Keldysh action of a multiterminal time-dependent scatterer
I. SnymanInstitut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands Y. V. Nazarov
Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands
共Received 15 January 2008; revised manuscript received 19 March 2008; published 14 April 2008兲
We present a derivation of the Keldysh action of a general multichannel time-dependent scatterer in the context of the Landauer–Büttiker approach. The action is a convenient building block in the theory of quantum transport. This action is shown to take a compact form that only involves the scattering matrix and reservoir Green’s functions. We derive two special cases of the general result, one valid when reservoirs are character-ized by well-defined filling factors, the other when the scatterer connects two reservoirs. We illustrate its use by considering full counting statistics and the Fermi-edge singularity.
DOI:10.1103/PhysRevB.77.165118 PACS number共s兲: 73.23.⫺b, 73.50.Td, 05.40.⫺a
I. INTRODUCTION
The pioneering works of Landauer1 and Büttiker2–4 lay the foundations for what is now known as the scattering ap-proach to electron transport. The basic tenet is that a coher-ent conductor is characterized by its scattering matrix. More precisely, the transmission matrix defines a set of transparen-cies for the various channels or modes in which the electrons propagate through the conductor. As a consequence, conduc-tance is the sum over transmission probabilities. Subse-quently, it was discovered that the same transmission prob-abilities fully determine the current noise, also outside equilibrium, where the fluctuation-dissipation theorem does not hold.5,3
Indeed, as the theory of full counting statistics6,7 共FCS兲 later revealed, the complete probability distribution for out-comes of a current measurement is entirely characterized by the transmission probabilities of the conductor. The fact that the scattering formalism gives such an elegant and complete description inspired some to revisit established results. Thus, for instance, interacting problems such as the Fermi-edge singularity8,9共FES兲 was recast in the language of the scatter-ing approach.10–13The scattering approach has further been successfully employed in problems where a coherent con-ductor interacts with other elements, including, but not re-stricted to, measuring devices and an electromagnetic environment.14–17It is also widely applied to study transport in mesoscopic superconductors.18
Many of these more advanced applications are unified through a method developed by Feynman and Vernon for characterizing the effect of one quantum system on another when they are coupled.19The work of Feynman and Vernon dealt with the effect of a bath of oscillators coupled to a quantum system. It introduced the concept of a time contour describing a propagation first forward and then backward in time. By using the path-integral formalism, it was possible to characterize the bath by an “influence functional” that did not depend on the system that the bath was coupled to. This functional was treated nonperturbatively. A related develop-ment was due to Keldysh.20While being a perturbative dia-grammatic technique, it allowed for the treatment of general systems and shared the idea of a forward and backward time contour with Feynman and Vernon.
In general, the Feynman–Vernon method expresses the dynamics of a complex system in the form of an integral over a few fields共t兲. Each part of the system contributes to the integrand by a corresponding influence functional Z关兴 or, synonymously, a Keldysh action A关兴=ln Z关兴. Thus, the Keldysh action of a general scatterer can be used as a building block. In this way the action of a complicated nano-structure consisting of a network of scatterers can be con-structed. As in the case of classical electronics, a simple set of rules, applied at the nodes of the network, suffice to de-scribe the behavior of the whole network.21,22
The essential element of the approach is that the fields take different values on the forward and backward parts of the time contour. One writes this as ⫾共t兲, where ⫹ 共⫺兲 corresponds to the forward共backward兲 part of the contour. The Keldysh action for a given subsystem is evaluated as a full nonlinear response of the subsystem to the fields⫾共t兲. 关See Eq. 共6兲 below for the precise mathematical definition.兴
Applications involving the scattering approach require both the notion of the nonperturbative influence functional and the generality of Keldysh’s formalism. Until now, the combination of the Feynman–Vernon method with the scat-tering approach was done on a case-specific basis: only those elements relevant to the particular application under consid-eration were developed. In this paper, we unify previous de-velopments by deriving general formulas for the Keldysh action of a general scatterer connected to charge reservoirs.
The time-dependent fields+共t兲 and−共t兲 parametrize two
HamiltoniansH+共t兲 and H−共t兲 that govern forward and
back-ward evolutions in time, respectively. Since we are in the framework of the scattering approach, these field-dependent Hamiltonians are not the most natural objects to work with. Rather, depending on where the fields couple to the system, it is natural to incorporate their effect either in the scattering matrix of the conductor or in the Green’s functions of the electrons in the reservoirs: the fields affecting the scattering potential inside the scatterer are incorporated in a time-dependent scattering matrix. Since the fields⫾for forward and backward evolutions are different, the scattering matri-ces for forward and backward evolutions differ. The effect of the fields perturbing the electrons far from the scatterer is incorporated in the time-dependent Green’s functions of the
electrons in distant reservoirs. A bias voltage field applied across a conductor can conveniently be ascribed to either Green’s functions of the reservoirs or to a phase factor of the scattering matrix. The same holds for the counting fields en-counterd in the theory of full counting statistics. There are, however, situations where our hand is forced. For instance, in the example of the Fermi-edge singularity, which we will discuss in Sec. VI, the time-dependent fields have to be in-corporated in the scattering matrices.
Previous studies of the Keldysh action concentrate on situations where the fields ⫾ can be incorporated in the reservoir Green’s functions. These studies therefore assume stationary, contour-independent scattering matrices while al-lowing for a time dependence and/or time-contour depen-dence of the electron Green’s functions. Early works共Refs.
23and24兲 used an action of this type to analyze Coulomb
blockade phenomena. Later, the same action was understood in a wider context of arbitrary Green’s functions.21,25 In this form, it has been used to treat problems involving, for ex-ample, interactions and superconductivity. The action em-ployed in these studies corresponds to Eq.共4兲 and can readily
be derived in the context of a nonlinear sigma model of disordered metals.26
The main innovation of the present work is to generalize the action to contour- and time-dependent scattering matri-ces. The only assumption we make is that scattering is in-stantaneous: we do not treat the delay time an electron spends inside the scattering region realistically.
The resulting scattering matrices associated with forward and backward evolutions are combined into one big matrix sˆ. It has a kernel s共␣; c , c
⬘
; t兲␦␣,␣⬘␦共t−t⬘
兲, where the Keldysh indices␣,␣⬘
苸兵+,−其 refer to the forward and backward parts of the time contour, c,c⬘
are integers that refer to channel space, and t , t⬘
are time indices that lie on the real line. The forward 共backward兲 scattering matrix sˆ+共−兲 with kernel sˆ(␣= +共−兲;c,c
⬘
; t) obeys the usual unitarity condition sˆ⫾†sˆ⫾= 1. With the aide of these preliminary definitions, our main result is summarized by a formula for the Keldysh action.A关sˆ兴 = Tr ln
冋
1 + Gˆ2 + sˆ 1 − Gˆ
2
册
− Tr ln sˆ−. 共1兲 In this formula, Gˆ is the Keldysh Green’s function character-izing the reservoirs connected to the scatterer.27 It is to be viewed as an operator with kernel G共␣,␣⬘
; c ; t , t⬘
兲␦c,c⬘, where the indices carry the same meaning as in the definition of sˆ. This formula is completely general as demonstrated by the following:共1兲 It holds for time-dependent scattering matrices that differ on the forward and backward time contours.
