Non-equilibrium current cumulants and moments with a point-like defect

0.35 MB·

by  Mihail Mintchev

January 2016
Non-equilibrium Current Cumulants
and Moments with a Point-like Defect
6
1
0
2

n Mihail Mintchev
a
J
Istituto Nazionale di Fisica Nucleare and Dipartimento di Fisica, Universit`a di Pisa, Largo

8
Pontecorvo 3, 56127 Pisa, Italy

]
h
Luca Santoni
c
e
m Scuola Normale Superiore and Istituto Nazionale di Fisica Nucleare, Piazza dei Cavalieri
7, 56126 Pisa, Italy
-
t
a
t Paul Sorba
s
.
t
a LAPTh, Laboratoire d’Annecy-le-Vieux de Physique Th´eorique, CNRS, Universit´e de
m
Savoie, BP 110, 74941 Annecy-le-Vieux Cedex, France
-
d
n
o
c
[
Abstract

1
v
9 We derive the exact n-point current expectation values in the Landauer-Bu¨ttiker
1
non-equilibriumsteadystateofamultiterminalsystemwithstargraphgeometryanda
8
1 point-like defect localised in the vertex. The current cumulants are extracted from the
0 connected correlation functions and the cumulant generating function is established.
.
1 We determine the moments, show that the associated moment problem has a unique
0
solution and reconstruct explicitly the corresponding probability distribution. The
6
1 basic building blocks of this distribution are the probabilities of particle emission and
: absorption from the heat reservoirs, driving the system away from equilibrium. We
v
i derive and analyse in detail these probabilities, showing that they fully describe the
X
quantum transport problem in the system.
r
a
IFUP-TH 1/2016
LAPTH-069/15
1 Introduction
Current fluctuations represent a fundamental characteristic feature of non-equilibrium quan-
tum transport. A complete information about these fluctuations is provided by the cumu-
lants of the particle current, which generalise the quadratic noise fluctuations to n 3
n
C ≥
currents. For this reason the study of the sequence : n = 1,2,... attracted much
n
{C }
attention in last two decades. Following the fundamental work of Khlus [1] and Levitov,
Lesovik and Chtchelkatchev [2]-[4], there has been a series of contributions studying various
systems [5]-[14] and different non-equilibrium situations. Several examples are discussed in
the proceedings [15] as well as in the review papers [16]-[18] and the references therein.
✤R ✜
2
✤R ✜
1
✣......✂✂ ✢
. . ✂✂✂✌L!✠2....L..!!1✣✢
L . !
✤R ✜......✲i ✛S ✘
i .
. .
✣✢ ✚L❅❅✙■......
N
❅❅
✤R ✜
N
✣✢
Figure 1: N-terminal junction with scattering matrix S.
In this paper we consider the particle current fluctuations for general class of point-like
defects. More precisely we investigate a system with N terminals with the geometry of a
star graph as shown in Fig. 1. Each of the N semi-infinite leads is attached at infinity
to the a heat reservoir R with (inverse) temperature β and chemical potential µ . The
i i i
defect, which drives the system out of equilibrium, is localised in the vertex of the graph
and is described by a N N unitary scattering matrix S. Although relatively simple, the
×
system in Fig. 1 represents a remarkable laboratory for studying a large class of intriguing
quantum phenomena. The deep relation between the particle density cumulants and the
R´enyi entanglement entropies in an equilibrium configuration at zero temperature has been
investigated in [19]. The study of the first cumulant of the particle and heat currents away
from equilibrium shows [20] that the junction transforms heat to chemical potential energy
and vice versa, depending on the parameters of the heat baths. The explicit form of the
second cumulant in the scale invariant limit reveals [21] the existence of nonlinear effects,
2
C
which lead to reduction or enhancement of the particle and heat noises in certain ranges of
the chemical potentials. In what follows we pursue further the investigation of the junction
in Fig. 1, adopting the following strategy. In a quantum field theory framework we first
derive in explicit and closed form the particle current cumulants for generic n. We use
n
C
this information for reconstructing both the cumulant and the moment generating functions.
