This thesis consists of two parts, both centred around the theme of deformations of the canonical (anti-)commutation relations. In the first part, we provide an accessible and self-contained, yet complete, account of the recent partial resolution of the isomorphism problem concerning the q-Gaussian von Neumann algebras of Bozejko and Speicher by the free monotone transport method of Guionnet and Shlyakhtenko. We cover the necessary background in free probability, describe the construction of q-Gaussians and prove some of their elementary properties due to Ricard and Voiculescu, and provide a detailed proof of the existence of free monotone transport and its application to q-Gaussians. Some special attention is given to the commutative case and its links to random matrices, large deviations, and optimal transportation. The second part starts with a brief review of the physics of quons, which are the particles in the Fock space realisation of the q-Gaussians. Afterwards, we turn to parafermions and their importance to edge modes and topological phases. The Fock parafermion operators due to Cobanera and Ortiz are introduced, which allow for a Fock representation of parafermions. Then, we study Fock parafermions as fundamental degrees of freedom, considering both Potts-like Hamiltonians in the spirit of Calzona et alia, and simple tight-binding Hamiltonians in the spirit of Rossini et alia. This includes mappings of Fock parafermions to electrons and mixed Fermion-Boson systems, the exact ground states for the Potts-like models, analytical evidence that the tight-binding models generally have conformal charge c = 1, and first steps towards constructing their phase diagram.
This thesis consists of two parts, both centred around the theme of deformations of the canonical (anti-)commutation relations. In the first part, we provide an accessible and self-contained, yet complete, account of the recent partial resolution of the isomorphism problem concerning the q-Gaussian von Neumann algebras of Bozejko and Speicher by the free monotone transport method of Guionnet and Shlyakhtenko. We cover the necessary background in free probability, describe the construction of q-Gaussians and prove some of their elementary properties due to Ricard and Voiculescu, and provide a detailed proof of the existence of free monotone transport and its application to q-Gaussians. Some special attention is given to the commutative case and its links to random matrices, large deviations, and optimal transportation. The second part starts with a brief review of the physics of quons, which are the particles in the Fock space realisation of the q-Gaussians. Afterwards, we turn to parafermions and their importance to edge modes and topological phases. The Fock parafermion operators due to Cobanera and Ortiz are introduced, which allow for a Fock representation of parafermions. Then, we study Fock parafermions as fundamental degrees of freedom, considering both Potts-like Hamiltonians in the spirit of Calzona et alia, and simple tight-binding Hamiltonians in the spirit of Rossini et alia. This includes mappings of Fock parafermions to electrons and mixed Fermion-Boson systems, the exact ground states for the Potts-like models, analytical evidence that the tight-binding models generally have conformal charge c = 1, and first steps towards constructing their phase diagram.
The study of electrical currents in a regime where a description as a continuous flow fails and the particle nature becomes important, but where notions such as particle density or currents continue to be meaningful, will be at the heart of the present dissertation. Given such a mesoscopic system through which electrons flow, either actively through quantum pumping or passively due to a potential difference, we investigate quantities such as transferred charge, noise, current fluctuations, and relations between them. More generally we shall examine the statistics of particle transport. All discussions will concentrate on compact devices transferring charge between two one-dimensional reservoirs (the leads) of independent electrons. Firstly, we consider the generating function of the moments of transferred charge and we prove that it is given by the determinant of a single-particle operator. It is formally equivalent to a formula of Lesovik and Levitov, but is shown to be well-defined in the situations under consideration, whereas the former is not, in general. The mechanism leading to regularization is typical for mesoscopic systems: we consider a limiting case that reveals the fundamental structure of the problem. In fact, as suited for an open system, the leads contain an infinite number of charges. It follows mathematically that the second quantization ought to be carried out upon the Fermi sea and not upon the vacuum, a procedure known as the GNS construction. We shall rely heavily on operator algebraic techniques. The set of observables, the C*-algebra of canonical anticommutation relations, admits different representations for systems in the thermodynamic limit, which are described by their reduced one-particle density matrix. An important question will be the existence of operators such as the charge or the time evolution in a given representation, which induce so-called Bogoliubov