共2兲 It holds for multiterminal devices with more than two reservoirs.
共3兲 It holds for devices such as Hall bars, where particles in a single chiral channel enter and leave the conductor at different reservoirs.
共4兲 It holds when reservoirs cannot be characterized by stationary filling factors. Reservoirs may be superconducting or contain “counting fields,” which couple them to a dynami-cal electromagnetic environment or a measuring device.
When the reservoirs can indeed be characterized by filling factors fˆ共兲, the Keldysh structure can explicitly be traced out to yield
A关sˆ+,sˆ−兴 = Tr ln关sˆ−共1 − fˆ兲 + sˆ+fˆ兴 − Tr ln sˆ−. 共2兲
In this expression, operators retain their channel structure and their time structure. In “time” representation, fˆ is the Fourier transform to time of the reservoir filling factors and, as such, has a kernel f共c;t,t
⬘
兲␦c,c⬘diagonal in channel space and depending on two times. In the stationary limit, this formula immediately reduces to the Levitov formula for low-frequency FCS.7Another formula that can be derived from Eq.共1兲 is valid
for two terminal devices and a stationary, time-contour-independent scattering matrix but allows for arbitrary Green’s functions in the terminals. Each terminal may still be connected to the scatterer by an arbitrary number of chan-nels. We denote the two terminals left共L兲 and right 共R兲. In this case, the reservoir Green’s function has the form
Gˆ =
冉
G ˇ L 0 0 Gˇ R冊
channel space , 共3兲where GL共R兲 have no further channel space structure. The matrix structure in Keldysh and time indices共indicated by a check sign兲 is now retained in the trace, but the channel structure is traced out. Thus, the following is obtained:
A关⫾兴 =12
兺
n
Tr ln
冋
1 + Tn兵GˇL关⫾兴,GˇR关⫾兴其 − 24
册
. 共4兲In this expression, the field dependence⫾is shifted entirely to the Keldysh Green’s functions GLand GR of the left and right reservoirs. This formula makes it explicit that the con-ductor is completely characterized by its transmission eigen-values Tn.
The structure of the paper is as follows. After making the necessary definitions, we derive Eq.共1兲 from a model
Hamil-tonian. The derivation makes use of contour ordered Green’s functions and the Keldysh technique. Subsequently, we de-rive the special cases of Eqs.共2兲 and 共4兲.
We conclude by applying the formulas to several generic setups and verify that results agree with the existing litera-ture. Particularly, we explain in detail how the present work is connected to the theory of full counting statistics and to the scattering theory of the Fermi-edge singularity.
II. DERIVATION
We consider a general scatterer connecting a set of charge reservoirs共Fig.1兲. We allow the scatterer to be time
depen-dent. A sufficient theoretical description is provided by a set of transport channels interrupted by a potential that causes interchannel scattering. We consider the regime where the scattering matrix is energy independent in the transport en-ergy window. Since transport is purely determined by the scattering matrix, any model that produces the same scatter-ing matrix gives identical results. Regardless of the actual
microscopic detail, we may therefore conveniently take the Hamiltonian of the scatterer to be
H = vF
兺
m,n冕
dzm †共z兲兵− i␦ m,nz+ um,n共z兲其n共z兲 + Hres+HT, 共5兲 whereHresrepresents the reservoirs, andHTtakes account oftunneling between the conductor and the reservoirs. The scattering region and the reservoirs are spatially separated. This means that the scattering potential umn共z兲 is nonzero only in a region z−⬍z⬍z+, while tunneling between the
res-ervoirs and the conductor only takes place outside this re-gion. Note that in our model, scattering channels have been “unfolded,” so that instead of working with a channel that confines particles in the interval 共−⬁,0兴 and allowing for propagation both in the positive and in the negative direc-tion, we equivalently work with channels in which particles propagate along共−⬁,⬁兲, but only in the positive direction. Hence, to make contact with most physical setups, we con-sider −z and z to refer to the same physical position in a channel but opposite propagation directions.
We consider the generating functional
Z = eA= Tr
冋
T+exp再
− i冕
t0 t1 dtH+共t兲冎
⫻0T−exp再
i冕
t0 t1 dtH−共t兲冎
册
, 共6兲in which H⫾ is obtained fromH by replacing umn共z兲 with arbitrary time-dependent functions umn⫾共z,t兲. In this expres-sions,T+exp andT−exp respectively refer to time-ordered
共i.e., largest time to the left兲 and anti-time-ordered 共i.e., larg-est time to the right兲 exponentials. In the language of Feyn-man and Vernon,19this is known as the influence functional.
It gives a complete characterization of the effect that the electrons in the scatterer have on any quantum system that they interact with. Furthermore, the functionalZ generates expectation values of time-ordered products of operators as follows. Let Q be an operator
Q =
兺
mn
冕
z− z+dzm
†共z兲qmn共z兲n共z兲. 共7兲
Choose umn⫾共z,t兲=umn共z兲+⫾共t兲qmn共z兲. Then
冓
T-冉
兿
j=1 M Q共tj兲冊
T+冉
兿
k=1 N Q共tk⬘
兲冊
冔
=兿
j=1 M冉
− i ␦ ␦−共tj兲冊
兿
k=1 N冉
i ␦ ␦+共tk⬘
兲冊
兩Z关兴兩=0. 共8兲By merging the power of the Keldysh formalism of contour-ordered Green’s functions with that of the Landauer scatter-ing formalism for quantum transport, we obtain an expres-sion forZ in terms of the Keldysh Green’s functions in the reservoirs and the time-dependent scattering matrices associ-ated with uˆ⫾共z,t兲.
The argument will proceed in the following steps: 共1兲 First, we introduce the key object that enables a sys-tematic analysis of Z, namely, the single particle Green’s function g of the conductor. We state the equations of motion that g obeys.
共2兲 We define the Keldysh action A=lnZ and consider its variation␦A. We discover that␦A can be expressed in terms of g.
共3兲 We therefore determine g inside the scattering region in terms of the scattering matrix of the conductor and its value at the edges of the scattering region, where the reser-voirs impose boundary conditions.
