Interactions and disorder in one dimension: from quantum Hall regime to many-body localization

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Thesis

Reference

Interactions and disorder in one dimension: from quantum Hall regime to many-body localization

GOREMYKINA, Anna

Abstract

Le progrès expérimental a frayé une voie pour réaliser des systèmes à N-corps cohérents avec un niveau de contrôle sans précédent, fournissant des aperçus dans la physique des systèmes à fortes corrélations. Motivé par ces avances, nous explorons deux directions dans la physique des systèmes unidimensionnels. Le premier concerne la nature d'excitations collectives au bord d'un système dans le regime de Hall quantique entier et adresse partiellement les conclusions expérimentales déconcertantes sur la cohérence électronique [PRB 93, 035420, 2016] et le flux d'énergie perdu [PRL 105, 056803, 2010]. Deuxièmement, nous étudions la nature de la transition de la localisation à N-corps. Nous présentons un premier groupe de renormalisation (GR) analytiquement soluble qui capture toutes les propriétés connues de la transition et prévoit son caractère de Kosterlitz-Thouless. Les prédictions faites dans ce travail sont en accord avec plusieurs approches différentes, y compris plusieurs GRs phénoménologiques et des études analytiques.

GOREMYKINA, Anna. Interactions and disorder in one dimension: from quantum Hall regime to many-body localization. Thèse de doctorat : Univ. Genève, 2019, no. Sc. 5357

DOI : 10.13097/archive-ouverte/unige:121672 URN : urn:nbn:ch:unige-1216728

Available at:

http://archive-ouverte.unige.ch/unige:121672

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

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Département de physique théorique Professeur E. V. Sukhorukov

Interactions and disorder in one dimension: from quantum Hall regime to many-body localization

THÈSE

présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention Physique

par

Anna Goremykina

de Russie

Thèse N

^o

5357

GENÈVE

2019

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Let me start with thanking Thierry Giamarchi, Patrice Roche, Maksym Serbyn and my advi- sor Eugene Sukhorukov for finding time in their schedules and agreeing to be in the committee for my defence as well as providing valuable comments on my work.

I am happy that along these almost five years road I had the right people by my side who helped me to survive until this moment when I write my own acknowledgements. Eugene gave me a chance to study together with my husband and has always been patient with me especially when I was stuck, frustrated and needed to see a bigger picture. It definitely changed the way I think about physics now. I am also grateful to Eugene for supporting my desire to explore many-body localization and allowing me to spend a year working on it at IST Austria. This year at IST where I worked under the supervision of Maksym Serbyn was invaluable experience. I cannot thank Maksym enough for all the support both in scientific and life matters; for teaching me new ideas, methods and broadening my physics scope; for making me talk more to people and boosting my confidence. And most importantly for inspiring and encouraging me to move further. Maksym has also given me a fantastic opportunity to spend three weeks at KITP in Santa-Barbara where I was exposed to such a vibrant environment of cool physicists. Also my stay would not have been that fun without the care and guidance I have received there from Maksym and his wife Dasha; from friendly support of Romain Vasseur and Sid Parameswaran.

I am grateful to Romain for patiently answering all my questions and sharing his knowledge and ideas which helped me immensely to develop intuition and understanding of the physics we were working on. Also, the abstract of my thesis in french would not be as perfect as it is without his help. Many thanks go to Philipp Dumitrescu for being such a great teacher in the numerics business. Without his careful guidance I would not have dared to write my first C++

code in such a short time period. The help and constant advice I received both from him and Maksym made my experience in the numerics less frightening and in the end enjoyable.

I would like to thank my colleagues at UNIGE Artur Slobodeniuk, Edvin Idrisov, Dario Ferraro and Ivan Levkivskyi for the clarifying discussions at the first stages of my PhD; and for helping me and my husband to adapt to life in Geneva. I am also grateful to Ivan for introducing me to hiking and for letting me discover the beauty of Vallais and the Jura mountains. I was also enjoying the regular meetings of Thierry Giamarchi’s group and profited a lot from them.

I would like to particularly thank Thierry’s group member Michele Filippone for interesting discussions and his friendly attitude. To Dmitry Abanin I am thankful for his advice when I needed it most. I would like to thank the current and former members of Dmitry’s group, Wen

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Wei Ho, Louk Rademaker and Ivan Protopopov for helpful discussions and especially Ivan for his willingness to share his knowledge on many occasions.

The year at IST will be unforgettable due to the time I spent with the members of Maksym’s group Rita Davydova, Rao Peng and Alex Michailidis, both in the office and outside. Rita, thanks for your super sketches and for getting me out to Vienna, when I wanted to hide in my room. And for just making the routine in the office more fun. Maciek Adamowski, thanks for friendly chats in the lonely halls of IST at weekends and our bouldering sessions. Peter Vlasov and the company, for bringing the Russian spirit to IST.

For taking care of all the administrative procedures and for being very understanding I would like to thank Cécile Jaggi-Chevalley, Francine Gennai-Nicole and Angela Stark-Sanchez. I am also very grateful to the department of theoretical physics at UNIGE and to Antonio Riotto for supporting my decision to spend a year at IST and giving me extra time to wrap up my work.

Big thanks goes to my friends for helping me to survive through some exhausting and de- pressing periods: Natasha Rebrova, Theresea Barrett, Olya Poroykova, Katya Kravchenko and especially Vardan Kaladzhyan for always being there for me 24/7; my classmates Tamara Kur- dyaeva and Margo Drevnitskaya for rare but heart-warming skypes and catch-ups. I would like also to appreciate the friendly atmosphere of the CrossfitGVA gym, where I have spent many long and happy hours. Speaking of sports, I am so lucky to have had a chance to learn skiing and snowboarding at the regular samedis de ski with the university, as well as at the university ski camps. It was such a pleasure spending so much time in the sunny mountains with a nice company and it will remain something I will always remember.

My infinite gratitude goes to my parents, Liudmila and Sergey, who have given me all their love and support in all my endeavours. I have always felt cared for. Moreover, my father’s attempt to teach me maths and physics resulted in my deep love for these subjects and defined a path I decided to pursue. Мам, пап, спасибо! Finally, I would like to immensely thank my husband and best friend Artem Borin, with whom we have spent the last years hand in hand doing our PhDs, for teaching me, for supporting and bringing the best in me.

