{'fraction_non_alphanumeric': 0.04743718592964824, 'fraction_numerical': 0.014994974874371859, 'mean_word_length': 4.31879409878127, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
High-temperature Aharonov-Bohm-Casher interferometer

We study theoretically the combined effect of the spin-orbit and Zeeman interactions on the tunneling electron transport through a single-channel quantum ring threaded by magnetic flux. We focus on the high temperature case (temperature is much higher than the level spacing in the ring) and demonstrate that spin-interference effects are not suppressed by thermal averaging. In the absence of the Zeeman coupling the high-temperature tunneling conductance of the ring exhibits two types of oscillations: Aharonov-Bohm oscillations with magnetic flux and Aharonov-Casher oscillations with the strength of the spin-orbit interaction. For weak tunneling coupling both oscillations have the form of sharp periodic antiresonances. In the vicinity of the antiresonances the tunneling electrons acquire spin polarization, so that the ring serves as a spin polarizer. We also demonstrate that the Zeeman coupling leads to appearance of two additional peaks both in the tunneling conductance and in the spin polarization.


I. INTRODUCTION
Quantum interferometers based on lowdimensional electronic nanosystems proved to be very powerful tools in studying coherent mesoscopic phenomena [1][2][3][4] . The simplest example of such an interferometer is a single-channel ballistic quantum ring tunnel-coupled to leads and threaded by the magnetic flux Φ (see Fig. 1). The dependence of the conductance of this setup G on Φ encodes important information about the phase coherence of the tunneling electrons. In particular, the interference of electron waves, propagating in the ring clockwise and counterclockwise, results in the Aharonov-Bohm (AB) oscillations 5,6 of G with Φ. The oscillation period is given by the flux quantum Φ 0 = hc/e. The shape and amplitude of the AB oscillations depend essentially on the strength of the tunneling coupling and on the relation between the temperature T and the level spacing in the ring ∆.
For T ≪ ∆ and weak tunneling coupling there are narrow resonant peaks in the dependence G(Φ). 7 The positions of the peaks depend on the electron Fermi energy 7 and the strength of the electron-electron interaction. 8 Remarkably, the interference effects are not entirely suppressed by thermal averaging, and the resonant behavior of G(Φ) survives for the case T ≫ ∆. Specifically, the high-temperature conductance of the noninteracting ring with weak tunnel coupling to the contacts exhibits sharp antiresonances at φ = 1/2 + N, where φ = Φ/Φ 0 is the dimensionless flux and N is an arbitrary integer number (see Fig.2). 9,10 The electron-electron interaction leads to appearance of a fine structure of the antiresonances: each antiresonance splits into a series of narrow peaks, whose widths are governed by dephasing. 10 The question which we address in this paper is the role of the spin-orbit (SO) and Zeeman interaction in the high-temperature tunneling transport through the ballistic single-channel ring with noninteracting electrons.
