Geometric spin dephasing on a ring

Zero temperature geometric spin dephasing on a ring in presence of an Ohmic environment


We study zero temperature spin dynamics of a particle confined to a ring in presence of spin orbit coupling and Ohmic electromagnetic fluctuations. We show that the dynamics of the angular position are decoupled from the spin dynamics and that the latter is mapped to certain correlations of a spinless particle. We find that the spin correlations in the direction (perpendicular to the ring) are finite at long times, i.e. do not dephase. The parallel (in plane) components for spin do not dephase at weak dissipation but they probably decay as a power law with time at strong dissipation.


Spin relaxation and scattering Spin-orbit coupling, Zeeman and Stark splitting, Jahn-Teller effect Phases: geometric; dynamic or topological

1 Introduction

Due to recent advances in semiconductor technology, it became possible to isolate and manipulate spins of individual electrons [1, 2]. For efficient spin manipulation, however, slow spin decay is needed. Spin decay in mesoscpopic devices is generated by two major sources: Hyperfine interaction with nuclear spins [3] is responsible for spin decay in most materials. However, spin-orbit (SO) coupling can also induce spin relaxation, and under certain conditions, phonon-[4] or electromagnetic field-induced SO relaxation [7] can dominate the decay[5, 6]. As shown in Ref. [7] two-photon (or two-phonon) processes lead to geometrical spin relaxation even in the absence of external field and, as pointed out recently, this mechanism can become even dominant in hole-doped systems [8, 9].

Here we make an attempt to understand, whether the above-mentioned geometrical spin relaxation can survive even at temperature. Although Ohmic electromagnetic fluctuations were found to lead to a vanishing spin relaxation rate at  [7], the results of Ref. [7] are not conclusive, since they allow for non-exponential relaxation, common in Ohmic systems. To address this issue more rigorously, we consider a ring geometry. Studying a ring is, however, not of pure theoretical interest; high quality semiconductor rings [10, 11] can in fact also be used as quantum spin qubits [11], and usefulness of these devices depends on spin dephasing, a topic under active experimental study [5, 6, 8, 12, 13].

There are two types of spin-orbit coupling in two-dimensional electron systems: the Rashba interaction induced, e.g., by an electric field perpendicular to a two-dimensional (2D) layer  [14], and the Dresselhaus coupling induced by bulk inversion asymmetry [15]. Our aim is to study, how these couplings influence spin coherence for an electron confined to a ring, in the presence of Ohmic fluctuations. We shall first derive the appropriate Hamiltonian for a confined electron, and show that the presence of the spin does not influence the orbital motion of the confined electron, which is governed exclusively by fluctuations of the external electric field. The dynamics of the spin, on the other hand, is determined by the orbital motion of the electron, and has a topological character. We find that for weak dissipation the spin does not dephase, but certain spin components are reduced by fluctuations. For strong dissipation, however, we find that certain components of the spin probably relax even at temperature, due to the disordering of the orbital degrees of freedom [16]. The relaxation we find is, however, not exponential but of a power law, typical of confined particles at temperature  [17, 18].

2 Hamiltonian

Let us start by projecting a 2D spin-orbit Hamiltonian on a ring, a procedure which is not entirely trivial [19, 20]. In addition to the kinetic terms, the 2D Hamiltonian consists of a potential that confines the particle to a ring of radius , with . We write the total Hamiltonian in polar coordinates as where


Here , are spin operators, is the electron mass and and denote the and components of the momentum. The term is the Rashba coupling while denotes the Dresselhaus coupling. Labeling the radial eigenstates of by and their energies by , our aim is to project on the subspace, , while keeping terms up to order , a procedure that involves some subtleties. First we rewrite by introducing and ,


As noticed by Meijer et al. [19], for any state that is radially localized near , one has . Therefore, to first order in the SO coupling, does not give a contribution to the projected Hamiltonian. Nevertheless, as previously overlooked, cannot be ignored: localization on a scale implies and hence 2nd order perturbations in do give a contribution, , since . The next order contributions scale as , and vanish in the limit , similar to all higher order terms in the perturbation series.

Perturbation theory to 2nd order yields therefore the projected spin and angle dependent effective Hamiltonian


The sum in Eq. (3) can be evaluated analytically by making use of a sum rule [22, 23], and the second term of Eq. (3) simply becomes .

Introducing the vector via , and with the dimensionless Rashba and Dresselhaus couplings defined as and , we can finally rewrite our effective ring Hamiltonian in the limit as


We remark, in particular, that the term in the effective Hamiltonian of Ref. [20] is exactly canceled by the 2nd order terms. As a consequence, Eq. (4) possesses a conserved ”momentum”, .

