Semiclassical theory of ballistic transport through chaotic cavities with spin-orbit interaction

Semiclassical theory of ballistic transport through chaotic cavities with spin-orbit interaction

Jens Bolte*

Institut für Theoretische Physik, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany Daniel Waltner^†

Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany 共Received 20 April 2007; revised manuscript received 28 June 2007; published 16 August 2007兲 We investigate the influence of spin-orbit interaction on ballistic transport through chaotic cavities by using semiclassical methods. Our approach is based on the Landauer formalism关IBM J. Res. Dev. 1, 223 共1957兲; 32, 306 共1988兲兴and the Fisher-Lee relations 关Phys. Rev. B 23, 6851共1981兲兴, appropriately generalized to spin-orbit interaction, and a semiclassical representation of Green’s functions. We calculate conductance coef- ficients by exploiting ergodicity and mixing of suitably combined classical spin-orbit dynamics, and making use of the Sieber-Richter method关Phys. Scr., T T90, 128共2001兲; Phys. Rev. Lett. 89, 206801共2002兲兴and its most recent extensions. That way, we obtain weak antilocalization and confirm previous results obtained in the symplectic ensemble of random matrix theory.

DOI:10.1103/PhysRevB.76.075330 PACS number共s兲: 73.23.⫺b, 71.70.Ej, 72.15.Rn, 03.65.Sq

I. INTRODUCTION

Ballistic transport through chaotic cavities realized as quantum dots in semiconductor heterostructures has been a central issue in mesoscopic physics for many years. The uni- versal transport properties observed in this context can be described on a phenomenological level by random matrix theory¹ 共RMT兲. The same applies to disordered systems, where averages over impurities can be shown to be equiva- lent to random matrix averages. This not being possible for individual, clean cavities, theoretical explanations of the RMT connection have been provided making use of semi- classical methods, which are based on the Landauer formalism^2,3 and semiclassical representations of Green’s functions. This approach⁴ leads to questions that are closely analogous to problems arising in semiclassical explanations of universal spectral correlations in classically chaotic quan- tum systems. Recent progress in the latter context is based on the seminal work of Sieber and Richter⁵ and its extensions.^6–8This method has been adapted^9–12to be able to successfully explain conductance coefficients, including the effect of weak localization, i.e., a decrease of conductance at zero magnetic field. Further studies have been devoted to analyses of the universality of conductance fluctuations,^12,13 of shot noise,^12,14–16 and of general higher moments.¹⁷ 共For an overview see, e.g., Ref.12.兲

In the work mentioned, transport properties were consid- ered for ballistic, nonrelativistic electrons, neglecting their spin. In the emerging field of semiconductor based spin electronics¹⁸共spintronics兲, however, one requires an efficient control of the spin dynamics associated with electrons in nonmagnetic semiconductors. This purpose calls for studies of transport properties in systems with sufficiently strong spin-orbit couplings, i.e., where the dwell time is much larger than the spin relaxation time. In contrast to previous theories neglecting the spin, there one would expect appro- priate classical spin-orbit dynamics to produce weak antilo- calization, i.e., an enhancement of the conductance at zero magnetic field. This prediction is also obtained on the phe- nomenological level provided by RMT, where a half-integer

spin requires the symplectic, as opposed to the orthogonal, circular ensemble. On this ground, one expects universal conductance fluctuations and other transport properties also to be affected by the presence of spin-orbit interactions.^1,19A first semiclassical approach²⁰to these questions employs the semiclassical representation of the Green’s function in spin- orbit coupling systems derived in Ref.21and considers the first order of the semiclassical Sieber-Richter expansion. It, moreover, assumes a randomization of spin states, which is shown to be responsible for weak antilocalization.

In this paper, our goal is to extend the results of Ref.20to all orders of the Sieber-Richter expansion and to base the semiclassical estimates entirely on dynamical properties of suitably combined classical spin-orbit dynamics.²² These then replace the randomization hypothesis of spin states made in the analytic part of Ref. 20. In order to determine the spin contribution to transmission amplitudes, we closely follow an analogous calculation introduced in the context of semiclassical explanations of spectral correlations in quan- tum graphs with spin-orbit couplings.^23,24We also comment on shot noise and on the variance of conductance fluctua- tions.

As our model, we consider a two dimensional cavity with two straight, semi-infinite leads with hard walls. Apart from boundary reflections, particles with mass m, charge e, and spin s move freely within the leads and are subjected to a magnetic field and to spin-orbit interactions inside the cavity.

Although the relevant case of electrons enforces the spin to bes= 1 / 2, we deliberately allow for general spin s. Below, this will allow us to point out characteristic differences be- tween integer and half-integer spins. The Hamiltonian gov- erning the dynamics in the cavity reads

Hˆ = 1

2m

冋

^{pˆ −ecA共xˆ兲}

册

^{2+sˆ ·C共xˆ ,pˆ兲. 共1兲}

Here,Ais the vector potential for an external magnetic field andCcontains all couplings of the translational degrees of freedom to the spin operatorsˆ. This may be a Zeeman cou-

pling, or any type of spin-orbit interactions, including Rashba and Dresselhaus couplings. Moreover, in order to model the hard walls, we require Dirichlet conditions at the boundaries of the cavity and of the leads.

The paper is organized as follows: Section II is devoted to a generalization of the Landauer formalism and the Fisher- Lee relations to systems with spin-orbit interaction. Then, we present semiclassical representations ofS-matrix elements in that case. In Sec. III, we first introduce ergodicity and mixing conditions that include a classical spin-orbit interaction. This is followed by our calculation of the conductance in two ways: in the configuration-space and in the phase-space ap- proach. In Secs. IV and V, we then outline how our approach can be extended to calculate shot noise and conductance fluctuations, respectively. An appendix contains a calculation whose result is central to the phase-space approach employed in Sec. III.

