Building upon Dyson's fundamental 1962 article known in random-matrix theory as 'the threefold way', we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds. The primary data of the classification are a Nambu space of fermionic field operators which carry a representation of some symmetry group. Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms. After reduction, the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V, a space of endomorphisms in End(V+V*), a bilinear form on V+V* which is either symmetric or alternating, and one or two antiunitary symmetries that may mix V with V*. Every such set of block data is shown to determine an irreducible classical compact symmetric space. Conversely, every irreducible classical compact symmetric space occurs in this way. This proves the correspondence between symmetry classes and symmetric spaces conjectured some time ago.
Building upon Dyson's fundamental 1962 article known in random-matrix theory as 'the threefold way', we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds. The primary data of the classification are a Nambu space of fermionic field operators which carry a representation of some symmetry group. Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms. After reduction, the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V, a space of endomorphisms in End(V+V*), a bilinear form on V+V* which is either symmetric or alternating, and one or two antiunitary symmetries that may mix V with V*. Every such set of block data is shown to determine an irreducible classical compact symmetric space. Conversely, every irreducible classical compact symmetric space occurs in this way. This proves the correspondence between symmetry classes and symmetric spaces conjectured some time ago.
The goal of this diploma thesis is to develop a field theory, for a specific class of disordered bosonic systems, in order to study, whether disordered bosons generically \nhave universal statistical properties, similar to fermions. \nNow, if disordered bosons show system independent statistical features, the evident \nquestion is whether the bosonic universality classes are different from the fermionic \nones. For fermions, the classification in terms of symmetric spaces is a largely algebraical one, which happens in a complexified setting before compact or non-compact \nreal forms are specified. Hence one can rightly state that all effective bosonic non-linear \nσ-models are already classified within the ten fold way together with the fermionic ones. \nAnd indeed, in the system studied in [LSZ06], well known universal GUE statistics \nwere found in the bulk and Lück finds GOE statistics in [Lü09], both for systems of \ndisordered bosons. \nBut for bosons non-linear σ-models are not everything. A subtle point about noninteracting, i.e. quadratic, bosonic systems is stability. One single particle state \nof negative energy for a bosonic system immediately leads to an unbounded many \nparticle Hamiltonian. For fermionic systems, a finite number of such single particle \nstates is of no concern, due to the Pauli principle. This important difference leads \nto the well studied Gaussian ensembles not providing feasible distributions of bosonic \nHamiltonians. If, however, more complicated probability distributions are taken into \naccount, one has to be prepared to face a more complicated effective model than a \npure non-linear σ-model in the end. In this sense, an interesting question is whether \nbosonic systems if they show universal behaviour lead to novel universality classes. \nA hint to those might be the unusual density of states near zero energy that was \nfound in [LSZ06] or [GC02] and [GA04]. On physical grounds, one can expect the \ncommutation relations of bosons to have important effects at low frequencies, which \nfits to the observations of [GC02] and [GA05]. \nIn this thesis, we will develop and study a specific model, which is the next step \nafter the work of Lück, Sommers and Zirnbauer, [LSZ06], towards a more realistic \ndescription of a disordered system of bosonic degrees of freedom. As in [LSZ06], our \nmodel is still purely random, i.e. there is no limit of an underlying pure system and \nwe also stay with a model without global symmetries, such as time reversal or charge \nconjugation. But we turn from the homogeneous, zero-dimensional model to a spatially \nextended one and implement the physically important feature of locality. Furthermore, \nwe add an additional parameter to tune the number of modes per volume. In particular \nwe can hereby enforce a macroscopical number of zero modes. The goal is to study the \ndensity of states of the new model and therefore to develop an effective field theory. \nThroughout we will pay special attention to the region near zero frequency and watch \nout for unusual, possibly universal features.
We provide a systematic treatment of the tenfold way of classifying fermionic systems that naturally allows for the study of those with arbitrary N-body interactions. We identify four types of symmetries such systems can possess, which consist of one ordinary type (usual unitary symmetries), and three non-ordinary symmetries (such as time reversal, charge conjugation and sublattice). Focusing on systems that possess no non-trivial ordinary symmetries, we demonstrate that the non-ordinary symmetries are strongly constrained. This approach not only leads very naturally to the tenfold classes, but also obtains the canonical representations of these symmetries in each of the ten classes. We also provide a group cohomological perspective of our results in terms of projective representations. We then use the canonical representations of the symmetries to obtain the structure of Hamiltonians with arbitrary N-body interactions in each of the ten classes. We show that the space of N-body Hamiltonians has an affine subspace (of a vector space) structure in classes which have either or both charge conjugation and sublattice symmetries. Our results can help address many open questions including the topological classification of interacting fermionic systems. (C) 2017 Elsevier Inc. All rights reserved.
