Introduction

Electrons subject to a cyclic motion in mesoscopic loops reveal a whole series of quantum effects, both of fundamental and practical interest. A charge circulating a magnetic flux line gathers a quantum phase leading to the Aharonov-Bohm (AB) effect1, which demonstrates the distinct role played by electromagnetic potentials in quantum physics. Moreover, the AB effect is topological, meaning that it does not depend on the particulars of the loop’s geometry as long as the magnetic flux line is enclosed. AB phases are nothing but an example of the geometric phases formalized by Berry2, Simon3, and Wilczek & Zee4 in the early 1980s, which have become increasingly influential in many areas, from condensed-matter physics and optics to high-energy and particle physics and fluid mechanics to gravity and cosmology5.

As for the spin degree of freedom, its coupling to magnetic and/or electric fields can lead to rich dynamics and corresponding spin quantum phases with significant consequences. One example relevant to this work is the Aharonov-Casher effect6, the electromagnetic dual of the AB effect, due to the spin coupling to electric fields, i.e., spin-orbit interaction (SOI). Here, we focus on the combined action of Rashba and Dresselhaus SOI in polygonal and circular circuits. In semiconducting systems, the former is due to the lack of structural inversion symmetry, whereas the latter is due to the lack of bulk inversion symmetry7.

To dig into the problem of spin dynamics in curved mesoscopic circuits, we frame the discussion in terms of field and spin textures. By field texture, we refer to the geometry displayed by the magnetic field that couples to the spin carrier in a loop circuit, the main characteristic of which is to be inhomogeneous in direction. These fields can be external (e.g., inhomogeneous magnetic fields interacting through Zeeman coupling) or internal (e.g., effective magnetic fields emerging from SOI in curved circuits). By spin texture, we refer to the geometric shape defined by the local quantization axis of spin eigenstates along the circuit, represented in the Bloch sphere. Both field and spin textures coincide in the so-called adiabatic limit, where spin eigenstates are locally aligned with the driving field. This is the limit in which Berry geometric phases are formulated2. However, reaching this limit may be either hard (due to the large fields required) or impossible (due to discontinuities in the field textures that spins cannot follow). Aharonov & Anandan8 generalized the concept of geometric phases to the case of non-adiabatic dynamics. For 1/2 spins, the geometric phase equals the solid angle subtended by the spin texture (times −1/2).

Several works study the correlation between field and spin textures in loop circuits. It is well established that for regular (discontinuity-free) field textures, the spin dynamics is determined by the relative magnitude of two characteristic frequencies: the Larmor frequency of spin precession ωs and the orbital frequency of carrier propagation ω09,10,11, where the adiabatic limit corresponds to ωs/ω0 ≫ 1. Some works have studied how this limit is approached12. By assuming fully adiabatic spin dynamics in circular circuits, Lyanda-Geller demonstrated13 that a topological transition in a flat field texture (from a circular field texture to an oscillating one) would lead to a discontinuity in the spin Berry phase (a sudden π shift) with observable consequences in electronic transport. Later works14,15 showed that this topological effect takes place far from the adiabatic limit.

Moreover, it has been acknowledged that the geometric curvature of a circuit can play a critical role in spin dynamics16,17,18,19,20. For example, in polygonal Rashba loops where effective field texture discontinuities at the highly curved vertices force the spin carriers to respond in a strongly non-adiabatic fashion16,21,22,23,24. This has been proven to have dramatic consequences for the correlation between field and spin textures: while field and spin textures are typically well correlated in circular Rashba loops (i.e., it usually takes a topological change in the driving field texture to produce a topological transition in the spin texture). For square Rashba loops, it has been shown that small perturbations in the field texture (created by an in-plane Zeeman field) can induce significant changes in the topological characteristics of the spin textures23,25,26. Here, we show how similar changes can be achieved by purely electrical means ∣ without introducing magnetic fields.

Our study focuses on square and ring loops suitable for experimental realization25,27,28. These experiments are realized on arrays of many interferometric loops where only one single (quasi-one-dimensional) orbital mode appears to contribute to quantum interference due to the decoherence experienced by relatively slow propagating higher modes. This justifies the use of strictly one-dimensional (1D) model circuits in this work.

