Transport in Magnetically Doped One-Dimensional Wires: Can the Helical Protection Emerge without the Global Helicity?

We study the phase diagram and transport properties of arbitrarily doped quantum wires functionalized by magnetic adatoms. The appropriate theoretical model for these systems is a dense one-dimensional Kondo Lattice (KL) which consists of itinerant electrons interacting with localized quantum magnetic moments. We discover the novel phase of the locally helical metal where transport is protected from a destructive influence of material imperfections. Paradoxically, such a protection emerges without a need of the global helicity, which is inherent in all previously studied helical systems and requires breaking the spin-rotation symmetry. We explain the physics of this protection of the new type, find conditions, under which it emerges, and discuss possible experimental tests. Our results pave the way to the straightforward realization of the protected ballistic transport in quantum wires made of various materials.


Introduction
Protected states, which are important elements for nanoelectronics, spintronics and quantum computers, attract evergrowing attention of physicists. A certain protection strongly reduces effects of material imperfections, including backscattering and localization, and provides a possibility to sustain the ballistic transport in relatively long samples.
We focus on another recently predicted and very promising possibility to realize protected transport in quantum wires functionalized by magnetic adatoms. The corresponding theoretical model is a dense 1D Kondo lattice (KL): the 1D array of local quantum moments-Kondo impurities-interacting with conduction electrons. KLs have been intensively studied in different contexts, starting from the Kondo effect and magnetism to the physics of TIs and Tomonaga Luttinger liquids (TLLs) [10,11,. The physics of KL is determined by the competition between the Kondo screening and the Ruderman-Kittel-Kosuya -Yosida (RKKY) interaction, as illustrated by the famous Doniach's phase diagram [38]. We have recently predicted that the 1D RKKY-dominated KL with magnetic anisotropy of the easy-plane type will form a

Theoretical model
We start from the standard KL Hamiltonian: Here ψ n ≡ {ψ n,↑ , ψ n,↓ } T are electron annihilation (ψ † n -creation) operators; ρ n = ψ † n ψ n ; S n are quantum spins with magnitude s; σ ≡ {σ x , σ y , σ z } are Pauli matrices; t and μ are the electron hopping and the chemical potential; summation runs over lattice sites. We assume that sJ K < D = 2t and consider low temperatures, T → 0.

Method
We proceed in several steps. Firstly, we find classical spin configurations minimizing the free energy. Secondly, we identify degrees of freedom whose fluctuations are gapped, including gapped fermionic and spin variables (|m| and α in equation (4) below) and integrate out the gapped variables perturbatively. Remaining spin fluctuations (described by vectors e a in equation (4)) receive the fully quantum mechanical treatment. This approach is justified by the separation of scales: the shortest scale is of order of the inverse Fermi momentum, 1/k F . It is present in the spin ordering and must be much smaller then the coherence length ζ of the gapped variables. We have performed the self-consistency check which confirms that ζ 1/k F and, thus, justifies the validity of our theory.