The random matrix theory (RMT) can be used to classify both topological phases of matter and quantum chaos. We develop a systematic and transformative RMT to classify the quantum chaos in the colored Sachdev-Ye-Kitaev (SYK) model first introduced by Gross and Rosenhaus. Here we focus on the two-colored case and the four-colored case with a balanced number of Majorana fermions N. By identifying the maximal symmetries, the independent parity conservation sectors, the minimum (irreducible) Hilbert space, and especially the relevant antiunitary and unitary operators, we show that the color degrees of freedom lead to novel quantum chaotic behaviors. When N is odd, different symmetry operators need to be constructed to make the classifications complete. The two-colored case only shows the threefold Wigner-Dyson way, and the four-colored case shows the tenfold generalized Wigner-Dyson way which may also have nontrivial edge exponents. We also study two- and four-colored hybrid SYK models, which display many salient quantum chaotic features hidden in the corresponding pure SYK models. These features motivate us to develop a systematic RMT to study the energy level statistics of two or four uncorrelated random matrix ensembles. The exact diagonalizations are performed to study both the bulk energy level statistics and the edge exponents and to find excellent agreement with our exact maximal symmetry classifications. Our complete and systematic methods can be easily extended to study the generic imbalanced cases. They may be transferred to the classifications of colored tensor models, quantum chromodynamics with pairings across different colors, quantum black holes, and interacting symmetry protected (or enriched) topological phases.
The tenfold way is a classification scheme for the building blocks of free fermion systems. More precisely, it classifies the isomorphism classes of spaces of equivariant free Hamiltonians in irreducible fermion systems with symmetries. This classification scheme naturally leads to the K-theoretical classification of topological phases of matter, known as the periodic table of topological insulators and superconductors. Topological K-theory is represented by spectra KU and KO, and in this article we present realizations of these spectra in terms of time evolution operators of irreducible free fermion systems with symmetries, with explicit formulas for the structural suspension maps. We introduce a geometric definition of the space of weakly interacting time evolution operators, as the complement of the cut locus of the subspace of free operators. Our main result is that spectra KUwi and KOwi of weakly interacting time evolution operators deformation retract to KU and KO. We thus have a stable homotopy theoretical proof that the tenfold way is stable to weak interactions.
Although classification for free-fermion topological superconductors (TSC) is established, systematically understanding the classification of 3D interacting TSCs remains difficult, especially those protected by crystalline symmetries like the 230 space groups. We build up a general framework for systematically classifying 3D interacting TSCs protected by crystalline symmetries together with discrete internal symmetries. We first establish a complete classification for fermionic symmetry protected topological phases (FSPT) with purely discrete internal symmetries, which determines the crystalline case via the crystalline equivalence principle. Using domain wall decoration, we obtain classification data and formulas for generic FSPTs, what are suitable for systematic computation. The four layers of decoration data (n1,n2,n3,ν4) characterize a 3D FSPT with symmetry Gb×ω2Z2f, corresponding to p+ip, Kitaev chain, complex fermion, and bosonic SPT layers. Inspired by previous works, a crucial aspect is the p+ip layer, where classification involves two possibilities: anti-unitary and infinite-order symmetries (e.g., translation). We show the former maps to some mirror FSPT classification with the mirror plane decorated by a p+ip superconductor, while the latter is determined by the free part of H1(Gb,ZT), corresponding to weak TSCs. Another key point is the Kitaev chain decoration for the anti-unitary symmetries, which differs essentially from unitary ones. We explicitly obtain formulas for all three layers of decoration (n2,n3,ν4), which are amenable to automatic computation. As an application, we classify the 230 space-group topological crystalline superconductors in interacting electronic systems.
Quantum mechanical systems with some degree of complexity due to multiple scattering behave as if their Hamiltonians were random matrices. Such behavior, while originally surmised for the interacting many-body system of highly excited atomic nuclei, was later discovered in a variety of situations including single-particle systems with disorder or chaos. A fascinating theme in this context is the emergence of universal laws for the fluctuations of energy spectra and transport observables. After an introduction to the basic phenomenology, the talk highlights the role of symmetries for universality, in particular the correspondence between symmetry classes and symmetric spaces that led to a classification scheme dubbed the 'Tenfold Way'. Perhaps surprisingly, the same scheme has turned out to organize also the world of topological insulators.