In this manuscript, we study the development of spin textures and their response due to the combined action of Rashba and Dresselhaus SOI in polygonal circuits. The field textures produced by Rashba and Dresselhaus SOIs are contained within the circuit’s plane. Their topology depends on the SOI components’ relative magnitude, which can be controlled electrically in semiconducting nanostructures28,29,30,31,32. We find that, for specific circuit orientations, the spin textures respond with a regular pattern of topological transitions as a function of the SOIs without requiring a topological change of the driving field texture. This means that for any SOI setting, it is always possible to change the topological characteristics of the corresponding spin texture by shifting to a different setting in its vicinity. We also show how this manifests in the conductance of the circuits. There are several possible approaches to studying quantum transport in mesoscopic systems, such as the recursive Green’s function approach33 and the tight-binding method34. In this work, we address the problem by employing a quantum network (QN) technique16,35,36,37,38,39,40. We have recently generalized this QN technique to account for Abelian and non-Abelian gauge fields, including Rashba SOI and Zeeman fields23,24. Here, we incorporate Dresselhaus SOI and study its interplay with the Rashba SOI41.

Results

Model and formalism

We consider a 2DEG in the presence of Rashba and Dresselhaus SOIs42,43; both these terms are linear in momentum p (here, we neglect cubic corrections to Dresselhaus SOI41,43). Within this 2DEG, we realize single-mode 1D quantum wires along the \(\hat{{{{{{{{\boldsymbol{\gamma }}}}}}}}}\) direction with respect to the crystallographic axes of the 2DEG ∣ see Fig. 1a. The quantum wire Hamiltonian then reads:

$$\hat{{{{{{{{\mathcal{H}}}}}}}}}=\frac{{p}^{2}}{2{m}^{* }}+\frac{\hslash {k}_{{{{{{{{\rm{R}}}}}}}}}}{{m}^{* }}p\,(\hat{{{{{{{{\boldsymbol{\gamma }}}}}}}}}\times \hat{{{{{{{{\boldsymbol{z}}}}}}}}})\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}+\frac{\hslash {k}_{{{{{{{{\rm{D}}}}}}}}}}{{m}^{* }}p\,\hat{\overline{{{{{{{{\boldsymbol{\gamma }}}}}}}}}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}},$$
(1)

where kR and kD are the Rashba and the Dresselhaus SOI strengths (in inverse-length units), respectively, \(\hat{{{{{{{{\boldsymbol{\gamma }}}}}}}}}=(\cos \gamma ,\sin \gamma ,0)\) is the unit vector specifying the direction of the quantum wire, and \(\hat{\overline{{{{{{{{\boldsymbol{\gamma }}}}}}}}}}=(\cos \gamma ,-\sin \gamma ,0)\) is the y-reflected \(\hat{{{{{{{{\boldsymbol{\gamma }}}}}}}}}\) (see Supplementary note 1https://doi.org/10.1038/s42005-023-01308-8 for additional details). In Eq. (1), σ is the vector of the Pauli matrices associated with the electron spin, p is the momentum along the quantum wire, m* is the effective electron mass of the 2DEG and ℏ the reduced Planck constant.

Fig. 1: Effective spin-orbit interaction fields for wire, ring, and square geometry.
figure 1

a Sketch of a quantum wire along direction \(\hat{{{{{{{{\boldsymbol{\gamma }}}}}}}}}\) with its associated effective magnetic fields due to Rashba BR (red arrow) and Dresselhaus BD (blue arrow) SOIs. b Evolution of the effective magnetic field BSO = BR + BD for an electron moving counterclockwise around a ring for different values of the Rashba (kR) and Dresselhaus (kD) coupling strengths. Rashba (red) and Dresselhaus (blue) effective magnetic fields of a counterclockwise propagating spin carrier due to SOI for the case of the square circuit with c η = 0 and d η = π/4.

Full size image

The Rashba and Dresselhaus SOI terms in Hamiltonian (1) can be unified as

$$\hat{{{{{{{{\mathcal{H}}}}}}}}}=\frac{{p}^{2}}{2{m}^{* }}+\frac{\hslash \kappa }{{m}^{* }}p\,\hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}},$$
(2)

where

$$\kappa =\sqrt{{k}_{{{{{{{{\rm{R}}}}}}}}}^{2}+{k}_{{{{{{{{\rm{D}}}}}}}}}^{2}+2{k}_{{{{{{{{\rm{R}}}}}}}}}{k}_{{{{{{{{\rm{D}}}}}}}}}\sin (2\gamma )},$$
(3a)
$$\theta =\arg \left[({k}_{{{{{{{{\rm{R}}}}}}}}}\sin \gamma +{k}_{{{{{{{{\rm{D}}}}}}}}}\cos \gamma )+{{{{{{{\rm{i}}}}}}}}(-{k}_{{{{{{{{\rm{R}}}}}}}}}\cos \gamma -{k}_{{{{{{{{\rm{D}}}}}}}}}\sin \gamma )\right].$$
(3b)