From these data we finally recover the associated probability distribution, which captures
the microscopic characteristic features of the system. In this respect we explicitly determine
1
the probability p of a particle with given energy to be emitted from the heat reservoir R
ij i
and absorbed by R . It turns out that p are nontrivial in both cases i = j and i = j. Our
j ij
6
investigation covers all point-like defects in the vertex, which are compatible with a unitary
time evolution in the bulk generated by the Schro¨dinger Hamiltonian. We call these systems
Schro¨dinger junctions.
The paper is organised as follows. In the next section we first recall the form of the par-
ticle current in the presence of a point-like defect. Afterwards we derive the exact connected
n-point current correlation functions in the Landauer-Bu¨ttiker non-equilibrium steady state.
The zero-frequency limit of these functions defines the cumulants . The cumulant gener-
n
C
ating function (λ) in the general case with N-terminals is reconstructed in section 3. We
C
establish here also the explicit form of (λ) in the scale invariant limit and describe the main
C
properties of the cumulants and in this regime. The last part of section 3 provides a
3 4
C C
comparison with other results in the subject. In section 4 we derive the moments in a single
energy channel, the associated probability distribution and the probabilities p mentioned
ij
above. Our conclusions are collected in section 5, whereas the appendices contain some
technical details.
2 Current correlation functions and cumulants
2.1 Particle current in the presence of defect
The system in Fig. 1 is localised on a star graph with coordinates (x,i),: x 0, i =
{ ≤
1,...,N , where x denotes the distance from the vertex and i labels the leads. The main
}
object of our investigation is the particle current j(t,x,i) flowing along the leads of the
junction. In order to determine j(t,x,i) one should fix both the dynamics in the bulk and
the boundary conditions in the vertex of the graph. We consider1
1
i∂ + ∂2 ψ(t,x,i) = 0, (2.1)
t 2m x
(cid:18) (cid:19)
with the boundary condition
N
lim [η(I U) +i(I+U) ∂ ]ψ(t,x,j) = 0, (2.2)
ij ij x
x 0− −
→ j=1
X
where U is a N N unitary matrix and η R is a parameter with dimension of mass.
× ∈
Equation (2.2) parametrises all self-adjoint extensions of the bulk Hamiltonian ∂2 to the
− x
whole graph and gives rise to non-trivial one-body interactions, which are described by the
scattering matrix [22]-[24]
[η(I U) k(I+U)]
S(k) = − − , (2.3)
−[η(I U)+k(I+U)]
−
1The natural units ~=c=kB =1 are adopted throughout the paper.
2
k being the particle momentum. We stress that the scattering matrices (2.3) provide a phys-
ical description of all point-like contact interactions among the leads, which are compatible
with a unitary time evolution in the bulk of the system. This is the fundamental requirement
selecting the class of defects considered in this paper.
The solution of (2.1,2.2) is given by
N dk k2
ψ(t,x,i) = ∞ e iω(k)tχ (k;x)a (k), ω(k) = . (2.4)
− ij j
2π 2m
j=1 Z0
X
where
χ(k;x) = e ikxI+eikxS( k) (2.5)
−
−
and a (k), a (k) : k 0, i = 1,...,N generate the standard canonical anticommutation
{ i ∗i ≥ }
relation algebra. With the above definitions the particle current takes the form2
i dk dp
j(t,x,i) = ∞ ∞ eit[ω(k) ω(p)]
−
2m 2π 2π
Z0 Z0
N
a (k) χ (k;x)[∂ χ ](p;x) [∂ χ ](k;x)χ (p;x) a (p). (2.6)
× ∗l ∗li x im − x ∗li im m
l,Xm=1 n o
Using the orthogonality and completeness of the system χ(k;x) : k 0,x 0 one can
{ ≥ ≥ }
prove [25] the operator Kirchhoff rule
N
j(t,0,i) = 0, (2.7)
i=1
X
which is a simple but fundamental feature of the currents flowing in the junction.