共4兲 This allows us to express the variation of the action in terms of the reservoir Green’s functions Gin共out兲and the scat-tering matrix s of the conductor.
共5兲 The variation␦A is then integrated to find the action A and the generating functional Z.
A. Preliminaries: Definition of the Green’s function
The first step is to move from the Schrödinger picture to the Heisenberg picture. To shorten the notation, we define two time-evolution operators:
U⫾共tf,ti兲 = T+exp
再
− i冕
ti tf
dt
⬘
H⫾共t⬘
兲冎
. 共9兲 Associated with every Schrödinger picture operator, we de-fine two Heisenberg operators, one corresponding to evolu-tion with each of the two HamiltoniansH⫾:Q⫾共t兲 = U⫾共tf,ti兲†QU⫾共tf,ti兲. 共10兲
In order to have the tools of the Keldysh formalism at our disposal, we need to define four Green’s functions as fol-lows: gm,n++共z,t;z
⬘
,t⬘
兲 = − eATr兵U+共t1,t0兲T+关n+† 共z⬘
,t⬘
兲m+共z,t兲兴 ⫻0关U−共t1,t0兲兴†其, u u Gin Gin Gin Gout Gout Gout 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 z− z+FIG. 1. We consider a general scatterer connected to reservoirs. The top figure is a diagram of one possible physical realization of a scatterer. Channels carry electrons toward and away from a scatter-ing region共shaded dark gray兲, where interchannel scattering takes place. Reservoirs are characterized by Keldysh Green’s functions
Gin共out兲. These Green’s functions also carry a channel index in order to account for, among other things, voltage biasing. In setups such as the the quantum Hall experiment where there is a Hall voltage,
Ginwill differ from Gout, while in an ordinary QPC, the two will be identical. The bottom figure shows how the physical setup is repre-sented in our model. Channels are unfolded so that all electrons enter at z−and leave at z+.
gm,n+−共z,t;z
⬘
,t⬘
兲 = eATr兵U+共t1,t0兲m+共z,t兲0n− † 共z⬘
,t⬘
兲 ⫻关U−共t 1,t0兲兴†其, gm,n−+共z,t;z⬘
,t⬘
兲 = eATr兵U+共t1,t0兲n+† 共z⬘
,t⬘
兲0m−共z,t兲 ⫻关U−共t 1,t0兲兴†其, gm,n−−共z,t;z⬘
,t⬘
兲 = eATr兵U+共t 1,t0兲0T-关n−† 共z⬘
,t⬘
兲m−共z,t兲兴 ⫻关U−共t 1,t0兲兴†其. 共11兲Here, the symbolT+orders operators with larger time argu-ments to the left. If permutation is required to obtain the time-ordered form, the product is multiplied with 共−1兲n, where n is the parity of the permutation. Similarly,T-
anti-time-orders with the same permutation parity convention. The Green’s functions can be grouped into a matrix in Keldysh space as follows:
gm,n共z,t;z
⬘
,t⬘
兲 =冉
gm,n++共z,t;z
⬘
,t⬘
兲 gm,n
+−共z,t;z
⬘
,t⬘
兲gm,n−+共z,t;z
⬘
,t⬘
兲 g−−m,n共z,t;z⬘
,t⬘
兲冊
. 共12兲 Notation can be further shortened by incorporating channel indices into the matrix structure of the Green’s function, thereby defining an object g¯共z,t;z⬘
, t⬘
兲. The element of g¯ that is located on row m and column n is the 2⫻2 matrix gm,n.The Green’s function satisfies the equation of motion
兵it+vFiz−vF¯u共z,t兲其g¯共z,t;z
⬘
,t⬘
兲 −冕
dt⬙
⌺共z;t − t⬙
兲g¯共z,t⬙
;z⬘
t⬘
兲 =␦共t − t⬘
兲␦共z − z⬘
兲1¯. 共13兲The delta functions on the right of Eq.共13兲 encode the fact
that due to time ordering, gmn
++
and gmn
−−
have a step structure, 1
vF
共z − z
⬘
兲␦冉
t − t⬘
−z − z⬘
vF
冊
␦mn+ f共z,t;z
⬘
t⬘
兲, 共14兲 where f is continuous in all its arguments. The self-energy⌺共z;兲 = − iG¯in共兲
2c
共z−− z兲 − iG¯out共兲
2c
共z − z+兲 共15兲
results from the reservoirs and determines how the scattering channels are filled. It is a matrix in Keldysh space. The time cis the characteristic time that correlations survive in the region of the conductor that is connected to the reservoirs before the reservoirs scramble them. G¯in共out兲共兲 is the reser-voir Green’s functions, where electrons enter 共leave兲 the scattering region, summed over reservoir levels, and
normal-ized to be dimensionless. This form of the self-energy can be derived from the following model for the reservoirs: we imagine every point z in a channel m outside共z−, z+兲 to ex-change electrons with an independent Fermion bath with a constant density of states. The termsHres andHT are
ex-plicitly Hres=
兺
m冕
dE冕
z⑀共z−,z+兲 dzEam †共E,z兲am共E,z兲, HT=兺
m cm冕
dE冕
z⑀共z−,z+兲 dzm †共z兲am共E,z兲 + am†共E,z兲m共z兲, 共16兲 where the tunneling amplitude cmcharacterizes the coupling between the reservoir and the channel m. More general res-ervoir models need not be considered, since, as we shall see shortly, the effect of the reservoirs is entirely contained in boundary conditions on the Green’s function g¯ inside thescatterer. This boundary condition does not depend on mi-croscopic detail, but only on the reservoir Green’s functions
G¯in共out兲.
We do not need to know the explicit form of the reservoir Green’s functions yet. Rather, the argument below exclu-sively relies on the property of G¯in共out兲 that it squares to
unity:27
冕
dt⬙
G¯ 共t − t⬙
兲in共out兲G¯ 共t⬙
− t⬘
兲in共out兲=␦共t − t⬘
兲1¯. 共17兲 A differential equation similar to Eq.共13兲 holds for g¯†.B. Varying the actionA
We are now ready to attack the generating functionalZ. For our purposes, it is most convenient to considerA=lnZ. We will call this object the action. Our strategy is as follows: we will obtain an expression for the variation␦A resulting from a variation uˆ共z,t兲→uˆ共z,t兲+␦uˆ共z,t兲 of the scattering
po-tentials. This expression will be in terms of the reservoir filling factors fˆ and the scattering matrices associated with
uˆ共z,t兲. We then integrate to find A.