Hamiltonian (2) describes an equivalent system where an electron moves along a quantum wire subject to an effective SOI with strength κ. This SOI term can be interpreted as an effective magnetic field \({{{{{{{{\boldsymbol{B}}}}}}}}}_{{{{{{{{\rm{SO}}}}}}}}}=2\hslash \kappa p/(g{{{{{\mathrm{\mu}}}}}} {m}^{* })\hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}\), where g is the g-factor, μ is the Bohr magneton and \(\hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}=(\cos \theta ,\sin \theta ,0)\). In Fig. 1b, we show the effective magnetic field texture experienced by spin carriers following circular trajectories for various values of the Rashba and Dresselhaus SOIs. The arrows indicate the evolution of BSO when an electron moves counterclockwise (CCW) in a circular trajectory. The field texture can be characterized topologically in terms of the winding number ω around the z axis ∣ see Eq. (5). The winding number changes depending on the relative strength between the Rashba and Dresselhaus SOIs; i.e., it is ω = 1 for kR > kD and ω = − 1 for kR < kD. This change of the winding at the critical line kR = kD is reflected in the spin texture of the polygon eigenstates, but as shown in the Section “Topological characterization”, the spin textures develop out-of-plane components that lead to richer winding patterns. In Fig. 1c, d, we present the SOI field texture for the two different orientations of the square circuit. In polygonal structures, the effective field exhibits discontinuities at the vertices, but such sharp changes of direction are smoothened in realistic setups where the circuits are realized by lithographic etching of a 2DEG. If the vertices of the square are treated as slightly rounded arcs24, the evolution of the field texture on B space is equivalent to that of the ring (see Fig. 1b).

The QN problem is solved by fixing a wave function for each 1D quantum wire satisfying the Dirichlet boundary conditions. The overall solution is obtained by imposing the general boundary condition on the set of wires composing the QN ∣ see below. The wave function of a quantum wire can be written as38,39:

$${{{{{{{\boldsymbol{\Psi }}}}}}}}(r)=\frac{{{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}\kappa r\hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}}}{\sin (kL)}[\sin k(L-r){{{{{{{{\boldsymbol{\Psi }}}}}}}}}_{\alpha }+\sin (kr){{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}\kappa L\hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}}{{{{{{{{\boldsymbol{\Psi }}}}}}}}}_{\beta }],$$
(4)

where Ψα and Ψβ are the spinors evaluated at the quantum wire boundaries α and β, \(k=\sqrt{2{m}^{* }\epsilon /{\hslash }^{2}+{\kappa }^{2}}\), r is the coordinate along the wire, L is the length of the wire and ϵ is the energy. The exponent in the prefactor of Eq. (4) accounts for the spin precession due to the effective magnetic field created by the two SOIs. When the Rashba and Dresselhaus SOI strengths are equal, and γ = 3π/4, both SOI terms cancel in the Hamiltonian, the effective magnetic field vanishes, and the energy dispersion turns spin degenerate44,45. In this case, κ vanishes, so the SU(2) rotation disappears from the wave function. This means that the spinor evolves along the wire as a free particle.

The spin-carrier dynamics in a QN can be solved by considering two general boundary conditions: first, the continuity of the wave function to each node of the QN, and second by requiring the conservation of the probability current at the same points36.

Equation (4) together with the conservation of the current at the vertices provides the values of the wave function at the vertices Ψα, and Ψ(r) by extension.

We can evaluate the transport properties by supplementing the QN with an additional connection to external leads. The extension of the method is explained in the Methods section.

