transmission coefficient in Eq. (4) can be replaced by its averaged value . The contribution to the conductance by the gapped electrons as function of temperature is shown in Fig. 2 (green line). At low temperatures it is proportional to .

In the opposite limiting case, the magnetic field adiabatically changes from zero in the leads to the finite value in the wire. The transmission coefficient in Eq. (4) can be taken equal to unity above the gap and zero below the gap . The conductance is given in this case by

 G−≈G0[1−tanh(b/2T)], (6)

and is shown in Fig. 2 (red line). At low temperatures the conductance is described by the activation law .

We also considered numerically an intermediate case of a smoothly varying Overhauser field . The coordinate dependence is modeled as follows,

 B(x)=B2(tanhxl+tanhL−xl). (7)

It turns out that if the length over which the Overhauser field varies satisfies , the numerically obtained conductance is close to the result for the ideal transmission given by Eq. (6) (see Fig. 2). Thus, in the following we can assume that the transmission is ideal , and the Overhauser field does not depend on the position (or adiabatically vanishes in the leads).

iii.2 Bosonization

It is instructive to obtain the same result for the conductance of non-interacting electrons in bosonization representation. The bosonized Hamiltonian density reads GiamarchiBook (); VoitRepProg1995 ()

 H=vF2π∑ν=ρ,σ[(∂xθν)2+(∂xϕν)2]−bπacos[√2(ϕρ−θσ)], (8)

where the conjugate bosonic fields , describe the charge (spin) sector, and is a short-distance cut-off.

For the sake of simplicity we assume that the Overhauser field is position-independent (as discussed in the previous section).

It is convenient to introduce bosonic fields , corresponding to the gapped (gapless) branches,

 ϕ∓ =ϕρ∓θσ√2,θ∓=θρ∓ϕσ√2. (9)

In the new variables the Hamiltonian can be rewritten as a sum of two independent Hamiltonian densities, , for gapped and gapless modes,

 H+(x) =vF2π{(∂xθ+)2+(∂xϕ+)2}, (10) H−(x) =vF2π{(∂xθ−)2+(∂xϕ−)2}−bπacos(2ϕ−). (11)

The first term is a standard LL Hamiltonian, while is exactly the sine-Gordon Hamiltonian.

The charge current is related to the bosonic field by

 j=j++j−, (12) j±=∂tϕ±/π, (13)

and according to the Maslov-Stone approach MaslovStonePRB1995 (), the full conductance can be extracted from the retarded Green functions via the Kubo formula,

 G=e2π2limω→0ω[GR+(ω)+GR−(ω)]. (14)

The definitions of the Green functions are given in Appendix B. We will use also the Matsubara version of Eq. (14), which is given by analytical continuation,

 G=e2π2¯ω[GM+(x,x,¯ω)+GM−(x,x,¯ω)]∣∣∣i¯ω→ω+i0,ω→0. (15)

Currents corresponding to the gapped and gapless modes commute with each other, so their contribution can be calculated independently, i.e., . The gapless fields yield the conductance , while the gapped ones give rise to . From now on we focus on the gapped modes.

The Hilbert space for the sine-Gordon model consists of the vacuum sector (with vacuum state and fluctuations around it) and the sectors with different number of kinks, antikinks, and the bound states Rajaraman1982 (); GoldstoneJackiwPRD1975 (). The one-kink sector is orthogonal to the vacuum sector and consists of the following. (i) kink-particle states with mass , momentum , and energy ; (ii) scattering states of the kink-particle and ‘mesons’ (fluctuations around the kink) with asymptotic momenta .

The sine-Gordon kink-particles are important for describing the transport for the gapped modes since these kinks carry electric current and their topological charge corresponds to the electric charge.

The mass of the kink ZamolodchikovIntJModPhysA1995 () in the non-interacting case is related to the Overhauser field energy by which is in agreement with the fermionic picture (the one-kink sector corresponds to electron states above the gap). In this section we will restrict our study to the one-kink sector, disregarding multi-kink states, which can be justified at low temperatures .