We start by writing ␦A = − ivFeA
兺
m,n冕
t0 t1 dt冕
dz关␦un,m+ 共z,t兲具m †共z兲n共z兲典 +共t兲 −␦un,m− 共z,t兲具m†共z兲n共z兲典−共t兲兴, 共18兲 where 具m†共z兲n共z兲典+共t兲 = Tr冋
T+exp再
− i冕
t t1 dt⬘
H+共t⬘
兲冎
m†共z兲n共z兲T+exp再
− i冕
t0 t dt⬘
H+共t⬘
兲冎
0T-exp再
i冕
t0 t1 dt⬘
H−共t⬘
兲冎
册
,具m†共z兲n共z兲典−共t兲 = Tr
冋
T+exp再
− i冕
t0 t1 dt⬘
H+共t⬘
兲冎
0T-exp再
i冕
t0 t dt⬘
H−共t⬘
兲冎
m†共z兲n共z兲T-exp再
− i冕
t t1 dt⬘
H−共t⬘
兲冎
册
. 共19兲C. Expressing␦A in terms of the Green’s function g
In terms of the defined Green’s functions, the variation ␦A becomes ␦A = ivF
兺
m,n冕
t0 t1 dt冕
dz关␦un,m+ 共z,t兲gm,n++共z,t − 0+;z,t兲 +␦un,m− 共z,t兲gm,n−−共z,t + 0+;z,t兲兴 = ivF冕
t0 t1 dt冕
dz Tr关␦¯u共z,t兲g¯共z,t + 0k;z,t兲兴. 共20兲 The object␦¯ is constructed by combining the channel anduKeldysh indices of the variation of the potential. The trace is over both Keldysh and channel indices. The symbol 0krefers to the regularization explicitly indicated in the first line, i.e., the first time argument of g++共z,t−0+; z , t兲 is evaluated an infinitesimal time 0+⬎0 before the second argument, while in g−−共z,t+0−; z , t兲, the first time argument is evaluated an
infinitesimal time 0+after the second. This is done so that the
time ordering共anti-time-ordering兲 operations give the order of creation and annihilation operators required in Eq.共18兲.
It proves very inconvenient to deal with the 0k regulariza-tion of Eq.共20兲. It is preferable to have the first time
argu-ments of both g++and g−−evaluated an infinitesimal time 0+
before the second. Taking into account the step structure of
gˆ++, we have g ¯共z,t + 0k ;z
⬘
t⬘
兲 = g¯共z,t − 0+;z⬘
,t⬘
兲 + 1 vF ␦冉
t − t⬘
−z − z⬘
vF冊
1ˆ冉
1 −ˇ3 2冊
. 共21兲 Here,3 is the third Pauli matrix冉
1 0 0 − 1冊
acting in Keldysh space. The equations of motion allow us to relate g¯共z,t−0+; z
⬘
, t⬘
兲 for points z and z⬘
inside thescatter-ing region where u¯ is nonzero to the value of g¯ at z−where
electrons enter the scatterer. For z艋z
⬘
and t艋t⬘
, the equa-tions of motion giveg ¯
冉
z,t +z − z − vF − 0;z⬘
,t +z⬘
− z − vF冊
= s¯共z,t兲g¯共z−,t − 0+;z⬘
−,t⬘
兲s¯†共z⬘
,t⬘
兲, 共22兲 where s ¯共z,t兲 = Z exp再
− i冕
z− z dz⬙
¯u冉
z⬙
,t +z⬙
− z − vF冊
冎
. 共23兲 The symbol Z indicates that the exponent is ordered along the z axis, with the largest coordinate in the integrand to the left. Note that the potential u¯ at position z is evaluated at thetime instant t +共z−z−兲/vF that an electron entering the
scat-tering region at time t reaches z. Often the time dependence of the potential is slow on the time scale共z+− z−兲/vF,
repre-senting the time a transported electron spends in the scatter-ing region, and u¯共z,t+z−zv−
F 兲 can be replaced with u¯共z,t兲. This
is, however, not required for the analysis that follows to be valid.
Substitution into Eq.共24兲 yields
␦A =vF
冕
dt Tr关w¯共t兲g共z−,t − 0+;z−,t兲兴 −冕
dt lim t⬘→t ␦共t − t⬘
兲Tr冋
w¯共t兲1ˆ冉
1 −ˇ3 2冊
册
, 共24兲 with w ¯共t兲 = − i冕
z− z+ dzs¯†共z,t兲␦¯u冉
z,t +z − z − vF冊
s ¯共z,t兲 = s¯†共t兲␦¯s共t兲. 共25兲 In this equation, z+is located where electrons leave thescat-terer. Importantly, here, Tr still denotes a trace over channel and Keldysh indices. We will later on redefine the symbol to include also a trace over the 共continuous兲 time index, at which point the second term in Eq.共24兲 will 共perhaps
decep-tively兲 look less offensive, but not yet. In the last line of Eq. 共25兲, s¯共t兲=s¯共z+, t兲 is the 共time-dependent兲 scattering matrix.
We sent the boundaries t0and t1, over which we integrate in
the definition of the action, to −⬁ and ⬁, respectively, which will allow us to Fourier transform to frequency in a moment. The action remains well defined as long as the potentials u+
and u−only differ for a finite time.
