Noise and interactions in mesoscopic systems far from equilibrium

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Thesis

Reference

Noise and interactions in mesoscopic systems far from equilibrium

BORIN, Artyom

Abstract

This thesis is devoted to developing of an universal description of strongly interacting mesoscopic systems out of equilibrium. The nature of universalities we are looking for is twofold. Firstly, our goal is to find novel and simple dependences of physical observables on the experimental parameters. Secondly, we are looking for an universal theoretical approach that can be applied to a class of seemingly different strongly interacting physical problems. To achieve this generality, we rely on the bosonization technique, which allows one to account for interactions exactly. However, it is not trivial to apply this technique to out-of-equilibrium systems, because an injection of non-equilibrium electrons is naturally described at the fermion level, whereas the interactions are very elegantly incorporated within the bosonic framework. In this thesis, we discuss an approach that treats out-of-equilibrium corrections as perturbations. This allows us to stay within the framework of bosonization and to study non-equilibrium systems.

BORIN, Artyom. Noise and interactions in mesoscopic systems far from equilibrium . Thèse de doctorat : Univ. Genève, 2019, no. Sc. 5317

DOI : 10.13097/archive-ouverte/unige:115679 URN : urn:nbn:ch:unige-1156797

Available at:

http://archive-ouverte.unige.ch/unige:115679

Disclaimer: layout of this document may differ from the published version.

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UNIVERSIT É DE GEN ÈVE Département de physique théorique

FACULT ´E DES SCIENCES Professeur E. V. Sukhorukov

Noise and interactions in mesoscopic systems far from equilibrium

TH` ESE

pr´ esent´ ee ` a la Facult´ e des Sciences de l’Universit´ e de Gen` eve pour obtenir le grade de Docteur ` es sciences, mention Physique

par

Artem Borin

de Russie

Th` ese N

^o

5317

GEN` EVE

Atelier de reproduction ReproMail

2019

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Acknowledgments

First of all, I would like to acknowledge the contribution of my supervisor Prof. Eugene Sukhorukov. In the spring of 2014 me and my wife were applying to many European condensed matter groups to start our PhD at the same place. It was Eugene who gave us a chance to join his theoretical group together. After successful passing through the interview we were able to continue our research in the world class University of Geneva. Speaking about scientific side of our collaboration, Eugene exposed me to his extraordinary physical intuition and unique style, along with giving me contemporary theoretical knowledge about mesoscopic physics. Overall Eugene proposed me ideas for three physical problems, which helped me to develop a deep understanding of the subject of this thesis. We were solving the problems using pure analitical tools, which was very insightful for me as a PhD student.

I would also like to thank Prof. Dmitry Abanin, who collaborates with me starting from the early 2016.

My work with Dmitry was indeed complementary to my primary research both in terms of topics and techniques. Dmitry exposed me to many trending topics of the contemporary research, which is really helpful for building up the confidence of a young researcher.

I want to thank my advisor during my Bachelor and Master studies, Prof. Kirill Nagaev, who accepted me to his group in the winter of 2011/2012 after I was not able to continue my research in another group. He helped me to realize my Bachelor thesis in less than half a year, he helped me to develop basic understanding of the field of condensed matter physics and under his supervision I was able to publish my first paper to my favorite journal Phys. Rev. B.

I would like to say the words of gratitude to my colleagues, who worked at some point in Eugene’s group:

Edwin Idrisov, Dario Ferraro, Artur Slobodeniuk, Ivan Levkivskyi, Iurii Chernii, each of whom contributed to my growth as a physicist. I would like to thank the group of Thierry Giamarchi for inviting our group to participate in their weekly group meetings and other activities organized by their group. In particular I want to thank Michele Filippone for his will to help me with scientific and career advice. I also benefited a lot from the scientific discussions with the members of the group of Prof. Abanin: Ivan Protopopov, Wen Wei Ho, Louk Rademaker. I am also thankful to DPT along with DQMP for organizing various kinds of interesting scientific talks.

I want to thank the secretaries of our department Francine Gennai-Nicole, C´ecile Jaggi-Chevalley, Angela Stark-Sanchez, who handled all the bureaucracy very efficiently, which can be appreciated even further after hearing horrific stories about the secretaries at other universities. I want to acknowledge the IT support from Andreas Malaspinas and Jacques Rougemont.

I am grateful to Thierry Giamarchi, Vadim Cheianov, Frederic Pierre, and Eugene Sukhorukov for agree- ing to be on the committee for my thesis defense.

Last but not least I want to thank my wife Anya Goremykina, whose support was the key thing that made it possible for me to complete my PhD.

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Chapter 1 Introduction

The advances in fabrication of nanodevices open up a way to explore novel phenomena ofmesoscopic physics, the branch of physics that deals with the systems that have experimentally achievable dimensions (∼1µm) and at the same time the properties of which are affected by quantum correlations. This definition determines a major challenge of this field: how can one realistically describe a system constituted of billions of particles/degrees of freedom? The only possible way to understand systems of such a complexity is to develop an effective description that allows to determine key factors that encode the properties of the system and reduces the problem to a few relevant degrees of freedom.

This thesis is devoted to developing an universal description of strongly interacting mesoscopic systems out of equilibrium. The nature of universalities we are looking for is twofold. On the one hand, our goal is to find novel and simple dependences of physical observables on the experimental parameters. In this case considering strongly interacting systems out of equilibrium provides experimentalists with additional control parameters, i.e., new energy scales, which can be helpful for a deeper physical understanding of such systems. As an example, this energy scales can be introduced by applying different chemical potentials to the subsystems. The universal behaviour typically develops, when observables are measured at energies close to different Fermi levels.