For the sake of simplicity, in this section we will also disregard scattering states . Since in the non-interacting case mesons have a finite mass, this can be justified in the same limit of low temperatures. The effect of mesons on the conductance will be discussed in Sec. IV.1, when we consider the more general interacting case.

iii.2.1 Vacuum sector

In order to consider the system in the vicinity of the vacuum state we replace the cosine term in the Hamiltonian with a quadratic one, using a self-consistent harmonic approximation GiamarchiBook (),

 bπacos(2ϕ−)→const−Δ22πvFϕ2−, (16) Δ2=2bvFae−2⟨ϕ2−⟩. (17)

Following the Maslov-Stone approach MaslovStonePRB1995 () we assume infinitesimal dissipation in the leads, so that the Matsubara Green function vanishes far away from the contacts . The Matsubara Green function in the wire can be calculated straightforwardly,

 GM¯ω(x,x′)=−π2√¯ω2+Δ2exp{−√¯ω2+Δ2|x−x′|vF}. (18)

Since the Matsubara Green function is finite in the low-frequency limit , the Kubo formula, see Eq. (15), yields zero conductance contribution from the gapped modes.

iii.2.2 One-kink sector

Here we will follow a method for calculating low-temperature correlation functions described in Ref. Essler2005, . First, we expand out the thermal trace

 (19)

where is the inverse temperature, and are kink and anti-kink states with momentum and and energies and , respectively, stands for the partition function. We will restrict ourselves to the low temperature limit and disregard higher-order sine-Gordon solitons. The first (“vacuum”) term in Eq. (19) yields zero contribution to the current as discussed in the previous section.

The two-point correlator in the kink state can be expressed via the matrix elements of as

 (20)

The matrix elements can be calculated in the quasiclassical limit GoldstoneJackiwPRD1975 (); MussardoNuclPhysB2003 () () as the Fourier transformation of the static kink solution ,

 ⟨P∣∣ϕ−∣∣P′⟩=b∫dxei(P−P′)xϕK(x)≈πi(P−P′)cosh[γ(P−P′)]+π2δ(P−P′). (21)

where is the width of the kink, and . The latter term with the delta-function will result in a divergent but time-independent contribution to the correlation functions, and therefore will vanish after taking the time derivative.

Now we can perform the integration over , in Eqs. (19)–(20) and obtain a contribution to the Matsubara Green function from the kink-sector,

 ⟨ϕ−(¯ω,x)ϕ−(−¯ω,x)⟩k=12¯ωe−b/T. (22)

Similarly, there will be the equal contribution from the anti-kink sector. The total conductance by the gapped modes can be calculated by the Kubo formula (15),

 G−=2G0e−b/T,

which agrees with the low-temperature expansion of Eq. (6) obtained in the fermionic representation.

Iv Interacting wire

In the general case with interactions the LL is described by charge and spin interaction parameters , . In the following we put . The LL parameter varies from zero for strong (unscreened) electron repulsion to for non-interacting electrons. In order to treat the leads correctly, we assume similarly to Ref. MaslovStonePRB1995, that the interaction parameter depends on the coordinate , and there is no interaction in the leads at , . The Hamiltonian density describing interacting electrons in the Overhauser field is given by BrauneckerPRB2009 (); MengEPLJB2014 ()

 (23)

In terms of the gapped and gapless fields , and dual fields , defined by Eq. (9) the Hamiltonian density takes the form

 H=vF2π{(∂xθ−)2+(∂xθ+)2+K−2ρ+12[(∂xϕ−)2+(∂xϕ+)2]+K2ρ−122(∂xϕ1∂xϕ2)}−bπacos(2ϕ−). (24)

After integrating out dual fields , the Euclidean action consists of the sine-Gordon action describing gapped modes,

 S−=1πvF∫dxdτ{(∂τϕ−)22+v2+(x)(∂xϕ−)22−bvFacos(2ϕ−)}, (25)

a standard LL action describing gapless modes,

 S+ =1πvF∫dxdτ{(∂τϕ+)22+v2+(x)(∂xϕ+)22}, (26)

and in the general interacting case there also appears a coupling between the gapped and the gapless modes,

 S+− =1πvF∫dxdτv2−(x)∂xϕ+∂xϕ−, (27)

where . Note that the cross term vanishes in a non-interacting system ().

The cosine term in Eq. (25) is relevant in the renormalization-group (RG) sense and leads to the gap in the spectrum. The renormalized gap is given by BrauneckerPRB2009 ()

 Δ=b(lξa)(1−Kρ)/2, (28)

with the correlation length of the gapped modes .