The effect of the SO interaction on the properties of one-dimensional (1D) and quasi onedimensional systems, in particular 1D quantum wires and rings, has attracted much attention.  It is known that in a 1D noninteracting wire with an arbitrary spatial dependence of the spin-orbit coupling, the spin degree of freedom may be excluded by a unitary transformation. 18 However, this is not the case for a multiply-connected 1D system such as a single-channel quantum ring. Though such a ring is actually a 1D system, the interference between spin parts of two counter-propagating electron waves makes the problem less trivial. Indeed, the rotation of electron spin in the built-in SO magnetic field results in a spin phase shift between clockwise and counterclockwise waves, which is a manifestation of the Aharonov-Casher (AC) effect. 37,38 The AC phase is the spin analog of the orbital AB phase. More precisely, the AC phase is additional with respect to AB phase and exists even at zero external magnetic field (Φ = 0). An important consequence is the existence of the AC oscillations of zero-field conductance G(0) with the strength of the SO coupling. The AC oscillations were intensively discussed theoretically [12][13][14][15][16][17]21,[24][25][26][27][29][30][31][32][33]35 and their signatures were observed experimentally. 22,23 Another consequence, especially important from the point of view of possible applications, is that the unpolarized incoming electron beam acquires polarization after passing through the ring, so that the ring may serve as a spin polarizer. The latter effect was recently discussed in a number of publications [15][16][17]24,26,27,29,32,33 mostly concerned with the study of the zerotemperature case. The finite temperature effects were also analyzed on the basis of numerical simulations. 17,26,32 Here, we develop an analytical theory of the spin-dependent transport through a ballistic single-channel ring with two symmetric contacts focusing on the high temperature case, T ≫ ∆. It will be shown that the spin-selective properties of the discussed setup survive thermal averaging. We will see that the SO interaction (in the absence of the Zeeman coupling) leads to the splitting of the high-temperature conductance antiresonances ( Fig. 2) into two dips (Fig. 4a), the distance between dips being proportional to the AC phase. In the vicinities of the dips the tunneling electrons acquire polarization P(φ) [see Fig. 4b, Eqs. (33) and (42)]. The vector P(φ) lies in the (x, z) plane formed by two axes: the axis x, which connects contacts a and b and the axial symmetry axis of the ring (z axis perpendicular to the ring's plane). At zero external field (φ = 0) the conductance G(0) exhibits series of sharp AC antiresonances [see Eq. (44)]. We also demonstrate that taking into account the Zeeman interaction leads to two effects (see Fig. 5): (i) emergence of additional antiresonances in T (φ) and P(φ); (ii) appearance of nonzero polarization in y direction.

II. RING WITH SPINLESS ELECTRONS
We start with a brief discussion (see also Ref. 10) of the high-temperature conductance of the ring with spinless electrons. The purpose of this section is to introduce methods, which will be later generalized for the spinful case. The conductance is calculated using the Landauer formula where is the energy-averaged value of the transmission coefficient T 0 (φ, E) and f (E) is the Fermi-Dirac function.
We consider both contacts to be identical and describe them with the following S−matrix: which relates the amplitudes of three outgoing waves in the channels (1 ′ , 2 ′ , 3 ′ ) to the ones in the three incoming channels (1, 2, 3) (see Fig. 3).
In the simplest case of a point tunneling contact one can express all the elements of the S matrix in terms of single real parameter γ: 7 The case of weak tunneling coupling corresponds to γ ≪ 1. The point metallic-like contact is described by γ ∼ 1.
To find the transmission coefficient we calculate the sum of the amplitudes of all the trajectories that correspond to electron passing through the ring from contact a to contact b. The summation of the amplitudes will be performed in the following way. Each of the trajectories consists of the odd number 2n + 1 of semicircles connecting the contacts. The length of the trajectory is given by L n = πR(2n + 1), where R is the radius of the ring. Let us denote the sum of the amplitudes of all trajectories with a given length L n (including trajectories with different number of backscatterings by contacts) as β n exp(ikL n ), where k = √ 2mE/ is the electron wave vector. The total transmission amplitude reads so that the transmission coefficient is given by We notice now that due to the condition T ≫ ∆ the terms in Eq. (6) corresponding to n = m vanish after the averaging over E within the temperature window. Hence, the expression for the averaged transmission coefficient becomes Next, we write β n = β + n + β − n , where β + n (β − n ) corresponds to trajectories ending with lower (upper) semicircle. It is easy to write the recurrence equations for β + n and β − n : where the matrixÂ 0 =Â 0 (φ) is given bŷ Physically, the elements of matrixÂ 0 are the amplitudes of four different trajectories of length 2πR, starting and ending on the contact b. From Eqs. (7) and (8) we find where ⊗ denotes the direct (Kronecker) product of matrices and represent the amplitudes of shortest counterclockwise and clockwise trajectories, respectively. Using Eqs. (4), (9), (10), and (11) after some algebra we get the following expression for the transmission coefficient: 10 In the almost closed ring with weak tunneling coupling, γ ≪ 1, Eq. (12) can be wellapproximated with the function [Eq. (13) is valid for 0 < φ < 1]. This dependence is shown in Fig. 2. We see that there is a sharp antiresonance at φ = 1/2 and T 0 (1/2) = 0. The physics behind this behavior can be explained as follows. 7 For each trajectory there exists a corresponding mirrored (with respect to the x-axis) trajectory. The sum of the amplitudes of these two trajectories is proportional to e ikLn (e i(2|m|+1)πφ + e −i(2|m|+1)πφ ), where m is a difference between the number of full clockwise and counterclockwise revolutions, |m| ≤ n. At φ = 1/2 this sum turns to zero for any k. Thus, the destructive interference at φ = 1/2 survives the thermal averaging.