3 Spectrum

The eigenstates and eigenenergies of (4) can be analytically computed for , when the system is rotationally invariant and therefore is also conserved. The Hamiltonian can then be written as


with a unit vector. The energy spectrum and the eigenvalues can then easily be found by constructing common eigenstates of the two commuting operators, and . For , and have eigenvalues and , respectively, with and an integer. The spectrum is , and the eigenstates are of the form , with denoting spin coherent states, defined through the usual relation, [24]. The wave functions can be explicitly expressed as


with defined as . The states are related by time reversal, and their energies equal, . For the ground state has .1

4 Dissipation

Having understood the properties of an isolated ring, we now couple the motion of the particle to the coordinate of a dissipative environment, i.e., we consider the total Hamiltonian as . Throughout most of this paper we shall assume that describes the coupling to a Caldeira-Leggett (CL) environment, appropriate for small rings in an Ohmic (metallic) environment.2 Then represents the random force generated by the environment, , and the Fourier transform of the environment correlations is , with the dimensionless friction coefficient.

Figure 1: Evolution of the spin (coherent state) while the electron makes a circle, . The initial state is rotated around an axis by an angle .

The corresponding equations of motion for are


Hence, as a consequence of the simple form of , Eq. (4), the dynamics of in the dissipative environment are not affected by the spin-orbit couplings. This decoupling allows us to describe the evolution by a path integral, where for each trajectory the spin dynamics follow from Eq. (4)


Viewing as a ”time” variable, these dynamics correspond to a spin precession around a ”time” dependent magnetic field . Note that switching to ”Schrödinger” picture, the spin coherent states have a simple evolution, too. Apart from a phase, they evolve as , where is the vector solution of (8), i.e. . In particular, the vector can also be shown to satisfy this equation.

In terms of the spin operators, Eq. (8) is solved as a simple linear mapping , with a rotation matrix. The rotation matrix , corresponding to the particle going once around the ring by , is of special interest. We denote by the unit vector its axis of rotation and by the corresponding rotation angle (see Fig. 1). In particular, for we find that the angle is independent of the initial value, , and is typically incommensurate with .

5 Mapping to a spinless system

For a given evolution, , we can obtain the evolution of the spin part of the wave function from Eq. (8), which is described by a unitary operator, . Here the Hamiltonian to describe the (”time”) evolution of the spin is . We proceed to study the case . Then, as in the standard NMR rotating field problem, the spinor transformation to the ”rotating frame” cancels the ”time” () dependence, and amounts in replacing . For this leads to the evolution operator, . Using now the expression of we find that the evolution of the spin states has a particularly simple form


where denote the eigenvalues of the momentum . After a rotation the state picks up an incommensurate phase, . Note that the semiclassical evolution involves a similar incommensurate angle, , as discussed below Eq. (8) (see also Fig. 1).

Making use of the decoupling of orbital and spin degrees of freedom, we can construct a mixed path integral formalism (to be detailed in Ref. [23]), where the spin is treated in an evolution operator formalism, while the orbital motion of the particle is developed in a path integral formalism. The full evolution for a given environment history is then obtained as:


Note that in this equation is a non-compact variable, and an additional integration over the environment configurations has to be carried out in the end. Importantly, the action describes a particle on the ring in the presence of dissipation for a given environment history, and is independent of the spin evolution.

For we can make use of Eq. (9) and obtain a particularly simple path integral representation for the spin evolution. Consider spin correlations with an initial density matrix built from one of the two Kramers degenerate ground states of and , having momenta with . Using Eqs. (10) and (9) for the forward and backward spin evolutions, we find an exact mapping of the spin correlations onto a superposition of equilibrium correlations of spinless particles on a ring with a flux (in units of quantum flux):


Here again, is a non-compact variable within to be used within the path integral representation of the spinless problem. Note that the bath still couples to hence we expect that (11) depends only on the noninteger part of . For we obtain the following identity for an initial density matrix, ,


For the same result holds with all subscripts of reversing sign. For the correlations, on the other hand, we obtain


and for the same holds with . Notice that the degeneracy point, , corresponding to fluxes represents a special case, and is not studied here.

While the correlation function of the -component of the spin, , obviously contains a constant non-decaying piece, the correlation function contains only phase correlation functions with . There is some evidence that these correlations decay in time. In particular from the XY lattice model [16] and from small expansion [25]. Correlations with incommensurate were studied in a related system of dissipative Josephson junctions [27], and found to decay algebraically. Further evidence is for large , as discussed below. To further appreciate these correlations we have evaluated the path integrals in (11) analytically for , and surprisingly, we find . As a consequence, for the correlation function contains a piece which does not oscillate. As discussed below, though reduced, this part seems to survive for very weak dissipation, , while it apparently decays algebraically for strong dissipation, .