Conductance pattern

In this section, we study the transport properties of square and ring loops by applying the QN formalism. Rings are modeled as regular polygons of perimeter P with a large number of edges (N ≫ 1)16 such that the Fermi wavelength and the spin precession length of the carriers are much larger than the edges’ length L = P/N. This overcomes the problem of dealing with approximate solutions for rings subject to Rashba and Dresselhaus SOI46,47. These restrictions do not apply to square loops. Moreover, mesoscopic experiments are typically carried out in the so-called semiclassical regime where the Fermi wavelength of the carriers is much smaller than the loops’ perimeter, such that k ≫ 2π/P25,26,27,48. We calculate the ballistic conductance of these polygonal structures by using the Landauer approach (See Supplemental Material at https://doi.org/10.1038/s42005-023-01308-8 for additional details)49 [see Eq. (9)].

In Fig. 2, we show the conductance maps as a function of the dimensionless SOI intensities kRP/2π and kDP/2π for different interferometric loops. Figure 2a, b shows the conductance for the case of square-shaped loops with different orientations, with η the rotation angle measured from the “diamond” configuration depicted in Fig. 2a (corresponding to the square’s sides forming an angle of π/4 with respect to the crystallographic axes). In Fig. 2c, instead, we show the conductance corresponding to a ring-shaped loop. For all configurations, the conductance shows a symmetric behavior with respect to the critical line kR = kD along which the Rashba and Dresselhaus SOIs have the same strength. This can be understood in terms of the Hamiltonian in Eq. (2). If the strengths of the interactions are interchanged, kR ↔ kD, the value of the equivalent effective SOI strength κ for a given edge remains unchanged, but the direction of the effective magnetic field BSO becomes \({\theta }^{{\prime} }=3\pi /2-\theta\). This means that the BSO field texture of the polygon is mirrored with respect to the same \({\hat{r}}_{-}=(\hat{x}-\hat{y})/\sqrt{2}\) direction for all edges, so the conductance pattern remains unchanged under the kR ↔ kD transformation.

Fig. 2: Conductance for the square and the ring geometries.
figure 2

Conductance G in units of 2e2/h as a function of the dimensionless Rashba and Dresselhaus spin-orbit interaction for a square for different orientations a η = 0, b η = π/4 and a c ring with perimeter P. In all the panels, the dashed line corresponds to the critical line, where kR = kD. The left (right) dot in the sketches of the polygonal structures indicates the position of the input (output) lead.

Full size image

The most interesting case is presented in the case of Fig. 2a: the conductance presents a checkerboard pattern. In the absence of Dresselhaus SOI, the minima of the conductance are presented every kRP/2π = 2n + 1 with \(n\in {{\mathbb{N}}}_{0}\)16,23,24. A similar behavior is observed in the absence of Rashba SOI with kR replaced by kD. The combined presence of the two SOIs adds an overall shift of the conductance minima by a factor of 2π. The resulting checkerboard pattern was first reported in ref. 40. In the next section, we elaborate on this by studying the geometric properties of spinors in terms of winding numbers.

By rotating the square with respect to the crystallographic axes (Fig. 2b) we find something remarkable: the checkerboard conductance pattern disappears. Instead, we find a conductance map that looks similar to that of a ring-shaped loop (Fig. 2c) except for a period-doubling due to strongly non-adiabatic processes at the square vertices hindering spin-phase development16,50.

From Fig. 1c, d, we observe that the SOI field textures present discontinuities at the vertices of the squares. For η = 0, Fig. 1c, the discontinuities have an angle π/2 for any relative field strength (except when Rashba and Dresselhaus SOIs are equal and a persistent spin helix is set up). As η increases, field discontinuities are softened, spin scattering is discouraged, and destructive spin interference is suppressed: optimal π/2 field texture discontinuities at the vertices persist only along the Rashba and Dresselhaus axes in Fig. 2b, where a definite interference pattern in the conductance survives as discussed in refs. 23,24,39 for the Rashba case. In Supplementary Note 2 (See Supplemental Material at https://doi.org/10.1038/s42005-023-01308-8 for additional details), we show how the conductance and the winding number of the propagating spin modes evolve from the checkerboard pattern in Fig. 3a, b to the simpler structure in Fig. 3c, d upon changing the orientation of the square.

Fig. 3: Comparison of the winding number and conductance for the square and the ring geometries.
figure 3

Winding number ω (left column) and winding number overlapped with the conductance in units of 2e2/h (right column) for a square (a, b), a square rotated π/4 from the initial configuration (c, d), and a ring (e, f) subject to Rashba and Dresselhaus spin-orbit interaction. The left (right) dot in the sketches of the polygonal structures in b, d, and f, indicates the position of the input (output) lead. The insets in e represent zoom in the anomalous winding in the critical line for better visualization of its structure.