On the other hand, we are looking for an universal theoretical approach that can be applied to a class of seemingly different strongly interacting physical problems. This connection, apart from being insightful, is also practical, since it allows simple generalization of the results for a single physical system to the whole class of problems. To achieve this generality we rely on the bosonization technique, which allows one to account for interactions exactly. Thus, different regimes of coupling strength can be studied including the case, where the non-perturbative effects of interaction become crucial. The bosonization technique is commonly used to describe 1D systems, where interaction is inherently strong. In 1D systems it is no longer possible to separate the charge degrees of freedom, which are affected by interaction, and the degrees of freedom related to the electron-hole excitations. In 1D both of these excitation types have linear spectrum and are strongly coupled in contrast to, e.g., 3D metallic systems, where these degrees of freedom are separated by the energy gap. The bosonization technique allows one to express electron-hole excitations as plasmons, reducing a problem with strong interaction to an exactly solvable one. Interestingly, the intuition coming from the bosonization approach extends beyond 1D systems. In systems, where the plasmonic degree of freedom is decoupled from others, the bosonic description also allows solving these problems naturally for an arbitrary coupling. Two such examples, where the solution is available for an arbitrary interaction strength are: (i) the problem of Fermi edge singularity (FES), where the charge of the impurity is strongly coupled to the surrounding transport channels; (ii) the inelastic tunneling in the tunneling device coupled to the fluctuating charge mode of the electric circuit.

It turns out that it is not trivial to apply this technique to the non-equilibrium systems. This complexity is the consequence of the fact that typically in experiments the electrons are injected into a system from the biased electrodes, i.e., the natural language to describe a non-equilibrium state is fermionic, while the interaction has a simple form, when thebosonicdescription is used. Luckily, there already exist conceptually similar techniques, that can be unified under the name of a non-equilibrium bosonization, which can treat systems with arbitrary non-equilibrium distribution. However, these techniques are applicable only to a certain class of systems. Therefore, in this thesis we discuss an alternative approach that treats the non- equilibrium corrections as a perturbation. This allows us to extend previous results to a wider class of systems and to new regimes. At the same time we keep connection to the previously studied techniques,

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where it is possible. Even though the effects discussed in this thesis appear in various mesoscopic setups, we mainly focus on their realization in quantum Hall (QH) systems mostly due to the high level of control that can be achieved in the corresponding experiments.

The body of this thesis is separated into 4 chapters. Chapter 2 is devoted to the discussion of the relevant theoretical techniques that are either used or discussed in this thesis. In Chapter 3 we consider the simplest type of a non-equilibrium effect that depends on a multiple energy-scales, namely, cotunneling between QH edge channels through a quantum dot (QD) in a FES regime, i.e., QD strongly interacts with edge channels.

In this case all the involved subsystems are in local equilibrium. The universal dependence of the tunneling current on the energy scales of the problem is derived.

In Chapter 4 we discuss the case of theweaklynon-equilibrium states. In particular, we consider 4 different systems (integer and fractional QH, Luttinger liquid (LL), QD in FES regime), where weak tunneling from the biased electrode introduces non-equilibrium correlations. We study either how these systems evolve into stationary prethermal states or how the additional fluctuations modify the transport properties of the system.

In this chapter we also comment on the connection of our approach to the technique of non-equilibrium bosonization that does not require tunneling to be weak.

In Chapter 5 we illustrate how by adopting the bosonic description one obtains results for a new regime even in the case of a well-studied effect. In particular, we focus on the drag effect, the effect of the induction of the charge current in one conductor by the current in another one due to the interaction. This phenomenon is typically studied in the context of quasiparticle scattering, which is accounted for at the lowest order in interaction. However, we discuss a new proposal of treating this effect as scattering of the quasiparticles off the collective bosonic mode of the non-equilibrium system. The advantage of this approach is that it allows us to go beyond the lowest order in interaction. In particular, we consider the next order in coupling and discuss the cases, where the corresponding contribution is dominant. In the end of the chapter, we comment on how the interaction can be studied non-perturbatively in the case of strong interaction, which is an extremely hard problem on the fermionic level.

Most of the work presented in this thesis is published in scientific journals. Throughout the thesis, we set ~= 1 and the charge of an electrone=−1.

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Chapter 2 Bosonization

2.1 Equilibrium bosonization in 1D

2.1.1 Fermionic description

In this section we review the technique of the equilibrium bosonization in 1D, which proved to be very powerful tool in solving the problems of 1D strongly interacting fermions. Elaborate introductions on this topic include Refs. [1,2,3,4]. In this chapter we follow the lines of the discussion of Ref. [2].

Let us start with one-dimensional fermions. The corresponding creation and annihilation operators at a momentum k c^†_k,ck satisfy the fermionic commutation relations

{c^†_k, c_k⁰}=δ_kk⁰. (2.1)

At this point we treat momentumkto be discrete and the system, thus, is of finite sizeW. We do it in order to avoid unnecessary infinities below, e.g., the total charge in the system. If one further assumes that the effective low-energy description is sufficient for a given physical problem, the kinetic part of the Hamiltonian of the fermions can be linearized, leading to the following expression:

H_kin =v_F

Λ

X

k=−Λ

kc^†_kc_k, (2.2)

wherevF is the velocity of the fermions close to the Fermi point. We also introduce an ultraviolet cut-off Λ, which determines the range of the applicability of the linearized theory. We expect that physical quantities will not be affected by the cut-off at low-energies.

To introduce the interaction terms we need to define the operator of the local electronic densityρ(x).

It can be expressed in terms of the fermionic creation and annihilation operators at a point x ψ^†(x),ψ(x), which are defined by the relation:

ψ(x) = 1

√W X

k

cke^ikx. (2.3)

In terms of this operators Eqs. (2.1,2.2) take the form

{ψ^†(x), ψ(x⁰)}=δ(x−x⁰), H_kin=−iv_F Z

dxψ^†(x)∂_xψ(x). (2.4) Then, one can introduce the naive definition of density ρ(x) =ψ^†(x)ψ(x). This definition should be regularized, since its expectation value is divergent even for a ground state (GS) due to the contributions of the ultraviolet momenta. The regularized quantity can be defined as

ρ(x) = :ψ^†(x)ψ(x) : =ψ^†(x)ψ(x)− hψ^†(x)ψ(x)iGS. (2.5) This subtraction is perfectly physical. As an example, it can be attributed to the background compensating positive charge. Also, this definition preserves the correct commutation relation [ψ(x), ρ(x⁰)] =ψ(x)δ(x−x⁰), which means that operator ρ(x) corresponds to the density of the particles that are annihilated by the operator ψ(x).

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At this point we can introduce the interaction term to the Hamiltonian Hint=

Z Z

dxdx⁰U(x−x⁰)ρ(x)ρ(x⁰), (2.6)

whereU(x−x⁰) is the Coulomb interaction potential. Luckily, throughout this thesis the exact form of this potential is not important.