D. Relating g inside the scattering region to g at reservoirs: Imposing boundary conditions implied by reservoirs
Our task is now to find g¯共z−, t − 0+; z−, t兲. Because of the
t − t
⬘
dependence of the self-energy, it is convenient to trans-form to Fourier space, whereg
¯共z,;z−,
⬘
兲 =冕
dtdt⬘
eitg
¯共z,t;z−,t
⬘
兲e−i⬘t⬘ ,G ¯
in共out兲共兲 =
冕
dteitG¯ 共t兲in共out兲. 共26兲In frequency domain, the property that G¯in共out兲 squares to unity is expressed as G¯in共out兲共兲2= 1¯. 关Due to the standard
conventions for Fourier transforms, the matrix elements of the identity operator in energy domain is 2␦共−
⬘
兲.兴 The equation of motion for z⬍z−reads再
− i + vFz+G¯in共兲
2c
冎
g
¯共z,;z−,
⬘
兲 = 0. 共27兲There is no inhomogeneous term on the right-hand side be-cause we restrict z to be less than z−. We thus find
g ¯共z−− 0+,;z−,
⬘
兲 = ei⌬z/vF exp冋
−G ¯ in共兲 2lc ⌬z册
⫻g¯共z−−⌬z,;z−,⬘
兲. 共28兲Here, the correlation length lcis the correlation timec mul-tiplied by the Fermi velocityvF. By using the fact that G¯ 共兲in
squares to unity, it is easy to verify that
exp
再
−G ¯ in共兲 2lc ⌬z冎
=1 + G ¯ in共兲 2 exp冉
− ⌬z 2lc冊
+1 − G ¯ in共兲 2 exp冉
⌬z 2lc冊
. 共29兲 Since spacial correlations decay beyond z−, g¯共z−−⌬z,;z−,
⬘
兲 does not blow up as we make ⌬z larger. Fromthis we derive the condition 关1 + G¯
in共兲兴g¯共z−− 0+,;z−,
⬘
兲 = 0. 共30兲Transformed back to the time domain, this reads
冕
dt⬙
关␦共t − t⬙
兲 + G¯in共t − t⬙
兲兴g¯共z−− 0+,t⬙
;z−,t⬘
兲 = 0.共31兲 We can play the same game at z+, where particles leave the
scatterer. The equation of motion reads
再
− i + vFz+共z − z+兲 G¯out共兲 2c冎
g ¯共z,;z+,⬘
兲 = 2␦共z − z⬘
兲␦共 − ⬘
兲. 共32兲 This has the general solutiong ¯共z,;z
⬘
,⬘
兲 = exp再
iz − z⬘
vF −关共z − z+兲共z − z+兲 − 共z⬘
− z+兲 ⫻共z⬘
− z+兲兴G ¯ out共兲 2lc冎
冋
¯g共z⬘
− 0 +,⬘
;z⬘
,⬘
兲 +2 vF 共z − z⬘
兲␦共 − ⬘
兲册
. 共33兲 We will need to relate the Green’s function evaluated at z⬍z+ to the Green’s function evaluated at z⬎z+, and so we
explicitly show the inhomogeneous term. The same kind of argument employed at z−then yields the condition
关1 − G¯ out共兲兴
冋
¯g共z+− 0+,;z+,⬘
兲 + 2 vF ␦共 − ⬘
兲册
= 0, 共34兲 where the inhomogeneous term in the equation of motion is responsible for the delta function. In time domain, this reads冕
dt⬙
关␦共t − t⬙
兲 − G¯out共t − t⬙
兲兴冋
¯g共z+− 0+,t⬙
;z+,t⬘
兲+ 1
vF
␦共t
⬙
− t⬘
兲册
= 0. 共35兲 It remains for us to relate g¯共z+− 0+, t +z+−z−vF ; z
+, t
⬘
+z+−z−vF 兲 to g
¯共z−− 0+, t ; z−, t
⬘
兲. This is done with the help of Eq. 共22兲,from which follows
g ¯
冉
z+− 0+,t +z +− z− vF ;z+,t⬘
+z +− z− vF冊
= s¯共t兲g¯共z−− 0+,t;z−,t⬘
兲s¯†共t⬘
兲. 共36兲We substitute this into Eq.共35兲, multiply from the right with s
¯共t
⬘
兲 and from the left with s¯†共t兲. If we define G¯ out⬘
共t,t⬘
兲 = s¯†共t兲G¯out共t−t
⬘
兲s¯共t⬘
兲, the resulting boundary condition is冕
dt⬙
关␦共t − t⬙
兲 − G¯out⬘
共t − t⬙
兲兴冋
¯g共z−− 0+,t⬙
;z−,t⬘
兲+ 1
vF
␦共t
⬙
− t⬘
兲册
= 0. 共37兲E. Finding the variation of the action in terms of the reservoir Green’s functions and the scattering matrix
At this point, it is convenient to incorporate time into the matrix structure of the objects G¯in, G¯out
⬘
, and g¯. The resultingmatrices will be written without overbars. Thus, for instance,
s will denote a matrix diagonal in time indices, whose entry
共t,t
⬘
兲 is ␦共t−t⬘
兲s¯共t兲. Similarly, the 共t,t⬘
兲 entry of Gin共out兲 isG¯in共out兲共t−t
⬘
兲. Also, let g−be the matrix whose共t,t⬘
兲 entry isg ¯共z−− 0+, t ; z−, t
⬘
兲. In this notation, G in 2= G out⬘
2= I and Eqs.共31兲 and共37兲 read 共I + Gin兲g−= 0, 共I − Gout⬘
兲共g−+ 1/vF兲 = 0. 共38兲These two equations uniquely determine g−as follows: from
the first of the two equations, we have
0 = Gout
⬘
共I + Gin兲g−= −共I − Gout⬘
兲g−+共I + Gout⬘
Gin兲g−.共39兲 In the first term, we can make the substitution −共I−Gout
⬘
兲g−g−= − 1 vF 1 I + Gout
⬘
Gin 共I − Gout⬘
兲 = 1 vF 共1 − Gin兲 1 Gout⬘
+ Gin 共40兲 and the last line follows from the fact that Gin2= Gout⬘
2= I. We have taken special care here to allow for different reservoir Green’s functions at z−, where particles enter the conductor, and at z+, where they leave the conductor. In order to pro-ceed, we must now absorb the difference between the two Green’s functions in the scattering matrix. We define ⌳ through the equationG ¯
out=⌳−1Gin⌳ 共41兲
and drop the subscripts on the Green’s functions by setting
G⬅Gin. Substituting back into Eq. 共24兲 for the variation of
the action yields
␦A = Tr
冋
␦s⬘
共1 − G兲 1Gs
⬘
+ s⬘
G册
− Tr关␦sˆ−共sˆ−兲†兴, 共42兲
where the trace is over time, channel, and, in the first term, Keldysh indices. The operator s
⬘
is related to the scattering matrix s through s⬘
=⌳s.F. Integrating the variation to find the actionA
We now have to integrate ␦A to find A. This is most conveniently done by working in a basis where G is diago-nal. Since G2= 1, every eigenvalue of G is ⫾1. Therefore,
there is a basis in which
G =
冉
I 00 − I
冊
. 共43兲In this representation, s
⬘
can be written ass
⬘
=冉
s11⬘
s12⬘
s21
⬘
s22⬘
冊
. 共44兲Here, the two indices of the subscript have the following meaning: the first refers to a left eigenspace of G, the second to a right eigenspace. The subscript 1 denotes the subspace of eigenstates of G with eigenvalue 1. The subscript 2 refers to the subspace of eigenstates of G with eigenvalue −1. In this representation, 共1 − G兲 1 Gs
⬘
+ s⬘
G=冉
0 0 0 共s22⬘
兲−1冊
, 共45兲 so that ␦A = Tr关␦s22⬘
共s22−1兲⬘
兴 − Tr关␦sˆ−共sˆ−兲†兴, 共46兲 and thus A = Tr ln s22⬘
− Tr ln s−, eA=共Det s−兲−1Det s22⬘
. 共47兲In these equations, s−is the scattering matrix associated with
H−as defined previously. Its time structure is to be included
in the operations of taking the trace and determinant.