II. PRELIMINARIES

We follow the usual approach to obtain semiclassical ap- proximations to transmission by employing the Landauer formalism^2,3 and introducing semiclassical representations for Green’s functions. In the absence of spin-orbit interac- tions, this procedure is well established.^25–27Here, we briefly describe the extensions required by the presence of spin-orbit interactions共see also Ref.20兲.

A. Landauer formalism with spin

The Landauer formalism provides a link between conduc- tance coefficients, as defined through

In=

兺

m

gnmVm, 共2兲

andS-matrix elements. In Eq.共2兲, the indices label the leads, Vm is the voltage applied at lead m, and In is the current through leadn. Here, the number of leads may be arbitrary.

AnS-matrix elementS_␣

n␣_m⬘

nm is defined as the transition ampli- tude between an asymptotic incoming state in the lead m, characterized by the collection␣m

⬘

of its quantum numbers, to an asymptotic outgoing state in the lead n, accordingly characterized by␣n.

In Refs.26and27, the Landauer formalism was derived from the Schrödinger equation in linear response theory, making use of an appropriate Kubo-Greenwood formula. We first remark that an inclusion of spin, interacting with the translational degrees of freedom via a Zeeman, Rashba, or Dresselhaus coupling, into this method causes no problems.

Although the current density is modified, its conservation in the form required for the Kubo-Greenwood expression of the conductivity to hold is indeed guaranteed. One then obtains for transmission共i.e.,m⫽n兲

gnm= −e² h

冕

0

⬁

dEf_␤

⬘

共E兲

兺

␣n,␣_m⬘ 兩S_␣

n␣_m⬘

nm 兩², 共3兲

and for reflection共i.e.,m=n兲

gnn=e² h

冕

0

⬁

dEf_␤

⬘

共E兲

冋

^{共2s+ 1兲Nn−␣}

^兺

^n,␣n⬘ 兩S_␣

n␣_n⬘

nn 兩²

册

^{. 共4兲}

Here,Nnis the number of open channels in the leadn共with- out spin degeneracy兲 at energy E, and f_␤共E兲 denotes the Fermi distribution function at inverse temperature ␤^{. Of} course, this requires the spin quantum number sto be half- integer.

In the next step,S-matrix elements have to be related to Green’s functions G共x,x

⬘

^,E兲. These satisfy the following equations:

再

^2m1

^冋

^{pˆ −ecA共xˆ兲}

^册

^{2+sˆ ·C共xˆ ,pˆ兲−E}

冎

^G共x,x

^⬘

^{,E兲=␦共x−x}

^⬘

^兲

共5兲 and

再

^2m1

^冋

^pˆ

^⬘

^+ecA共xˆ

^⬘

^兲

^册

^2−E

冎

^G共x,x

^⬘

^,E兲

+C^*共xˆ

⬘

^,pˆ

⬘

兲G共x,x

⬘

^,E兲sˆ =␦共x−x

⬘

兲. 共6兲 The unusual form of the second equation is dictated by the fact that G共x,x

⬘

^,E兲 is a Hermitian 共2s+ 1兲⫻共2s+ 1兲 matrix in spin space. In the following, we will always choose ad- vanced Green’s functions, fully characterized by Eqs.共5兲and 共6兲as well as the condition that, asymptotically in the leads, they contain only outgoing contributions.

As in the case without spin,²⁶ one can express the S-matrix elements in terms of the共advanced兲Green’s func- tion. To this end, one replaces the disorder potential U共x兲 occurring in Ref.26with the spin-orbit interaction term. Due to the Hermiticity of the coupling, one can then proceed as in Ref.26. Up to a global phase factor, form⫽n this yields

S_␣

n␣_m⬘

nm =2ប²

im

冑

W^kam^nkW^am⬘n

冕

0 W_n

dyn

冕

0 W_m

dy_m

⬘

^sin

冉

^anW^␲n^yn

冊

⫻sin

冉

^am

^⬘

^W␲m^ym

^⬘ 冊

^{G␴␴⬘共xn,xm}

^⬘

^{,E兲, 共7兲}

and form=n

S_␣

n␣_n⬘

nn =2ប² im

冑

^kanka n⬘ W_n

冕

0

W_n

dyn

冕

0 W_n

dy_n

⬘

sin

冉

^anW␲n^yn

冊

⫻sin

冉

^an

^⬘

^W␲n^yn

^⬘ 冊

^{G␴␴⬘共xn,xn}

^⬘

^{,E兲+␦␣n␣n⬘. 共8兲}

Here, we have introduced coordinatesxn=共xn,yn兲, wherexn

艌0 is a longitudinal, outward running coordinate in the lead nand 0艋y_n艋W_nis the corresponding transversal coordinate 共see also Fig. 1兲. The transversal quantum number is an

= 1 , . . . ,Nn with associated wave number ka_n

=

冑

2mE/ប²−a_n²␲^2/Wn2. The number Nn of open transversal channels then is the largest integer an that leaves the wave number real. Moreover,␴= −s, . . . ,sis a spin index such that altogether␣n=共E,an,␴兲.

We remark that in Eqs.共7兲and共8兲the pointsx_n,x_m

⬘

^{can be} chosen anywhere in the respective leads. For later conve- nience, we take them on the connection of the leads to the cavity, i.e., withxn= 0 =x_m

⬘

^.