2.1.2 Bosonic description

Surprisingly, it is possible to rewrite the kinetic part of the Hamiltonian in terms of the product of densities as well. To do it, let us consider the commutator of the operatorρk, which is defined as the Fourier transform of the density operator:

ρ_k≡ 1 W

Z

dxe^ikxρ(x) = 1 W

X

k⁰

:c^†_k+k0c_k0 :. (2.7)

Due to the hermiticity of the density operator, we get the property ρk=ρ^†_−k. Also this operator obeys the following commutation relation:

[ρk, ρk⁰] = 0, k6=−k⁰, (2.8)

which is natural, given its bosonic nature as a product of two fermionic operators. In the case ofk =−k⁰ one arrives to the following expression

[ρk, ρ−k] = 1 W²



 X

k⁰<−Λ

(nk⁰−k−nk⁰) + X

k⁰≥−Λ

(nk⁰−k−nk⁰)



, (2.9)

wherenk =c^†_kck. If we study low-energy physics, then there are no fluctuation atk∼ −Λ and the states are always occupied at that scale. Thus, the summation can be performed by shifting the variable in the second term, which results in

[ρ_k, ρ_−k] = sign(v_F) k

2πW. (2.10)

Having this commutator and noting that the following commutation relation holds [H_kin, ρ_k] = v_Fkρ_k we can rewrite the kinetic part of the Hamiltonian as

Hkin=πvF

Z

dxρ²(x). (2.11)

The same form of the kinetic and the interaction (2.6) terms inspires the transition from the description in terms of fermions to the one in terms of bosons. The crucial relation in the bosonization technique is the following identity:

ψ(x) = 1

√aexp

"

iϕ+ 2πiρ0x+X

k

2πi k ρke^ikx

#

, (2.12)

where ϕis the operator canonically conjugated to the zero mode ρ0 anda= 2π/Λ. This relation allows to preserve fermionic commutation relations (2.4) as well as the correct commutation with the density operator.

The bosonic fieldφ(x) can be introduced in order to rewrite relation (2.12) in a simpler form:

ψ= 1

√ae^iφ. (2.13)

This form presents the central object of the bosonization. The introduced field has the following property ρ(x) = 1

2π∂xφ(x), (2.14)

which then defines the commutation relations:

[φ(x), φ(x⁰)] = sign(v_F)πsign(x−x⁰). (2.15)

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In terms of this field the total Hamiltonian H≡H_kin+H_int can be written as H =

Z dxdx⁰

8π² V(x−x⁰)∂_xφ(x)∂_x⁰φ(x⁰), (2.16) which is now quadratic (free). The interaction kernel is defined asV(x−x⁰) =U(x−x⁰) + 2πvFδ(x−x⁰).

The problem of computing of the observable is typically reduced to the evaluation of some correlator.

In this thesis we deal with the states that are close to equilibrium and, therefore, it is important to be able to perform an averaging at equilibrium. To do this, we need to rewrite our theory in terms of free bosonic modes, which would allow us to compute any equilibrium correlator from the bosonic occupation number. From the expression (2.10) we see that bosonic creation and annihilation operators are defined by a^†_k =p

2πW/kρ_k anda_k =p

2πW/kρ_−k, which diagonalize the Hamiltonian (2.16) H = 1

2π X

k

kV(k)a^†_kak+W

2 V(0)ρ²₀ (2.17)

and are related to φ(x) by

φ(x) =ϕ+ 2πρ₀x+X

k

r2π k

ha_ke^ikx+a^†_ke^−ikxi

. (2.18)

At this point the computation of the correlators becomes a technicality. Below we illustrate the power of the equilibrium bosonization by computing the ground state (zero temperature) correlators for various setups that are touched upon in the next chapters of the thesis. All setups appear as the generalization of the theory presented in this section to the case, by combining more than one fermionic mode.

2.1.3 Applications

Chiral fermions

Let us start with the case of the chiral fermions. We are going to consider the correlator of the form K(x, t) =ihψ^†(x, t)ψ(0,0)i, which, for example, appears in the expression for the tunneling density of states n() =R

dte^−itK(x, t). Computing this correlators directly on the fermionic level gives

K(x, t) =i

Z dkdk⁰

2π e^iv^F^kt−ikxhc^†_kck⁰i= 1 2π

1

(vFt−x)−i0, (2.19)

where the fermionic occupation factor ishc^†_kck⁰i=δ(k−k⁰)θ(EF/vF−k) and we took Fermi energyEF to be zero.

The same result can be obtained on the bosonic level. Using the identity (2.13), the representation in terms of free oscillators (2.18) and the Baker–Campbell–Hausdorff formula we obtain

ln[K(x, t)] =h(φ(x, t)−φ(0,0))²i= Z Λ

0

dk

k(e^ikx−ikvt−1) = ln[Λ(v_Ft−x)], (2.20) which reproduces the correct expression. However, what is more important, the bosonic approach is applicable beyond the free-fermionic limit. In particular, if we are interested in the long wave-lengths of the excitations, much longer than the screening length of the Coulomb interaction, then, the dispersion of V(k) in Eq. (2.17) can be neglected and the same expression for the correlatorK(t) is preserved upto the renormalization of the velocity.

Edge of the integer quantum Hall effect at the filling factor ν = 2

In case of the integer quantum Hall effect there are two co-propagating plasmonic mode at the edge that can interact with each other. The most general Hamiltonian in this case is given by

H =X

ij

Z dxdx⁰

8π² Vij(x−x⁰)∂xφi(x)∂x⁰φj(x⁰), (2.21)

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with i, j= 1,2 corresponding to the edge modes. Let us again focus on the universal limit of this Hamilto- nian. If we consider long wave-length limit, then the dispersion of the interaction can be neglected and the Hamiltonian can be diagonalized by the rotation

φ˜i=Rijφj, (2.22)

with some rotation matrix

Rij =

cosα sinα

−sinα cosα.

(2.23) The Hamiltonian in terms of the new fields is given by

H =X

i

Z dxdx⁰

4π v˜i∂xφ˜i(x)∂x⁰φ˜i(x⁰), (2.24) with ˜vi being the velocities of the eigenmodes, which preserve the original sign.