III. RING WITH SPINFUL ELECTRONS
In the spinful case the Hamiltonian is given bŷ whereĤ is the kinetic energy, D ϕ = ∂/∂ϕ + iφ, is the Zeeman term, ω Z is the Zeeman splitting energy in the external field B parallel to z axis, andĤ SO corresponds to the SO coupling. We assume that the SO interaction is described by the Rashba Hamiltonian, which for the case of a straight wire looksĤ SO = α[n×σ] p. Here n is the unit vector parallel to built-in electric field,σ is the vector of the Pauli matrices, α is the constant of the SO interaction, and p is the electron momentum. In a curved wire, n depends on the coordinate, and the Hamiltonian becomes 13,14,18 where {. . .} stands for the anticommutator. For a ring with axially symmetric built-in field, n = (cos ϕ cos θ, sin ϕ cos θ, sin θ), we find from Eq. (17) Here ϕ is the angle coordinate of the electron in the ring, θ is the angle between effective SOinduced magnetic field B eff (this field is proportional to α[p × n]) and the z axis. The coefficient ξ = αmR/ entering Eq. (18) is the dimensionless parameter characterizing the strength of SO interaction. Physically, ξ is the angle of spin rotation in the local field B eff during the time on the order of R/v F , where v F is the Fermi velocity. In the simplest case θ = 0, ξ is proportional to the angle of the spin rotation after passing around the ring [see Eqs. (29) and (33)].
We study the problem quasiclassically assuming that k F R ≫ 1 and α ≪ v F (or, equivalently, ξ ≪ k F R). Within this approximation the combined effect of the SO and Zeeman interaction is fully described by the rotation of the electron spin in the field B + B eff , which varies along the electron trajectory. 13,14 Using Eqs. (14)-(18) we find (see Appendix A) the matrices of the spin rotationM ± a→b = cos πδ ± e iϑ±σy ± i sin πδ ±σz , (19) HereM + i→j (M − i→j ) describes spin rotation for an electron passing a semicircle from contact i to contact j with zero winding number in counterclockwise (clockwise) directions,M T denotes the transpose of a matrixM , and The strength of the Zeeman coupling is characterized by dimensionless parameter Ω Z = ω Z R/2v F . 36 The sum of the amplitudes of the trajectories having length L n , initial spin state |χ i , and final spin state |χ f is given by χ f |β n |χ i , whereβ n are now 2 × 2 matrices. The amplitude of transmission through the ring with spin state changing from |χ i to |χ f is given by The transmission coefficient reads The electrons passing through the ring acquire spin polarization. For the case of unpolarized incoming electron beam the spin polarization is calculated as and, therefore, is also expressed in terms ofβ n .
To findβ n we separate trajectories into two groups (just as in the previous section), writinĝ β n =β + n +β − n , whereβ + n andβ − n satisfy the recurrence equations, analogous to Eq. (8): The block matrixÂ is given bŷ The ma-trixM + (M − ) describes spin rotation after passing a full circle starting from contact b and propagating in counterclockwise (clockwise) direction. The matrixK (K ′ ) is the spin rotation matrix for an electron, which starts from the contact b, then propagates along the lower (upper) shoulder of the interferometer and returns back to the contact b after single backscattering on the contact a.