Note that in the representation where G is diagonal, the following holds: 1 + G 2 + s
⬘
1 − G 2 =冉
I s12⬘
0 s22⬘
冊
. 共48兲 Due to the upper-共block兲-triangular structure, it holds that Det s22⬘
= Det关1+G2 + s⬘
1−G2 兴, leading to our main resultA = Tr ln
冋
1 + G2 + s
⬘
1 − G2
册
− Tr ln s−, 共49兲 where it has to be noted that many matrices have the same determinant as the above. Some obvious examples include冉
I 00 s22
⬘
冊
=共1 + G兲/2 + 共1 − G兲s⬘
共1 − G兲/4,冉
I 0s21
⬘
s22⬘
冊
=共1 + G兲/2 + 共1 − G兲s⬘
/2. 共50兲 III. TRACING OUT THE KELDYSH STRUCTUREUp to this point, the only property of G that we relied on was the fact that it squares to identity. Hence, the result关Eq. 共49兲兴 holds in a setting that is more general than that of a
scatterer connected to reservoirs characterized by filling fac-tors.共The reservoirs may, for instance, be superconducting.兲 In the specific case of reservoirs characterized by filling fac-tors, the following holds:
G¯ 共兲 =
冕
d2e
−i
冉
1 − 2fˆ共兲 2fˆ共兲2 − 2fˆ共兲 − 1 + 2fˆ共兲
冊
. 共51兲 Here, fˆ共⑀兲 is the diagonal in channel indices and fm共⑀兲 is the filling factor in the reservoir connected to channel m. We will also assume that electrons enter and leave a channel from the same reservoir, so that Gin= Goutand, hence, s⬘
= s. We recall as well that the Keldysh structure of the scattering matrix iss =
冉
sˆ+ 00 sˆ−
冊
. 共52兲
Here, sˆ⫾ have channel and time 共or, equivalently, energy兲 indices. sˆ⫾is diagonal in time indices, with the entries on the time diagonal the time-dependent scattering matrices corre-sponding to an evolution with the HamiltoniansH⫾.
With this structure in Keldysh space, we find
eA= Det
冉
1 +共sˆ+− 1兲fˆ −共sˆ+− 1兲fˆ 共sˆ−− 1兲共fˆ − 1兲 sˆ−共1 − fˆ兲 + fˆ冊
Det
冉
1sˆ−−1
冊
.共53兲 We can remove the Keldysh structure from the determinant with the aid of the general formula
Det
冉
A BC D
冊
= Det共AD − ACA−1B兲 = Det共DA − CA−1BA兲.
共54兲 Noting that in our case the matrices B and A commute, so that CA−1BA = CB, we have
eA= Det兵关sˆ−共1 − fˆ兲 + fˆ兴关1 + 共sˆ − 1兲fˆ兴 − 关sˆ−共1 − fˆ兲 + fˆ − 1兴
⫻共sˆ+− 1兲fˆ其Det共sˆ−−1兲 = Det关sˆ−共1 − fˆ兲 + sˆ+fˆ兴Det共sˆ−−1兲.
共55兲
IV. AN EXAMPLE: FULL COUNTING STATISTICS OF TRANSPORTED CHARGE
A determinant formula of this type appears in the litera-ture of full counting statistics7 of transported charge. This formula can be stated as follows: the generating function for transported charge through a conductor characterized by a time-independent scattering matrix sˆ is
Z共兲 = Det关1 + 共sˆ−†sˆ− 1兲fˆ兴, 共56兲
where sˆ is a scattering matrix, modified to depend on the counting fieldthat, in this case, is time independent. 共The precise definition may be found below.兲
As a consistency check of our results, we apply our analy-sis to rederive this formula. We will consider the most gen-eral setup, where every scattering channel is connected to a distinct voltage-biased terminal. To address the situation where leads connect several channels to the same terminal, the voltages and counting fields associated with channels in the same lead are set equal.
The full counting statistics of charge transported through a scatterer in a time interval t is defined as
Z共,t兲 = 具eiHt
e−iH−t典. 共57兲
In this equation, the HamiltonianHis given by
H=vf
兺
m,n冕
dzm†共z兲兵− iz␦m,n+ um,n共z兲其n共z兲 +兺
m mIm共z0兲, 共58兲where Im共z0兲 is the current in channel m at the point z0,
which is taken to lie outside the scattering region. The full counting statistics is thus generated by coupling the counting fieldmto the current operator in a channel m.
Explicitly, the current operator in channel m is given by
Im共z0兲 = vF关m
†共z
0兲m共z0兲 −m
†共− z
0兲m共− z0兲兴. 共59兲
To understand this equation, recall that the coordinates z0and −z0 in channel m refer to the same point in space, but oppo-site propagation directions.
The presence of current operators in Eq.共58兲 can be
in-corporated in the potential by defining a transformed poten-tial
um,n共兲共z兲 = um,n共z兲 +␦m,n m
2 关␦共z − z0兲 −␦共z + z0兲兴. 共60兲 Introducing counting fields that transform H0→H is thus
achieved by transforming u→u共兲.
The calculation of the full counting statistics has now been cast into the form of the trace of a density matrix after forward and backward time evolutions controlled by differ-ent scattering potdiffer-entials. Our result 关Eq. 共55兲兴 is therefore
applicable, with
sˆ⫾=Z exp
冉
− i冕
z−z+
dzuˆ共⫾兲共z兲
冊
= e⫿iˆ/2s0e⫾iˆ/2= s⫾.共61兲 In this equation,ˆ is a diagonal matrix in channel space, with
entries␦m,nm. Substitution into Eq.共55兲 gives
Z共兲 = Det关1 + 共sˆ−†sˆ− 1兲fˆ兴, 共62兲
in agreement with the existing literature.7
V. TRACING OUT THE CHANNEL STRUCTURE
A large class of experiments and devices in the field of quantum transport is based on two terminal setups. In such a setup, the channel space of the scatterer is naturally parti-tioned into a left and a right set, each connected to its own reservoir. We are generally interested in transport between left and right as opposed to internal dynamics on the left-or right-hand sides. The scattering matrices have the general structure sˆ⫾= X
冉
r t⬘
t r⬘
冊
X −1, X =冉
XL ⫾ XR⫾冊
. 共63兲Here, r共r
⬘
兲 describes left 共right兲 to left 共right兲 reflection, while t共t⬘
兲 describes left 共right兲 to right 共left兲 transmission 共t is not to be confused with time兲. These matrices have no time or Keldysh structure but still have a matrix structure in the space of left or right channel indices. The operators XL⫾共兲 and XR⫾共兲 have a diagonal Keldysh structure 共denoted by the superscript⫾兲 and a diagonal time structure 共here indicated byto avoid confusion with the transmission matrix t兲. They do not have an internal channel structure and, as a result, the Keldysh action is insensitive to the internal dynamics on the left- or right-hand sides. Our shorthand for the Keldysh scat-tering matrix will be XsX−1, where we remember that s has no Keldysh structure.We now consider the square of the generating functional
Z and employ the first expression we obtained for it 关Eq.