Full size image

For the case of a ring, Fig. 2c, the conductance presents a fishbone structure with minima as a function of the Rashba SOI following the sequence predicted theoretically in ref. 50. Similar behavior is observed as a function of the Dresselhaus SOI term. However, in the presence of both SOI terms, the conductance behavior is more intricate.

Interestingly, the conductance remains constant along the critical line regardless of the orientation or number of edges of the polygon [see Supplementary note 3 (See Supplemental Material at https://doi.org/10.1038/s42005-023-01308-8 for additional details) for the additional case of a hexagonal and octagonal loop]. When the Rashba and Dresselhaus SOIs have the same strength, the effective SOI field decouples from the momentum and points always in the same direction44,45. Moreover, the spin precession angle only depends on the distance traveled along the \({\hat{r}}_{+}=(\hat{x}+\hat{y})/\sqrt{2}\) direction. This effect is known as persistent spin helix51, and in recent years several experiments claimed to have achieved this effect32,52,53,54,55. Since all paths contributing to the transmission amplitude begin and end at the same points, the spin precession along each path is the same, and the resulting interference is always constructive.

Topological characterization

We can characterize spin and field textures topologically in terms of (integer) winding numbers around the z axis. This quantity is defined as

$$\omega =\frac{1}{2\pi }\int\nolimits_{0}^{P}d\ell \left(\widehat{{{{{{{{\boldsymbol{n}}}}}}}}}\times \frac{d\hat{{{{{{{{\boldsymbol{n}}}}}}}}}}{d\ell }\right)\cdot \hat{{{{{{{{\boldsymbol{z}}}}}}}}}$$
(5)

with \(\hat{{{{{{{{\boldsymbol{n}}}}}}}}}(\ell )\) a in-plane unit vector and 0 ≤ ℓ ≤ P a linear parametrization of the circuit’s perimeter. For the field texture, we identify \(\hat{{{{{{{{\boldsymbol{n}}}}}}}}}(\ell )\) with \(\hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}(\ell )\) in Eq. (2). This means that ω = 1 for dominating Rashba SOI and ω = −1 for dominating Dresselhaus SOI (see Fig. 1b). The transition occurs at the critical line kR = kD. As for the spin texture, this is given by \(\hat{s}(\ell )=\langle \Psi (\ell )| {{{{{{{\boldsymbol{\sigma }}}}}}}}| \Psi (\ell )\rangle\). In this case, \(\hat{{{{{{{{\boldsymbol{n}}}}}}}}}\) is identified with the normalized projection of \(\hat{s}(\ell )\) on the xy-plane.

Spin textures developed in Rashba and Dresselhaus squares present a periodic, checkerboard-like pattern alternating positive and negative windings ∣ see Fig. 3a. This pattern contrasts with the simplicity of the field texture driving the spin dynamics discussed in Section “Model and Formalism”, demonstrating the possibility of producing topological transitions in the spin texture by slightly tuning the SOI fields. This means that the spin winding can change from clockwise (CW) to CCW and vice versa without changing the winding of the field, except for the particular orientation of η = π/4, Fig. 3c, where the field and spin textures stay fully correlated. Figure 3b shows that the winding pattern is fully correlated with the conductance up to a period-doubling.

In Fig. 4A, we present a series of spin textures undergoing a topological transition along segment A in Fig. 3a, corresponding to a square loop. We find that a positive winding texture becomes negative by collapsing into a flat texture (subtending no solid angle and no geometric phase) at the critical line24.

Fig. 4: Spin texture for the square and the ring geometries.
figure 4

The spin texture of a propagating mode in the Bloch sphere (up) and its azimuthal projection (down) for different spin-orbit interaction strengths. The texture is evaluated for three values of the winding number (ω), covered by the arrows A–D in Fig. 3, with corresponding strengths of SOI taken from the back of the arrow to the tip. In each panel, the color indicates the circulation of the local spin states as the carrier propagates along the perimeter from red to violet.