B. Semiclassical Green’s function and transmission amplitudes In order to proceed further, one requires a semiclassical representation for the Green’s function defined in Eqs. 共5兲 and共6兲. In Ref.21, this was achieved through an asymptotic expansion in powers of Planck’s constantបfor the quantum propagator generated by the Hamiltonian共1兲which yielded, after a Fourier transformation, a respective semiclassical ex- pansion for the Green’s function. The range of validity of this procedure follows from the observation that since the spin operatorsˆ is linear in ប, the energy scale of the spin- orbit interaction term becomes small as compared to the ki- netic term in the limitប→0. This condition is equivalent to the spin-precession length being large compared to the Fermi wavelength. In semiconductor heterostructures, this require- ment is usually fulfilled.

The semiclassical representation for the Green’s function obtained in Ref.21reads

G共x,x

⬘

^,E兲 ⬃

兺

␥共x,x⬘^兲

A_␥共x,x

⬘

^,E兲exp„共i/ប兲S_␥共x,x

⬘

^,E兲…, 共9兲

as ប→0. The sum extends over all classical trajectories

␥共x,x

⬘

^兲 generated by the classical Hamiltonian

H₀共x,p兲= 1

2m

冋

^{p−ecA共x兲}

册

^{2 共10兲}

共plus reflections from hard walls兲 that run from x

⬘

^{to x at} energy E. Choosing 共x,x

⬘

^兲=共xn,xm

⬘

兲 as in Eqs.共7兲and 共8兲, the relevant trajectories are those that enter the cavity at lead m and leave through lead n. Moreover, S_␥共x,x

⬘

^{,E兲 is the} classical action of the trajectory, and the leading order of the amplitudeA_␥共x,x

⬘

^{,E兲 reads}

A_␥共x,x

⬘

,E兲= e^{−i共␲/2兲␯␥}

iប

冑

2␲iប

冑

C_␥D_␥共x

⬘

^,p

⬘

,t兲关1 +O共ប兲兴. 共11兲 Here,␯␥is a Maslov index of the trajectory␥^{, and}

C_␥ª

冨

^det

冢

^{⳵⳵⳵⳵xx22}

^⬘

^{⳵SS⳵x␥␥E}

^⬘

^{⳵⳵⳵⳵x22ES⳵SE2␥␥}

冣冨

^{. 共12兲}

The contribution of the spin is, in leading semiclassical or- der, completely contained in the spin-transport matrix D_␥共x

⬘

^,p

⬘

^,t兲. This is the spin-s representation of the spin propagatord_␥共x

⬘

^,p

⬘

^,t兲, which is defined as a solution of the equation

d

dtd_␥共x

⬘

^,p

⬘

^{,t兲+ i}

2C„X共t兲,P共t兲…·␴^d␥共x

⬘

^,p

⬘

^{,t兲= 0, 共13兲} with initial condition d_␥共x

⬘

^,p

⬘

^{, 0兲= 1. Here, (X共t兲,P共t兲) is} the point in phase space of the classical trajectory␥^{at timet.}

Its initial point at time t= 0 is 共x

⬘

^,p

⬘

兲. Moreover, ␴ ^{is the} vector of Pauli spin matrices. Therefore,d_␥is an SU共2兲ma- trix that can be seen as a propagator for the spin along the classical trajectory␥.

Upon dividing the trajectory␥into two pieces␥1and␥2, such thatt=t₁+t₂, the spin propagator is clearly multiplica- tive. SinceD_␥arises from a group representation, it inherits this multiplicative property from the propagator, i.e.,

D_␥共x

⬘

^,p

⬘

^,t1+t₂兲=D_␥

2„X共t₁兲,P共t₁兲,t₂…D_␥

1共x

⬘

^,p

⬘

^,t1兲. 共14兲 This relation will be used extensively in Sec. III.

In order to obtain a semiclassical representation of trans- mission amplitudes, we insert expression 共9兲 into Eq. 共7兲.

Then, the integrals overyandy

⬘

, respectively, are evaluated, asymptotically asប→0, with the method of stationary phase.

In this context, we stress the following important observa- tion: The number of accessible transversal states共including spin兲in thenth lead is 共2s+ 1兲Nn=共2s+ 1兲关

冑

2mEWn/共␲ប兲兴, where关x兴 denotes the integer part of x苸R. We choose the widthsWnof the leads to formally shrink proportionally toប in this limit共compare also Ref. 15兲and hence setWn=W˜

nប, to the effect that the sine factors in Eqs.共7兲and共8兲contrib- ute rapidly oscillating phases. These have to be taken into account when determining stationary points of the total phases in the integrals. The condition of stationary phase hence imposes the following restrictions on the transversal momenta:

p

⬘

_y^{= − ⳵S␥}

⳵^ym

⬘

^{= ±} a_m

⬘

␲

W˜

m

共15兲

and

p_y=⳵^S_␥

⳵^yn

= ⫿a_n␲ W˜

n

, 共16兲

upon entry and exit, respectively, of the trajectories. If the points of entry and exit are free of magnetic fields, and thus p=mx˙ at these points, one can characterize the trajectories in terms of the angles ␪ ^and ␪

⬘

, under which they enter and leave the cavity with respect to the longitudinal directions of the leads 共see also Fig. 2兲. These angles are related to the FIG. 1. Sketch of the geometry.

transversal momenta 关Eqs. 共15兲 and 共16兲兴 through sin␪

=py/

冑

2mE and sin␪

⬘

^=py

⬘

^/

冑

2mE. If one wished to keep the widths of the openings fixed, however, the method of station- ary phase would enforce the conditions p_y

⬘

^{= 0 =p}y upon the trajectories, thus leading to semiclassical expressions differ- ent from the ones we use henceforth. In order to keep con- ditions equivalent to Eqs.共15兲and共16兲, one then would have to consider large transversal quantum numbers. This could be achieved by introducing rescaled quantum numbers˜a_n=anប, kept at a fixed value. In that context, the limit of large num- bers Nn of open transversal channels is required 共see also Ref.16兲.