transformations. These prerequisites at hand, we shall pose the problem of counting statistics clearly, prove that the regularized determinant emerges naturally, and show that it describes the transport of integer charges with non-negative probabilities. Secondly, we introduce a model of instantaneous scattering in which various examples can be computed explicitely. In particular, we show that the above theorem applies for scattering through a smoothly operating pump over a finite time interval. We also exhibit situations where the propagator does exist but the generating function is ambiguous. In a more computational approach, we then turn to the relation of individual moments of transferred charge to current correlators. It turns out that the former can indeed be expressed as integrals of the latter over time, provided the current operators are ordered according to Matthews’ T∗ ordering. It is given by first arranging the charge operators according to their time indices, and only then taking the time derivatives, which induce contact terms whose singular support lies along time coincidences. We shall give two derivations of this result, because the first one is transparent, and the second one is related to contradictory computations in the literature where the system is considered to be measured by an external spin. In the model of constant instantaneous scattering, we then argue from the Lesovik-Levitov determinant that the transport can be interpreted in terms of a binomial process in the long time limit. In a parallel computation, we concentrate on the third cumulant as defined by integration of current correlators and show how the contact terms of the T∗ ordering play an essential role in providing the expected binomial result, too. To close this thesis, we suggest an algebraic framework that lays the foundations for a rigorous discussion of the Casimir effect. The problem of infinites arising in the comparison of two physical situations for a bosonic field is interpreted as the non-equivalence of states on the algebra of canonical commutation relations. We exhibit sufficient conditions on the one-particle Hamiltonians which ensure that the Casimir energy is finite and give an expression for it. We discuss its formal equivalence with the result obtained from the standard mode summing method.
CAR on Fock spacesThis chapter is devoted to the study of Fock representations of the canonical anticommutation relations, a basic tool of quantum many-body theory and quantum field theory.It is parallel to Chap. 9, where Fock CCR representations were studied.The basic framework is almost the same as in Chap.9. Throughout this chapter Z is a Hilbert space, called the one-particle space.We will consider the Fock CAR representation acting on the fermionic Fock space Γ a (Z).As in Sect.1.3, we introduce the space Y = Re(Z ⊕ Z) := {(z, z) : z ∈ Z}, which will serve as the dual phase space of our system.Recall that in the bosonic case we equipped Y with the structure of the Kähler space consisting of the anti-involution j, the Euclidean scalar product • and the symplectic form ω: j(z, z) := (iz, iz), (13.1) (z, z) • (w, w) := 2Re(z|w), (13.2) (z, z)•ω(w, w) := 2Im(z|w) = -(z, z) • j(w, w).(13.3)In our presentation of the fermionic case, we will need the Kähler anti-involution j.The symplectic form ω will not be used.Instead of the scalar product (13.2) we will use another scalar product,Again we will avoid identifing Z with Y. CY is identified with Z ⊕ Z by the mapY # , the space dual to Y, is canonically identified with Re(Z ⊕ Z) by using the scalar product (9.2), and CY # is identified with Z ⊕ Z. Fock CAR representationConsider the fermionic Fock space Γ a (Z).Recall that, for z ∈ Z, a * (z), resp.a(z) denote the corresponding creation, resp.annihilation operators defined in Sect.3.4.
Suppose that we have a state ψ on the polynomial algebra generated by the fields ϕ ( y ) satisfying the CCR or CAR relations. For simplicity, assume that ψ is even , that is, vanishes on odd polynomials. Clearly, this state determines a bilinear form on Y given by the “2-point function” We say that a state ψ is quasi-free if all expectation values can be expressed in terms of (17.1) by the sum over all pairings. This chapter is devoted to a study of (even) quasi-free states, both bosonic and fermionic. This is an important class of states, often used in physical applications. Fock vacuum states belong to this class. It also includes Gibbs states of quadratic Hamiltonians. Representations obtained by the GNS construction from quasi-free states will be called quasi-free representations . They are usually reducible. Many interesting concepts from the theory of von Neumann algebras can be nicely illustrated in terms of quasi-free representations. Quasi-free states can be easily realized on Fock spaces, using the so-called Araki–Woods, resp. Araki–Wyss representations in the bosonic, resp. fermionic case. Under some additional assumptions, in particular in the case of a finite number of degrees of freedom, these representations can be obtained as follows. First we consider a Fock space equipped with a quadratic Hamiltonian. Then we perform the GNS construction with respect to the corresponding Gibbs state. Finally, we apply an appropriate Bogoliubov rotation.