Let us now evaluate the correlator of the electrons in one of the edge channels,i= 0 for concreteness:

K(x, t) =ihψ₀^†(x, t)ψ0(0,0)i=Y

i

he^−i[R⁻¹^]⁰ⁱ^φ^˜ⁱ^(x,t)e^i[R⁻¹^]⁰ⁱ^φ^˜ⁱ^(0,0)i= 1 2π

Y

i

1

(˜v_it−x−i0)^([R⁻¹^]⁰ⁱ⁾². (2.25) The important observation about this result is that it coincides upto the prefactor with the free-fermionic one, if x= 0. It turns out that this holds for any number of the interacting co-propagating channels [2].

Fractional quantum Hall

Let us now consider a slightly different case of the fractional quantum Hall effect with the filling factor ν = 1/(2n+ 1),n∈N. In contrast to previous examples, in this case the bosonic theory emerges from the effective topological theory rather than from microscopic theory. The Hamiltonian in this case is given by

H =πv ν

Z

dxρ²(x), (2.26)

with the density operator

ρ(x) =

√ν

2π∂xφ(x) (2.27)

and the commutation relations for the bosonic field

[φ(x), φ(x⁰)] =iπsgn(x⁰−x). (2.28)

The electronic operator in this case is constructed to preserve the correct exchange statistics as well as to correspond to the correct charge

ψ∝eⁱ

√

ν⁻¹φ. (2.29)

Therefore, the effective field theory allows to compute the fermionic correlator K(x, t)∝ 1

(x−vt)^ν⁻¹. (2.30)

Spinless Luttinger liquid

In the case of spinless Luttinger liquid the original fermions are no longer chiral. The fermionic operator is decomposed asψ(x) =e^ikxψR(x) +e^−ikxψL(x) into the right- and left-moving electrons that correspond to two Fermi points. One can bosonize these fields separately introducing two bosonic fields φi(x), i=R, L, with the commutation relations

[φ_i(x), φ_j(x⁰)] =±iπsign(x−x⁰)δ_ij, (2.31) where the±corresponds to the right-, left-moving bosons respectively. The kinetic part of the Hamiltonian is given by

Hkin=vF

4π X

i

Z

dx(∂xφi(x))². (2.32)

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The interesting thing happens, when interactions are introduced, since they lead to the mixing of these two fields. This happens since the density operator that enters interaction contains both of the fermionic fields ρ(x) = :ψ^†(x)ψ(x) :. However, in the case of the excitation with small momentum the number of right- and left- moving electrons is preserved and the only possible interaction terms are ∝(∂xφi(x))², which is simple renormalization of the original velocity, and ∝ (∂xφR(x)∂xφL(x)), which leads to the mixing. The total Hamiltonian can be then written as

H = ˜vF

4π Z

dx

(∂xφR(x))²+ (∂xφL(x))²+ 2g2

˜

v_Fπ∂xφR(x)∂xφL(x)

, (2.33)

where ˜vF is the renormalized Fermi velocity and g2 is an interaction parameter that corresponds to the mixing of the modes. This Hamiltonian can be diagonalized by the Bogolubov transformation that have the form

˜

a_R=a_Rcoshθ−a^†_Lsinhθ,

˜

a_L=a_Lcoshθ−a^†_Rsinhθ,

where ai,˜ai are old and new bosonic annihilation operators. Choosingθaccording to K=e^2θ=

s

1−g2/˜vFπ

1 +g₂/˜v_Fπ, (2.34)

where Kis the so called Luttinger liquid parameter, one diagonalizes the Hamiltonian H =v˜F

4π Z

dxX

i

(∂xφ˜i(x))². (2.35)

Finally one can evaluate correlation functions. For instance

hψ_R^†(x, t)ψR(0,0)i ∝ he⁻ⁱ^cosh^{θ( ˜}^φ^R^(x,t)−^φ^˜^R^(0,0))ihe⁻ⁱ^sinh^{θ( ˜}^φ^L^(x,t)−^φ^˜^L^(0,0))i ∝ 1

(x−˜vFt)^cosh²^θ(x+ ˜vFt)^sinh²^θ. (2.36) Arbitrary four-point correlator

It this thesis we also encounter more complex objects than two-point correlators. Namely, if one considers the forth order perturbations over the ground state, then the correlators involving four fermionic operators would appear. Of course, any kind of averaging can be performed over simple Gaussian state (ground state of a free theory in our case). To illustrate this let us compute arbitrary four-point correlator

heîξφâ^(X¹⁾eîζφâ^(X²⁾e^−iζφâ^(X³⁾e^−iξφâ^(X⁴⁾i ∝ (2.37) 1

(i(X1−X4) + 0)^ξ²(i(X2−X3) + 0)^ζ²

(i(X₁−X₂) + 0)^ξζ(i(X₃−X₄) + 0)^ξζ (i(X1−X2) + 0)^ξζ(i(X3−X4) + 0)^ξζ,

where X_i =v_Ft_i−x_i. The structure of this correlator is simple: one can notice the appropriate ordering of the arguments on the right hand side as well as that the sum of the coefficients in exponents on the left hand side is zero, which is necessary to have the non-zero average.

2.2 Non-equilibrium bosonization

In order to understand the motivation behind the approach developped in this thesis we want to discuss the competing method. There exist a number of theoretical techniques that can be united under the name of non-equilibrium bosonization. These techniques allow to study strongly interacting electronic systems for an arbitrary non-equilibrium distribution as an initial condition. Non-equilibrium bosonization allows to simplify the problem of evaluation of some non-equilibrium electronic quantity in a strongly interacting system, e.g. Green function. Such a problem can be reduced to the evaluation of the Fredholm determinant over the single particle degrees of freedom. This approach was developed in Refs. [5, 6, 7, 8, 9]. In this section we follow the lines of Ref. [7]. We want to outline the key details of this technique, its advantages and disadvantages as well as create a ground for making a connection to the perturbative methods that we study in this thesis.