Separating the slow and the fast variables
To describe an effective low energy theory, it is convenient to focus on the regime |J K | < |μ| t where we can linearize the dispersion relation and introduce right-/left moving fermions, ψ ± , in the standard way [68]. In the continuum limit, the fermionic Lagrangian reads Here v F is the Fermi velocity, ν is the chiral index which indicates the direction of motion, ∂ ν is the chiral derivative, τ is the imaginary time. According to Doniach's criterion, the RKKY interaction wins in 1D when the distance between the spins is smaller then a crossover scale: ξ s < ξ ϑ 0 J 2 K /T K ; here ξ is the lattice spacing, ϑ 0 is the density of states at E F ; T K is the Kondo temperature. We study this RKKY-dominated regime. For simplicity, we assume ξ s = ξ.
Following references [46,66,67], we keep in the Lagrangian of the Kondo exchange interaction only the backscattering terms governing the physics of the dense 1D KL: where R ≡ ψ + , L ≡ ψ − . L bs contains the fast 2k F -oscillations which must be absorbed into the spin configuration. We perform this step using the path integral approach where the spin operators are replaced by integration over a normalized vector field decomposed as Here q 2k F ; {e 1 , e 2 , m} is an orthogonal triad of vector fields whose coordinate dependence is smooth on the scale 1/k F , |e 1,2 | = 1. Angle α and constants b, θ must be chosen to maintain normalization |S/s| = 1. Equation (4) is generic; it allows for only three possible choices of the constants which, in turn, reflect the band filling f, see appendix A. After inserting equation (4) into equation (3), we select the non-oscillatory parts of L bs for these three cases, and take the continuous limit. This yields the smooth part of the Lagrangian density: HereJ ≡ sJ K √ 1 − m 2 /2; σ ± = (σ 1 ± iσ 2 )/2; we expressed vectors e 1,2 via matrix g ∈ SU(2), see appendix B. g is a smooth function of {x, τ }; it governs the new rotated fermionic basis Equation (5) assumes a staggered configuration of spins at half-filling, ↑↓, which was studied in reference [46]. The spin sector of the half-filled KL is an antiferromagnet where the spins fluctuate around the Neel order with a finite correlation length. Equation (6) reflects two spins up-two spins down configuration, ↑↑↓↓, which agrees with the spin dimerization tendency observed numerically in reference [58] at quarter-filling. Equation (7) is a rotationally invariant counterpart of the helical spin configuration discovered in references [66,67] in the anisotropic KL at incommensurate fillings. Spins fluctuate around this configuration. Detailed derivation of their effective action is presented in reference [69]. A simplified version of equation (7) was used in reference [41] for analyzing magnetic properties of KLs. Below, we refer to equations (5) and (6) at α = 0 as 'commensurate configurations' and to equations (6) and (7) at α = π/4 as 'general configurations'. We note that the low energy physics of KLs with the 1/4-and 3/4-filling is equivalent in our model. Therefore, we will discuss only 1/4-filling and do not repeat the same discussion for the case of the 3/4-filling.

Results
Let us start from the presentation of our results at the simplified and transparent semiclassical level.

Fermionic gap
The backscattering described by equations (5)-(7) opens a gap in the spectrum of the rotated fermionsR,L. It decreases their ground state energy: the larger the gap, the greater is the gain in the fermionic kinetic energy. Since the spin degrees of freedom do not have kinetic energy, the minimum of the ground state energy is achieved by maximizing the fermionic gap. This indicates that |m| is the gapped variable with the classical value m 0 = 0. Below, we use m 0 for the semiclassical part of the discussion. The KL contains two fermionic sectors which can have different gaps depending on the band filling and the spin configuration. The gaps can be found from equations (5)- (7): generic filling: The gain in the fermionic ground state energy reads see appendix C. ϑ 0 = 1/πv F for the 1D Dirac fermions. Let us now analyze various band fillings.

Special commensurate fillings, insulating KLs
At f = 1/2, 1/4, we have to decide which spin configurations-the commensurate ones (equation (5) for f = 1/2 and equation (6) with α = 0, π/2 for f = 1/4) or the generic configuration-minimize the ground state energy. Using equations (9)-(12), we obtain with E ≡ ϑ 0 ξJ 2 . In both commensurate cases, the commensurate configuration wins, the conduction band of such KLs is empty and, hence, they are insulators, as expected. Note that, at quarter-filling, the minimum of δE GS is provided by α = 0 which means that α is gapped.

Vicinity of special commensurate fillings, collinear metal and heavy TLL
Let us consider fillings which are slightly shifted from the special commensurate cases. To be definite, we analyze an upward shift; a downward shift can be studied in much the same way. Equations 13 and 14 suggest that the commensurate spin configuration remain energetically favorable even close to the commensurate filling. The wave vector q of the spin modes remains commensurate, equations (5) and (6), and is slightly shifted from 2k F : 2k F − q ≡ Q 1/ξ with q = 2πf/ξ and f = 1/2, 1/4. This case is described in terms of Dirac fermions with a non-zero chemical potential: see appendix D. Backscattering by the commensurate spin configuration opens a gap below the chemical potential. The electrons with energies 0 < E v F Q/2 are pushed above the gap, figure 1, and have (almost) parabolic dispersion: see equation (D4). Since this new phase possesses a partially filled band it is a metal. Its metallic behaviour originates from the (almost) collinear spin configuration whose classic component is governed by only one slowly rotating vector, e.g. e 1 . We will reflect this fact by referring to such phases as 'collinear metals' (CMs). A detail description of CMs is presented in reference [69]. Let us mention here that spin modes can mediate repulsion between the conduction electrons and, for energies , they form a repulsive and spinful TLL characterized by a new Fermi momentum k * F = Q/2. If the effective repulsion is strong enough, TLL becomes heavy. Such TLL has been observed numerically in reference [65]. 1D nature makes repulsive CMs very sensitive to spinless impurities: even a weak disorder easily drives it to the localized regime with suppressed dc transport [70].