共49兲兴, which retains the Keldysh structure in the determinant, Z2= Det
冋
1 + G 2 + XsX −11 − G 2册
2 Det s†. 共64兲 Here, we exploited the fact that sˆ− acts on half of Keldyshspace together with the fact that sˆ+= sˆ−, i.e., s has no Keldysh
structure, to write exp 2 Tr ln sˆ−= Det s. We now shift X to act on G and define
Gˇ = X−1GX, P =1 + G ˇ
2 , Q =
1 − Gˇ
2 . 共65兲
The operators P and Q are complementary projection opera-tors, i.e., P2= P, Q2= Q, PQ = QP = 0, and P + Q = I. Because of this, it holds that Det共P+sQ兲=Det共P+Qs兲. Thus, we find
Z2= e2A= Det共Ps†+ sQ兲. 共66兲
The left channels are all connected to a single reservoir, while the right channels are all connected to a different res-ervoir. This means that the reservoir Greens function has a channel space structure
Gˇ =
冉
G ˇL
GˇR
冊
, 共67兲
where GLand GRhave no further channel space structure. At this point, it is worth explicitly stating the structure of opera-tors carefully. In general, an operator carries Keldysh indi-ces, indices corresponding to left and right, channel indices
within the left or right sets of channels, and time indices. However P, Q, and s are diagonal or even structureless, i.e., proportional to identity in some of these indices. Let us de-note Keldysh indices with k , k
⬘
苸兵+,−其, left and right with ␣,␣⬘
苸兵L,R其, channel indices within the left or right sets with c , c⬘
苸Z, and time t,t⬘
苸R. Then P has the explicit formP共k,k
⬘
;␣,␣⬘
;c,c⬘
;t,t⬘
兲 = P共k,k⬘
;␣;t,t⬘
兲␦␣,␣⬘␦c,c⬘. 共68兲 The projection operator Q has the same structure. The scat-tering matrix s has the structures共k,k
⬘
;␣,␣⬘
;c,c⬘
;t,t⬘
兲 = s共␣,␣⬘
;c,c⬘
兲␦k,k⬘␦共t − t⬘
兲. 共69兲 We now use the formulaDet
冉
A BC D
冊
= Det共A兲Det共D − CA−1B兲
to eliminate the left-right structure from the determinant.
Z2=
冉
PLr †+ Q Lr PLt†+ QRt⬘
PRt⬘
†+ QLt PRr⬘
†+ QRr⬘
冊
= Det共PLr† + QLr兲Det关PRr⬘
†+ QRr⬘
−共PRt⬘
†+ QLt兲共PLr†−1+ QLr−1兲共PLt†+ Qrt⬘
兲兴 = Det共PLr†+ Q Lr兲 a Det关PR共r⬘
†− P Lt⬘
†r†−1t†兲 + 共r⬘
− QLtr−1t⬘
兲QR− PR共PLt⬘
†r†−1t⬘
+ QLt⬘
†r−1t⬘
兲QR兴. b 共70兲Here, it is important to recognize that the reflection and transmission matrices commute with the projection operators
PL,R and QL,R. Furthermore, notice that, in term b, the pro-jection operator PRalways appears on the left of any product involving other projectors, while QR always appears on the right. This means that in the basis where
PR=
冉
I 0
0 0
冊
, QR=冉
0 00 I
冊
, 共71兲term b is the determinant of an upper block-diagonal matrix. As such, it only depends on the diagonal blocks, so that the term PR共...兲QR may be omitted. Hence,
b = Det关PR共r
⬘
†− PLt⬘
†r†−1t†兲 + 共r⬘
− QLtr−1t⬘
兲QR兴.共72兲 Now we invoke the so-called polar decomposition of the scattering matrix,28
r = u冑1 − Tu
⬘
, t⬘
= iu冑Tv,
t = iv
⬘
冑Tu
⬘
, r⬘
=v⬘
冑
1 − Tv, 共73兲where u, u
⬘
,v, and v⬘
are unitary matrices and T is a diag-onal matrix with the transmission probabilities Tnon thedi-agonal. We evaluate term a in the basis where PLand QLare diagonal to find
a = Det
冉
u⬘
†
冑
1 − Tu† 00 u冑1 − Tu
⬘
冊
= Det共I冑
1 − T兲, 共74兲 where I = PL+ QL= PR+ QL is the identity operatorI共k,k
⬘
; c , c⬘
; t , t⬘
兲=␦k,k⬘␦c,c⬘␦共t−t⬘
兲 in Keldysh, channel, and time indices. For term b, we findb = Det
冋
PR冉
冑
1 − T + PLT
冑
1 − T冊
+冉
冑
1 − T + QL冑
T1 − T
冊
QR册
. 共75兲 Combining the expressions for a and b, we findZ2= e2A= Det关1 − T共PRQ
L+ PLQR兲兴. 共76兲 Using the fact that PL共R兲=共1+GˇL共R兲兲/2 and QL共R兲=共1 − GˇL共R兲兲/2 and taking the logarithm, we finally obtain the remarkable result
A =1
2
兺
nTr ln
冋
1 +Tn 4共兵GˇL,GˇR其 − 2兲
册
. 共77兲This formula was used in Ref. 14 to study the effects on transport of electromagnetic interactions among electrons. In Ref.16, the same formula was employed to study the output of a two-level measuring device coupled to the radiation emitted by a quantum point contact共QPC兲.
VI. FERMI-EDGE SINGULARITY
In this section, we show how our formulas apply to a phenomenon known as the Fermi-edge singularity. The sys-tem under consideration is one of the most elementary ex-amples of an interacting electron system. The initial analysis8,9relied on diagrammatic techniques rather than on the scattering approach or the Keldysh technique, and was confined to equilibrium situations. Several decades later, the problem was revisited in the context of the scattering approach.12,13 An intuitive derivation of a determinant for-mula was given. Here, we apply our approach to confirm the validity of this previous work. We find exact agreement. This highlights the fact that the determinant formulation of the FES problem is also valid for multichannel devices out of equilibrium, an issue not explicitly addressed in the existing literature.