Collecting now all terms that emerge in the stationary phase calculation finally leads to the following leading semi- classical contribution to theS-matrix elements:

S_␣

n␣_m⬘

nm ⬃

兺

␥共␪,␪⬘^兲

B_{␥共␪,␪⬘兲}D_{␥共␪,␪}^␴␴⬘_⬘兲exp„共i/ប兲S_{␥共␪,␪⬘兲}…, 共17兲 where the sum extends over all trajectories that run from lead mthrough the cavity to leadnand are characterized by con- ditions 共15兲 and 共16兲, expressed in terms of the angles of entry and exit. The explicit form of the factorB_{␥共␪,␪⬘兲} is the same as if there were no spin present,²⁶

B_{␥共␪,␪⬘兲}=

冑

2W^i␲m^បWn

sgn共±a_m

⬘

兲sgn共±a_n兲

兩cos␪^cos␪

⬘

^M_{␥共␪,␪}²¹ _⬘兲兩^1/2exp

冠

^i␲

冉

^±aWm

^⬘

m^ym

^⬘

+±anyn

Wn

−1

2␮␥共␪,␪⬘^兲

冊冡

^{. 共18兲}

Here, M_{␥共␪,␪}²¹ _⬘兲 is an element of the monodromy matrix of

␥共␪,␪

⬘

^兲that arises from the matrix appearing in Eq.共12兲by a restriction to the phase-space directions transversal to the trajectory. Furthermore,␮␥共␪,␪⬘^兲 is a modified Maslov index that contains the index␯_{␥共␪,␪⬘兲} ^{from Eq.}共11兲 and additional phases resulting from the stationary phase calculation of the integrals overynandy_m

⬘

^.

The above result关Eq. 共17兲兴 primarily refers to transmis- sion amplitudes共n⫽m兲, but can be carried over to the case of reflection共n=m兲. The reason for this is that the additional term␦␣_n␣_n⬘in Eq.共8兲is canceled by the contribution of direct trajectories in the opening of the leadn that never enter the cavity.⁴

The ultimate goal being a semiclassical calculation of the conductance coefficients关Eqs.共3兲and共4兲兴, one therefore re- quires the evaluation of double sums,

兩S_␣

n␣_m⬘

nm 兩²⬃

兺

␥共␪,␪⬘^兲

兺

␥⬘^共␪,␪⬘^兲

B_␥B_␥^*_⬘D_␥^␴␴⬘D_␥^␴␴_⬘^⬘*exp„共i/ប兲共S_␥−S_␥⬘兲…, 共19兲 over classical trajectories. This will be the task for the rest of this paper.

To simplify the calculations, from now on we restrict our attention to the case of two leads. With an incoming wave in the lead m= 1, we are thus dealing with the transmission coefficientg₂₁ and the reflection coefficientg₁₁. To this end, we will determine the transmission matrixS²¹and the reflec-

tion matrix S¹¹, leading to the transmission and reflection coefficients

T=

兺

␣2,␣₁⬘ 兩S_␣

2␣₁⬘

21 兩², R=

兺

␣2,␣₂⬘ 兩S_␣

2␣₂⬘

22 兩², 共20兲

respectively. Hence, at zero temperature the current through lead 2 is

I₂=e²

h兵TV1+关R−共2s+ 1兲N2兴V2其, 共21兲 whereT andRare taken at the Fermi energy EF. Together with the conditiong₂₁+g₂₂= 0, expressing that equal voltages at both leads produce no current, this yields the relation

I₂=e²

hT共V1−V₂兲. 共22兲

III. SEMICLASSICAL CALCULATION OF CONDUCTIVITY COEFFICIENTS

The calculation of the double sum关Eq.共19兲兴over classi- cal trajectories requires input from dynamical properties of the associated classical system. With spin-orbit interactions present, one therefore first has to identify an appropriate clas- sical system. Moreover, ergodic properties of the classical system imply necessary ingredients for the further calcula- tion. The diagonal contribution to the double sum is evalu- ated with a sum rule,^9,20whereas the nondiagonal terms are evaluated following the Sieber-Richter method.^5,9,15,20

A. Classical spin-orbit dynamics

The classical dynamics that enters the semiclassical rep- resentation 共9兲 consists of two parts:²¹ the motion of the point particle generated by the Hamiltonian 共10兲, including elastic reflections from hard walls, and the spin that is driven by this motion according to Eq.共13兲. These contributions can be combined into a single dynamics on a spin-orbit phase space.²²The relevant classical trajectory is(X共t兲,P共t兲,g共t兲), with initial condition共x

⬘

^,p

⬘

^,g

⬘

^{兲att= 0. Here,g苸SU共2兲and} g共t兲=d_␥共x

⬘

^,p

⬘

^,t兲g provides the spin part of the combined motion. We remark that this description of spin appears quantum mechanical. However, by passing to expectation values of the spin operatord_␥^{† 1}₂␴^d␥in normalized spin states

␹共Heisenberg picture兲, the spin variable becomes a unit vec- tor 具_␹,d_␥^{† 1}₂␴^d␥␹典. Hence, the spin part of the combined phase space is a unit sphere. The two views of the spin motion, either on SU共2兲 or on a unit sphere, are, in fact, equivalent.²¹ In both cases, we will therefore speak of clas- sical spin-orbit dynamics.