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2.2.1 Bosonic correlations from boundary conditions

First of all, let us write the equation of motion for the bosonic field φ(x) that is generated by Eq. (2.16):

∂_tφ(x, t) =− 1 2π

Z

dx⁰V(x−x⁰)∂_x⁰φ(x⁰, t). (2.38) This equation of motion should be supplemented with the boundary condition. It can be done by specifying the current j(0, t)≡j(t) at the pointx= 0, since the bosonic field φ(x, t) is related to the currentj(x, t) by the relation j(x, t) =∂tφ(x, t)/2π, which follows from the continuity equation on the charge. Thus, the boundary condition is given by

∂tφ(0, t) = 2πj(t). (2.39)

The linear differential equation (2.38) with the boundary condition (2.39) can be solved using Green’s function method. The solution is given by

φ(x, t) = Z

dt⁰G(x, t−t⁰)φ(0, t⁰), (2.40) where G(x, t−t⁰) is dependent on the specific form of the interaction. In the case of low-energy excitations this kernel takes a simple formG(x, t−t⁰) =δ(t−t⁰−x/v), wherev is the velocity of the excitations. The field at pointx= 0 is given by

φ(0, t) = 1 2π

Z t

−∞

j(t⁰)dt⁰ ≡ 1

2πQ(t). (2.41)

We identified it with the total charge Q(t) transferred through the pointx= 0 upto the timet. Therefore, in the case of local kernel the simple identity holds

φ(x, t) = 1

2πQ(t−x/v). (2.42)

As we discussed in the previous section the typical quantities of interest are of the form

χ(t, λ) =he^−λφ(t)e^λφ(0)i. (2.43) What is important here is that the averaging in this case is done not necessarily over the ground or thermal state. The goal of the non-equilibrium bosonization is to find the way to evaluate this quantity for a more general state. This correlator can be rewritten as

χ(t, λ) =he^{−λQ(t)/2π}e^λQ(0)/2πi, (2.44) which take the form of the generator of the full counting statistics [10], the generalization to the quantum case of the classical moment generator of the transmitted charge χcl(λ)≡ he^iλQi=R

dQP(Q)e^iλQ. In the long-time limit one can extract the current cumulants as

∂_iλⁿχ(t, λ)|λ=0/t=hhjⁿii. (2.45) Therefore, the problem of the evaluation of the correlator is reduced to the calculation of the statistics of the transferred charge. Below we discuss, how this statistics can be determined.

2.2.2 Evaluation of the FCS for a quantum noise

Let us highlight that the quantity (2.45) can be evaluated only for limited number of cases. Below we discuss two cases when this can be achieved.

Fredholm determinant

If the transport of the electrons is considered (Fig. 2.1), then, according to Ref. [10], the evaluation of the FCS generator can be reduced to the evaluation of the Fredholm determinant as

χ(t, λ) =det(1−F+U_2πλF), (2.46)

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Figure 2.1: An electronic scattering problem with scattering matrixS. The green rectangle marks the region, where the charge statistics is studied.

where F is the diagonal in energy basis matrix with elements being the Fermi distribution functions in the incoming scattering channels. The matrix U_λ=S_λ⁻¹S_λ is obtained from, the scattering matrix

S_λ=

r t

˜t[10,t(eîλ/2−1) + 1] re˜ îλ/2[10,t(eîλ/2−1) + 1]

(2.47) and can be evaluated to be

Uλ= 1 0

0 1

+ 10,t e^iλ−1

T rt^∗ r^∗t (1−T)

, (2.48)

whereT =|t|²is the tunneling probability and 1_0,tis equal to unity at the interval [0,1] and is zero otherwise.

The role of this matrix is to “count” electrons, which end up in the channel of interest (red rectangle in Fig.2.1) after scattering.

The are several ways to evaluate the determinant (2.46). Its asymptotic long-time behaviour is evaluated in Ref. [10] and is given by a result corresponding to a classical process

lnχ(t, λ) = t 2π

Z

dE ln 1 +T[e^−2πiλ−1]fL(1−fR) +T[e^2πiλ−1]fR(1−fL)

, (2.49)

where f_i, i=L, R is a distribution function in the left, right reservoirs respectively. In the limit of a large voltage compared to a temperatureV T this result turns into the formula for binomial process

χ(t, λ) = (1 +T[e^2πiλ−1])^{V t}. (2.50) However, one should find the logarithmic corrections to this result, if the quantum correlations of a Fermi see are important.

Another approach to evaluate the determinant, which we use in Chapter4, is to make an expansion in the tunneling probability, which leads to the following expression:

lnχ(t, λ) =T r([U_2πλ−1]F). (2.51)

Poissonian process

If one considers the case of rare tunneling and focuses solely on the long-time asymptotics, then the classical expression for the FCS can be used. In particular, in Ref. [11] this was done for the case of fractional quantum Hall effect, where no expression in terms of single-particle determinant exists. Namely, given the average charge current hIiand the effective charge of the particle e^∗, the non-equilibrium part of the FCS generator is given by

lnχ(t, λ)' hIit

e^∗ (e^iλ−1). (2.52)

2.2.3 Evaluation of the FCS for a classical noise and stochastic path integral

In previous subsection we demonstrated how the FCS can be calculated in a long-time limit either for free fermions or for rare tunneling. However, it is possible to calculate this quantity at all times and for any strength of tunneling, if the classical regime is considered. This can be achieved by employing the stochastic path integral (SPI) technique[12, 13]. Below we present a sketch of the derivation of this quantity. Let us

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R

_L

Q R

_R

μ

_L

C μ

_R

Q

Q _L Q _R

Figure 2.2: An electrical circuit on the left and its equivalent network on the right.

consider the electric circuit presented in Fig.2.2on the left. We will calculate the FCS of the fluctuations of the chargeQon the capacitor. This fluctuations appears due to the current fluctuations through the left and right resistance, which are defined by the corresponding cumulant generating functions H_i(λ, Q), i=L, R that have the property

∂_iλⁿHi(λ, Q)|λ=0=hhj_iⁿii, i=L, R. (2.53) We also assume dependence of the cumulant generating on the capacitor chargeQ, which should be the case, since the change of this charge affects the voltage drop across the resistors. To simplify our consideration of this problem it is convenient to study the equivalent network model of the circuit (Fig. 2.2in the right).

