научная статья по теме CROSSOVERS BETWEEN SUPERCONDUCTING SYMMETRY CLASSES Физика

Текст научной статьи на тему «CROSSOVERS BETWEEN SUPERCONDUCTING SYMMETRY CLASSES»

Pis'ma v ZhETF, vol. 94, iss. 3, pp. 240 - 245

© 2011 August 10

Crossovers between superconducting symmetry classes

V. A. Koziy, M. A. Skvoitsov L.D. Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia

Moscow Institute of Physics and Technology, 141700 Moscow, Russia Submitted 20 June 2011

We study the average density of states in a small metallic grain coupled to two superconductors with the phase difference tt, in a magnetic field. The spectrum of the low-energy excitations in the grain is described by the random matrix theory whose symmetry depends on the magnetic field strength and coupling to the superconductors. In the limiting cases, a pure superconducting symmetry class is realized. For intermediate magnetic fields or couplings to the superconductors, the system experiences a crossover between different symmetry classes. With the help of the supersymmetric cr-model we derive the exact expressions for the average density of states in the crossovers between the symmetry classes A-C and CI-C.

Introduction. Energy levels in small metallic particles with chaotic electron dynamics are random numbers. It is generally accepted that their spectral statistics in the ergodic regime is described by the random matrix theory (RMT) [1]. For disordered grains, this had been proved by Efetov [2] with the help of the supersymmetry technique [3], while for quantum billiards in the absence of disorder this statement is usually referred to as the Bohigas conjecture [4].

In the RMT, a system is characterized solely by its symmetry. In the application to condensed matter, the standard three Wigner-Dyson ensembles (orthogonal, unitary and symplectic) [5] describe level statistics in small metallic grains in the presence or absence of the time-reversal and spin-rotation symmetries [3].

Recently, the Wigner-Dyson classification had been extended to superconducting [6] and chiral [7] symmetry classes, which arise when the Hamiltonian possesses an additional symmetry with respect to changing the sign of the energy (counted from the Fermi energy). With the appearance of a selected energy point, in the su-perconducting/chiral classes even the average density of states (DOS), (p(E)), becomes a nontrivial function of the energy. This should be contrasted to the standard Wigner-Dyson ensembles where (p(E)) = 5= const and the first nontrivial quantity is the pair correlation function R2(uj) = S2(p(E + uj)p(E)) - 1.

The symmetry classes (three Wigner-Dyson, four superconducting and three chiral) correspond to the limits when various symmetries are either present or completely broken. In the intermediate cases, the system experiences a crossover between different symmetry classes. The pair correlation function in the crossover between the orthogonal and unitary classes was obtained in Refs. [8, 9]:

= 1

sin2 X

r

dX

sin Xx X

-oa'

p 1

/ dp ps'mpx eai* , (1)

Jo

where x = -kw/5 and a is the symmetry-breaking parameter. Equation (1) interpolates between the orthogonal (a = 0) and unitary (a = oo) results. The pair correlator in known also in the symplectic-unitary crossover [10, 11], its form being similar to Eq. (1).

The purpose of this Letter is to theoretically study crossovers between superconducting classes.

We will calculate the average density of states in a small diffusive metallic grain coupled to two superconducting terminals through tunnel barriers, see Fig. 1. The terminals have the phase difference ir ensuring the

© H

N

9 = 0

9 = m

Fig. 1. A normal-metal dot coupled to two superconducting terminals with the phase difference 7r, in a magnetic field. NS interfaces are characterized bv the set of transparencies №}

absence of the minigap in the excitation spectrum [12]. A magnetic field H is applied to the system. The spinrotation symmetry is assumed to be intact. We will be interested in the ergodic regime, E -C -Em, where .Etii = D/L2 is the Thouless energy, D is the diffusion constant, and L is the grain size.

x

e

S

S

Under these conditions, the excitation spectrum in the grain can be described in terms of the RMT in the crossover region between the four symmetry classes shown in Fig. 2.

Class AI (Orthogonal)

(4, 4)

H

Class A (Unitary)

(2, 2)

Supercond. class CI

(3, 1)

H

Supercond. class C

(2, 0)

Fig. 2. Crossovers between spin-symmetric symmetry classes driven by the magnetic field (if) and coupling to superconductors (A). The dimensions of the FF- and BB-sectors of the supersymmetric cr-model for the average density of states are shown by (hfjAb)

Mapping to Efetov's cr-model. First attempts of field-theoretical description of hybrid NS systems [13-17] inspired by the identification of superconducting symmetry classes [6] have used the Bogolyubov-de Gennes (BdG) Hamiltonian as the starting point,

^BdG =

H

.A*

(2)

where H is the single-particle Hamiltonian, and A(r) is the pairing field. The average quasiparticle DOS,

<p(£,r)> = -lm<trel(r)>/7r,

(3)

is expressed in terms of the retarded Green function of the BdG Hamiltonian,

Gl = (E

■ 'HßdG + Î0)~

(4)

which is then represented as a functional integral over an 8 x 8 supermatrix field Q acting in the direct product FB ® N ® PH of the Fermi-Bose (FB), Nambu (N) and Particle-Hole (PH) spaces (spin-symmetric case is considered).