Ergodicity is a concept developed for closed systems. It can, however, be suitably extended to open systems of the kind under consideration here. To this end, one divides the configuration space Q of the device into a closed part Q_c, consisting of the cavity with the leads truncated and the openings closed, plus the infinite leads. From now on, we suppose the shape of the closed part to form a chaotic bil- liard, ensuring ergodicity of the motion inside the cavity.

Then, ␳共t兲 is the probability for a typical trajectory to stay within the cavity at least up to timet. For large times,

␳共t兲 ⬃exp共−t/␶兲, t→⬁, 共23兲

with inverse dwell time 1

␶⁼ ប

mA共N1+N₂兲, 共24兲 in which A denotes the area of the closed part Q_c. For the associated part of phase space, we also introduce the volume

⌺共E兲=

冕

Q_c

d²x

冕

R²

d²p␦„E−H₀共x,p兲…= 2␲^mA 共25兲 of the energy shell. This expression has no integration over the spin part, since the Hamiltonian is independent thereof, and an integration over SU共2兲with respect to Haar measure dgyields 1.

For the open system, the concept of ergodicity has to be modified in that the possibility of a trajectory to leave the cavity must be taken into account. When the motion inside

the cavity is ergodic, this leads to the following relation be- tween phase-space averages and time averages over typical spin-orbit trajectories:

冓 ^冕

^0Tdtf„X共t兲,P共t兲,g共t兲…

冔

⬃ 1

⌺共E兲

冕

0 T

dt␳共t兲

冕

Q_c

d²x

冕

_R^2d2p

冕

_SU共2兲^{dgf共x,p,g兲}

⫻␦„E−H₀共x,p兲…, 共26兲 as T→⬁. Here, f is an arbitrary function on the combined phase space and具¯典denotes an average over initial condi- tions. This relation, which properly reflects the chaotic nature of the combined classical spin-orbit motion, provides the ba- sis for further use of dynamical properties in the calculation of the sum关Eq. 共19兲兴over classical trajectories.

The stronger mixing property, which we also assume to hold henceforth, means that correlations of two spin-orbit observablesf andh decay, i.e.,

limt→⬁

冕

Q_c

d²x

冕

_R^2d2p

冕

_SU共2兲^dgh„X共t兲,P共t兲,g共t兲…f共x,p,g兲␦„E−H₀共x,p兲…

= 1

⌺共E兲

冕

Q_c

d²x

冕

R²

d²p

冕

SU共2兲dgh共x,p,g兲␦„E−H₀共x,p兲…

冕

Q_c

d²x

⬘ 冕

R²

d²p

⬘ 冕

_SU共2兲^dg

^⬘

^f共x

^⬘

^,p

^⬘

^,g

^⬘

^{兲␦„E−H0共x}

^⬘

^,p

^⬘

^兲….

共27兲

We stress that ergodicity and mixing of the spin-orbit dynam- ics do not simply follow from chaotic properties of the un- derlying translational motion. They require, in addition, that the spin-orbit coupling is sufficiently complex in order to exclude trivial spin motions.

B. Transmission and reflection coefficients in the configuration-space approach

In the first step, we calculate the leading semiclassical contribution to transmission and reflection coefficients from Eq.共19兲, averaged over a small energy window, by using the configuration-space approach. Such a calculation has been performed previously,²⁰ however, with a sum rule that only takes the particle motion into account. The spin contribution was built in subsequently, assuming that traces of products of spin-transport matrices can be replaced by certain averages.

Here, we reproduce the result obtained in Ref.20by using a sum rule for the complete spin-orbit dynamics that follows from Eq.共26兲. Thus, we base the assumptions made in Ref.

20on a firm dynamical ground.

As ប→0, the terms in the double sum 关Eq. 共19兲兴 are highly oscillatory, except for contributions withS_␥=S_␥⬘. Ge- nerically, if no symmetries are present, this only occurs for

the diagonal␥

⬘

⁼␥. In the event that time-reversal invariance is not broken, however, the time-reversed trajectory␥⁻¹ has the same action as␥. Of course,␥⁻¹ is only among the tra- jectories to be summed over in the case of reflection 共n= 1

=m兲when, moreover,␪=␪

⬘

, i.e., only forS_␣

1␣₁⬘

11 witha₁=a₁

⬘

^. All further terms are oscillatory, with a decreasing impor- tance of their contribution, after averaging over an energy window, when the action differences increase. Below, we calculate the two leading contributions to the quantity

␴,␴

兺

⬘^=−s

s

兩S_␣

n␣_m⬘

nm 兩²⬃

兺

␥,␥⬘

B_␥B_␥^*_⬘Tr共D_␥D_␥^†_⬘兲exp„共i/ប兲共S_␥−S_␥⬘兲…

共28兲 for systems with time-reversal invariance: 共i兲 the diagonal contribution in which the sum over␥

⬘

is restricted to␥

⬘

⁼␥ 共for transmission兲or␥

⬘

^=␥±1共for reflection兲and共ii兲the one- loop contribution in which the sums over␥ and␥

⬘

^{are con-} fined to so-called Sieber-Richter pairs共see also Ref.20兲.