Full size image

In the case of ring loops, Fig. 3e, f, we find that the spin textures have a dominant tendency to follow the driving field texture by sharing its topological characteristics. Still, we find a fishbone pattern of anomalous winding ∣ see insets in Fig. 3e, where we observe that the winding gets values larger than ±1. To shed some light on it, it is useful to assume a dominating Rashba SOI with kR ≫ kD and work in the rotating frame where the radial Rashba texture is uniform (with an oscillating Dresselhaus perturbation). We notice that the fishbone pattern meets the Rashba axis at points \({k}_{{{{{{{{\rm{D}}}}}}}}}P=\pi \sqrt{4{j}^{2}-1}\) with j integer. This coincides with the Rabi condition for spin resonance in the rotating frame. As the magnitude of Dresselhaus perturbation increases, the resonance condition changes by undergoing a so-called Bloch-Siegert shift15,56. Something similar happens along the Dresselhaus axis. Close to the resonance condition, complex spin textures emerge with anomalous winding. Figure 4B–D illustrates the winding transitions taking place in these textures. We notice that, in contrast to the case of square loops, a winding transition does not require a full collapse of the spin texture with vanishing geometric phases. Still, in both square and ring geometries, the spin winding is antisymmetric with respect to the critical line along which the driving field changes topology.

Conclusions and outlook

We demonstrate how the geometry of SOI circuits can be used to manipulate the carriers’ spin state. Effective SOI field textures are built by introducing circuit sections of different curvatures steering the carriers’ momentum. In this way, highly curved vertices in polygon circuits act as effective spin-scattering centers for the carriers. This can be achieved by purely electrical means (without introducing magnetic fields that break time-reversal symmetry), in contrast to other proposals23,25.

For square circuits subject to Rashba and Dresselhaus SOI, we find that the topological characteristics of the spin textures can be manipulated with relative ease by electric control of the SOIs in semiconducting nanostructures28,29,30,31,32. This contrasts with the case of ring circuits where, as a general rule, a topological change in the field texture is required to induce a corresponding change in the spin textures. Still, this restraint can be overcome by tuning the SOI to satisfy the spin resonance conditions where complex spin textures develop. Moreover, additional possibilities for spin control appear by in-plane rotation of square circuits with respect to the crystallographic axes.

Remarkably, these topological features leave an imprint on the quantum conductance of the circuits, which can be addressed experimentally. We find a correlation between the spin-dependent conductance and a winding number associated with the propagating spin modes. This demonstrates that conventional conductance measurements can reveal the geometrical properties of the spin-carrier states.

Methods

Formalism for quantum transport

Here we present the QN formalism used to study the transport properties of polygonal QNs, Fig. 5. Semi-infinite input and output leads are attached to the network’s vertices for the transport measurements. Each lead consists of a quantum wire with two spin channels. The leads are not subjected to any interaction, so they are characterized at zero temperature by the Fermi energy and a wave vector k. We assume that the channels behave like incoherent sources, so there is no phase relationship between the different input channels57.

Fig. 5: Various interferometric geometries.
figure 5

Sketch of the polygonal structures considered for the quantum transport: a square, b hexagon, c octagon, d ring.

Full size image

In a system with Nin (Nout) input (output) channels, if an electron is injected through input channel σ with wavenumber k, the wave function alongside the channels can be written as

$${\Psi }_{{{{{{{{\rm{in}}}}}}}},{\sigma }^{{\prime} }}(r)={{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}kr}{\delta }_{{\sigma }^{{\prime} }\sigma }+{r}_{{\sigma }^{{\prime} }\sigma }{{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}kr},$$
(6a)
$${\Psi }_{{{{{{{{\rm{out}}}}}}}},{\sigma }^{{\prime} }}(r)={t}_{{\sigma }^{{\prime} }\sigma }{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}kr},$$
(6b)

where r is the position measured from the edge, and it is negative for input leads and positive for output leads. Here \({r}_{{\sigma }^{{\prime} }\sigma }\) and \({t}_{{\sigma }^{{\prime} }\sigma }\) are the channel-resolved reflection and transmission coefficients, respectively, so that \(\mathop{\sum }\nolimits_{{\sigma }^{{\prime} }}^{{N}_{{{{{{{{\rm{in}}}}}}}}}}{\left\vert {r}_{{\sigma }^{{\prime} }\sigma }\right\vert }^{2}+\mathop{\sum }\nolimits_{{\sigma }^{{\prime} }}^{{N}_{{{{{{{{\rm{out}}}}}}}}}}{\left\vert {t}_{{\sigma }^{{\prime} }\sigma }\right\vert }^{2}=1\). The indices σ and \({\sigma }^{{\prime} }\) specify the lead and the spin state of the channel. We define the total transmission and reflection coefficients of a channel σ as