Quantum phase transition at generic filling
The CM becomes less favorable when |Q| increases: the energy of the electrons in the TLL, figure 1. If |Q| is large enough, such that E p E , the minimum of the ground state energy is provided by the generic spin configuration, equation (7). Equalizing the leading part of E p with E, we can estimate the critical value of Q at which the spin configuration changes:

Generic incommensurate fillings, locally helical metal
The remaining case of generic filling, equation (7), is the most promising for transport because rotated fermions are gapped only in one helical sector, e.g. {R ↓ ,L ↑ }, and the second helical sector, {R ↑ ,L ↓ }, remains gapless, see equation (11) and figure 1. The semiclassically broken helical symmetry is restored by the fluctuations: the rotating matrix field g(x, τ ) slowly changes in space and time around the underlying spin spiral and, therefore, the global helicity cannot appear, see figure 1. Hence, one can describe properties of the new phase only in terms of the local helicity. Simultaneously, there are no sectors of the physical fermions, {R σ , L σ }, which can be found from the inverse of the rotation equation (8), which are either gapless or globally helical. To emphasize the underlying locally helical spin configuration, we refer to this phase as 'locally-helical metal'.
Since neither the spins nor the physical charge carriers in the locally-HM possess the global helicity, one can surmise that they are not a platform for a protected transport. This is, however, incorrect since the most significant property of the locally-HMs is that they inherit protection of the ballistic transport from those HMs where SU(2) symmetry is broken and the global helicity emerges [66,67]. The absence of the global helicity in lMHs is reflected in the gapped nature of the spin excitations [69].

Origin of protection
Let us explain the physics of the seemingly counterintuitive protected transport in the locally-HMs. The density and backscattering operators are invariant under g-rotation: R † R =R †R , L † L =L †L , R † L =R †L . The low energy physics is governed by fields whose correlation functions decay as power law. To obtain them, we project the fields on the gapless sector, i.e., average over the high energy gapped modes. For example, components of the charge density are: ρ(2k F ) is absent because it would correspond to a single particle elastic backscattering between the gapless and gapped fermions which is not allowed. This is the direct consequence of the (local) spin helix which gaps out only one helical fermionic sector. Thus, the HM is the 4k F -CDW phase. This fact has two important consequences: (i) the (local) spin helix shifts the Friedel oscillations of the charge density from 2k F to 4k F , which is indistinguishable from 4(k F − π/2ξ) due to the lattice periodicity; (ii) even more importantly, it drastically reduces backscattering caused by spinless disorder.
To illustrate the 2nd statement, we introduce a random potential of spinless backscattering impurities which couples to the 2k F -component of density L dis = V 2k FR †L + h.c. Here V 2k F is a smooth 2k F -component of the random potential. Since the charge response function of locally-HM at the 2k F wave-vector is non-singular, backscattering can occur only via many particle processes with much smaller amplitude. Averaging over the gapped fermions, we find: L dis 2 V 2 k F 2 /Δ (gen) R † ↑L ↓ + h.c. see appendix E. If the helical gap is large enough, Δ (gen) =J V 2k F , backscattering and all disorder effects are parametrically suppressed.