Due to the unitarity of the spin-transport matrices, in the diagonal case, terms with␥

⬘

⁼␥yield a spin contribution of Tr共D_␥D_␥^†兲= 2s+ 1. Thus, the diagonal contribution to Eq.共28兲

can immediately be obtained from the respective result with- out spin,^4,9

冓

^␴,␴

^兺

^{⬘s=−s兩S␣nmn␣m⬘兩diag2}

冔

^{⌬E⬃N2s1++ 1N2. 共29兲}

In the case of reflection共n= 1 =m兲witha₁=a₁

⬘

, an additional diagonal contribution arises from the terms with␥

⬘

^{=␥−1, if} time-reversal invariance is unbroken. Its spin contribution is Tr共D_␥D_␥†−1兲= Tr共D_␥²兲. One hence requires a suitable sum rule that incorporates the combined classical spin-orbit motion.

For this purpose, we choose the function f„X共t兲,P共t兲,g共t兲…= 1

m␦„␽共t兲−␪…␦„x共t兲…关⌰„y共t兲…−⌰„y共t兲

−W₁…兴Tr„␲s关g共t兲g共0兲⁻¹兴…² 共30兲 in Eq.共26兲. Here,␲s共g兲 denotes the spin-srepresentation of g苸SU共2兲,␽ is the angular variable in planar polar coordi- nates forp, and⌰共y兲is a Heavyside step function. An evalu- ation of Eq.共26兲with the function in Eq. 共30兲then leads to the sum rule共asT→⬁兲

␥,T

兺

_␥艋T

兩B_␥兩²Tr共D_␥²兲 ⬃ ␲ 2W˜

1

共− 1兲^2s 2␲^mA

冕

0

T

dt␳共t兲. 共31兲

After an average over a small window in energy this, to- gether with Eq.共29兲, finally yields the semiclassical result

冓

^␴,␴

^兺

^{⬘s=−s兩S␣111␣1⬘兩diag2}

冔

^{⌬E⬃2s+ 1 +N1共− 1兲+N22s␦a1a1⬘ 共32兲}

for the diagonal contribution to Eq. 共28兲. For s= 1 / 2, the right-hand side is 1 /共N₁+N₂兲.

Sieber-Richter pairs of trajectories are characterized by the fact that one trajectory possesses a self-crossing with a small crossing angle ␧, thus forming a loop. The partner trajectory then looks like the former one cut open at the self-crossing, but with the loop direction reversed and then glued together, such that the self-crossing is replaced by an almost-crossing共see Fig. 2兲. In principle, the trajectories in such pairs can have an arbitrary number of self-crossings, but the magnitude of their contributions to Eq. 共28兲 decreases with increasing action differences. These, in turn, grow with the number of places in which a self-crossing of one trajec- tory is paired with an almost-crossing of the corresponding partner trajectory. The most important共“one-loop”兲contribu- tion comes from pairs which differ in one crossing. In order to calculate the one-loop contribution, one requires the dis- tribution of the crossing angles␧for pairs of trajectories with loops of durationT,

PS共␧,T兲= 1

⌺共E兲

冕

Q_c

d²x

⬘ 冕

R²

d²p

⬘ 冕

T_min共␧兲 T

dtlpS共␧,T,tl兲.

共33兲 Here,p_S共␧,T,t_l兲is a density of crossing angles defined as

pS共␧,T,tl兲=

冕

0 T−t_l

dts兩J兩␦共E−H₀„P共ts兲…兲Tr„␲s兵g共t兲

⫻关g共0兲兴⁻¹其…²␦„␧−␬共ts,tl兲…␦„X共ts兲−X共ts+tl兲…, 共34兲 where␬共ts,tl兲 denotes the angle between the velocitiesv共ts兲 andv共ts+tl兲. Given a crossing angle␧, the minimal duration for a loop to close isTmin共␧兲. In chaotic systems, this quantity behaves likeTmin共␧兲=O共log␧兲as ␧→0.⁵Furthermore,

兩J兩=兩v共ts兲⫻v共ts+tl兲兩=兩v共ts兲兩兩v共ts+tl兲兩sin␬共ts,tl兲 共35兲 is a Jacobian, andt_s,t_ldenote the time along the trajectory up to the starting point of the loop and along the loop, respec- tively.

Assuming that the classical spin-orbit dynamics is not only ergodic, but also mixing, the distribution in Eq.共33兲can be calculated further. It can be identified as the left-hand side of an appropriate relation of the type of Eq.共27兲. The right- hand side then yields, as␧→0,

PS共␧,T兲 ⬃共− 1兲^2s

␲^A 2E

m sin␧

冋

^{T22−TTmin共␧兲+Tmin22共␧兲}

册

^.

共36兲 This expression differs from the respective one without spin that was obtained in Ref.5only by a factor共−1兲^2s, i.e., a sign in the case s= 1 / 2. With this information at hand, the one- loop contribution can be calculated as in the case without spin,⁹ finally yielding

冓

^␴,␴

^兺

^{⬘s=−s兩S␣211␣2⬘兩1-loop2}

冔

^{⌬E⬃−共N共− 1兲1+N2s2兲2. 共37兲}

This is in accordance with what has been obtained in Ref.20.

C. Transmission coefficients in the phase-space approach Higher orders in the “loop expansion” described above have been calculated previously for spectral form factors⁷as well as for conductance coefficients for systems without spin contributions.¹⁰ The approach taken in these papers utilizes trajectories in classical phase space and identifies the pairs of self-crossings and/or almost-crossings in configuration space

FIG. 2. A Sieber-Richter pair of trajectories.

as pairs of trajectories with almost-crossings in phase space, which differ in the way they are connected at the 共almost兲 crossings. This point of view opens the possibility for a clas- sification of the trajectory pairs in terms of their encounters.⁷ Here, we follow this phase-space approach and amend the previous result¹⁰ with the contribution of the spin-orbit cou- pling.