A recursive deformation of the boson commutation relation is introduced. Each step consists of a minimal deformation of a commutator [a,a°]=fk(... ;n̂) into [a,a°]qk+1=fk(... ;n̂), where ... stands for the set of deformation parameters that fk depends on, followed by a transformation into the commutator [a,a°]=fk+1(... ,qk+1;n̂) to which the deformed commutator is equivalent within the Fock space. Starting from the harmonic oscillator commutation relation [a,a°]=1 we obtain the Arik–Coon and Macfarlane–Biedenharn oscillators at the first and second steps, respectively, followed by a sequence of multiparameter generalizations. Several other types of deformed commutation relations related to the treatment of integrable models and to parastatistics are also obtained. The ‘‘generic’’ form consists of a linear combination of exponentials of the number operator, and the various recursive families can be classified according to the number of free linear parameters involved, that depends on the form of the initial commutator.
Let H1 be a finite dimensional complex Hilbert space. Let ψ↦Z(ψ) be a canonical anti-commutation relations (CAR) field over H1 acting irreducibly on a Hilbert space K. The ∗-algebra AH1 generated by the Z(ψ), ψ∈H1, is simply all operators on K. However, the CAR field endows AH1 with additional structure, and we are concerned with quantum operations acting in harmony with this structure. In particular, there is a "gauge" automorphism group generated by "second quantizing'' ψ↦eitψ. The fixed point algebra of the gauge group, GH1, is a sub-algebra of AH1 studied by Araki and Wyss. It contains the density matrices of an important class of states, the {\em gauge invariant Gaussian states}, SGIG.
Our focus is on semigroups {etL}t≥0 of quantum operations on AH1 that map SGIG into itself. Each etL is one-to-one, and our first main result is a structure theorem for such quantum operations on GH1 that map SGIG into itself.
We apply this to study semigroups of quantum operations on GH1 that map SGIG into itself. Our second main result is a structure theorem showing that they are parameterized by pairs (G,A) where G is a contraction semigroup generator on H1, and 0≤A≤−G−G∗. We then show that each of these semigroups has a natural extension to the full CAR algebra AH1. Further results are obtained under further assumptions on the pair (G,A).
In this thesis exotic phases of models preserving the symmetries of the one-dimensional quantum Ising and three-state Potts chains are considered, the natural habitats of Majorana fermions and parafermions, respectively. Along with numerical techniques, conformal field theory (CFT) is used to study the resulting physics. Chapter 1 gives an introduction to the field and explains the basics of the numerical techniques to be used throughout the thesis. CFT is briefly introduced in Chapter 2 before the key results necessary for the thesis are given. In Chapter 3, self-dual interactions are added to the critical Ising model, giving a range of different phases. A tricritical Ising point separates the Ising phase from an order-disorder coexistence phase on the chirally symmetric line, while a supersymmetric line provides the transition to an incommensurate phase in the general model. The model is solved exactly on a free-fermionic line giving insight into the points where only numerical analysis can be performed. This is supplemented by Appendix A, where the qualitative difference between similar perturbations is considered. Chapters 4 and 5 consider models with the symmetries of the three-state Potts model. Various phases are found including an order-disorder coexistence phase, a second critical Potts phase, a "not-A" phase, and a representation symmetry protected topological phase. The effect of lattice momentum on the stability of phases and the connection with conformal field theory operators is studied. Lattice realizations of several CFTs are found, such as the tricritical Potts CFT and a CFT expressible in two different forms, either in terms of a boson and fermion, or Potts and tricritical Ising CFTs.