The Euler-Lagrange equations for the action read

 ∂2τϕ−+∂x[v2+(x)∂xϕ−]+∂x[v2−(x)∂xϕ+]=2bvFasin2ϕ−, (29) ∂2τϕ++∂x[v2+(x)∂xϕ+]+∂x[v2−(x)∂xϕ−]=0. (30)

A static solution can be found by taking and expressing from Eq. (30): . The resulting equation for resembles the sine-Gordon equation,

 ∂x[c2∂xϕ−]=2bvFasin2ϕ−, (31)

with the effective “speed of light” which takes values from for non-interacting electrons to for strongly-interacting electrons.

The vacuum classical static solution is trivial . If the Overhauser field and the interaction parameter depend on the coordinate adiabatically, the classical static solution inside the wire for an (anti)kink with center at is given by

 ϕk(a)−(x)=2arctanexp⎧⎪⎨⎪⎩±x∫ξdx′δ0(x′)⎫⎪⎬⎪⎭. (32)

The plus sign corresponds to a kink solution, while the minus sign corresponds to an antikink solution, and is the soliton width.

The Matsubara current-current correlator can be expressed by using functional integration over the fluctuations in the vicinity of the classical static solutions

 (33)
 Z=∫Dδˇφe−SE[ˇϕ0+δˇφ]+∫Dδˇφe−SE[ˇϕk+δˇφ]+…

Here we use a short-hand notation . In the vicinity of a classical one-kink solution , we expand the fields as a sum of classical solution and fluctuations around it,

 ˇϕ(x,τ)=ˇϕk(x,ξ)+δˇφ(x−ξ,τ), (34)

and treat the center of the kink as a dynamical variable . However, this representation is redundant: shifts of both the collective coordinate and the Goldstone zero-mode describe a translation. In order to avoid double counting we have to impose the following constraint. The integration is performed only over the fluctuations orthogonal to the zero-mode . This can be done by the Faddeev–Popov technique GervaisSakitaPRD1975 (); Sakita1985 (). The integrals over the fluctuations near static kink solutions in Eq. (33) have to be rewritten as

 ∫Dδˇφe−SE[ˇϕk+δˇφ]⋯→∫DδˇφDξδ(Q[ξ])det(δQδξ)e−SE[ˇϕk+δˇφ]…, (35)

Using expansion (34) and relating the current to bosonic fields by Eqs. (12)–(13) we represent the current-current correlator as an (anti)kink-particle contribution and “background fluctuations” contribution

 ⟨j(τ)j(0)⟩=⟨j(τ)j(0)⟩k+⟨j(τ)j(0)⟩a+⟨δj(τ)δj(0)⟩, (36) ⟨j±(τ)j±(0)⟩k(a) =−1π2∂2τ⟨ˇϕk(a)±(x,τ)ˇϕk(a)±(x,0)⟩, (37) ⟨δj±(τ)δj±(0)⟩ =−1π2∂2τ⟨δˇϕ±(x,τ)δˇϕ±(x,0)⟩. (38)

The electric current does not depend on the coordinate . However, the calculations are easier if we calculate the correlators in the leads, taking in Eqs. (37)–(38), where the stationary classical solution for gapless modes turns to zero. In this case it is sufficient to calculate the correlators for the gapped field in Eqs. (36)–(38).

iv.1 Current carried by a single kink-particle

At finite but low temperatures a kink with the rest energy and the mass can be activated. The kink can propagate inside the wire carrying electric charge and interacting with the environment consisting of gapless and gapped modes of background fluctuations (see Appendix C for details). The spectrum of fluctuation modes is given by

 (ω±q)2=(c/δ0)2+2q2v2+±√(c/δ0)4+4q4v4−2. (39)

The plus sign corresponds to the gapped mode , while the minus sign corresponds to a gapless acoustic mode: at , at .

While the gapped mode leads to renormalization of the kink mass, which is described by Eq. (28), the coupling to the gapless mode causes an effective friction: the kink dissipates energy interacting with the gapless mesons. The mechanism resembles Caldeira-Leggett type dissipation CaldeiraLeggett1983 (), damping of Bloch walls in quasi-1D ferromagnets caused by interaction with spin waves BraunPRB1996 (). It is also resembles a mechanism of dissipation due to scattering of spinons in Wigner crystals MatveevWignerCrystals (). However, in contrast to spinons, kinks carry electric current, and the resulting temperature dependence of conductance is different.