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Le progrès dans les techniques expérimentales ont frayé une voie pour réaliser des systèmes à N-corps cohérents avec un niveau de contrôle sans précédent, fournissant des aperçus dans la physique des systèmes à fortes corrélations. Motivé par ces avances, ce manuscrit explore deux directions dans la physique des systèmes unidimensionnels. Le premier concerne la nature d’excitations collectives au bord d’un système dans un regime de Hall quantique (QH) entier et adresse partiellement les conclusions expérimentales déconcertantes sur la cohérence électronique [Phys. Rev. B 93, 035420, 2016] et le flux d’énergie perdu [Phys. Rev. Lett. 105, 056803, 2010]. Le deuxième sujet se concentre sur la nature de la transition de la localisation à N-corps (MBL), qui a été observée expérimentalement et dont l’existence en une dimension un a été mathématiquement prouvée avec des hypothèses mineurs. Ceci nous permet d’explorer les aspects divers de systèmes 1d en présence de fortes interactions et de désordre.

La première direction contient trois parties. D’abord, nous étudions le taux de transition tunnel d’un système mésoscopique d’états de bord QH à un îlot quantique à plusieurs niveaux.

Nous montrons comment l’application de l’approche de “scattering theory of bosonization” peut simplifier le problème, en aussi soulignant l’universalité de la singularité du bord de Fermi dans une telle approche. La partie suivante est dédiée à l’explication possible de l’expérience [Phys.

Rev. B 93, 035420, 2016] sur la cohérence d’un seul électron injecté dans un état de bord QH au facteur de remplissage ν = 2, qui persiste même pour des hautes énergies d’injection.

Ensuite, nous adressons une expérience [Phys. Rev. Lett. 105, 056803, 2010], qui presente un mystere du flux de l’énergie perdu injecté dans le bord d’un système au ν= 2 et qui n’a pas été expliqué jusqu’à present. Nous montrons que l’énergie perdue pourrait être expliquée par des modes neutres dissipatives supplémentaires se formant dans les régions compressibles du bord.

Finalement, nous abordons le sujet de MBL et nous présentons un premier groupe de renormalisation (GR) analytiquement soluble qui capture toutes les propriétés connues de la transition et prévoit son caractère de Kosterlitz-Thouless. Nous montrons alors que le flux de type de KT peut être déduit intuitivement à partir du mécanisme d’avalanches responsables de la transition. Les prédictions faites dans ce travail sont en accord avec plusieurs approches dif- férentes, y compris plusieurs GRs phénoménologiques et des études analytiques. Nous discutons comment la théorie peut être testée dans un de ces GRs.

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Overview

Advances in the experimental techniques have paved a way to engineering of the coherent many-body systems with unprecedented level of control providing insights into strongly- correlated physics. These systems include isolated ones, with custom-designed features, such as trapped ions [1–3], cold-atomic set-ups [4, 5] and nitrogen-vacancy centers [6, 7], to name a few. On the other hand, mesoscopic systems such as integer quantum Hall edge states enjoy long coherence times/lengths naturally, due to the insensitivity to disorder and the irrelevance of scattering. Although these systems fall short of real quantum computing applications, they already serve as quantum simulators, shedding light on the physics, which so far has been beyond analytical description [8] or discovering new phases of matter [9].

Motivated by these advances, this manuscript explores two directions in the physics of one- dimensional systems. The first one concerns the nature of collective excitations at the edge of an integer quantum Hall (QH) system and addresses partially the puzzling experimental findings on the electron coherence [10] and lost energy flux [11]. The second topic concentrates on the nature of the many-body localization (MBL) transition, which has been observed experimentally [12]

and whose existence in 1d has been mathematically proven with minor assumptions [13]. This allows me to explore the various aspects of 1d systems in the presence of strong interactions and disorder. This thesis is based on the following publications and a preprint:

1. A.S. Goremykina, E.V. Sukhorukov, “Fermi-edge singularity and related interaction induced phenomena in multilevel quantum dots”, Phys. Rev. B 95, 155419 (2017)

2. Anna S. Goremykina, Eugene V. Sukhorukov, “Coherence recovery mechanisms of quantum Hall edge states”, Phys. Rev. B 97, 115418 (2018)

3. Anna Goremykina, Romain Vasseur, and Maksym Serbyn, “Analytically Solvable Renor- malization Group for the Many-Body Localization Transition”, Phys. Rev. Lett. 122, 040601 (2019)

4. Philipp T. Dumitrescu,Anna Goremykina, Siddharth A. Parameswaran, Maksym Ser- byn, and Romain Vasseur, “Kosterlitz-Thouless scaling at many-body localization phase transitions”, Phys. Rev. B 99, 094205 (2019)

5. Anna Goremykina, Artem Borin, Eugene Sukhorukov, “Heat current in a dissipative quantum Hall edge”, arXiv:1908.01213 (2019)

We leave the specifics of the studied systems and problems to the introductions of respective chapters, while briefly outlining below the structure of the thesis. We start with the physics of integer QH edge states in the first part I, giving an introduction to the field and reviewing current experimental results, which have partially motivated my work. The main analytical tool in studying of these systems will be a bosonization technique which we recall in Ch.2. It is then followed by the study of Fermi-Edge singularity in the tunneling rate from the mesoscopic system of QH edge states to a multi-level quantum dot (QD). The low-energy character of

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the problem allows us to underline the universality of FES in the bosonization framework and consequently study it in a more complicated setting, as the tunneling to the excited states of the QD. The next Ch.4is devoted to the discussion and possible explanation of the experiment [10] performed in the group of P. Rocheon the coherence of a single electron injected into a QH edge state at the filling factor of ν = 2, which persists even for high injection energies. Next, in Ch. 5 we address the experiment [11] carried out in the group of F. Pierre, which revealed an already long standing puzzle of the lost energy flux injected in the edge at ν = 2. To attack this problem we study an effective low-energy model of a dissipative compressible strip [14] at ν = 1andν = 2. We show that the additional neutral mode, a.k.a. Aleiner-Glazman excitation

“living” in this strip, carries a portion of the energy flux quantum in addition to the energy transmitted by the usual magnetoplasmons.