The original problem8,9 was formulated for conduction electrons with a small effective mass and valence electrons with a large effective mass, bombarded by x rays. The x rays knock one electron out of the valence band, leaving behind an essentially stationary hole. Until the hole is refilled, it interacts through the Coulomb interaction with the conduc-tion electrons. The x-ray absorpconduc-tion rate is studied. Abanin and Levitov reformulated the problem in the context of quan-tum transport, where an electron tunnels into or out of a small quantum dot that is side-coupled to a set of transport channels.
We prefer to consider a slightly simpler setup that exhibits the same physics. The setup is illustrated in Fig.2. A QPC interacts with a charge qubit. The shape of the QPC constric-tion depends on the state of the qubit. The Hamiltonian for the system is
H = H1兩1典具1兩 + 共H2+兲兩2典具2兩 +␥共兩1典具2兩 + 兩2典具1兩兲. 共78兲
The operatorsH1共H2兲 describe the QPC electrons when the
qubit is in state兩1典 共兩2典兲. They differ by a potential energy
term, describing the pinching off of the QPC constriction depending on the state of the qubit. We may take both Hamiltonians to be of the form关Eq. 共5兲兴 that we wrote down
for a general scatterer. The energy is the qubit level split-ting, an experimentally tunable parameter. The QPC may or may not be driven by a voltage bias V.
QPC electrons do not directly interact with each other, but rather with the qubit. This interaction is the only qubit relax-ation mechanism included in our model. We work in the limit ␥→0, where the inelastic transition rates ⌫12,21between
qu-bit states are small compared to the energies eV and. In this case, the qubit switching events can be regarded as indepen-dent and incoherent.
Now consider the qubit transition rate⌫21from state兩1典 to
兩2典 as a function of the qubit level splitting . To lowest order in the tunneling amplitude␥, it is given by
⌫21= 2␥2Re
冕
−⬁ 0 dei lim t0→−⬁ expA共兲, expA共兲 = Tr关eiHˆ2e−iHˆ1共−t0兲0e−iH
ˆ
1t0兴. 共79兲
This is the usual Fermi golden rule. The timeover which we integrate can be interpreted as the time when the qubit switches from兩1典 to 兩2典. The trace is over QPC states, and0 is the initial QPC density matrix. We see that the expression for⌫21contains an instance of the Keldysh actionA that we
have calculated. The correspondence requires us to set
H+共t兲 = H
1+共H2−H1兲共t −兲共− t兲,
H−共t兲 = H
1. 共80兲
In order to conform to the conventions of the existing litera-ture, we writeZ in the form where the Keldysh structure has been removed关Eq. 共4兲兴:
A共兲 = Tr ln关sˆ−共1 − fˆ兲 + sˆ+共兲fˆ兴 − Tr ln sˆ−. 共81兲
In this formula, sˆ−is the scattering matrix corresponding to
H−=H
1 when the qubit is in state 兩1典. It is proportional to
identity in time indices. The scattering matrix sˆ+共兲
corre-sponds to H+. It is still diagonal in time indices, but the
diagonal elements sˆ+共兲t are time dependent. If we take the time it takes an electron to traverse the conductor to be much shorter than other time scales, such as the attempt rate of charge transfers, then
sˆ+共兲t= sˆ1+共sˆ2− sˆ1兲共t −兲共− t兲, 共82兲
where sˆ2 is the scattering matrix associated with H2 when
the qubit is in state兩2典. This expression first appeared in Ref.
12. In the language of the original diagrammatic treatment of the FES problem,8,9it represents the total closed loop contri-bution.
We may also write this closed loop contribution as
eA共兲= Det关1 + 共sˆ1†sˆ2− 1兲⌸ˆ共兲fˆ兴, 共83兲
where⌸ˆ is a diagonal operator in time domain with a kernel that is a double step function,
a 1 2 V ε b 1 2 V ε
FIG. 2. 共Color online兲 A schematic picture of the system con-sidered. It consists of a charge qubit coupled to a QPC. The shape of the QPC constriction, and hence its scattering matrix, depends on the state of the qubit. A gate voltage controls the qubit level split-ting. There is a small tunneling rate␥ between qubit states.
⌸共兲t,t⬘=共− t兲共t −兲, 共84兲 and the scattering matrices sˆ1and sˆ2no longer have any time
structure. We may work in the channel space basis where
sˆ1†sˆ2 is diagonal. Its eigenvalues are eik. Suppose we are in zero-temperature equilibrium, then the filling factor f is the same in every channel. In the Fourier transformed energy basis, f is simply a step function:
f,⬘=␦共 −
⬘
兲共− 兲. 共85兲 Thus, one findseA=
兿
kDet关1 + 共eik− 1兲⌸ˆ共兲fˆ兴. 共86兲 This determinant contains no channel structure any more. Operators only have one set of indices共time or, after Fourier transform, energy兲. ⌸ˆ is a projection operator, diagonal in time domain, while fˆ is a projection operator in energy do-main. Such a determinant is known as a Fredholm determi-nant.
The resulting transition rate is8,9,12
⌫21共兲 =共−⑀兲 1 兩⑀兩
冉
兩兩 Eco冊
␣ , 共87兲where Ecois a cutoff energy of the order of the Fermi energy
measured from the bottom of the conduction band. The ex-ponent␣is known as the orthogonality exponent. It may be calculated by analytically evaluating the Fredholm determi-nant with the Wiener–Hopf method. It is given in terms of the scattering matrices as11,12
␣= 1 42兩Tr ln 2共s 1 †s 2兲兩, 共88兲
with the trace being over channel indices. Inspired by the work of Abanin and Levitov,12,13 we considered the case where the QPC is driven by a voltage bias. The results of our study may be found in Ref.17.
VII. CONCLUSION
In this paper, we have derived several expressions for the Keldysh action A for a general multiterminal, time-dependent scatterer. This object is defined as the共logarithm of the兲 trace of the density matrix of the scatterer after an evolution forward and backward in time with different Hamiltonians: eA= Tr
冋
T+exp再
− i冕
t0 t1 dtH+共t兲冎
⫻0T−exp再
i冕
t0 t1 dtH−共t兲冎
册
. 共89兲Our main result is a compact formula for the action in terms of reservoir Green’s functions and the scattering matrix of the scatterer 关Eq. 共1兲兴. We have shown how to explicitly
perform the trace over Keldysh indices when reservoirs are characterized by filling factors. Thus, we obtained a formula 关Eq. 共2兲兴 generalizing the Levitov counting statistics formula.
We have also explicitly performed the trace over channel indices for a two terminal scatterer关Eq. 共4兲兴. In this case, we
demonstrated that the Keldysh action only depends on the scattering matrix through the eigenvalues of the transmission matrix. To illustrate the utility of the Keldysh action, and confirm the correctness of our results, we considered full counting statistics and the Fermi-edge singularity. We found that our results agree with the existing literature.
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