In order to calculate a contribution to the conductance due to the motion of kinks we integrate out fluctuations and obtain an effective low-energy Euclidean action for the collective coordinate (see Appendix C for details of derivation),

 SeffE[ξ]=ΔT+T∑¯ωξ(¯ω){M¯ω22+M2η|¯ω|}ξ(−¯ω). (40)

The summation over Matsubara frequencies is performed. The first term is responsible for the activation law exponent; the second term describes the free motion of the kink, while the third term corresponds to an Ohmic-like friction caused by the interaction between a kink and the gapless fluctuation modes. The friction coefficient is given by

 η =4TΔc2δ0v2−v3+. (41)

The Matsubara Green function for the collective coordinate in the limiting case reads

 DM(¯ω) =1M(¯ω2+η|¯ω|). (42)

In order to avoid subtleties arising from proper analytic continuation in Matsubara technique, it is convenient use the Keldysh path-integral approach. The retarded (advanced) Green function for the collective coordinate can be extracted from the Matsubara Green function by analytic continuation,

 DR(A)(ω) (43) DR(τ) =1M1−e−ητηΘ(τ). (44)

The Keldysh Green function can be obtained using the fluctuation-dissipation theorem

 DK(ω)=[DR(ω)−DA(ω)]cothω2T=−2iMωηη2+ω2cothω2T, (45)
 DK(τ)−DK(0)≈2iπMTτ2arctan1ητ. (46)

In the absence of the friction the Keldysh Green function reads

 DK(τ)−DK(0)=iMTτ2. (47)

The Green function for the bosonic fields can be obtained by Keldysh functional integration

 iGαβ(x,x,t,t′)=⟨TKϕ−(r,tα)ϕ−(r′,t′β)⟩=∫Dξϕk(x,ξ(t))ϕk(x,ξ(t′))exp[iS[ξ]]=∫Dξdqdq′(2π)2ϕk(q)ϕk(q′)eiqx+iq′x×exp[−iqξ(t)−iq′ξ(t′)]exp(iS[ξ]), (48)

where the indices , indicate whether the times , are taken on the upper or on the lower branch of the Schwinger-Keldysh contour, and refers to the Fourier transformation of the static kink solution (32).

The functional integration is performed in the Appendix D. Finally, we express the retarded Green for bosonic fields at coincident coordinates inside the wire in terms of the Green functions of the collective coordinate ,

 GR(x,x,τ)=G++(x,x,τ)−G+−(x,x,τ)=4πΘ(τ)√Mβ×DR(τ)−DR(0)√i(DK(0)−DK(τ))√2   ⎷  ⎷1−(DR(τ)−DR(0))2(DK(τ)−DK(0))2+1. (49)

The conductance is related to low-frequency current–current correlator by Eq. (14), and, hence, can be extracted from the retarded Green function in time-representation at large times,

 Gk=e−Δ/Tlimτ→+∞GR(τ). (50)

The activation law exponent arises due to the rest energy of the kink. First consider the important limiting case , . In the absence of friction, the Green functions for collective coordinate grow infinite at ,

 DR(τ)|η→0=τMΘ(τ), (51) DK(τ)−DK(0)|η→0=iMTτ2. (52)

The kink and equal anti-kink contributions to the conductance are obtained straightforwardly from Eq. (50),

 Gk=Ga=G0e−Δ/T. (53)

Thus, we see that in the absence of the friction the only effect of interactions is the renormalization of the gap .

The situation differs in the general case . Now the retarded Green function for the collective coordinate is finite at infinite times, , but the Keldysh Green function (in the limit of infinite length ) is still infinite, . Therefore, Eq. (50) yields zero conductance.

This can be easily understood, since the Ohmic-like friction causes an internal resistivity, and we may expect that at large the total conductance will drop to zero with the increase of the length .

The result for finite but large wire length can be easily estimated. Since the collective coordinate is bounded inside the wire , the Keldysh and retarded Green functions must be bounded as well, , . Therefore, we assume that the Green functions grow until they reach their asymptotic value of order of . This gives a cut-off parameter at large times with

 τR=ΔL2ℏc2,τK=Lc√ΔT. (54)

If the cut-off time and the friction are large enough (this occurs at low temperatures, when the gap is large in comparison to ), the conductance is suppressed by friction,

 Gk=G0√TΔcη(T)Le−Δ/T,ητ∞≫1. (55)

In the opposite limit , the friction becomes insignificant, and the conductance is the same as in the non-interacting case.