$${T}_{\sigma }=\mathop{\sum}\limits_{{\sigma }^{{\prime} }}{\left\vert {t}_{\sigma {\sigma }^{{\prime} }}\right\vert }^{2},\qquad {R}_{\sigma }=\mathop{\sum}\limits_{{\sigma }^{{\prime} }}{\left\vert {r}_{\sigma {\sigma }^{{\prime} }}\right\vert }^{2}.$$
(7)

where the sum runs over the input channels. The total transmission (reflection) is given by the sum of the transmission (reflection) coefficients of the output (input) channels,

$$T=\mathop{\sum}\limits_{\sigma }{T}_{\sigma }=\mathop{\sum}\limits_{\sigma {\sigma }^{{\prime} }}{\left\vert {t}_{\sigma {\sigma }^{{\prime} }}\right\vert }^{2}$$
(8a)
$$R=\mathop{\sum}\limits_{\sigma }{R}_{\sigma }=\mathop{\sum}\limits_{\sigma {\sigma }^{{\prime} }}{\left\vert {r}_{\sigma {\sigma }^{{\prime} }}\right\vert }^{2}$$
(8b)

The zero temperature conductance G based on the Landauer formula reads33:

$$G=\frac{{e}^{2}}{h}{{{{{{{\rm{Tr}}}}}}}}\,\left[t{t}^{{{{\dagger}}} }\right]=\frac{{e}^{2}}{h}T.$$
(9)

Equation (9) sets an upper limit for the conductance, which is bounded by the number of input channels, such that G ≤ Nine2/h.

The wave function of the QN satisfies boundary conditions at the vertices, which ensure the continuity (uniqueness) of the wave function and the conservation of the probability current. In an isolated QN, imposing the continuity of the wave function and conserving the probability current yields a set of linear homogeneous equations where the variables are the values of the wave function at the vertices. This allows us to study the spectral properties of the QN. When adding the external leads, the system’s energy is fixed by the Fermi energy of the leads. Due to the first term in Eq. (6a), the set of equations becomes inhomogeneous, with a unique solution for T and R.

In a system with Rashba and Dresselhaus SOIs, the wave function of a wire is described by the values it takes at the nodes Ψα [see Eq. (4)]. The single-valuedness of the wave function at the nodes is automatically satisfied by this equation. In addition, imposing the continuity of the wave function at the vertices connected to external leads allows writing the reflection and transmission coefficients of the leads in terms of Ψα. Therefore, the number of unknown variables equals the number of vertices V. The conservation of probability current at the nodes allows one to write V equations, which fix the values of Ψα, and consequently the reflection/transmission coefficients. Notice that the presence of SOI modifies the definition of probability current58. This is accounted for by the extended derivative:

$$D=\frac{\partial }{\partial r}+{{{{{{{\rm{i}}}}}}}}{k}_{{{{{{{{\rm{R}}}}}}}}}(\hat{{{{{{{{\boldsymbol{\gamma }}}}}}}}}\times \hat{{{{{{{{\boldsymbol{z}}}}}}}}})\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}+{{{{{{{\rm{i}}}}}}}}{k}_{{{{{{{{\rm{D}}}}}}}}}\hat{\overline{{{{{{{{\boldsymbol{\gamma }}}}}}}}}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}=\frac{\partial }{\partial r}+{{{{{{{\rm{i}}}}}}}}\kappa \hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}.$$
(10)

The conservation of probability current at a node is given by the sum of the outgoing extended derivatives of the wave function, which must be equal to zero. For a generic node α, the continuity of probability current reads

$$\mathop{\sum}\limits_{\langle \alpha ,\beta \rangle }{\left.D{{{{{{{{\boldsymbol{\Psi }}}}}}}}}_{\alpha ,\beta }(r)\right\vert }_{r = 0}=0,$$
(11)

where the sum ∑〈α, β〉 runs over all nodes β which are connected to α. This equation can be rewritten in terms of Ψα and Ψβ. For internal nodes, it reads

$${{{{{{{{\boldsymbol{M}}}}}}}}}_{\alpha ,\alpha }{{{{{{{{\boldsymbol{\Psi }}}}}}}}}_{\alpha }+\mathop{\sum}\limits_{\langle \alpha ,\beta \rangle }{{{{{{{{\boldsymbol{M}}}}}}}}}_{\alpha ,\beta }{{{{{{{{\boldsymbol{\Psi }}}}}}}}}_{\beta }=0,$$
(12)