Besides the particle current operator, we have to fix also the state for evaluating the
current expectation values. The physical setting, presented in Fig. 1, is nicely described
by the Landauer-Bu¨ttiker (LB) [26, 27] non-equilibrium steady state Ω , defined in terms
β,µ
of (β ,µ ) and S(k). A simple and intuitive way [25] to construct this state is to use the
i i
scattering matrix S(k) in order to extend the tensor product of Gibbs states, relative to
the reservoirs R at the level of asymptotic incoming fields, to the outgoing fields. The
i
state, obtained in this way, has both realistic physical properties [26, 27] and interesting
mathematical structure [25, 28, 29]. The basic expectation values of a (k), a (k) in Ω ,
{ i ∗i } β,µ
which are needed in what follows, are reported in appendix A.
2.2 Current cumulants in the LB state
Let L be an arbitrary but fixed lead and let us consider the n-point correlation function
i
i(t ,x ,...,t ,x ) = j(t ,x ,i) j(t ,x ,i) , (2.8)
Wn 1 1 n n h 1 1 ··· n n iβ,µ
2The ∗ stands for Hermitian conjugation.
3
of the current (2.6), where denotes the expectation value in the LB state Ω . The
β,µ β,µ
h···i
n-th cumulant in L is defined by the connected part of (2.8),
i
i(t ,x ,...,t ,x ) = j(t ,x ,i) j(t ,x ,i) conn. (2.9)
Cn 1 1 n n h 1 1 ··· n n iβ,µ
For n = 1 the correlators (2.8,2.9) coincide and have the following well known [26, 27] time
and space independent form
N
dω 1
i = i = ∞ δ S (√2mω) 2 d (ω), d (ω) = . (2.10)
W1 C1 Z0 2π Xl=1 (cid:16) il −| il | (cid:17) l l 1+eβl(ω−µl)
The situation complicates for n 2. First of all the correlators (2.8,2.9) depend on the
≥
time differences tˆ t t : k = 1,...,n 1 , which reflects the invariance under time
k k k+1
{ ≡ − − }
translations of the LB state. Moreover, since the defect violates translation invariance in
space, (2.8,2.9) depend separately on all the coordinates x : l = 1,...,n . It is clear that
l
{ }
dealing with this large number of variables becomes complicated with growing of n. Also, it
turns out that most of them are marginal for the particle transfer we are interested in. One
possibility [15]-[18] to get rid of some space-time variables is the replacement
T
j(t ,x ,i) dt j(t ,x ,i), l = 1,...n, (2.11)
l l l l l
7−→ ∀
Z0
in (2.8,2.9). The operation (2.11) obviously simplifies the time dependence. Instead of the
(n 1) time variables tˆ , one has now only one, namely T. The final step in this scheme is
k
−
to study the system for T large enough. Unfortunately, in the presence of defect the above
procedure solves the problem only partially, because the x -dependence persists.