The crossover between these regimes can be roughly described by taking the limit at finite instead of in Eq. (50),

 Gk=2G0√TM1−e−ητ∞Mη√2πMTτ2∞arctan1ητ∞e−Δ/T. (56)

iv.2 Background fluctuations

iv.2.1 Vacuum fluctuations

First, we calculate the Matsubara Green functions for fluctuations around the vacuum solution. In order to do this we insert a point source into the action, then the solution of the Euler-Lagrange equations can be represented as . The Euler-Lagrange equations for small fluctuations are given by

 ¯ω2δφ−−∂x(v2+∂xδφ−+v−∂xδφ+)+W2(x)δφ−=πvFJ−δ(x−x′), (57)
 ¯ω2δφ+−∂x(v2+∂xδφ++v−∂xδφ−)=πvFJ+δ(x−x′), (58)

where the potential varies adiabatically, and following Ref. MaslovStonePRB1995, we assume that the interaction parameter , , in the leads. Since the current in the wire does not depend on the coordinate, the point can be chosen arbitrary. We take for the sake of simplicity.

Solving Eqs. (57)–(58) in the limit , similarly to Ref. MaslovStonePRB1995, , we obtain

 δφ−=O(¯ω0)J−+O(¯ω0)J+, (59) δφ+=O(¯ω0)J−+(π2¯ω+O(¯ω0))J+. (60)

Therefore for the Matsubara Green functions in the leads, in the limit , we obtain

iv.2.2 Fluctuations around the kink solution

Now we expand the fields around the static soliton solution . We again insert a point source into the action and solve the Euler-Langrange equations which have the form of Eqs. (57)–(58), but with the potential .

Similarly to Ref. MaslovStonePRB1995, , in the limit and if , the result does not depend on the specific form of ,

 δφ−=O(¯ω0)J−+O(¯ω0)J+, (61) δφ+=O(¯ω0)J−+(π2¯ω+O(¯ω0))J+. (62)

For the Matsubara Green functions in the leads, in the limit , we obtain

The correlators for the fluctuations around the antikink solution are the same.

Finally, for the contribution to the conductance from background fluctuations we obtain

 G=e2π2lim¯ω→0¯ωπ2¯ω+2π2¯ωe−ΔT+…1+2e−ΔT+…+O(¯ω)=G0. (63)

The contribution is temperature-independent and does not depend on interaction parameters inside the wire.

iv.3 Dilute soliton gas approximation

At higher temperatures the one-kink approximation is not valid: a larger number of kinks or anti-kinks can be activated. In order to extend the theory to higher temperatures we assume that the soliton gas is dilute and that we may disregard interactions between solitons.

We describe a configuration of the -soliton gas by collective coordinates and labels , where denotes whether the -th soliton is a kink () or anti-kink ().

The asymptotic form of the classical solution is given by

 ϕl,ξ(x)=N∑k=1ϕlk(r,ξk), (64)

where is a classical solution for a (anti-) kink located at .

The Green functions are given by

 iGαβ(t,t′)=1Z∞∑N=0e−NβΔ×∑l∫(N∏k=1Dξkexp{iS[ξk]})ϕl,ξ(tα)(x)ϕl,ξ(t′β)(x′), (65)

where are indices in Keldysh space. The partition function is defined as

 Z =∞∑N=0e−NβΔ∑l∫(N∏k=1Dξkexp{iS[ξk]}). (66)

The integration and summation are straightforward and yield a simple expression,

 Z=11+e−βΔ. (67)

The integrations over in Eq. (65) can be reduced to one-kink Green functions , and the summation over can be easily performed,

 iGαβ(t,t′)=−2dlnZd(βΔ)G(1)αβ=2e−βΔ1+e−βΔG(1)αβ. (68)

In comparison to the one-kink approximation the conductance acquires an extra factor , and the total conductance is given by

 G=G0+2G0e−Δ/T1+e−Δ/T√TΔcηLe−Δ/T, (69)

when , and the one-soliton expression for crossover to the non-interacting conductance given by Eq. (56) is replaced by

 G=G0+2G0e−Δ/T1+e−Δ/T√TM1−e−ητ∞Mη√2πMTτ2∞arctan1ητ∞. (70)

V Temperature dependence of the gap

We have derived the conductance for a fixed value of the gap . However, the gap itself is temperature dependent and is given by Eq. (28). The Overhauser field also depends on the temperature BrauneckerPRB2009 (); MengEPLJB2014 (),

 BOv=IAm(T)2, (71)

where , with , is an order parameter. At zero temperature, the nuclear spins are polarized with , while at temperatures above , , the order is destroyed, and . The order parameter at temperatures near can be estimated as (see Appendix