Then the distribution of the charge Qi that is transferred from thei-th reservoir over a time interval ∆tis given by

P_i(Q_i,∆t) = Z dλ

2πe^iλQⁱ^+Hⁱ^(λ,Q)∆t, i=R, L. (2.54)

This expression is only valid for an intermediate ∆t, i.e., larger than the correlation time of the fluctuations in the resistorsτ0and smaller than the response time of the circuitτc, over which the variation of the charge occurs (τ0∆tτc). The charge conservation between time pointsnandn+ 1, separated by the interval

∆t, implies that

Qn+1−Qn=QR+QL. (2.55)

Enforcing this equation of motion by introducing the corresponding delta-function to the probability distri- butions (2.54) we can get a one-step propagator for the charge evolution

U(Q_n+1, Q_n) = Z

dλe^−iλ(Qⁿ⁺¹^−Qⁿ^)+∆t(H^L^(λ,Qⁿ^)+H^L^(λ,Qⁿ⁾⁾, (2.56) which is defined by the following identity

P(Qn+1) = Z

dQnU(Qn+1, Qn)P(Qn), (2.57)

where P(Qn) is the probability distribution of the charge on the capacitor at time point n. Having the propagator we can now construct the path integral and compute averages. In particular, let us compute the FCS for the charge fluctuations over the time t

Z(χ) =heχ(Q(t)−Q(0))i. (2.58)

This quantity is then given by the following path integral Z(χ) =

Z

DQDλexp(S), (2.59)

S = Z

dt⁰h

−λQ˙ +HL(Q, λ) +HR(Q, λ) + (χ/C)Θ(t⁰)Qi

. (2.60)

Evaluating this integral in the stationary point approximation we obtain the quantity of interest. In particular, one can obtain the cascade corrections[14] to the correlators of the third order and higher due to the circuit effects. Such calculations are present in Chapter5.

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Chapter 3 Cotunneling

3.1 Introduction

In contrast to the systems with the Fermi liquid like behavior, the electron-electron interactions become large in QD based devices. The Coulomb blockade is the most prominent manifestation of this effect.[15] Its main feature is the appearance of the gap in the QD excitation spectra due to the finite energy required in order to add electron or hole to a QD. This energy is of the order of charging energyE_C =e²/2C, wheree is the electron charge, andCis the capacitance of the dot, and it can be controlled by the gate voltage. If in addition the interaction of the charge on the QD with metallic leads is taken into account, this results in the power-law dependence of the tunneling density of states at the energy close to the Fermi level. This effect is analogous to the one that leads to the singularity in X-ray absorption spectra in metals¹and is often referred to as the Fermi edge singularity. It has been extensively studied both experimentally[17,18,19,20,21] and theoretically [22,23,24,25,26,27,28,29,5] in various systems.

Because of the large charging energy the dominant contribution to the transport through a QD is sequential tunneling,[15] i.e., the electron (hole) enters the dot only if the previous one has left it. This leads to the correlation of incoming and outgoing currents, which can be seen, e.g., in the suppression of the zero-frequency noise power[30, 31, 32]. The Coulomb blockade is not very sensitive to the size of the QD, which only affects the value of the charging energy. Therefore, any mesoscopic QD demonstrates Coulomb blockade effect at sufficiently small temperatures.

In contrast, the FES is a more delicate phenomenon. It arises only if the number of transport channels of the QD is of the order of one[16]. To be more precise, the exponent of the power-low energy dependence of the density of states at low energies is inversely proportional to the number of scattering channels. Since the large QDs are typically coupled to a large number of scattering channels, the FES effect is suppressed in such systems. It has been recently proposed to circumvent this difficulty by attaching large QDs to the quantum Hall channels,[33] as it has been recently implemented experimentally.[34]

Recent breakthrough on the theoretical side, namely, the development of the non-equilibrium bosonization approach,[7] enabled one to study the FES phenomenon far away from equilibrium.[35] One of the key results of this study is the universal dependence of the tunneling current on both parameters: the energy of the level on the dot and the power of the non-equilibrium noise, which is controlled by an additional voltage source and provides the second energy scale. Here we propose an alternative approach, where the second energy scale is introduced by keeping the leads at equilibrium. Namely, we focus on the cotunneling regime away from the QB resonance, where the sequential tunneling is suppressed, and the transport is dominated by simultaneous tunneling of two electrons or holes via the dot.[36] Such a process depends on two energy scales: the energy of the level on the QD and the bias between contacts.

Using the same approach that was developed in Ref. [35] we obtain the universal power-law behavior of the tunneling current in two limits: in the case of small bias between the contacts, and in the case where the Fermi level in one contact is close to the energy level on the QD. We obtain new power-law exponents that depend on the scattering phases of the electrons in the contacts, thus establishing the connection to the

“classical” FES effect.

The rest of the chapter is organized as follows: In Sec. II we introduce the model of a QD tunnel coupled to one-dimensional electron channels and present the Hamiltonian of the system using the bosonization

1For an early review, see [16]

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technique. In Sec. III we present the formal perturbation theory to second order in tunneling, derive the expression for the cotunneling current, and discuss its analytical structure. In Sec. IV we obtain new FES exponents and discuss their physical significance. Finally, in Sec. V we present conclusions. For the sake of simplicity we work in the units wheree=~= 1.

3.2 Model

In order to grasp the main features of the effect, we model the transport contacts to the QD by one- dimensional electronic channels. This model provides an effective description of metallic leads,[37] and can also be applied to quantum Hall systems,[38] such as in the recent experiment [34]. It is well known,[1] that one-dimensional electronic systems can be described either in terms of the electrons or in terms of collective excitations (plasmons). FES is a non-perturbative effect in the electron-electron interactions. Since the Hamiltonian of one-dimensional channels, when expressed in plasmon fields, preserves its quadratic form in the presence of the interactions, we adopt this approach to study the FES.

In this chapter we concentrate on the regime of the elastic cotunneling,[36] which can be realized in relatively small dots with few levels and relatively low biases. Thus, the minimal model for the FES includes tunneling coupling of the dot to two electron channels, and Coulomb coupling to arbitrary number of channels (see Fig. 3.1). The Coulomb interaction should be considered non-perturbatively, while tunneling, in contrast, is the smallest perturbation. The corresponding Hamiltonian can be written as follows:

H=H₀+H_d+H_i+H_t, (3.1)

where the Hamiltonian of the electronic channels reads H0=π

Z

dxX

a

vaρ²_a(x) +1

2 Z

dxdyX

ab

ρa(x)Vab(x, y)ρb(y). (3.2)

Here, the first part describes the free propagation of the charge densitiesρ_a(x) with the speedsv_a and the second part accounts for the density-density interactions characterized by the Coulomb potentialsVab(x, y).

For the particular set up shown in Fig. 3.1, experimentally studied in Ref. [34], a = L, R, U, D, thus, enumerating left, right, up and down channels respectively. Note that only left and right channel, denoted byLandR are coupled to the QD by tunneling.