For the shortest trajectories we havê The matrix elements of matrixT can be written in a form, analogous to Eq. (10): In the following sections we will use Eqs. (19)- (27) to calculate the full transmission coefficient and the spin polarization for a ring with the SO and Zeeman interactions.
Now we will make use of the important feature of SO interaction: if the electron travels along a certain trajectory and then returns to the initial point along the same trajectory moving in the opposite direction, its spin returns to the original state. This implies that [one can easily check Eq. (31) using Eqs. (19) and (28)]. These properties essentially simplify further calculations. The block matrixÂ is now fully expressed in terms of matrixM Next, we writeM where ρ is the vector of spin rotation for counterclockwise propagation around the ring (starting from contact b). From Eqs. (19), (28), (29), and (30) we find ρ = 4πδ(e x sin ϑ − e z cos ϑ).
For ξ ≫ 1, the frequency of spin precession in the field B eff is much larger than the orbital frequency v F /R and the direction of the spin follows adiabatically the direction of B eff . In this case Eq. (29) simplifies 2πδ = 2πξ − π(1 − cos θ), for ξ ≫ 1. (36) Thus, in the adiabatic limit the AC phase separates into two parts: 13 dynamical contribution 2πξ and geometrical SO Berry phase 39 π(1 − cos θ) which is the half of the solid angle subtended by B eff when electron passes the full circle. 40 In order to find the recurrence equations forβ ± n we now introduce the spinorsχ ↑↓ = exp(∓iπ|δ|)(M + a→b ) −1 χ ↑↓ , which are transformed to χ ↑↓ when the electron propagates from contact a to contact b [the phase multiplier exp(∓iπ|δ|) is added for convenience]. Using Eq.(25) we get χ ↓ |β n |χ ↑ = χ ↑ |β n |χ ↓ = 0, so thatβ n can be written asβ For β ↑ n = χ ↑ |β n |χ ↑ we get the recurrence equations HereÂ 0 (φ) is given by Eq. (9) and We see that the quantities β ↑± n satisfy the same recurrence equations as the ones in the spinless case [see Eq. (8)] with the replacement φ with φ + |δ|. One can easily show that the recurrence equations for β ↓ n = χ ↓ |β n |χ ↓ are given by Eqs. (38) and (39) with the replacement φ+|δ| with φ − |δ|. As a result, we find where T 0 is the transmission coefficient of the spinless electrons given by Eq. (9). The expressions for the full transmission coefficient and the spin polarization become where It is worth noting that Eq. (41) is in agreement with the general theorem, relating any transport property of 1D system with the SO interaction with the same property without the SO interaction. 11 The dependencies of the conductance and the spin polarization on magnetic flux are schematically depicted in Fig.4. As seen, there are two dips (per period) in the function T (φ), corresponding to φ = 1/2 ± δ + N. At these two points the incoming electrons with spin states described, respectively, byχ ↓ andχ ↑ are totally blocked by the destructive interference. Therefore, the tunneling current becomes fully polarized in the direction of ρ for φ = 1/2 ± δ + N.
We see that SO-induced splitting of the resonances is proportional to the AC phase 2πδ. Eqs. (41) and (43) reveal coexisting of two types of oscillations: the AB oscillations with φ and AC oscillations with δ. Importantly, AC oscillations of tunneling conductance exist even in the case of zero external field. Indeed, for φ = 0, we have Here we took into account that T 0 (δ) is an even function. Thus, transmission coefficient exhibits the AC oscillations with the period δ = 1. For the case of almost closed ring, γ ≪ 1, the oscillations have the form of sharp antiresonances periodic in δ.