In hybrid NS systems, Andreev reflection off the order parameter field A couples the states with opposite energies, E and —E. So, the Nambu-Gor'kov Green function Q¡j essentially involves a pair of the retarded and advanced normal-metal Green functions, G§ and Ga:e. In the absence of the superconducting pairing field, A(r), correlations between the latter are conveniently described by Efetov's supersymmetric cr-model

[2, 3] of the orthogonal symmetry class, with an 8 x 8 superfield Q acting in the direct product FB eg RA eg TR of the Fermi-Bose (FB), Retarded-Advanced (RA) and Time-Reversal (TR) spaces (again we assume no spin interactions).

Thus, in studying the proximity effect in the normal part of a hybrid system, it is tempting to reformulate the field theory of Refs. [13-17] in the language of Efetov's supersymmetric cr-model. Provided that the inverse proximity effect in the superconducting regions can be neglected (rigid boundary conditions), Andreev scattering of normal electrons off the superconducting terminal will be viewed as an effective boundary condition at the NS interface mixing the R and A components of the field Q. Such a description is close in spirit to the scattering approach [18].

The average local DOS is given by the functional integral over the normal-metal region [17]:

(p(E, r)> = ^Re J sti(kAQ)ers^-s^DQ(r), (5) where SnfQ] is the bulk action:

Sd = ^-J drsti{D(VQ + ieA[T3,Q])2+ UEAQ},

(6)

and the action Sr [Q] = St1 [Q] + 5r2 [Q] describes NS interfaces [3, 19]:

= 4$>trln[l + e-2^a)Q<a)]. (7)

Here v is the DOS per one spin projection at the Fermi level, A is the vector potential, Q^ labels the Q field at the boundary with the a-th superconductor, and the NS interface is specified by transmission coefficients Ti = 1/cosh2/3,-, with i labelling open channels. The

field Q satisfies Q2 = symmetry constraint

and is subject to an additional

Q = CQTCT

(8)

In the NS cr-model for ((?§) [17], the matrices A, 73 and C are given by the first column of Table. An exact mapping to Efetov's cr-model is realized by the similarity transformation Q ^ VQVwith the matrix

V =

/ -Ifb 0 0 0 \

0 0 0 Ifb

0 Ifb 0 0

V 0 0 -k fb 0 y

(9)

Basic matrices in the two versions of the cr-model

NS cr-model [171

Efetov's cr-model [31

Space

PB ® N ® PH

FB 0 RA 0 TR

A

T3

C

tra

" z N ' z

tph

J x

" z T™ 0

where the inner (outer) grading corresponds to the PH (N) space, and k = diag(l, —1)fb (we follow notations of Ref. [3]). Conjugation by V simultaneously transforms the matrices A, 73 and C from NS representation to Efetov's representation given by the last column of Table. This provides an exact mapping between the NS cr-model for Green function of the BdG Hamiltonian, (£?!), to the standard Efetov's orthogonal cr-model for the product (G^G^E). On such a mapping both the structure of the manifold and the cr-model action get reproduced. We emphasize that this mapping takes place only in the normal part of a hybrid NS system, where the pairing amplitude A = 0.

To complete the formulation of the model we have to specify the Q matrix in the bulk of a superconductor, Qs- It has a familiar form parameterized with the help of the spectral angle 9 s = arctan(j A¡E) as

Qs = A cos 9s + S sin 9s-

(10)

The most nontrivial ingredient of the mapping from the SN cr-model to Efetov's cr-model is the form of the matrix S. In the initial NS representation [17] it is just the Pauli matrix in the Nambu space: cr^. Conjugating by V we get it in Efetov's representation:

0

%H Wb 0 )RA

We see that superconducting boundary conditions olate supersymmetry": the matrix £

(H)

acts as crfAcr

vi-TR

in the FF-block and as afAaJR in the BB-block. This is the reason why a nontrivial DOS can be obtained by integration (5) over the standard orthogonal cr-model manifold.

Zero-dimensional limit. In the ergodic regime, E -C Eth, the functional integral (5) is dominated by the zero mode, Q(r) = const. We will be interested in the average global DOS normalized by the inverse mean quasiparticle level spacing 5 = (2fV)^1:

(,o(E))=sJ(p(E,r))dr. (12)

This quantity can be written as an integral over a single 8x8 supermatrix Q:

(e(E)) = ^Re J sti(kAQ)e-s^DQ, (13)

with the action consisting of three terms:

IT fy

S[Q] = -strAQ - -str(r3Q)2

str(ÊQ)2. (14)

Here x = irE/5, and the symmetry breaking parameters a and 7 are given by

a = 7ïuDe2 / A2 dx

7 = (15)

One can estimate a ~ where 4> is the flux

through the grain, (f>0 is the flux quantum, and g ~ ~ Erh/S 1 is the dimensionless grain conductance. The last term in the action (14) is written in the tunneling limit, Tj -C 1, and the

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