ϵ

Δμ

Figure 3.1: A particular example of a more general system considered in this chapter is schematically shown.

This set up has been experimentally studied in Ref. [34] using QH edge states at filling factorν = 2. The QD is coupled by the Coulomb interaction (green line) to the surrounding transport channels. The electrons tunnel from the left voltage-biased channel to the right grounded channel through the virtual states of the QD.

The QD Hamiltonian can be written asHd=0d^†d, wheredis the electron annihilation operator, and0

is the bare single-particle energy of the QD. The effect of FES arises due to the Coulomb interaction between

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the electron on the QD and those in the channels. The corresponding term in the Hamiltonian is given by Hi =d^†d

Z dxX

a

Ua(x)ρa(x), (3.3)

where U_a are the Coulomb potentials. We note, that the potentialsU_a andV_ab do not need to be specified because of the universality of FES, as we demonstrate below.

Finally, the tunneling Hamiltonian transfers electrons between the channels in the vicinity of the QD and the QD level, H_t = d^†P

aτ_aψ_a(0) +h.c., where ψ_a(x) are the electron annihilation operators in the channels, andτ_a are the amplitudes of tunneling. To make the connection between the electron and plasmon descriptions, we follow the standard bosonization procedure [1] and introduce the set of bosonic fieldsφ_a(x), which are defined by ρa(x) = _2π¹∂xφa(x). They satisfy standard commutation relations [∂xφa(x), φb(y)] = 2πiδabδ(x−y). The electron operators are related to the bosonic fields byψa(x)∝e^iφ^a^(x). Therefore, the tunneling Hamiltonian can be written as

Ht=d^†X

a

τae^iφ^a⁽⁰⁾+h.c. . (3.4)

As a first step, we follow the procedure outlined in Ref. [35] and apply the unitary transformationU ≡e^iS that removes the interaction term (3.3) in the total Hamiltonian. Here

S=d^†d Z

dxX

a

σa(x)φa(x), (3.5)

and the functions σ_a(x) are determined by the integral equation Ua(x) +

Z

dx⁰X

b

Vab(x, x⁰)σb(x⁰) = 0. (3.6)

On the one hand, such a choice of the functions σa(x) follows from the requirement of the cancellation of the term Hi in the Hamiltonian. On the other hand the functions σa(x) turn out to be equal to electron densities accumulated in the channels to screen the extra unite charge on the dot. From this point of view, the Eq. (3.6) simply states, that all the channels are grounded (see, however, the discussion below).

After the introduced above unitary transformation the tunneling Hamiltonian acquires the following form:

H˜t=d^†X

a

τaeⁱ^R^dx^P^b^σ^b^(x)φ^b^(x)e^iφ^a⁽⁰⁾+h.c. . (3.7) It is natural to assume, that the screening charges are accumulated in the vicinity of the pointx= 0. Thus, the simplification arises in the low-energy limit, where the wave length of plasmons exceeds the size of the QD, and leads to the universal behavior attributed to the FES. In this limit, we can approximate the fields in Eq. (3.7)φa(x)≈φa(0), which in our particular case leads to the expression

H˜t=AL+AR+h.c., (3.8)

with the elementary tunneling operators

AL =τLd^†eîφ^L⁻ⁱ^Pâ^ηâ^φâ, (3.9a) AR=τRd^†eîφ^R⁻ⁱ^Pâ^ηâ^φâ, (3.9b) and ηa ≡ −R

dxσa(x) being the absolute values of the charges accumulated in the channels as a result of screening of the extra electron on the QD.

Finally, it is worth mentioning, that the unitary transformation shifts the energy level at the dot Hamil- tonian: Hd→H˜d=d^†d, where

=0+X

a

Z

dxUa(x)σa(x). (3.10)

This shift, obviously, arises due to the Coulomb interaction of an electron on the dot with the induced charge densities in the channels.

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3.3 Perturbation theory

We start with the standard Fermi golden rule expression for the cotunneling current at zero temperature:

I= 2πX

m

|hm|Tˆ|0i|²δ(Em−E0), (3.11)

where |mi are the eigenstates of the total Hamiltonian in the absence of tunneling. Since the first-order sequential tunneling processes are forbidden due to the energy gap in the cotunneling regime, we use the general expression for the tunneling transfer operator ˆT in terms of the time-ordered exponent: [39]

Tˆ = d dt

Tˆte⁻ⁱ

Rt

−∞dt⁰H_t(t⁰)

|t=0, (3.12)

which gives the familiar expression, when expanded to second order in tunneling Hamiltonian in the energy domain,

I= 2πX

m

|hm|Aˆ^†_LRˆ₀Aˆ_R|0i|²δ(E_m−E₀), (3.13) where ˆR0≡(E0−H0−Hd+i0)⁻¹is the retarded resolvent. However, in the present context it is convenient to keep the time representation (3.12) and replace the delta-function by an integral over time. As a result, we arrive at the following expression:

I= Z ∞

−∞

dt Z t

−∞

dt1

Z 0

−∞

dt2hAˆ^†_L(t1) ÂR(t) Â^†_R(0) ÂL(t2)i, (3.14) where the averaging is taken over the ground state.

Next, we substitute tunneling operators (3.9) into the Eq. (3.14) and use the gaussianity of the fieldsφa

to evaluate the average. Since the fluctuations of fields in different channels are independent, the average splits in the product of the four-point correlators of the form

heîξφâ^(t¹⁾eîζφâ^(t)e^−iζφâ⁽⁰⁾e^−iξφâ^(t²⁾i ∝ eîδâL^∆µ(ξ(t²^−t¹^)−ζt)

(i(t1−t2) + 0)^ξ²(it+ 0)^ζ² ×(i(t1−t) + 0)^ξζ(−it2+ 0)^ξζ

(it1+ 0)^ξζ(i(t−t2) + 0)^ξζ . (3.15) Here ξ and ζ are arbitrary numbers and δ_aL is a Kronecker delta. The exponential oscillation function originates from the correlator of the fieldsφ_L and accounts for the fact that the left channel is biased.