B. Interplay of spin-orbit and Zeeman interactions
Next we discuss the role of the Zeeman interaction. Taking this interaction into account requires much more tricky calculations. The point is that the properties (31) are no longer valid when the time reversal symmetry is broken. Consequently, the elements of block matrixÂ can not be expressed in terms of a single rotation matrix. In principle, the expressions for T and P can be derived from (27), where both averaging over the temperature window and summation over winding number n are already done. This equation, indeed, turns out to be very useful for numerical simulations. However, the analytical expres-sions obtained with the use of (27) for the case of arbitrary γ turn out to be very cumbersome and we do not present them here. We restrict ourselves with the analytical study of the almost closed ring, γ ≪ 1. For this case, the calculations presented in the Appendix B yield where and These results are shown in Fig. 5. We see that in the presence of the Zeeman interaction instead of the two antiresonances there are four ones (per period) corresponding to the flux values 1/2 ± δ + N and 1/2 ± δ ′ + N. In the vicinity of each antiresonance the outgoing electrons are polarized. Importantly, the Zeeman coupling induces nonzero polarization in y direction. One may notice that the dependence P y (φ) is qualitatively different from dependencies P x (φ) and P z (φ). First of all, the peaks in P y (φ) are asymmetric [see Eq. (52)] in contrast to the peaks in P x (φ) and P z (φ). Secondly, all four resonances in P y (φ) have the same amplitudes. Next, we consider some limiting cases. For weak Zeeman coupling, |Ω Z | ≪ max{1, |ξ|}, one c + ≈ 1/2 + ξ cos θ 1/4 + ξ cos θ + ξ 2 , s + ≈ ξ sin θ 1/4 + ξ cos θ + ξ 2 .
In the strong Zeeman coupling limit, |Ω Z | ≫ max{1, |ξ|}, we obtain We see that in both limiting cases there are two deep antiresonances, the positions of which are controlled by the strength of the SO interaction, and two small ones with the positions controlled by the Zeeman coupling.
The competition between spin-orbit and Zeeman coupling is clearly seen in the case ξ ≫ 1, Ω Z ≫ 1, and arbitrary relation between ξ and Ω Z . In particular, the amplitudes of peaks in T (φ) are given by As follows from these equations, for ξ ≈ Ω Z all four antiresonances have the same amplitudes. The dependencies of the positions of four antiresonances on Ω Z with fixed ξ is shown in Fig. 6. It is noteworthy that in the special case θ = π/2, the distance between the deep antiresonances tends to zero when the strength of the Zeeman coupling increases, so that T (φ) → T 0 (φ) when Ω Z → ∞. This is also illustrated in Fig. 7 where T (φ) is plotted for θ = π/4 and θ = π/2. mind that such fluctuations are on the order of Ω Z T /E F , one can conclude that the inhomogeneous broadening should be taken into account when Ω Z T /E F becomes larger than γ.

IV. SUMMARY
In this paper we calculated the hightemperature transmission coefficient T (φ) and the spin polarization P(φ) of the electrons tunneling through a single-channel ring with the Rashba SO interaction threaded by magnetic flux. We obtained analytical expressions for T (φ) and P(φ) valid for arbitrary strength of the tunneling coupling. Both T (φ) and P(φ) reveal coexistence of two types of periodic oscillations: the Aharonov-Bohm oscillations with magnetic flux and the Aharonov-Casher oscillations with the strength of SO interaction. For weak tunneling coupling, the oscillations have the form of the sharp antiresonances periodic in φ and δ. Specifically, in the absence of the Zeeman coupling there are two antiresonances (per period) in the depen-dence T (φ) (instead of one antiresonance in the spinless case). In the vicinity of each antiresonance, the electron beam passing through the ring acquires strong spin polarization directed in the (x, z) plane formed by axial symmetry axis of the ring (z axis) and the line connecting two contacts. We also discussed the influence of the Zeeman interaction on the interference picture and showed that two additional antiresonances appear both in T (φ) and P(φ). Also, the Zeeman coupling leads to appearance of nonzero spin polarization in y direction.