In the second part we switch the topic to the many-body localization phase transition, for which we present a concise introduction in Ch.6. There we discuss approaches to studying the transition and present in Ch.7a first analytically solvable renormalization-group scheme which captures all the known properties of the transition. In addition, the RG obtained here predicts the Kosterlitz-Thouless (KT) universality class of the localization transition. In Ch. 8we show that the KT type flow can be discovered under a couple of assumptions from a general scaling theory following from the avalanche mechanism of the transition [15]. In addition we discuss how the theory can be tested, concentrating particularly on one of the previous approaches [16]

in Sec.8.3. Finally, we put the results of this work in the context of current studies in Sec.8.4.

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Part I

Mesoscopic integer quantum Hall systems

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Chapter 1 Introduction

It is well known that a 2D electron gas subject to strong enough magnetic fields reveals the quantum Hall effect [17, 18] – a quantization of conductance σ_xy = νe²/h, with the filling factor ν=n_el/n_LL describing the number of filled Landau levels. The Landau level degeneracy n_LL = 1/2π`²_B relates one state to the area of size 2π`²_B = eB/h (`_B – a magnetic length), while the magnetic flux through such area corresponds to the flux quantumh/e. Thus, another meaning of the filling factor is the number of electrons per flux quantum.

The ideas and theories [19] developed to understand such a “simple” result play key roles in the modern condensed matter physics. Among all of the interesting properties of the QH effect, we are particularly interested in the edge states resulting from the bending of Landau levels at the boundaries of a sample. An intuitive picture of the edge states is given by their semi-classical analogue – the so called skipping orbits, where the electrons drift in the applied magnetic field and the electric field produced by the boundary potential. It thus defines the direction of the drift E×B: opposite at the two edges of the sample. The edge states are the only ones to carry the current flow, as the bulk is gapped out. Each state contributes I = e²/hV to the current, where eV is the potential difference between the source and drain electrodes. Thus, as formulated by Büttiker [20], the quantization of the conductance can be easily interpreted from the point of view of the edge transport. Notably, such a theory is very rigorous, i.e. neither disorder nor scattering hinders the QH effect. The forward scattering on the impurities at the edge are unimportant, as it simply translates into adding an additional phase¹. The states at the opposite edges are also protected from the backscattering [20] as they are separated by an incompressible bulk. Therefore, these 1D chiral, ballistic and coherent channels became tempting candidates for the electron “rails” in the possible quantum interference experiments. It took some 20 years to design and perform such experiments allowing for the careful manipulation of the edge states and a study of their properties. For instance, in 2003, a fabrication of the electron analogue of the Mach-Zender interferometer (MZI) in the ν = 1 QH regime was reported. A basic structure of the interferometer is presented in Fig. 1.1a. A source state is split by the QPC1, such that the electrons can follow the lower and upper arms of the interferometer to interfere at the QPC2. The phase difference between the two paths is tuned by the modulation gate MG, which controls the length of the lower edge channel and thus the

1In what follows we concentrate only on the IQH. In the fractional QH regime, where there can be edge modes moving in both directions, disorder is essential to have a quantized conductance [21].

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a) b)

c)

Figure 1.1: (a) A microscope view of the MZI interferometer realized in a 2DEG [22]. The injected state (solid line) is split by the QPC1 into the two paths which interfere at the QPC2.

The signal is then collected from the drain D2. The modulation gate MG controls the phase difference between the two paths, by varying the enclosed area via the Aharonov-Bohm effect.

(b) The “lobes" structure of the visibility at ν = 2 is compared to a simple decay of visibility with increased bias at ν = 1 in (c).

enclosed area between the two paths via the Aharonov-Bohm (AB) effect. One then measures the visibility of the AB oscillations as a function of the source bias. This is done in the following way. At the fixed value of the bias, the signal oscillates as a function of the MG voltage. One records the maximum I_max and minimum I_min values of the signal and finally plots the visibility V = ^I_I^max⁻^I^min

max+Imin as a function of source bias ∆µ. What exactly is the signal I ? Depending on the experiment it can either be a total current or a differential conductance δI/δ∆µ. In this particular experiment the contribution to the current comes from the electrons with the energies up to ∆µ. To get an energy independent visibility, or so to say construct a monoenergetic source, one measures the differential conductance. Figure1.1c demonstrates the outcome of the measurement, with the visibility decaying with the increased source voltage.

The non-monotonic decay can be trivially explained, as up to certain energies the interference current grows, while decreasing at higher energies due to the increase of scattering inside the interferometer. The derivative of such a function inevitably has a zero.

Subsequent experiments studied MZI at the ν = 2 regime [22–25], measuring the visibility of AB oscillations as a function of bias, temperature, variation of the filling factor and the transmission probabilities of the QPCs. One of the unexpected results was the lobe structure of the visibility as a function of bias, see Fig. 1.1b. This effect was elegantly accounted for by a long-range Coulomb interaction present at the edge [26]. This interaction leads to the separation of the edge excitation into the two modes, whose velocities and the charge they carry depend on the interaction strength (which is influenced by the edge states separation and the distance to the gate). Although the analysis can be held out at an arbitrary strength, the calculations show that the experimentally observable situation most likely corresponds to the case of strong interactions. Then one of the modes, the chargedone, carries all the charge and propagates with a much faster velocity then the neutralor dipolemode. It is the neutral mode

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5 that is responsible for the dephasing as it takes away the information about the state. But the two plasmons can constructively interfere at the second QPC, resulting in the lobe structure of visibility. We shall discuss this model in detail in the next chapter within the bosonization formalism. Before that let us note, that this simple model was extremely fruitful in explaining the varying outcomes of different experiments, related to MZI at ν = 2. Moreover, it was capable to predict the scaling of the coherence length `_ϕ with temperature consistent with the one observed in the experiment, i.e. `_ϕ ∼1/T.

Despite an evident success of this theory, there are still missing pieces in the puzzle of the edge physics. A nine year old experiment [11] showed that the energy of the injected state read out by the quantum dot, misses approximately 13%. Besides, the energy does not leak into the bulk, as it saturates at a certain distance and stops varying. Addressing this problem within the model of Ref. [26] and taking the dispersion in the spectrum of edge excitations (i.e.

considering interacting bosons) into account was not able to resolve the paradox [11]. This suggests that the nature of the edge is more rich and might include other degrees of freedom to which the energy could be redistributed.

The ideas formulated in the above two paragraphs are the main motivation for our study of the QH edge physics. The common picture of the edge states as free fermions at integer ν might be failing. In certain cases, a description in terms of collective excitations, bosons, may be adopted to correctly capture the physics [27]. We demonstrate such a case in Ch. 3, where we propose a set-up allowing for an experimental differentiation between the free fermion and the bosonic description at the edge of a lateral QD in the integer QH regime. This can be done by studying the tunneling rates to the excited levels of the QD.

In Ch. 4, we also go beyond a free fermion description in the attempt to explain another puzzling experimental finding [10] on the coherence of a single electron injected into the edge.

As it has been shown in [28], the observed effect cannot be attributed to the strong interaction between the edge channels within Ref. [26]. Thus, we come up with an effective model taking into account the presence of dissipation and/or dispersion observed in the spectrum of the neutral excitations [29]. While we do not provide microscopic mechanisms of the dissipation, we nevertheless demonstrate how it can significantly alter the interference pattern. Finally, after analysing the results of [28], we comment on the unexpected partial coherence remaining in the charged and neutral excitations, even though the electron is destroyed. We continue with Ch. 5, where we come up with a model of dissipation at ν = 1. In this minimalistic model, a single neutral mode residing in a dissipative compressible strip is shown to transmit additional part of energy flux, which upon further studies could be related to the situation of ν = 2, experimentally explored in Ref. [11].

In order to address these problems analytically, we turn to the bosonization framework in the next chapter, which will be the main workhorse in the description of the low lying excitations at the QH edge. We discuss the details of theν = 2 model of the QH edge [26] and the calculation of the correlation functions. Below we work in units where~=c=e= 1.

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Chapter 2 Bosonization

Bosonization is a basis transformation allowing to switch from 1D fermions to bosons. It is loved by how simple it can render certain problems, which demand substantial efforts when treated in terms of fermions. This is particularly pronounced in 1D where the Fermi-liquid theory fails due to the drastic effects of interactions. The bosonization technique has been exceptionally covered in many reviews [30–33] and textbooks [34–36]. Here we would like to outline the basic steps of bosonization of 1D chiral fermions, which is relevant for the integer quantum Hall.

2.1 Free fermions

We start with the case of ν chiral fermions, in a system of size L, described by the operators ψ_n(x) = 1

√L X

k

c_kne^ikx, {ψ_n(x), ψ⁰_n(y)}=δ_n,n0δ(x−y) (2.1)

with the creation/annihilations operatorsc^†_kn, c_knin a certain channeln satisfying the fermionic anti-commutation rules{c^†_kn, c_k0n⁰}=δ_k,k0δ_n,n0.

The single-particle Hamiltonian can be written as H⁰ = P

n

Pk=Λ

k=−Λε_k,nc^†_knc_kn, with the energies ε_kn and an introduced ultra-violet cutoff Λ. Since we are interested in the low-energy excitations the Hamiltonian can be diagonalized near the Fermi level

H⁰ =X

n k=Λ

X

k=−Λ

v_nkc^†_knc_kn, (2.2)

wherev_{F n}are the Fermi velocities in different channels. It acquires the following form in terms of the field operators ψ_n(x):

H0 =−iX

n

v_{F n} Z

dxψ_n^†(x)∂_xψ_n(x). (2.3)

Since only collective excitations “live” in 1D, the next step is to introduce the charge density operator ρ_n(x) = ψ_n^†(x)ψ_n(x). In the current form it contains the divergent contribution from

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2.2. Interactions and bosonization of fermions 7 the Dirac sea

hρ_n(x)i=L

Z dkdk⁰

(2π)²e^i(k⁰^−k)xhc^†_knc_k0ni= Z dk

2πhc^†_knc_kni. (2.4) However, this is not a problem, as we are studying the excitations above the Fermi level and the divergent contribution is constant. Thus, the definition of the density operator is modified to subtract this part

ρ_n(x)≡:ψ^†_n(x)ψ_n(x) := ψ_n^†(x)ψ_n(x)− hψ_n^†(x)ψ_n(x)i, (2.5) so that the density operator has finite values of its matrix values. This is known as the “normal ordering” procedure. A relation between ρ(x) and ψ_n^†(x⁰):

[ρ_n(x), ψ_n^†₀(x⁰)] = δ_nn0δ(x−x⁰)ψ_n^†(x) (2.6) reflects the charge locality, i.e. ψ_n^†(x⁰) creates a unit charge at the point x = x⁰ in the nth channel. Finally, let us write down a free fermionic correlation function, by also including the time variable in ψ_n(x, t) = √¹

L

P

kc_ke^ik(x⁻^vⁿ^t). Using hc^†_kc_k0i=δ(k−k⁰)θ(µ_n/v_n−k), where µ_n is a Fermi level in the nth channel, one gets a following correlation function

hψ_n^†(x, t)ψ_n(x⁰, t⁰)i=− i 2πv_n

e^iµⁿ^(τ⁻^τ⁰⁾

τ−τ⁰−i0, τ =t−x/v_n, τ⁰ =t⁰−x⁰/v_n. (2.7) We will show later that even in the presence of interactions thelocalcorrelation function in the chiral case has actually a free fermionic character. So let us move to something interacting.

2.2 Interactions and bosonization of fermions

The interacting part of the Hamiltonian reads H⁰ =R

dxU(x−x⁰)ρ(x)ρ(x⁰), where the interaction kernel U(x−x⁰)corresponds to the long-range Coulomb interaction. We do not need to indicate its exact form but we will comment later on its role.

A natural candidate for a boson is a charge density operator ρ_n(q) = _L¹ P

k : c^†_k−qc_k :, as a product of two fermions. If we now show that that is indeed the case, then the interaction Hamiltonian becomes quadratic in terms of bosons and thus easily diagonalizable. Indeed, the commutation relation acquires the form

[ρ_n(k), ρ_n0(k⁰)] = δ_n,n0

X

k2

1 L

hc^†_k

2−k⁰−kc_k₂i − hc^†_k

2−k⁰c_k₂_+ki

(2.8)

=δ_n,n0

δ_k,−k0

L X

k2

hc^†_k₂c_k₂i − hc^†_k₂_+kc_k₂_+ki

(2.9) Note that without the normal ordering we would have the indefinite expression in the line 2.8, corresponding to the subtraction of two infinities. However, in the present form the commutator is well defined and is found to be

[ρ_n(k), ρ_n0(k⁰)] =−δ_k,₋_k0δ_n,n0sgn(v_n) k

2πL, (2.10)

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assuming the quantization of momentum k = 2πn/L. Not surprisingly we have shown, that the density operators have the bosonic commutation relation (up to normalization). The goal now is to rewrite all the fermion operators in the new bosonic basis.

we start with the kinetic part of the Hamiltonian (2.3). It can be simply rewritten as H0 =πX

n

v_n Z

dxρ²(x), (2.11)

exploiting the commutation relation between the Hamiltonian (2.3) and the charge density [H⁰, ρ_n(k)] = v_nkρ_n(k). Therefore, the total (kinetic plus interacting so far) Hamiltonian is quadratic in terms of bosons and the system can be solved exactly.

In a similar manner one can rewrite the fermion operatorψ_n(x)in terms of bosons asψ_n(x)≈ exp(iP

k2π

kρ_ke^ikx), which can be inferred from the commutation relation (2.6). However, as the classical textbooks promptly emphasize, such an expression can not be correct sinceρ_kdoes not change the number of electrons. Thus, there must be a contribution from the total charge in the system both to the expression for the fermionic operator and the Hamiltonian.

Ignoring the technical details (see [36]), below we write down the correct relations. First, the fermion operator ψ_n(x) reads¹

ψ_n(x) = 1

√2πae^iφⁿ^(x), φ_n(x) =ϕ_n(x) +δφ_n(x), (2.12) zero mode: ϕ_n(x) =−ϕ⁽⁰⁾_n + 2πQ_nx/L, (2.13) fluctuations: δφ_n(x) = X

k>0

r2π kL

a_ke^ikx+a^†_ke^−ikx

. (2.14)

where we have introduced the boson operators a^†_k = q2πL

k ρ_−k and a_k = q2πL

k ρ_k, whose form is governed by [a_k, a^†_k₀] = δ_k,k0. So δφ_n(x) describe the electron-hole excitations, which we had already included. To account for the homogeneous part of the charge distribution and for its change, we have introduced thezero modes²inϕ_n(x). The operatorsϕ⁽⁰⁾n andQ_nare canonically conjugate [ϕ⁽⁰⁾n , Q_n] = i, where Q_n simply denotes the total charge Q_n =R

dxρ_n(x). Then the operatore^±iϕ⁽⁰⁾ⁿ indeed corresponds to the creation/annihilation of an electron as[Q_n, e^±iϕ⁽⁰⁾ⁿ ] =

±e^±iϕ⁽⁰⁾ⁿ . The linear part 2πQ_nx/L corresponds to the periodic boundary conditions and is responsible for a phase accumulation associated with the total charge.

The commutation relations for the fields φ_n(x) take the following form, in the case of an infinite system³

[φ_n(x), φ_n0(y)] = iπδ_n,n0sgn(x−y) sgn(v_n). (2.15)

1We mostly use the notations of [32].

2Alternatively, one can formulate the same idea in terms of the Klein factors [36].

3This expression fails for periodic boundary conditions. However, the periodicity is restored due to the zero modes [32].

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2.3. Correlation functions 9 To be general, let us also introduce the time dependence, which is straightforward

φ_n(x, t) = −ϕ⁽⁰⁾_n + 2πQ_n(x−v_nt)/L+X

k

r2π

kL a_ke^ik(x−vⁿ^t)+h.c.

. (2.16)

Then, one can show that the bosonic fields are in fact simply connected to the charge densities and the currents via

ρ_n(x) = ∂_xφ_n(x)

2π , j_n(x) = −∂_tφ_n(x)

2π . (2.17)

For instance, the average stationary current can be found from here to be hj_ni = v_nhρ_ni ≡ µ_n/2π, where µ_n is a chemical potential, so that it agrees with the Landauer formula.

A final remark, on the formula (2.12), is the role of the prefactor 1/√

2πa, which arises from a need to regularize the integrals overk in (2.14). Introducing the exponential cutoff e⁻^ka, one can check that including the prefactor guarantees the correct anti-commutation relations for ψ_n(x). In the further calculations we will often omit the prefactor for brevity.

In Chapter3we will recall the details of the scattering approach to bosonization, but for now let us go back to the Hamiltonian and appreciate its simple quadratic form:

H= 1 8π²

X

n,n⁰

Z

dxdyV_nn0(x−y)∂_xφ_n(x)∂_yφ_n0(y), (2.18)

where V_nn0(x−y) = U_nn0(x−y) + 2πv_nδ_nn0δ(x−y). Notably, the interaction in the channel reduces only to the modification of velocity⁴.