To be more precise, we consider trajectories that possess close self-encounters共in phase space兲, in which two or more short stretches of the trajectory are almost identical, possibly up to time reversal. These stretches are connected by long parts of the trajectory, which we call loops. We then form pairs共␥,␥

⬘

^兲of such trajectories in which␥and␥

⬘

^{are almost} identical 共up to time reversal兲 along the loops, but differ from each other in the way the loops are connected in the encounter region. In order to quantify these encounters, we introduce a vector vជ^{, whose}lth component,v_l, denotes the number of encounters withlstretches. Hence, the total num- ber of encounters is V=兺l艌2vl, with a total of L=兺l艌2lvl

stretches involved. In general, however, given a vector vជ, there will be N共vជ兲艌1 different trajectory pairs associated with it. These may, e.g., differ in the order the loops connect the encounters, or in the relative directions, in which the encounter stretches are traversed.

To reveal the phase-space structure of trajectory pairs and to compute their contributions to Eq. 共19兲, one introduces Poincaré sections, which cut the trajectories into pieces. In order to adapt this cutting to the sequence of encounters and loops, one chooses a Poincaré section in every of theVgiven encounter regions. We then denote by t_␣,j

⬘

^{, j}= 1 , . . . ,l_␣, ␣

= 1 , . . . ,V the times at which the encounter stretches pierce this section, and byt_enc^␣ , the duration of the encounters. To this cutting of the trajectories corresponds the splitting

D_␥=D_L+1DL¯D₁ 共38兲 of the spin-transport matrices which, with an obvious nota- tion, follows from the composition rule关Eq.共14兲兴. The spin transport along the partner trajectory then reads

D_␥⬘⬇D_L+1D_k

L

␩L¯D_k

2

␩2D₁. 共39兲

Here, ␩j= ± 1, depending on the relative orientation of the trajectory between the共j− 1兲st and the jth cutting of ␥ ^and

␥

⬘

, respectively, through the Poincaré section. We notice that at this point, time-reversal invariance enters crucially. More- over, the indiceskjtake care of the fact that in␥and␥

⬘

^{, the} loops may be traversed in different successions. Thus, the spin-dependent weights in Eq.共28兲for each pair of trajecto- ries are approximately given by

Tr共D_␥D_␥^†_⬘兲 ⬇Tr共DL¯D₂D_k

2

†␩2¯D_k

L

†␩L兲. 共40兲 The calculation of transmission amplitudes performed in Ref.10has now to be modified in that expressions共40兲must be included. To this end, we recall the strategy devised in Refs.7 and10: For each encounter, one introduces coordi- nates on the Poincaré section adapted to the piercing by the trajectories and the linear stability of the dynamics. In en- counter␣, the coordinates共sj␣,u^␣_j兲, j= 1 , . . . ,l_␣− 1, describe the separation of the共j+ 1兲st piercing from thejth one along

the stable and unstable manifolds, respectively, of the latter.

The total of L−V stable and unstable coordinates are then collected in the vectors 共s,u兲. In these coordinates, action differences of partner trajectories共approximately兲read as

⌬S=S_␥−S_␥⬘⬇

兺

_␣,j^{s␣juj␣. 共41兲}

Moreover, the requirement that encounters be close can then be expressed in terms of the condition兩sj␣兩,兩u␣j兩艋cwith some constantc, which yields the duration of an encounter,

t_enc^␣ ⬃ 1

␭ ln c²

maxi兵兩si兩其maxj兵兩uj兩其, t_enc^␣ →⬁. 共42兲 One then introduces a density w_T^spin共s,u兲 of encounters, weighted with the spin contribution, for trajectories of dura- tionTwith a given encounter structure specified by the vec- torvជ. In analogy to the case without spin,¹⁵this leads to the following approximation:

冓 ^兺

^␥

^兺

^{vជ N共vជ兲}

^冕

^{−cc ¯}

^冕

^{−ccdL−VudL−Vs}

⫻exp„共i/ប兲⌬S…w_T^spin共s,u兲兩B_␥兩²

冔

^{⌬E, 共43兲}

to the quantity T_a

2a 1⬘

nd ª

冓

^␴,␴

^兺

^{⬘s=−s兩S␣212␣1⬘兩2}

冔

^{⌬E−N2s1++ 1N2. 共44兲}

After summing over all possible values ofa₂,a₁

⬘

, this yields the nondiagonal contribution to the energy-averaged trans- mission amplitudeT 关compare Eqs.共20兲and共29兲兴.

The essential point now is to calculate the density w_T^spin共s,u兲. In the case without spin-orbit interaction, the cor- responding expression wT共s,u兲 was defined in Ref. 7 as a density of phase-space separations s and u similar to the density P共␧,T兲 with respect to␧ in the configuration-space approach. It was given as

w_T共s,u兲= 1

⌺共E兲

冕

Q_c

d²x

⬘ 冕

_R^2d2p

^⬘

^{␦„E−H0共x}

^⬘

^,p

^⬘

^兲…

⫻

冕

0

⬁

兿

j=1 L

dt_j⌰

冉

^T−␣=1

^兺

^{V l␣tenc␣ −}

^兺

^{j=1L tj}

冊

⫻_␣=1

兿

^V _t¹

enc

␣

冋 ^兿

^{j=2l␣ ␦„共X共t␣}

^⬘

^{j兲,P共t␣}

^⬘

^{j兲…−z␣j兲}

册

^.

共45兲 The average in the first line is over all possible initial points of the trajectory. In the second line, the integration extends over all loop durationstj; their lengths are constrained by the theta function. In order to prevent overcounting,⁷the product of all encounter durationst_enc^␣ is divided out. The last product guarantees that the position of the orbit at times when it pierces through the sections is fixed asz_␣j. This denotes the first point of the orbit in which it pierces through a certain

section plus the separation thereof as specified by the coor- dinates s and u. From Eq. 共45兲, one obtains w_T^spin共s,u兲 by including Tr共D_␥D_␥^†_⬘兲under the integral. Using that the dura- tions of encounters are semiclassically large, compare Eq.

共42兲, the result can be obtained in analogy to Eq.共34兲 by employing Eq.共27兲. The right-hand side then yields

w_T^spin共s,u兲 ⬇

冉

^T−␣=1

^兺

^{V l␣tenc␣}

冊

^L

⌺共E兲^L−V_␣=1

兿

^{V tenc␣ L!}

M_␥␥⬘, 共46兲

i.e., a factorization into the spin-independent part identical to wT共s,u兲and a spin contribution,

M_␥␥⬘ª

冕

_SU共2兲 ^¯

冕

_SU共2兲^dgL¯dg2

⫻Tr„␲s共gL¯g₂g_k

2

␩2†

¯g_k

L

␩L†兲…. 共47兲 In order to calculate Eq.共47兲, we follow the method devel-

oped in Refs.23and24for the spectral form factor of quan- tum graphs with spin-orbit interaction. In analogy to Theo- rem 6.1 of Ref.24, we find in the present context that

M_␥␥⬘=共2s+ 1兲

冉

^{共2s− 1+ 1兲2s}

冊

^{L−V. 共48兲}

This will be proven in the Appendix. We stress that this spin contribution, apart from the spin quantum number, only de- pends onL−V.

Equation 共44兲 can now be calculated in analogy to the case without spin.¹⁰Starting from Eq.共43兲, one employs the expressions for⌬Sfrom Eq.共41兲and forw_T^spin共s,u兲, the sum rule from Ref.9, and the survival probability␳共t兲, modified by replacing t with 关t−兺_␣=1^V 共l_␣− 1兲tenc␣ 兴 as in Ref. 10. This yields

T_a

2,a₁⬘

nd ⬇

冬

^{共2smA+ 1兲ប}

^兺

^{vជ N共vជ兲}

^冋

^L+1

^兿

ⁱ⁼¹

^冕

^0⬁dtiexp

^冉

^−t␶i

^{冊 册 冕}

^{−cc ¯}

^冕

^{−cc d关⌺共E兲兴L−VudL−VL−Vs␣=1}

^兿

^{V exp}

^冉

^{−tenc␣t␶enc␣ +បi⌬S}

^冊 冭

^⌬E

^冉

^{共− 1兲2s+ 12s}

^冊

^L−V

⬇共2s+ 1兲 N₁+N₂

兺

k=1

⬁

冉

^N1¹+N₂

冊

^k

冉

^{共− 1兲2s+ 12s}

冊

^kvជ,L−V=k

^兺

共− 1兲^VN共vជ兲. 共49兲

The integrals oversanduwere calculated in Ref.10, and the sum overvជ can be carried out with the recursion formula¹⁰

v

兺

ជ,L−V=k共− 1兲^VN共vជ兲=

冉

^{1 −}␤²

冊

^{k, 共50兲}

where␤= 1 if time-reversal symmetry is present and␤^{= 2 if} time-reversal symmetry is broken.

Finally, using these results in the case of time-reversal invariance, we obtain for the full transmission matrix, in- cluding also the diagonal part,

T_a

2,a₁⬘

nd + 2s+ 1

N₁+N₂⬇ 共2s+ 1兲²

共2s+ 1兲共N1+N₂兲− 1, 共51兲 in the case of half-integers, and

T_a

2,a₁⬘

nd + 2s+ 1

N₁+N₂⬇ 共2s+ 1兲²

共2s+ 1兲共N1+N₂兲+ 1, 共52兲 ifsis integer. Fors= 1 / 2, Eq.共51兲is identical with the one obtained using random matrix theory, in the circular sym- plectic ensemble.¹

These findings can now be compared with the respective results when time reversal is absent, thus revealing the be- havior of the transmission under a breaking of time reversal by, e.g., turning on a magnetic field. In that case,␤= 2 so that Eq.共50兲vanishes, implying via Eq.共49兲that only the diag- onal contribution survives. The difference ⌬T=T^共␤=1兲

−T^共␤=2兲 of the transmission coefficients is therefore

⌬T⬇ N₁N₂共2s+ 1兲

共N₁+N₂兲关共2s+ 1兲共N₁+N₂兲− 1兴, 共53兲 in the case of half-integers, and

⌬T⬇ −N₁N₂共2s+ 1兲

共N1+N₂兲共共2s+ 1兲共N1+N₂兲+ 1兲, 共54兲 ifsis integer. From these expressions, one immediately con- cludes that the transmission共i.e., conductivity兲 is enhanced at zero magnetic field共when time-reversal symmetry is re- stored兲, if the spin is half-integer; thus, weak antilocalization occurs. The only semiclassical derivation of weak antilocal- ization so far²⁰ was restricted to the one-loop contribution and employed asymptotics for largeN₁,N₂.

For integer spin, the above results predict weak localiza- tion. The latter property had previously been obtained in