l
In this paper we adopt an alternative strategy, which generalises to n 3 the definition
≥
(see e.g. [16]) of zero-frequency noise. For n 2 we consider the Fourier transforms
≥
i(x ,...,x ;ν) = ∞ dtˆ ∞ dtˆ eiν(tˆ1+ tˆn−1) i(t ,x ,...,t ,x ), i = i, i ,
Zn 1 n 1··· n−1 ··· Zn 1 1 n n Zn Wn Cn
Z Z
−∞ −∞ (2.12)
and perform the zero-frequency limit
i = lim i(x ,...,x ;ν). (2.13)
Zn ν 0+Zn 1 n
→
Wewillshowbelowthatinthislimitthex -dependencedropsoutand i dependsexclusively
l Zn
on the scattering matrix S and the heat bath parameters (β ,µ ). In fact, using the explicit
l l
form of the current (2.6) and the correlation function (A.82) in appendix A, after some
algebra one finds
Ti (ω)d (ω) Ti (ω)d (ω) Ti (ω)d (ω)
l1l1 l1 l2l1 l2 ··· lnl1 ln
Wni = Z0∞ d2ωπ l1,.X..N,ln=1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −TTiil1l2((ωω...))ddel1((ωω)) TTil2il2(ω(ω...)d)dl2(ω(ω)) ··...· TTiilnl2((ωω...))ddln((ωω)) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,(2.14)
(cid:12) − l1ln l1 − l2ln l2 ··· lnln ln (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) e 4 e (cid:12)
where3
Ti (ω) = δ δ S (√2mω)S (√2mω), (2.15)
lm li mi − li mi
d (ω) is the Fermi distribution (2.10) of the reservoir R and
l l
eβl(ω−µl)
d (ω) = 1 d (ω) = . (2.16)
l − l 1+eβl(ω−µl)
As expected, the connecteed part of (2.14) simplifies and can be conveniently written in
terms of traces involving the matrices
Ai TiD, Bi Ti(I D), D diag[d (ω),d (ω),...,d (ω)]. (2.17)
1 2 n
≡ ≡ − ≡
One finds
dω
i = ∞ Tr Ai , (2.18)
C1 2π
Z0
dω (cid:2) (cid:3)
i = ∞ Tr AiCi Ci Bi , n 2, (2.19)
Cn 2π σ1σ2 ··· σn−2σn−1 ≥
Z0 σ∈XPn−1 (cid:2) (cid:3)
where the sum runs over all permutations of n 1 elements and
n 1
P − −
Ai, σ < σ ,
Ci = − i i+1 (2.20)
σiσi+1 ( Bi, σi > σi+1.
The trace representation (2.18,2.19) of the current cumulants with a point-like defect
represents a first basic result of our study. It is worth stressing that the above derivation of
i is purely field theoretical and makes no use of any kind of cumulant generating function.
Cn
We will show in section 3 that this function can be uniquely reconstructed from (2.18,2.19).
2.3 The two-lead junction cumulants
In order to better illustrate the compact expressions (2.18,2.19), it is instructive report the
explicit form of the first few cumulants in the case N = 2. Without loss of generality we can
concentrate on the the cumulants 1 in the lead L . For notational simplicity we omit here
Cn 1
3Here and in what follows the bar indicates complex conjugation
5
and in what follows the apex 1 in 1. By means of (2.18)-(2.20) one gets
Cn
dω
∞
= τc , (2.21)
1 1
C 2π
Z0
dω
= ∞ τ(c τc2), (2.22)
C2 2π 2 − 1
Z0
dω
= ∞ τ2c (1 3c +2τc2), (2.23)
C3 2π 1 − 2 1
Z0
dω
= ∞ τ2[c 3c2 +12τc2c 2τc2(2+3τc2)], (2.24)
C4 2π 2 − 2 1 2 − 1 1
Z0
dω
= ∞ τ3c [1+30c2 15c (1+4τc2)+4τc2(5+6τc2)], (2.25)
C5 2π 1 2 − 2 1 1 1
Z0
dω
= ∞ τ3 c [1+15c (2c 1)] 2τc2[8+15c (9c 5)]+
C6 2π { 2 2 2 − − 1 2 2 −
Z0
+ 120τ2c4(3c 1) 120τ3c6 , (2.26)
1 2 − − 1}
where the following combinations
c (ω) d (ω) d (ω) c (ω) d (ω)+d (ω) 2d (ω)d (ω), (2.27)
1 1 2 2 1 2 1 2
≡ − ≡ −
have been introduced for convenience. Moreover, the transmission probability associated
with the S-matrix (2.3) is given by
2mω(η η )2sin2(θ)
τ(ω) = S (√2mω) 2 = 1 − 2 , θ [0,2π), (2.28)
| 12 | (2mω +η2)(2mω +η2) ∈
1 2
with
η ηtan(α ), α [ π/2,π/2), (2.29)
i i i
≡ ∈ −
(e 2iα1,e 2iα2) being the eigenvalues of the matrix U entering the boundary condition (2.2).
− −
The main properties of and have been discussed in [26, 27], whereas the non-linear
1 2
C C
dependence on the chemical potentials has been examined in detail in [21]. Before analysing
some , we will face the problem of deriving a generating function for the cumulants
n 3
C ≥
(2.18,2.19) and the associated probability distribution.