Quantum theory for smooth spin variables and self-consistency check
To complete the theory of the magnetically doped quantum wires, one must consider quantum fluctuations of smooth spin variables e 1,2 . They are described by using the heavy field-theoretical machinery of the nonlinear σ-model (nLSM). Its derivation is a lengthy task which is described in detail in reference [69]. Here, we very briefly recapitulate main steps of the derivation, give final answers, and argue that the fully quantum mechanical theory does not violate separation of scales, see section 3. The latter is especially important since it confirms validity of our approach and validates results described in the previous section at the simplified and transparent semiclassical level.
Derivation of the nLSM requires several steps: • One (i) integrates out gapped fermions and exponentiates the fermionic determinant; (ii) derives the Jacobian of the SU(2) rotation by the matrix g; (iii) selects smooth contributions from the Wess-Zumino term for the spin field [71]. The commensurate spin configurations generate also the topological term (see reference [46], section 16 of the book [71], and references therein). • The total Lagrangian, which is obtained by summing up the exponentiated fermionic determinant, the Jacobian, the Wess-Zumino contributions and the topological terms, is expanded in gradients of the matrix g and in small fluctuations of |m| around its classical value m 0 = 0. The commensurate spin configuration, which corresponds to 1/4-filling, requires also the expansion in fluctuations of α.
Here f = 1/2 at (or close to) the half-filling and f = 1/4 at (or close to) the quarter-filling; small dimensionless coupling constants, g 1/2 (4π/s) ϑ 0 ξJ log D/|J| 1 and g 1/4 g 1/2 / √ 8 1, determine small renormalized velocities of the spin excitations, c f = v F g f /4π v F . Smallness of g f and c f reflects the coupling between spins and gapped (localized) fermions. The integer k marks topologically different sectors of the theory.
The action S ( f ) corresponds to the well-known O(3)-symmetric nLSM in (1 + 1) dimensions with the topological term. It is exactly solvable [71][72][73] and possesses a characteristic energy E f ∼ |J|g −1 f exp(−2π/g f ) which governs a large spatial scale: The latter inequality confirms validity of our approach.
Locally-helical metals: the Largangian of the nLSM describing fluctuations of the spin variables in a locally-HM takes the following form: with g gen g 1/4 /4 1, c gen = v F g gen /π v F , and Ω (z) τ ≡ itr[σ z g −1 ∂ τ g]/2. This theory is anisotropic and has the SU(2)-symmetry, g → Mg, M ∈ SU(2). The time derivative is present only in the Ω z term. This points to a relatively short bare correlation length of spins which coincides with the UV cut-off of the theory. The latter is ∼ v F /J in our approach and does not violate the self-consistency requirement because v F /J ξ. The actual shortest scale of the theory is expected to be much larger if the anisotropy is irrelevant and L (gen) flows in the IR limit to the well-known SU(2) × SU(2)-symmetric nLSM. An example of such a behaviour is provided by the RG equations derived in reference [74] for the (2 + 1) dimensions. There is no counterargument against the irrelevance of the anisotropy in the (1 + 1) dimensions. Therefore, we arrive at a conclusion that the shortest spatial scale generated by L (hel) is v F /J k −1 F . Hence, the effective low energy theory of the locally-HM is a combination of the gapped spin excitations, described by nonlinear σ-model (18), and gapless fermions. These spin and charge sectors are coupled, see equation (8). However, their coupling cannot lead to closing the spin gap [75].
This concludes the self-consistency check of our approach and justifies qualitative results described in section 4 at the semiclassical level.

Possible numerical and experimental test of our theory
An important task for the subsequent research is to reliably detect different metallic phases in the 1D KLs (numerically) and in the magnetically doped quantum wires (experimentally). This requires to tune the band filling and the Kondo coupling. The key features distinguishing CM and locally-HM in numerics and experiments are as follows. The conductance of the CM is equal to the quantum G 0 = 2e 2 /h while the locally-HM must show only G 0 /2 conductance due to the helical gap. The CM is a spinful TLL whose charge and spin response functions have a peak at 2k * F ; k * F is the shifted Fermi momentum predicted by general theorems [76][77][78]. The locally-HM is the 4k F -CDW and has singular response in the charge sector. Since 4k F and 4k * F are indistinguishable on the lattice the response of the locally-HM does not show the shift k F → k * F . Unlike systems with broken SU(2) symmetry [66,67], the locally-HM, which we have considered, does not have singular response in the spin sector. Inasmuch as the CM responds to scalar potentials at 2k * F and the locally-HM-at 4k F , the spinless disorder potential has a profound difference with respect to transport in the CM and locally-HM phases. Namely, localization is parametrically suppressed in the locally-HM.
Detecting the CM is not difficult because it is generic at relatively large J K and filling away from 1/2, 1/4. The heavy TLL, which is formed by the interactions in the CM, has been observed in numerical results of reference [65]. However, J K was too large for finding the HM. The KL studied in reference [61] exhibits an unexpected 2k F -peak at small J K . Yet, the peak was detected in the spin susceptibility of six fermions distributed over 48 sites. So small KL cannot yield a conclusive support or disproof of our theory. A more comprehensive study of the larger KLs is definitely needed.
The thorough control of the system parameters is provided by the experimental laboratory of cold atoms where 1D KL was recently realized [79]. Experiments in cold atoms are, probably, the best opportunity to test our theory. However, modern solid-state technology also allows one to engineer specific 1D KL even in solid state platforms. It looks feasible to fabricate 1D KL in clean 1D quantum wires made, e.g., in GaAs/AlGaAs by using cleaved edge overgrowth technique [80] or in SiGe [81]. Magnetic adatoms can be deposited close to the quantum wire by using the precise ion beam irradiation. One can tune parameters of these artificial KLs by changing the gate voltage, type and density of the magnetic adatoms and their proximity to the quantum wire. Such a nano-engineering of 1D KL is essentially similar to the successful realization of topological superconductivity in atomic chains [82], in carbon nanotubes [83], and in Bi [84]. The experiments should be conducted at low temperatures, T Δ, E, where destructive thermal fluctuations are weak.