6 Strong dissipation limit

In the strong dissipation limit, , we can describe the evolution of the phase through a Langevin equation, and an expansion in is possible [17, 18]. In this limit, all correlation functions with are found to decay algebraically. Large perturbation theory yields that and the -component of the spin also decays algebraically, while the -component remains finite and does not decay. This result, however, holds only for up to times beyond which effects of renormalization of cannot be neglected. In a recent work [18] we have shown that in presence of a weak DC electric field there is a critical such that flows to which would indicate . In some sense the fluctuating spin corresponds to a time dependent flux, i.e. an electric field, though the correspondence is not precise.

7 Weak dissipation

The rather different behavior of and should already be manifest in the weak dissipation limit, where we can perform perturbation theory in the strength of the dissipation, . To do perturbation theory, we restrict ourselves to the case and , and use Abrikosov’s pseudofermion method to represent each spinor of (6) by a pseudofermion operator, . In this language the ring Hamiltonian becomes , while the interaction is expressed as


and standard field-theoretical methods can be used to evaluate physical quantities. A renormalization group analysis of the vertex function and the pseudofermions’ self-energy reveals that, although ultraviolet logarithmic divergencies appear in both quantities, they cancel and the dissipation parameter is in leading order, nevertheless, exactly marginal, and the mass of the particle remains also unrenormalized [23].

In this perturbative regime, fingerprints of a non-exponential spin decay should appear in the susceptibility, , which, in the absence of spin decay, should contain a Curie part. To compute , we first express the impurity spin operator in terms of pseudofermions as


with the matrix elements simply determined from the wave functions (6), as . The leading corrections to are shown in Fig. 2. Although all corrections shown contain logarithmic ultraviolet singularities, remarkably, all these singularities exactly cancel, and one finally obtains just a finite renormalization of the perpendicular Curie susceptibility

The prefactor originates from the coefficient of in Eq. (12), and accounts for -factor renormalization in the isolated ring. The correction , on the other hand, represents the environment-induced renormalization of the and components of the spin (g-factor). The above perturbative result and the survival of the Curie susceptibility indicates that the term decays to a reduced but non-zero value for small .

In contrast to the and components, the component of the susceptibility, , is found to remain unrenormalized by to leading order in the dissipation. These results imply that, for weak Ohmic dissipation, the only effect of dissipation is to slightly and anisotropically renormalize the -factor, but apart from that the spin behaves as a free spin, and does not decay.

Figure 2: Zero and leading order () corrections to the spin susceptibility. Continuous lines and wavy lines represent the pseudofermion and bosonic propagators, respectively, while the dots represent spin vertices. Logarithmic divergencies in the three diagrams above exactly cancel.

8 Case of

So far we discussed only the case . We show now that the system with both finite is equivalent to the Hamiltonian (14). Assume a state that is an eigenstate of . This state generates a ladder of states, , with integer by


Since , a sequence of time reversed states is also generated by the time reversal operator, : All these states are orthogonal since they correspond to different energy eigenvalues, and the environment couples the and states, exactly as for . The only difference is that , which is not known analytically, changes the factor in . Hence Eq. (14) is a correct representation also of the case.3

9 Conclusions

We derived the effective Hamiltonian of an electron confined to a ring within a 2-dimensional electron gas, in the presence of SO coupling, and subject to a dissipative environment. We have shown that the orbital motion of the particle decouples from the spin evolution, and correspondingly, spin decay has a geometric character [7]. For an Ohmic environment, we mapped the spin relaxation problem to that of a spinless particle on a ring pierced by a magnetic flux [Eqs. (12,13)]. We find that the component of the ground state spin is not affected by dissipation. The and in-plane spin components are, on the other hand, reduced by dissipation, but we find no dephasing for spin and weak dissipation. However, these components seem to dephase at large dissipation.

We should remark that these latter results are based on the assumption of Ohmic dissipation. The situation may, however, change for subohmic dissipation or noise, present in many systems. In this case, the decoupling of the spin and orbital motion and thus Eqs. (12,13) remain valid, however, for subohmic dissipation is a relevant perturbation, and even a small dissipation could possibly lead to the decay of the and spin components. This possibility, however, needs to be further explored.

We acknowledge useful discussions with B. Douçot, W. Zwerger and A. Zaikin. This research has been supported by the Hungarian Research Funds OTKA and NKTH under Grant Nos. K73361, TÁMOP-4.2.1/B-09/1/KMR- 2010-0002, and the EU-NKTH GEOMDISS project and by THE ISRAEL SCIENCE FOUNDATION (grant No. 1078/07).


  1. Here we used the phase convention of Ref. [24]. This construction can be generalized for larger spins.
  2. In a dirty metal environment, e.g., one needs the ring’s radius to be smaller than the mean free path.
  3. The matrix elements of the spin operators are nevertheless different, changing e.g. the overall coefficient in Eq. (15).


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