3 Cumulant generating function
We show in this section that in spite of the complicated explicit form of the cumulants
(2.18,2.19), there exists a relatively simple and compact generating function of (λ). It
C
it instructive to start by extracting the information encoded in (2.21-2.26) about (λ).
C
FollowingthepioneeringworkofKhlus[1], LesovikandLevitov[2,3],welookforagenerating
function in the form
dω
(λ) = ∞ ln 1+F (τ,d ,d ) eiλf(τ) 1 +F (τ,d ,d ) e iλf(τ) 1 , (3.30)
12 1 2 21 1 2 −
C 2π − −
Z0
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
6
where F , F and f are unknown functions. Using the standard definition of generating
12 21
function
= ( i∂ )n (λ) (3.31)
Cn − λ C |λ=0
and the information from the first three cumulants , and only, one can easily deter-
1 2 3
C C C
mine F , F and f. The simple result
12 21
1 1
F = (c +c √τ), F = (c c √τ), f = √τ . (3.32)
12 2 1 21 2 1
2 2 −
leads to the following generating function
dω 1 1
(λ) = ∞ ln 1+ (c +c √τ) eiλ√τ 1 + (c c √τ) e iλ√τ 1
2 1 2 1 −
C 2π 2 − 2 − −
Z0 (cid:20) (cid:16) (cid:17) (cid:16) (cid:17)(cid:21)
dω
∞
= ln 1+ic √τ sin(λ√τ)+c cos(λ√τ) 1 . (3.33)
1 2
2π −
Z0
(cid:8) (cid:2) (cid:3)(cid:9)
One caneasily check also that (3.33) reproduces perfectly the cumulants , and as well
4 5 6
C C C
and represents therefore a valid candidate for the final result in the case of two leads. The
expression (3.33) has been reported without derivation also by Lesovik and Chtchelkatchev
[4]. Our goal below will be to generalise (3.33) to the multi terminal junction in Fig. 1, thus
recovering the N = 2 formula as a special case. In the comments at the end of this section
we will briefly describe an alternative to (3.33), regarding a slightly different setup.
3.1 General result for N terminals
The argument in what follows is based on the fact that the particle transport in our system
can be separated in statistically independent processes with fixed energy. In fact, excitations
with different energies propagate in the graph in Fig. 1 in a fully independent way, because
the only interaction, localised in the vertex, leaves the energy invariant (see (A.84)). For
this reason one can focus first on a single energy channel ω, thus dealing with a system with
finite degrees of freedom. This fact significantly simplifies the problem and allows to derive
explicitly the single energy channel cumulant generating function i(λ) in the lead L . The
Cω i
final step is to integrate over all energies,
dω
i(λ) = ∞ i(λ), (3.34)
C 2π Cω
Z0
using at this stage the well known property that the total cumulant of a process, which can
be decomposed in statistically independent subprocesses, is the sum of the cumulants of each
of the latter.