where

$${{{{{{{{\boldsymbol{M}}}}}}}}}_{\alpha ,\alpha }=\mathop{\sum}\limits_{\langle \alpha ,\beta \rangle }\frac{{k}_{\beta ,\alpha }}{\tan {k}_{\beta ,\alpha }L}$$
(13a)
$${{{{{{{{\boldsymbol{M}}}}}}}}}_{\alpha ,\beta }=-\frac{{k}_{\beta ,\alpha }}{\sin {k}_{\beta ,\alpha }L}{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\kappa }_{\beta ,\alpha }L{\hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{\beta ,\alpha }\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}}.$$
(13b)

Here κβ,α and \({\hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{\beta ,\alpha }\) indicate the strength and direction of BSO for an electron traveling from vertex α towards vertex β, see Eq. (3).

Computation of the winding number

The model used to compute the winding number was built upon the one used in24 where we consider a regular polygon of N conducting sides of length L = P/N with P being the perimeter, which lies on the xy-plane. Each side connects the vertices α and β and it is oriented along directions \({\hat{\gamma }}_{\beta ,\alpha }\) (from α to β). The spin-carrier dynamics along each side are determined by Hamiltonian (2). The SOI terms can be interpreted as an effective in-plane magnetic field \({{{{{{{{\boldsymbol{B}}}}}}}}}_{{{{{{{{\rm{SO}}}}}}}}}=2\hslash \kappa p/(g{{{{{\mathrm{\mu}}}}}} {m}^{* })\hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}\) (see Eq. 3) coupled to the itinerant spins.

The solutions of the Schrödinger equation are plane waves propagating along each side from vertex α towards vertex β as

$$\left\vert \psi (r)\right\rangle ={{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}{k}_{{{{{{{{\rm{F}}}}}}}}}r}{{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}\kappa r{\hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{\beta ,\alpha }\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}}\left\vert \psi (0)\right\rangle ,$$
(14)

with kF the Fermi wavenumber. The first prefactor corresponds to the kinetic phase of the carrier associated with the dynamics of charged particles, while the second prefactor represents the spin phase due to spin precession. The propagation of a spin carrier from α to β is then fully determined by the phases \({k}_{{{{{{{{\rm{F}}}}}}}}}L+\kappa L{\hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{\beta ,\alpha }\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}\), and the spin evolution along one side is determined by the momentum-independent spin rotation operator:

$${R}_{\beta ,\alpha }=\exp \left[-{{{{{{{\rm{i}}}}}}}}\kappa L{\hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{\beta ,\alpha }\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}\right]$$
(15)

with \({R}_{\beta ,\alpha }^{{{{\dagger}}} }={R}_{\alpha ,\beta }\) due to time-reversal symmetry.

The full spin evolution along CCW and CW propagating paths from vertex 1 to vertex N is then given by the unitary operators

$${U}_{+}(N)={R}_{1N}\ldots {R}_{32}{R}_{21}$$
(16)

and

$${U}_{-}(N)={R}_{12}\ldots {R}_{N-1,N}{R}_{N1}$$
(17)

with \({U}_{-}(N)={U}_{+}^{{{{\dagger}}} }(N)\), see Fig. 6.

Fig. 6: Rashba and Dresselhaus spin-orbit interaction fields on a dodecagon geometry.
figure 6

One-dimensional polygon model with its associated effective magnetic fields due to Rashba BR (red arrows) and Dresselhaus BD (blue arrows) SOIs. It shows the effective fields corresponding to counterclockwise propagating spin carriers.

Full size image

The spin rotation operator allows us to obtain the xy-projection of the spin texture as \({\hat{{{{{{{{\bf{s}}}}}}}}}}_{x,y}(r)=\left\langle {\chi }_{s}\right\vert {{{{{{{\boldsymbol{{\sigma }}}}}}}_{x,y}}}\left\vert {\chi }_{s}\right\rangle\), then it is possible to compute the angle accumulated around the z axis by the itinerant spin state as the carrier propagates along each segment as the phase of the complex number sx(r) + isy(r).

Completing a CCW round trip, we obtain the accumulated angle around the z axis and, therefore the winding number.