Conclusions
We have studied the physics of quantum wires functionalized by magnetic adatoms with a high density and a small coupling between the itinerant electrons and local magnetic moments of the ad-atoms, |J K | E F . Their physics is determined by the RKKY interaction between the ad-atoms which results in a quite rich phase diagram. It includes: (i) the insulating phase which appears at special commensurate band filling, either 1/2, or 1/4, 3/4; (ii) spinful interacting metals which exist in the vicinity of that commensurate fillings; and (iii) the novel metallic phase at generic band fillings, see figure 1.
The third phase is our most important and intriguing finding. On one hand, the local spins form a slow varying in space and time spiral, which can yield a local helical gap of the electrons. On the other hand, the global helicity is absent because the spin-rotation symmetry is not (and cannot) be broken. The latter can result in an erroneous conclusion that a helicity-protected transport could not originate in these locally helical metals. That is not true: paradoxically, the locally helical phase inherits protection of the ballistic transport from those systems where the spin rotation symmetry is broken and the global helicity emerges. Protection of transport in locally-HMs has a simple physical explanation because they are the 4k F -CDW phase with the reduced 2k F response. This reduction is the direct consequence of the local helicity. It parametrically suppresses effects of a spinless disorder and localization. Thus, we come across the principally new type of emergent (partial) protection of transport caused by the interactions without a need of the global helicity. Our model and approach allow us to uncover the promising possibility for engineering the HM in the quantum wires and to identify the parameter range where the HM is formed, see figure 1. To the best of our knowledge, this gives the firstever example of such a protection in the system where the spin-rotation symmetry is not (and cannot be) broken. It would be interesting to study in the future how the direct Heisenberg interaction between the spins could modify out theory [85,86].
We believe that detecting the locally-HMs in numerical simulations and real experiments is the task of a high importance. Our results suggest how to tune the physical parameters, in particular the band filling and the Kondo coupling, such that the locally-HM could be realized. The fundamental sensitivity of the state and of the transport properties of the magnetically doped quantum wire to the band filling is especially important. It allows one to switch over normal and locally helical regimes of the conductor by varying a gate voltage. This can be used for creating fully controllable helical elements. Our theoretical prediction, that the backscattering is suppressed in the locally-HMs in spite of the absence of the global helicity, can pave the way towards flexible engineering principally new units of nano-electronics and spintronics with substantially improved efficiency. e i(qx+θ) = ± 1 ± i √ 2 : |s 0 | 2 + |s c | 2 + |s s | 2 2 = 1.
There are no other configurations which are compatible with decomposition equation (A1).

Appendix C. Ground state energy of the gapped 1D Dirac fermions
Consider 1D Dirac fermions with the inverse Greens function: Integrating out the fermions we find the partition function: Here Z 0 ≡ Z[Δ = 0],Ĝ 0 ≡Ĝ[Δ = 0] and Δ is assumed to be small. Using the expression for the free energy F = −T log[Z], we find that the gain of the energy, which is caused by the gap opening, reads as At T = 0 and in the continuous limit, this expression reduces to The UV divergence must be cut by the band width D. Thus, we obtain with the logarithmic accuracy:

Appendix D. Smoothly oscillating backscattering
The theory close to the special commensurate filling can be formulated in terms of Dirac fermions with a spatially oscillating backscattering described by Lagrangian: