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Entanglement properties of normal and topological states
Entanglement has, up to now, stayed away from our discussions on topological mate-rials. Yet, these non-local correlations are the fundamental di erence between quantum and classical systems, and at the core of the topological nature of the ground states. Historically, the study of entanglement was rst reserved to quantum information and to attempts to understand the consequences of quantum mechanics as a physical theory. The rst experiments, based on verifying the Bell’s inequality, illustrating the Einstein-Podolsky-Rosen paradox, were meant to check the validity and completeness of quantum mechanics. Nonetheless, the rushing development of quantum computing and information during the last decades has yield an increasing amount of new insights on the properties of entanglement, even in many-body systems. The study of the structure of entanglement has proved to be invaluable to describe (topological) phase transitions at zero temperature and (topological) ground states. Indeed, for such systems, the traditional study of corre-lation functions usually fail to properly describe the changes the system undergoes. The standard quantum correlations are not su cient to discriminate between di erent topo-logical phases, while the study of entanglement, through for example the entanglement entropy or the entanglement spectrum gives clear insights on the nature of the di erent phases.
This Chapter consists in an introduction to the notion of entanglement in quantum sys-tems, and more precisely in many-body physics. Section 2.1 starts by introducing the von Neumann entanglement entropy and other basic entanglement measures. After describing its global properties through a set of examples, we give an overview of the main ideas behind the study of entanglement in many-body systems, with a focus on gapless models. Section 2.2 is dedicated to the entanglement spectrum. It is especially useful for charac-terizing topological states that possess edge states, as the measure of the entanglement spectrum is based on the creation of arti cial edges. Finally, we present in Section 2.3 an introduction to the concept of bipartite uctuations. These uctuations can be used as a weak probe of entanglement, that has the added advantage to be experimentally measur-able. We give a short presentation of previous works on this topic, that focused on normal Luttinger liquids and other charge-conserving models, before devoting the next Chapter to our extension to topological models.
To go beyond the short review proposed in this Chapter, we refer the reader to Ref. .
Definition of the von Neumann Entanglement Entropy
At the core of quantum mechanics is the idea of non-local correlations. Such correlations correspond, in some sense, to shared information between the di erent parts of the system. While we may know perfectly a system S, quantum correlations and superposition may prevent us from knowing the state of any subpart of the system. The amount of shared information can be measured, just as in the classical case, by an entropy function. We will focus on this thesis on the von Neumann entanglement entropy (vNEE), and we will limit ourselves to the bipartite case, where we measure the entanglement between two di erent subsystems.
Let us consider a closed system S at zero-temperature, which is quite naturally in a pure state j i . Let A be a physical subpart of this system, we note A the rest of the system. The density matrix A entirely describes A and its correlations with A: A = Tr (j i h j): (2.1) j i h j is the projector on the state j i in the Hilbert space that describes the total system S, and the trace carries on all degrees of freedom (all sites for example) that do not belong to A. The associated vNEE is: SE(A) = TrA( A ln A): (2.2)
The logarithmic term is unique and properly de ned as A can be diagonalized and has all its eigenvalues between 0 and 1. It is the direct quantum equivalent of the classical Shannon entropy. It is also important to mention an additional set of properties that the vNEE veri es. Similarly to the Shannon entropy, it is actually uniquely speci ed by the following set of properties:
SE must be continuous (with respect to a continuous change in A or in A). It is invariant under local unitary transformations.
It must be additive for independent subsystems: if A = A1 A2 such that A = A1 A2 , then SE(A) = SE(A1) + SE(A2). It must be sub-additive in the general case: SE(A) SE(A1) + SE(A2).
The vNEE can also be extended to non-pure states, and in particular it can be de ned at non-zero temperature for thermal states. Cross-over from T = 0 to large temperature can be derived easily in the family of models that we will study (conformal models).
Though it is the most fundamental, the vNEE is not the only entropy-like function. If we relax the sub-additivity assumption, it is possible to de ne other functions that will have similar properties (but order di erently the entangled states).
Finally, we introduce the mutual information ISE of two disjointed regions A and B. ISE = SE(A) + SE(B) S E(A [ B) (2.10)
The mutual information can be straightforwardly extended to any entropy-like function by analogy with Eq. 2.10. It usually has the advantage of being free from arti cial divergences that can appear in continuum theories, at the price of generally requiring the computation of the multipartite entanglement entropy. In the case of the vNEE, it is additionally positive and monotonically increasing (if B C, ISE (A; B) ISE (A; C)). It is only exactly known for free, non-interacting fermions in one dimension. It has been shown that it is not only a function of the central charge, but also of more microscopic details of the theory, such as the Luttinger parameter of Luttinger Liquids[71, 72].
Entanglement entropy in many-body systems
The entanglement entropy reveals many properties of the many-body ground states. In this Section, we expose some of the main ideas on how to use entanglement entropy to characterize phases and critical theories. We are essentially interested in gapless phases, where anomalous logarithmic terms appear, that can be used to classify the di erent phases. After a rst general discussion, we focus on one and two-dimensional models.
Dependence on the boundaries
As we previously mentioned, the entanglement entropy of a subsystem A with the rest of the system essentially scales with the length of its boundaries. We rst derive some phenomenological physical arguments on the di erent scalings that can appear for di erent many-body systems.
Let us rst consider the case of gapped systems, i.e. systems where the excited states have a nite energy in the thermodynamic limit. The energy gap E, di erence of energy between the ground state and the rst excited states naturally de nes a length scale / E 1 that will govern the behavior of the system. In particular, one expects an exponential decrease in the correlations, with a characteristic length proportional to . Now let us consider d-dimensional region A of characteristic length lA such as represented in Figure 2.1. Then, the vNEE can be seen as a re-summation of the correlations between A and A: given the exponential decrease in these, its long range behavior should scale at best as: Z Z d~rk dr?e j@Ajf( ) = O(lA f( )); (2.11) where @A is the boundary of A, ~r? the local normal vector to the boundary, some numerical constant and f a non-universal function. This indeed corresponds to a dominant scaling exactly proportional to the boundaries between A and the rest of the system. For gapped systems, this area-law (by opposition to a volume-law where the entropy would scale as lAd) has been rigorously proven for both fermions and bosons[73, 74]. It is important to stress the importance of this result: if we take a random state of the Hilbert space HA, for dimHA HA, the vNEE should scale as ln dim HA / VA, the volume of A. Typical ground states of gapped Hamiltonian are very di erent from random states.
For gapless systems, the analysis is more complex. As the gap vanishes, the correlation length diverges. Typically, two-points correlation functions may scale now as power laws of the distance between the two points. For free non-interacting fermions.
Finally, we conclude by mentioning that additional sub-leading contributions may also appear when A has sharp corners and when the dimension of the Fermi surface is smaller than d 1. This is in particular the case for the Dirac metals that will be the subject of Sections 2.1.2 and 3.4.
Gapless systems in one dimension
We start by considering the one-dimensional limit of Eq. 2.13. Let us consider the vNEE of a segment of length lA in the middle of an in nite wire. Its boundaries are two zero-dimensional points. Correspondingly, we assumed that the Fermi surface consists in a set of M points.
On the other hand, in Kitaev’s wire (described in Section 1.1), on the lines = 2t; 6= 0, the gap closes only at either k = 0 or k = . The coe cient in front of the logarithmic term is now = 16 . Given that their vNEEs di er, these two gapless models are not equivalent, in other words, they are not in the same universality class.
In fact, the classi cation through the vNEE of one-dimensional gapless systems is much more generic. Many critical physical systems are described by a conformal eld theory (CFT)1. One-dimensional critical points can be described entirely by these CFTs.
It essentially labels the type of low-energy critical models and de nes its universality class. This central charge does not depend on the microscopic details of the theory, but only on its low-energy behavior. A change in the central charge means that a phase transition occurred. This also includes the interacting fermionic theories that we will study in Chapter 3. The two families that interest us in this thesis are the Ising universality class and the free bosons. The former corresponds to the Quantum Ising model in a transverse eld or the Kitaev’s chain at the critical point, where c = 12 and there is one free Majorana mode. The latter corresponds to normal free electrons, or alternatively to Luttinger liquids or a free scalar boson (described in Appendix E), where c = 1 and there is one free complex fermionic mode.
Entanglement spectrum and topological systems
Definition of the entanglement spectrum Definition
First introduced in Ref.  to study the Fractional Quantum Hall e ect, the entan-glement spectrum (ES) has become a common tool for studying topological systems. It is nothing but an extension of the entanglement entropy and of the study of the reduced density matrix of a system. Instead of computing the entropy, the ES reinterprets the density matrix A as a thermal Hamiltonian:
A = 1 e H A; (2.32) where is an e ective temperature taken to be 14, and HA is the so-called entanglement Hamiltonian. This reformulation is possible because the reduced density matrix stays def-inite positive. The ES is the set of eigenvalues « m = ln m of HA, where m have been de ned in Eq. 2.6. All « m are positive, given that m < 1. The vNEE can be evidently rewritten as a sum over the ES, that therefore o ers much more information. Computation of the ES is mathematically strictly equivalent to computation of all the Renyi entropies5.
The ES has many variants, with di erent properties, depending on the chosen cut A. Up to know, we chose a separation of the Hilbert space in terms of physical space, but any criterion can be used. We note in particular the possibility of choosing the so-called particle cut, where we integrate out a certain number of particles of the system, orbital cuts where orbitals are removed on each site, or cuts in momentum space to study critical phases. In the rest of this section, we will nonetheless focus on the real-space cut, that has the most direct application for the one-dimensional systems we study.
A rst di erence appears between entangled and non-entangled states directly in the ES: for non-entangled states, whatever the dimension of the Hilbert space of A, the en-tanglement spectrum is f0g, while entangled states lead to the appearance of multiple levels. Maximally entangled states will simply have D = min( dim A; dim A) degenerate entanglement states with entanglement energy ln D. Counting the number of low-energy quasi-degenerate eigenvalues and measuring the gap with eventual higher excited energies is therefore fundamental to describe the entanglement of the studied state.
Entanglement spectrum in topological systems
Generalities in d > 1.
The main idea behind the study of the entanglement spectrum is that it is somehow re-lated to the boundary physics of the original Hamiltonian. Take a topological system in a topological phase (gapped), with periodic boundary conditions (no edge states). Trac-ing out the rest of the system creates arti cial boundaries for the region A. For 2 and 3 dimensional models, it was conjectured in Ref. , and later proved in a variety of models by Ref. [107, 108] that the entanglement spectrum coincides with the spectrum of gapless edge states that would appear in an open system with the same geometry as A. In particular, one can identify CFT theories through their tower of states, and consequently identify the topological order of the system. Much more could be said on the properties of the entanglement spectrum in d > 1, but we will focus on d = 1 in the rest of the section.
Table of contents :
1 A general introduction to topology in Condensed Matter
1.1 Kitaev’s wire
1.1.2 Majorana fermions and edge states
1.1.3 Bogoliubov formalism and bulk topology
1.1.4 Quantum Ising model and locality
1.2 General band theory for non-interacting fermions
1.2.1 Single particle Hamiltonian in Fourier space
1.2.2 Two-band Hamiltonians and general solutions
1.3 Topological approach to band theory
1.3.1 Introduction to topology
1.3.2 Bulk hamiltonians and homotopy
1.3.3 Symmetry and classication
1.3.4 Relevant topological invariants for two-band Hamiltonians
1.4 Bulk-edge correspondence and edge states
1.4.1 D class for d = 1
1.4.2 BDI class for d = 1
1.4.3 D class for d = 2
1.5 Topical examples in 1D
1.5.1 The Su-Schrieer-Heeger model
1.5.2 Longer-range Kitaev’s wire
1.5.3 Rashba spin-orbit superconductors
1.6 Topical examples in 2D
1.6.1 The p + ip superconductor
1.6.2 A very brief overview of graphene
1.6.3 Haldane model for topological insulator
2 Entanglement properties of normal and topological states
2.1 Entanglement entropy
2.1.1 Denition of the von Neumann Entanglement Entropy
2.1.2 Entanglement entropy in many-body systems
2.2 Entanglement spectrum and topological systems
2.2.1 Denition of the entanglement spectrum
2.2.2 Entanglement spectrum in topological systems
2.3 Bipartite uctuations as a « weak » measure of entanglement
2.3.1 The problem of entanglement measure
2.3.2 General denition of bipartite uctuations
2.3.3 Bipartite charge uctuations in Luttinger Liquids
2.3.4 Beyond one dimension
3 Bipartite uctuations in non-interacting topological systems
3.1 Bipartite uctuations in 1D: generalities
3.1.1 Preliminaries on charge-conservation breaking
3.1.2 Bipartite uctuations in arbitrary non-interacting systems
3.1.3 The Fejer Kernel: properties and consequences
3.2 One-dimensional BDI class
3.2.1 BDI Hamiltonian and uctuations
3.2.2 Discontinuity of the derivative of the linear term
3.2.3 Logarithmic contributions and generalized uctuations
3.2.4 Conformal origin of the logarithm
3.2.5 Structure factor of the BCF
3.2.6 Bipartite uctuations in Kitaev’s model and the SSH chain
3.2.7 Bipartite uctuations in the extended Kitaev’s model
3.2.8 Extension to nite length and nite temperature
3.2.9 A brief comparison with other charge-based measurements
3.2.10 Beyond the two-band paradigm
3.3 One-dimensional D class
3.3.2 Cusp of the linear term
3.3.3 Logarithmic coecients
3.3.4 An example: extended Kitaev model
3.4 Bipartite uctuations in 2D Dirac materials
3.4.1 Geometric shape, Kernel functions and Sobolev spaces
3.4.2 Fluctuations for a single isotropic Dirac cone
3.4.3 Beyond the single isotropic Dirac cone
4 Beyond non-interacting systems: a complex interplay
4.1 Interactions and bipartite uctuations
4.1.1 Interacting systems and numerical simulations
4.1.2 Bosonization at the critical point
4.1.3 Bipartite uctuations and interactions
4.2 Model and simple limiting cases
4.2.1 Microscopic model
4.2.2 Phase diagram
4.2.3 The Hubbard model
4.3 Close to half-lling: bosonization and RG
4.3.1 Bosonization at half-lling
4.3.2 RG analysis
4.3.3 Characterization of the 4 Majorana phase
4.3.4 Behavior of the large g phases
4.3.5 Nature of the transitions
4.4 The Double Critical Ising phase
4.4.1 Mean-eld precursor to the DCI
4.4.2 Unraveling the ladder
4.4.3 Large g model
4.4.4 Numerical approach
5 Transport in topological systems
5.1 Motivation and summary
5.1.2 Summary of the main results
5.2 Model and bosonization
5.2.2 Tunelling term and number of Majoranas
5.2.3 Bosonization and Majorana fermions
5.3 Quantum Brownian Motion and Topological Kondo model
5.3.1 Quantum Brownian Motion
5.3.2 Far from charge degeneracy: the Topological Kondo limit
5.4 Charge degeneracy point: an exact mapping to the multichannel Kondo Model
5.5 Phase diagram at charge degeneracy
Conclusion and perspectives
A Résumé en francais
B Brief summary of abbreviations
C Brief summary on Hilbert-Sobolev spaces
D Details on microscopic computations
D.1 Brillouin Zone and reciprocal lattice
D.2 Diagonalization and correlation functions of the Rashba nanowires
D.3 Schrieer-Wol transformation in Kitaev’s ladder
E Luttinger liquids, bosonization and RG
E.1 Basics of Luttinger Liquids theory
E.2 Bipartite charge uctuations of a critical c = m bosonic model
E.3 RG analysis in bosonized theories
E.3.2 Derivation for the sine-Gordon model
E.3.3 Beyond the simple sine-Gordon model
E.3.4 Bosonization of the Kitaev-Hubbard model
F Conformal Theory computations
F.1 Bipartite charge uctuations for a c = 1 2 model
F.2 Bipartite charge uctuations for a c = 1 boson
G Topological Kondo model
G.1.1 RG equations for the Topological Kondo model
G.1.2 RG equations for the M-CKM
G.2 Kubo approach to conductance
G.2.1 Kubo formula
G.2.2 Application for the strong coupling limit far from charge degeneracy