2.3 Correlation functions

2.3.1 Chiral fermions

Let us now show how the free fermion correlation function (2.7) can be obtained within the bosonization approach. We start from the definition (2.12) and apply it to the case of the non-interacting channels, while using the Gaussian character of the theory:

hψ_n^†(x, t)ψ_n(x⁰, t⁰)i ≡ 1

2πae^iµⁿ^(τ−τ⁰⁾he^−iδφⁿ^(x,t)e^iδφⁿ^(x⁰^,t⁰⁾i (2.19)

= 1

2πae^iµⁿ^(τ⁻^τ⁰⁾exp hδφ_n(x, t)δφ_n(x⁰, t⁰)−1/2(δφ_n(x, t)²+δφ_n(x⁰, t⁰)²)i

(2.20)

= 1

2πae^iµⁿ^(τ⁻^τ⁰⁾exp(K_nn(x, t, x⁰, t⁰)). (2.21) Then, at zero temperature the correlation function K(x, t, x⁰, t⁰)reads

K_nn(x, t, x⁰, t⁰) = Z _∞

0

dk k

e⁻^ikvⁿ^(τ⁻^τ⁰⁾−1

e⁻^ka= ln a

a+iv_n(τ−τ⁰), (2.22) where, as noted above, 1/a is an ultra-violet cutoff regularizing the integration. One can immediately see that it results in the very expression Eq. (2.7), so that the free fermions and free bosons are two equivalent descriptions. However, in the case of interacting electrons

4This is known as ag4 process in the Luttinger liquid theory [36].

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the above result can be easily generalized in terms of bosons, which is not possible in the fermionic description. Let us demonstrate it on the example, which will be particularly useful in Chapter 4.

2.3.2 QH edge at ν = 2

In this case there are two co-propagating edge modes coupled by a Coulomb interaction:

H= 1 8π²

X

n,n⁰

Z

dxdyV_nn0(x−y)∂_xφ_n(x)∂_yφ_n0(y), n, n⁰ ∈[1,2]. (2.23)

If the interaction is non-local, one would diagonalize it in the k-space. For us, the more interesting case will be the local interactions. Although, the Coulomb interactionU(k)∼ln(ka) is long-range one can assume that it is screened at certain distances, defined, for instance, by a distance to metallic gates or air bridges, which form the structure of the experimental set-up.

Thus, at low energies U(k) can be taken as a constant, resulting in the real space locality.

Such a model of interaction, introduced in [26] was successful in explaining the lobe-structure of the MZI visibility and the temperature dependence of the coherence length, discussed in Ch. 1. Then, one can diagonalize this Hamiltonian by rotating the fieldsφ˜_n=S_nn0φ_n0 with the transformation

S_nn0 = cosα sinα

−sinα cosα

!

. (2.24)

So the correlation function, say in the first channel, acquires the form:

K₁₁(τ, τ⁰) = Y

n=1,2

a

a+iv_n(τ −τ⁰)

([S⁻¹]1n)²

. (2.25)

Interestingly, this correlator coincides (up to a prefactor) with a free-fermion one (2.22) when x=x⁰.

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Chapter 3 Fermi-edge singularity and related interaction effects in multilevel QDs

In the previous chapters we mentioned that interactions play crucial role in 1D and can not be neglected. Another famous effect arising from interactions is the Fermi-edge singularity (FES).

Originally discovered in the X-ray absorption spectra of metals [37], it represents a power- law divergence near the Fermi level. Mahan [38] has shown, building on the perturbation theory, that the FES exponent is defined by the finite state interaction between an electron near the Fermi level and a hole left behind. Already then it became clear that there is another contribution resulting from the orthogonality catastrophe, i.e. the orthogonality of the many- body wave functions with and without a local scattering potential, proven by Anderson [39].

Both Mahan and Anderson terms were obtained by Nozières and De Dominicis [40], who have also shown that the problem is exactly solvable at an arbitrary interaction strength.

In the more condensed matter context, the FES was shown to enhance the transition rate in the resonant tunneling [41], which was then measured in [42]. Experiments have explored the effect in various set-ups: in quantum wires [43], in the presence of random telegraph signals [44], in the tunneling through InAs QDs both via I-V curve [45] and shot noise measurements [46,47], and recently via lateral GaAs QD [48]. Theoretically, the effect has also been extensively explored [49–51], with a particular emphasis on 1D systems [52–56]. Finally, new methods were developed to study FES in the out-of-equilibrium set-ups [57–61].

Among experimentally accessible configurations the QH based systems might be a good plat- form for studying such pronounced effects of interaction as FES. Imagine a set-up, as pictured in Fig. 3.1, where a QD is interacting with a number of QH channels at ν = 1, where all but one are grounded. Varying a bias ∆µ between this channel, saymth one, and a level of a QD, one can observe the FES in the tunneling rate to a QD, which has a form [60]

Γ∝∆µ^α, α= 2q_m+X

q_n², (3.1)

where q_n < 0 denotes the charge induced in the nth channel. One can see that α is approximately inversely proportional to the number of scattering channels. Therefore, while the effect can be suppressed in systems where the number of channels is large, the QH based set-ups do

11

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not have such a drawback.¹ Note that tunneling between co-propagating QH edge channels is

0

𝜙₁

𝜙 𝜏

QD U 𝜙

2

𝜙₃ 𝜙₄

Δ𝛍

Figure 3.1: A scheme of one of the possible system set-ups. A quantum dot in the QH regime, described by the bosonic field φ₀, interacts with N = 4 QH edge channels at the filling factor ν = 2. The dashed line corresponds to tunneling, while the wavy lines depict Coulomb interactions. The rate of tunneling from one of the channels is studied as a function of the bias ∆µ between the channel and the dot.

suppressed [29,62] due to different spin projections. Hence only the counter-propagating ones are considered. However, besides that, there is no formal difference between these two cases.

Such a set-up is not artificial and has been implemented, for example, in [63], where the micrograph of the fabricated structure and a simplified scheme are presented in Fig. 3.2.

Figure 3.2: A scheme (a) of the experiment and a real micrograph (b) of the structure from [63]. The set- up comprises a MZI, whose dephasing, induced by the strong coupling with a QD, was studied.

In this chapter we would like to extend this picture, by studying the tunneling to the excited states of QDs in different regimes. Our approach is based on the scattering theory for bosons [32], which is a convenient method to treat QH systems. Being widely used [64–67] for its clarity and relative simplicity, this formalism remains relatively new. This novelty allows us to look at the FES from another point of view by studying it in the tunneling to the dot in the QH regime - an object extensively explored [68–71]. The benefit of such an approach is that in the low-energy limit the effect is universal and the theory can also be applied to a 3D metallic QD.