Since the time evolution of the dot operators is trivial ˆd(t) =e^−itd, their four-point correlator is just anˆ oscillatory factor. Knowing the expressions for all the averages that appear in Eq. (3.14) and manipulating with the time variables and the integration contours, we arrive at the following expression for the cotunneling current

I∝

∞

Z

−∞

dt

∞

Z

0

dt⁰

∞

Z

0

dt⁰⁰ e^i∆µte^−ε(2t⁰^+t⁰⁰⁾

(2t⁰+t⁰⁰+it)^1−β(it+ 0)^1−γ × t^0α(t⁰+t⁰⁰)^α

(t⁰+t⁰⁰+it)^α(t⁰+it)^α, (3.16) where the exponents are given by

α= (β+γ)/2, β= 2ηL−X

a

η_a², γ= 2ηR−X

a

η_a², (3.17)

and the parameter εis defined by the relation

ε≡+η_L∆µ−∆µ. (3.18)

This parameter represents the difference between the energy level of the dot, shifted by the voltage bias of the left channel, and the Fermi energy of this channel. The value of energy shift,ηL∆µ, has the transparent physical meaning: The voltage bias, applied to the left channel, induces the extra electron density ∆σa(x) =

∆µR

dx⁰V_aL⁻¹(x, x⁰) in the channel a, which follows from the Eq. (3.6) with zeroes on the right-hand side replaced by δaL∆µ. Replacing in the Eq. (3.10) σa with σa+ ∆σa and using again the Eq. (3.6) with the definition of ηL, one arrives at the desired value of the energy shift. Thusεis an effective size of the energy gap for the virtual state in the cotunneling process, where the dot is occupied by an extra electron.

At this point we want to make the following remark about the restriction on the screening fractionsη_a and, thus, on the exponents (3.17). According to the optical theorem P

aη_a = 1. However, this identity

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changes if some fraction of the QD chargeη_g is screened by metallic gates surrounding the QD. These gates may be considered as an ensemble ofn_geffectively one-dimensional scattering channels enumerated byiwith screening fractionsη_i. We consider these channels separately from those enumerated byadue to the different strength of the interaction, since the wave-length of the electrons in the metal is much smaller than the one in 2-dimensional electron gas. Therefore, the number of these channels is large, and necessarily ηi 1 for alli. Therefore, even though the total screening fraction of metal is finiteP

iηi=ηg, its contribution to the exponents (3.17) is negligible, since P

iη_i² scales as 1/ng. Therefore, below we neglect the contribution of the effective channels in the gates to the FES exponents. However, we keep in mind that the more general restriction on ηa holds: P

aηa = 1−ηg≤1.

All the integrals in Eq. (3.16) are convergent, since each time variable enters the integral with negative power and is integrated either with the oscillatory factor or with the exponential function, which decay rapidly at infinity. The analytical structure of the integrand as a function of t is presented in Fig.3.2. It has one branch cut that connects four branching points, which follows from the fact that the total power of the denominator as a function of t is integer. The integration over t from −∞to ∞allows one to deform the contour, and thus simplifies calculations.

t

Re t

i0

i(2t'+t'') i(t'+t'')

it'

Figure 3.2: The analytical structure of the expression under the integral (3.16) as a function of the variable t is schematically shown. The curly line represents the branch cut that connects four branching points.

In the case of free fermions, i.e., whereηa = 0 for all a, the integral (3.16) can be calculated exactly, leading to the well-known expression for the cotunneling current

I∝ ∆µ

ε(ε+ ∆µ), (3.19)

where, obviously, ε = −∆µ = ₀−∆µ. Below, we will use this result to compare to the interacting case. In general, the evaluation of the integral (3.16) as a function of two parameters, ∆µand ε, presents a challenge, and does not seem to be instructive. Indeed, only in the limits of the small bias, ∆µε, and large bias, ∆µε, one approaches the thresholds in the spectrum associated with two biased Fermi seas and thus expects the universal power-law behavior of the current. Therefore, in the rest of the chapter we will concentrate on finding the asymptotic forms of the integral (3.16).

3.4 FES exponents

We start with the low-bias regime and analyze the asymptotic behavior of the integral (3.16) for ∆µε.

Note, that in this regime ε ≈ , and we neglect the difference. We expect the linear dependence of the current on ∆µ, because in this regime the interactions are effectively screened. Indeed, in the course of electron tunneling from one channel to another, that occurs on the time scale 1/∆µ, quantum fluctuations of charge on the QD with characteristic time scale 1/ε are relatively fast and average to negligible values.

Thus, cotunneling in this case can be considered as tunneling of an electron through an effective potential barrier, that only depends on the energy gapε.

In order do find the asymptotic form of the integral (3.16), we integrate separately “fast” and “slow”

functions. We note that the exponential function limits the integrals over variables t⁰ and t⁰⁰ to small intervals of the size of the order of 1/ε around zero. On the other hand, slowly oscillating function of t limits the integral over t to large interval of the size of the order of 1/∆µ. Therefore, taking into account the relation (3.17) that connects the exponents, and keeping leading order terms in ∆µ/ε, we replace four

Noise and interactions in mesoscopic systems far from equilibrium

Thesis

Reference

Noise and interactions in mesoscopic systems far from equilibrium

BORIN, Artyom

BORIN, Artyom. Noise and interactions in mesoscopic systems far from equilibrium . Thèse de doctorat : Univ. Genève, 2019, no. Sc. 5317

DOI : 10.13097/archive-ouverte/unige:115679 URN : urn:nbn:ch:unige-1156797

Noise and interactions in mesoscopic systems far from equilibrium

TH` ESE

pr´ esent´ ee ` a la Facult´ e des Sciences de l’Universit´ e de Gen` eve pour obtenir le grade de Docteur ` es sciences, mention Physique

par

Artem Borin

de Russie

Th` ese N

5317

GEN` EVE

Atelier de reproduction ReproMail

2019

Acknowledgments

Contents

Chapter 1

Introduction

Chapter 2

Bosonization

2.1 Equilibrium bosonization in 1D

2.1.1 Fermionic description

2.1.2 Bosonic description

2.1.3 Applications

2.2 Non-equilibrium bosonization

2.2.1 Bosonic correlations from boundary conditions

2.2.2 Evaluation of the FCS for a quantum noise

2.2.3 Evaluation of the FCS for a classical noise and stochastic path integral

R

Q R

μ

C μ

Q

Q L Q R

Chapter 3

Cotunneling

3.1 Introduction

3.2 Model

ϵ

Δμ

3.3 Perturbation theory

t

Re t

i0

i(2t'+t'') i(t'+t'')

it'

3.4 FES exponents

Q _L Q _R