In order to obtain the particle current of a single energy channel ω 0, we modify the
≥
integration measure in the general expression of the particle current (2.6) according to
k2
dkdp dkdp(2π)2δ ω δ(k p), (3.35)
7−→ 2m − −
(cid:18) (cid:19)
7
which selects the contribution with energy ω. This operation leads to the simple time and
position independent expression
N
ji = a Ti (√2mω)a , (3.36)
ω ∗l lm m
l,m=1
X
where a ,a are standard fermionic oscillators:
{ i ∗i}
[a , a ] = δ , [a , a ] = [a , a ] = 0, a a = δ d (ω). (3.37)
i ∗j + ij i j + ∗i ∗j + h ∗i jiβ,µ ij j
Now, the generating function of cumulants in the channel ω and lead L is given by
i
i(λ) = ln eiλjwi . (3.38)
Cω h iβ,µ
The expectation value eiλjwi β,µ can be evaluated explicitly. The key points of the compu-
h i
tation, which leads to the final result
eiλjwi = det I+ eiλTi(√2mω) I D(ω) , D(ω) diag[d (ω),d (ω),...,d (ω)],
β,µ 1 2 n
h i − ≡
h (cid:16) (cid:17) i (3.39)
are given appendix C. In the lead L one finds
1
dω
(λ) = ∞ ln 1+iceff√τ sin λ√τ +ceff cos λ√τ 1 . (3.40)
C 2π 1 2 −
Z0
(cid:8) (cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:3)(cid:9)
Here
N
τ(ω) = τ (ω), τ (ω) S (√2mω) 2, (3.41)
i i 1i
≡ | |
i=2
X
is the total transmission probability between the lead L and the remaining N 1 leads L
1 i
−
and ceff are obtained from (2.27) by the substitution
1,2
N
τ (ω)
d (ω) deff(ω) i d (ω), (3.42)
2 −→ 2 ≡ τ(ω) i
i=2
X
which represents an effective distribution where d (ω) with i 2 are weighted by the ratios
i
≥
τ (ω)/τ(ω) [0,1]. As expected, the expression (3.40) reproduces (3.33) for N = 2.
i
∈
Summarising, we derived in explicit form the generating function of the cumulants (2.18,
2.19) for the Schro¨dinger junction with N > 2 leads. The novelty with respect to the two
terminal case (3.33) is the effective Fermi distribution (3.42), which captures the presence
of all N 1 > 1 reservoirs.
−
8
3.2 Scale invariant limit
The ω-integrationin (3.33,3.40) with general S-matrix of the form(2.3) cannot be performed
in a closed analytic form. For this reason it is instructive to select among (2.3) the scale-
invariant matrices, which incorporate the universal transport properties of the system [30]
while being simple enough to be analysed explicitly. The scale invariant (critical) elements
in the family (2.3) are fully classified [31] and belong to the orbits
S : U(N), S = diag( 1, 1, ..., 1) (3.43)
d ∗ d
{U U U ∈ ± ± ± }
of the adjoint action of the unitary group U(N) on the diagonal matrices S . As expected,
d
the critical S-matrices are ω-independent, which in the case β = β β allows to compute
1 2
≡
the integrals in (3.33,3.40) explicitly. In fact, introducing the variables
γ e βµj , Λ(λ) i√τ(γ γ )sin(λ√τ)+(γ +γ )cos(λ√τ), (3.44)
j − 2 1 2 1
≡ ≡ −
where τ = S 2 now is constant, one gets in the case N = 2
12
| |
1
(λ) = Li γ 1 +Li γ 1
C 2πβ 2 − 1− 2 − 2−
(
(cid:0) (cid:1) (cid:0) (cid:1) (3.45)
2 2
Li − Li − ,
2 2
− "Λ(λ) Λ2(λ) 4γ1γ2#− "Λ(λ)+ Λ2(λ) 4γ1γ2#)
− − −
p p
Li being the dilogarithm function.
2
The results of [21] suggest to investigate (3.45) as a function of
µ = (µ µ )/2, (3.46)
1 2
± ±
µ R playing the role of control parameter. For µ = 0 the expression (3.45) greatly
+ +
∈
simplifies in the low temperature limit β . In fact, one finds
→ ∞
µ
lim (λ) = | −| ln cos(λ√τ)+iε(µ )√τ sin(λ√τ) , (3.47)
β C |µ+=0 2π −
→∞
(cid:2) (cid:3)
ε being the sign function. The result (3.47) was derived for µ > 0 independently by
−
Levitov and Lesovik [3, 18] and provides therefore a valuable check on (3.45). For the first
few cumulants one gets from (3.47)
τµ τ(1 τ) µ τ2(τ 1)µ τ2(τ 1)(1 3τ) µ
1 = − , 2 = − | −| , 3 = − − , 4 = − − | −| .
C 2π C 2π C π C π
(3.48)
From (3.47) one infers that for any n 0
≥
, µ = 0, β , (3.49)
2n+1 1 +
C ∝ C → ∞
9