Therefore, we clarify a formalism of the scattering theory of bosons in Sec. 3.1 and demonstrate its potential in the application to the FES phenomenon. On the other hand, we explain that the theory has a clear physical meaning. Namely, the scattering states, constituting the basis for the boson fields, can be expressed in terms of certain positive charges in the low energy limit. Surprisingly, those turn out to be the charges (with a negative sign), induced in the channels around the QD when it is charged. Hence, the scattering problem of bosons

1If there is no strong screening from the metallic gates.

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3.1. Fermi - Edge Singularity from scattering theory 13 becomes deeply connected to the electrostatic one, which explains the universality of the FES phenomenon.

Then, in Sec. 3.2, we go beyond the low energy limit for the QD in QH regime, so that the scattering of the incoming wave in the channel leads to an excitation of collective modes in the QD. The nature of these excitations is fully governed by two parameters: the coupling constant σ =P

nq_n² and the dispersion of plasmons in the dot.

We first concentrate on the no dispersion case. Then, if there is no interaction, i.e. σ = 0, the tunneling rate behaves as a set of steps as a function of the bias. The steps correspond to the free fermion energy levels in the QD, implying that the bosonic and the fermionic pictures describe the same entity. If the interaction is “turned on", the steps become smeared, due to the finite width that the energy levels acquire. The width is proportional to the couplingσ, but it also grows quadratically with the number of the energy level. Thus, even if the interaction is small, it leads to a non-perturbative effect, which we are able to describe analytically due to the correspondence between free-fermion levels and boson resonances in this geometry and the chirality of the boson fields.

When there is interaction in the QD, the spectrum of plasmons acquires in general a weak dispersion. We then demonstrate splitting of the fermion levels (starting from the second one).

It originates from the shift between different single- and multi-plasmon processes corresponding to the excitation of a particular fermion level. The effect is mostly pronounced for the second level, on which we dwell in detail.

The following discussion returns to QDs with a linear spectrum for plasmons. In Sec. 3.3, we study the tunneling rate to a QD with two chiral edge channels with strong long-range interaction. Those can be decoupled into charged and neutral modes. By a proper choice of the bias only the lowest energy level of the charged mode can be excited. However, the heights of the steps stop being equal and acquire a universal structure which is a consequence of the strong interaction between the channels, leading to the charge fractionalization.

The previous case of a QD in QH regime immediately suggests that somewhat similar effects might be seen in the rate of tunneling to a 3D QD in the situation considered in the Sec. 3.4.

Indeed, this time there is again a separation between the charged and neutral modes, where by the latter we imply the energy levels of dimensional quantization. Using the formalism from Sec. 3.1, we analyze this situation and show the result to resemble the one of the Sec. 3.3. The difference is, however, in the fact that there is a direct coupling to the neutral modes so that the heights and the positions of the steps are arbitrary.

3.1 Fermi - Edge Singularity from scattering theory

Let us consider a QD of the characteristic size L in the quantum Hall regime with the filling factor ν = 1 interacting with N edge channels of a QH system at an integer filling factor, which we depict in Fig.3.1. Such a complex can be realized in a 2D electron gas, where using electrostatic gates and forming quantum point contacts one can create potential barriers, allowing some of the edge channels to pass while others to reflect [63]. To account for strong effects of interaction we first use the bosonization technique, discussed in Chapter 2. Next we

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describe the essence of the scattering theory for bosons, which is then applied to demonstrate the manifestation of the FES.

The Coulomb interaction might be screened in a complicated way, therefore we do not assume any particular form of density-density interactions. We only require that the sizeLof the QD is much smaller than the wave length of the density fluctuations in the edge channels (plasmons), thus taking into account the low-energy character of the FES. To bosonize the electrons in the QD, we consider its edge as a one-dimensional ring. We express the electron operators in the channels, ψ_n(x) and in the ring ψ₀(x), with the help of the boson fields φ_n(x) and φ₀(x):

ψ_n(x)∝e^iφⁿ^(x), n = 0, ..., N, (3.2) where the field φ_n(x) is related to the charge density operator ρ_n(x) = _2π¹ ∂_xφ_n(x). Next, we write down the Hamiltonian of the interacting fermions in terms of the new bosonic fields:

H =H⁰+H^int+H^t, (3.3)

where the free Hamiltonian is given by H0 = 1

4π X

n=0,..,N

v_n Z

dx{∂_xφ_n(x)}². (3.4)

We included the interaction part H^int = 1

8π² X

nn⁰

Z Z

dxdyU_nn0(x, y)∂_xφ_n(x)∂_yφ_n0(y) (3.5) with arbitrary electrostatic potentials U_nn0. Finally, the last term describes tunneling between the mth channel and the QD at some point x₀:

Ht=τ(A+A^†), A=eⁱ^{^φ⁰^(x⁰⁾⁻^φ^m^(x⁰⁾^}. (3.6) However, the exact position x₀ is of no interest because of the long wavelength limit, allowing to consider the bosonic field in a certain interaction region being independent of the coordinate.

Next, let us calculate the tunneling rate from one of the channels, say themth one, to the QD when a bias∆µis applied between them.² Notably, the change of the electro-chemical potential at the channel shifts the dot level due to the electrostatic interaction. The value∆µtakes this effect into account and will be written down explicitly later. Addressing the tunneling term as a perturbation, one can express the tunneling rate as the integral

Γ =|τ|² Z _∞

−∞

dt

A(0)A^†(t)

. (3.7)

Note, that such an approach enables us to use the fact that the remaining part of the Hamil- tonian, H₀ +H_int, has a quadratic form in the bosonic fields. Then the tunneling rate can be

2The rate of tunneling in the opposite direction follows from the electron-hole symmetry.

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3.1. Fermi - Edge Singularity from scattering theory 15 expressed in terms of their two-point correlators. The bosonic fields can be found from the equations of motion ∂_tφ_n(x) = i[H0+Hint, φ_n(x)], n = 0, ..., N that reveal

∂_tφ_n(t) +v_n∂_xφ_n(x) + + 1 2π

X

n⁰=0,..,N

Z

dyU_nn0(x, y)∂_yφ_n0(y) = 0. (3.8)

The solution may be presented as a sum of the zero mode ϕ_n(x, t) and the fluctuating part δφ_n(x, t), discussed in Chapter 2: φ_n(x, t) = ϕ_n(x, t) +δφ_n(x, t), where

ϕ_n(x, t) =−µ_nt+ϕ⁽⁰⁾_n (x), (3.9) δφ_n(x, t) =

Z _∞

0

√dω ω

N

X

n⁰=1

Φ_n0nω(x)e⁻^iωta_n0(ω) +h.c.