Let us finally briefly discuss some unsolved problems. In the above calculations we assumed that electrons are noninteracting. The detailed analysis of the effects caused by electron-electron interaction is out of scope of the current paper. Here we restrict ourselves to a very brief qualitative discussion (for the case T ≪ ∆, the role of the electron-electron interaction in a ring with SO interaction was discussed in Refs. 25). First of all, the interaction leads to a renormalization of the tunneling rate due to Luttinger-liquid correlations specific for purely 1D systems. This implies that in all equations derived above one should replace γ by renormalized tunneling rateγ (for discussion of renormalization of 3 × 3 matrix of the tunneling contact in the spinless case see Refs. [42][43][44][45][46]. This does not change our results much, since γ is a phenomenological parameter of the model. What is more important is that the interaction suppresses the interference between clockwise and counterclockwise propagating electron waves, thus leading to a broadening of the resonances. It is expected 47 that this broadening is negligible when interaction is so weak that where Γ 0 ϕ ≃ gT is the electron-electron scattering rate in the infinite single-channel wire 48 (here g is dimensionless constant characterizing the strength of the electron-electron interaction) Equations derived in the previous sections are valid (with the replacement of γ withγ) provided that inequality (55) is satisfied. In the opposite limiting caseγ∆ ≪ Γ 0 ϕ , the interaction should modify the structure of the resonant peaks in T (φ) and P (φ). One may expect two effects, predicted previously for the spinless case: 10 (i) all the resonances would acquire fine structure, splitting into a series of narrow peaks separated by distance g (ii) each of the peaks of the split structure would broaden within a width Γ ϕ /∆, where Γ ϕ is the phase breaking rate, which is expected to be much smaller than the bulk dephasing rate (Γ ϕ ≪ Γ 0 ϕ ) because of the charge and size quantization in the almost closed ring. More detailed analysis including rigorous calculation of Γ ϕ will be presented elsewhere. 47

V. ACKNOWLEDGMENTS
We thank I.V. Gornyi and D.G. Polyakov for valuable discussions. The work was supported by Russian Foundation for Basic Research (grants 11-02-00146 and 11-02-91346) and by programs of the Russian Academy od Science. The work of P.M. Shmakov was supported by Dynasty Foundation.
The probability W χi χ f can be found as It can also be written in a form, analogous to Eq. (10): Though Eq. (B4) is very useful for numerical analysis it yields very cumbersome analytical expressions for T and P. For the case γ ≪ 1 it turns out more convenient to use Eq. (B3) for deriving Eqs. (45)- (48). To this end, we first make a unitary transformation of the matrixÂ : described by the block matrix whereΛ ± are 2 × 2 matrices which correspond to two unitary transformations diagonalizing matri-cesM + andM − , respectively (the transformation, described by Eqs. (B5), (B6), exactly diagonalizes the matrixÂ in the case t b = 0, when A is a block matrix with blocks e −2πiφM + and e 2πiφM − ). For weak tunneling coupling, γ ≪ 1, and t b = 0, the matrixÂ ′ readŝ where λ ± = e ∓iπ(φ−δ±) , andλ ± = e ∓iπ(φ+δ±) . After the unitary transformation, Eq. (B3) is rewritten as follows: (B8) Here are the eigenstates of the spin rotation matriceŝ M + andM − , respectively, and The spinors (B9) and (B10) obey (µ = 1, 2) Since γ ≪ 1, the off-diagonal elements A ′ ij are small and should be taken into account only when they become comparable with the differences of corresponding diagonal elements: Thus we can neglect the backscattering for all φ, with the exception of the vicinities of the points, where λ + = λ − ,λ + =λ − , λ + = −λ − orλ + = −λ − . These equations correspond to the magnetic fluxes ±(δ + + δ − )/2 and ±(δ + − δ − )/2 + 1/2. Hence, the calculations can be performed in two steps. First, we solve the problem neglecting the backscattering. The obtained solution is valid everywhere except the vicinities of points defined above. Next, we assume that the flux is close to one of these points and take into account backscattering.