. (3.10)

Basically, the zero modes solve the system (3.8) in the zero frequency limit and satisfy the following equations:

µ_n=v_n∂_xϕ⁽⁰⁾_n (x) + 1 2π

X

n⁰

Z

dy∂_yϕ⁽⁰⁾_n₀ (y)U_nn0(x, y). (3.11) Thus, zero modes describe stationary charge densities and corresponding phase shifts, while the deviations are taken into account by the fluctuating part. We expressed the latter in the second- quantized form in the basis of the scattering states Φ_n0nω(x)with the creation and annihilation operators a^†_n₀(ω), a_n0(ω) satisfying the usual bosonic commutation relations. The scattering states diagonalize the HamiltonianH₀+H_intand satisfy specific boundary conditions. Namely, the scattering state Φ_n0nω(x) is described by an incoming plane wave in the channel n⁰ which then scatters into all the other channels n = 0, . . . , N. Thus, the scattering state presents the set of N + 1 functions, enumerated by the second index, while the first index enumerates the scattering states.

To find correlation functions entering the expression (3.7) for the tunneling rate, we exploit the low-energy limit and perform the perturbation expansion of the scattering states in vicinity of the QD in frequency:

Φ_nlω(x) = Φ⁽⁰⁾_nl (x) +iωΦ⁽¹⁾_nl (x). (3.12) It means that we look at the asymptotic behaviour of the scattering states in the regionx∼L, for which our approximation ωL/v 1 is valid. On the other hand, such an expansion may be understood as a way to describe how strong the scattering is. Specifically, the parameter ωL/v being small implies that almost the whole incident wave gets transmitted. Substituting now the expression (3.12) into (3.10) and into (3.8), we arrive at the system

0 =v_n∂_xΦ⁽⁰⁾_nl (x) + 1 2π

X

n⁰

Z

dy∂_yΦ⁽⁰⁾_nn₀(y)U_ln0(x, y), (3.13) Φ⁽⁰⁾_nl (x) =v_n∂_xΦ⁽¹⁾_nl (x) + 1

2π X

n⁰

Z

dy∂_yΦ⁽¹⁾_nn₀(y)U_ln0(x, y). (3.14)

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Obviously, Φ⁽⁰⁾_nl (x) = const are the solutions of (3.13). Particularly, for our scattering problem Φ⁽⁰⁾_m0(x) = ε₀, Φ⁽⁰⁾_nl (x) =δ_nl, n, l= 1, ..., N, (3.15) whereε₀needs to be defined. In fact, we wrote the solutions in such a form because substituting them into (3.14) brings us to the same kind of electrostatic equations (3.11) that identify the zero modes. Therefore, the physical meaning ofΦ⁽⁰⁾_nn₀ is that they define the chemical potentials in the channels. So let us find the dot potential ε₀, by treating the problem as an electrostatic one and formally write its solution

q_n=X

nn⁰

C_nn0µ_n0, (3.16)

whereq_n= _2π¹ R

dx∂_xϕ⁽⁰⁾_n (x), n= 0,1, ..., N are the charges in the channels and at the dot. The particular form of the interaction is of no interest and is generally described by a capacitance matrix C_nn0. Since all the channels are grounded, except for the mth one with µ_m = 1 and naturally ε₀, which arises due to the interaction, we get:

q_m =C_mm +C_m0ε₀. (3.17)

As there is no “charge” at the dot q₀ = 0, it is easy to define its “potential” ³

ε₀ =−C_m0/C₀₀. (3.18)

The meaning of this ratio is simple. If we consider a different problem, where a QD with a unit charge q₀ = 1 is screened by the grounded channels, we easily find from q_n = C_n0µ₀ that the charge induced in the nth channel is

q_n=C_n0/C₀₀. (3.19)

We then conclude that the “potential” ε₀, induced in the dot in the particular situation where the plane wave is incident in the channel m is the same up to a sign as the charge induced in this channel in the set-up when the dot is charged:

ε₀ =−q_m, q_m <0. (3.20)

Thus, in the low energy limit the scattering and the electrostatic problems are simply connected, which reflects the universality of the FES phenomenon. Finally, as it is mentioned above, the interaction between the biased channel m and the QD raises the dot’s energy level by µ₀ =−µ_mC_m0/C₀₀ [compare to Eq. (3.18)], so we denote the difference in their potentials as

∆µ=µ_m−µ₀.

3It is worth noting that the potentials µn in the electrostatic problem (3.16) are of 1/time dimensionality, while the charges are dimensionless. However, in the scattering problem (3.13)-(3.14) it is the potentials that are dimensionless and the charges get the time dimension. On the other hand, we are comparing the dimensionless potentialε0 and the chargeqn, so that the fact that they are equal is self-consistent.

Interactions and disorder in one dimension: from quantum Hall regime to many-body localization

Thesis

Reference

Interactions and disorder in one dimension: from quantum Hall regime to many-body localization

Interactions and disorder in one dimension: from quantum Hall regime to many-body localization

THÈSE

présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention Physique

par

Anna Goremykina

de Russie

Thèse N

5357

GENÈVE

2019

Overview

Contents

I Mesoscopic integer quantum Hall systems 2

II Many-body localization phase transition 56

Appendixes 81

Part I

Mesoscopic integer quantum Hall systems

Chapter 1 Introduction

a) b)

c)

Chapter 2 Bosonization

2.1 Free fermions

2.2 Interactions and bosonization of fermions

2.3 Correlation functions

2.3.1 Chiral fermions

2.3.2 QH edge at ν = 2

Chapter 3

Fermi-edge singularity and related interaction effects in multilevel QDs

3.1 Fermi - Edge Singularity from scattering theory