Symmetry groups are projectively represented in quantum mechanics, and crystalline symmetries are fundamental in condensed matter physics. Here, we systematically present a unified theory of quantum mechanical space groups from two complementary aspects. First, we provide a decomposition form for the space-group factor systems to characterize all quantum space groups. It consists of three factors, the factor system for the translation subgroup
L, an in-homogeneous factor system for the point group
P, and a factor connecting
L and
P. The three factors satisfy three consistency equations, which are exactly solvable and can completely exhaust all factor systems for space groups. Second, since factors systems are classified by the second cohomology group, we show the (co)homology groups for space groups can be derived from Borel's equivariant (co)homology theory, which leads to an algorithm that can compute all (co)homology groups for space groups. To demonstrate the general theory, we explicitly present quantum wallpaper groups with the
Z2 gauge group. Furthermore, as a primitive application, we find the time-reversal invariant quantum space groups with inversion symmetry can lead to a novel clifford band theory, where each band is fourfold degenerate to represent certain real Clifford algebras with topologically nontrivial pinor structures over the Brillouin zone. Our work serves as a foundation for exploring quantum mechanical space groups, and can find applications in spin liquids, unconventional superconductors, and artificial lattice systems, including cold atoms, photonic and phononic crystals, and even LC electric circuit networks.
The work can be considered as an essay on mathematical and conceptual structure of nonrelativistic quantum mechanics (QM) which is related here to some other (more general, but also to more special and “approximative”) theories. QM is here primarily reformulated in an equivalent form of a Poisson system on the phase space consisting of density matrices, where the “observables”, as well as “symmetry generators” are represented by a specific type of real valued (densely defined) functions, namely the usual quantum expectations of corresponding selfadjoint operators. It is shown in this paper that inclusion of additional (“nonlinear”) symmetry generators (i.e. “Hamiltonians”) into this reformulation of (linear) QM leads to a considerable extension of the theory: two kinds of quantum “mixed states” should be distinguished, and operator ‐ valued functions of density matrices should be used in the r ˆ ole of “nonlinear observables”. A general framework for physical theories is obtained in this way: By different choices of the sets of “nonlinear observables” we obtain, as special cases, e.g. classical mechanics on homogeneous spaces of kinematical symmetry groups, standard (linear) QM, or nonlinear extensions of QM; also various “quasiclassical approximations” to QM are all subtheories of the presented extension of QM - a version of the extended quantum mechanics (EQM). A general interpretation scheme of EQM extending the usual statistical interpretation of QM is also proposed. Eventually, EQM is shown to be (included into) a C -algebraic (hence linear) quantum theory. Mathematical formulation of these theories is presented. The presentation includes an analysis of problems connected with differentiation on infinite ‐ dimensional manifolds, as well as a solution of some problems connected with the work with only densely defined unbounded real‐valued functions on the (infinite dimensional) “phase space” which correspond to unbounded operators (generators) and to their nonlinear generalizations. Also “nonlinear deformations” of unitary representations of kinematical symmetry Lie groups are introduced. Possible applications are briefly discussed, and some specific examples are presented. The text contains also brief reviews of Hamiltonian classical mechanics, as well as of QM. Mathematical appendices make the paper nearly selfcontained.
Symmetry transformations have proved useful in determining the algebraic structure and internal dynamical properties of physical systems. In the quantum Rabi model, invariance under parity symmetry transformation has been used to obtain exact solutions of the eigenvalue equation and very good approximations of the internal dynamics of the interacting atom-light system. In this article, two symmetry operators, characterized as "duality" symmetry operators, have been introduced which transform the quantum Rabi Hamiltonian into duality conjugates. The parity and duality symmetry operators constitute an algebraically closed set of symmetry transformation operators of the quantum Rabi model. The closed
SU(2) Lie algebra provides the standard eigenvalues and eigenstates of the parity symmetry operator. It is established that Jaynes-Cummings and anti-Jaynes-Cummings operators are duality symmetry conjugates. Symmetric or antisymmetric linear combinations of the Rabi Hamiltonian and a corresponding duality conjugate yield the familiar spin-dependent force driven bosonic , coupling-only or quantized light mode quadrature-driven fermionic Hamiltonian. It is established that the effective bosonic, fermionic and coupling-only Hamiltonians are exact, not approximate forms of the quantum Rabi Hamiltonian as they have generally been interpreted. The effective bosonic form generates the dynamics of the light mode driven by the atomic spin-dependent force, while the fermionic form generates the dynamics of the atomic spin driven by the quantized light mode quadrature-dependent force, thus providing a complete picture of the quantum Rabi dynamics.
Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems. For a
Zp one-form symmetry in 3+1d these correspond to the Kramers-Wannier-Wegner duality
S, which is the gauging operation underlying non-invertible duality symmetries, and the stacking of a 1-form symmetry SPT
T. In the continuum, they form a central extension of
PSL(2,Z4) for
p=2, and of
SL(2,Zp) for odd primes
p, whose central elements are invertible theories with purely gravitational response. These central extensions are governed by a twisted, graded generalization of the Witt group of abelian anyon theories, which we determine. For
p=2 the resulting group is the single-qubit Clifford group, with duality and entangler acting as the Hadamard and phase gates. We realize this entire structure microscopically as quantum cellular automata (QCA) acting on a certain local operator algebra associated with a spin lattice Hilbert space on a cubic lattice. Specifically, our local operator algebra is built by starting with all local operators commuting with a
Zp 1-form symmetry, and taking the quotient by all the (local) 1-form symmetry generators. The central elements can always be extended to the full tensor product algebra with a uniquely defined QCA class. For
p=2 they are generated by the non-trivial semion QCA, and for odd prime
p they are generated by the non-trivial
Zp Clifford QCA. Consequently the lattice fusion rules reproduce the continuum ones only up to these QCAs and lattice translations, giving rise to fusion rules refined by QCAs.
In our previous work, we proposed a mathematical framework for -symmetric quantum theory, and in particular constructed a Krein space in which -symmetric operators would naturally act. In this work, we explore and discuss various general consequences and aspects of the theory defined in the Krein space, not only spectral properties and -symmetry breaking but also several issues, crucial for the theory to be physically acceptable, such as time evolution of state vectors, probability interpretation, uncertainty relation, classical–quantum correspondence, completeness, existence of a basis, and so on. In particular, we show that for a given real classical system we can always construct the corresponding -symmetric quantum system, which indicates that -symmetric theory in the Krein space is another quantization scheme rather than a generalization of the traditional Hermitian one in the Hilbert space. We propose a postulate for an operator to be a physical observable in this framework.
This paper develops a rigorous
C∗-algebraic framework for higher abelian quantum double models, generalizing Kitaev's construction to simplicial complexes of arbitrary finite dimension and local regularity. We fully characterize the frustration-free ground state space through an associated algebra of logical operators: its state space is shown to be homeomorphic to the space of frustration-free ground states, and to obey generalized canonical commutation relations. When the relevant homology and cohomology groups are finite, the logical algebra decomposes into a commutative factor and a full matrix algebra, thereby separating the classical and quantum parts of the frustration-free ground state structure. We further prove that the vanishing of these groups is necessary and sufficient for the core algebra of the model to form a Cartan pair with the full algebra of observables, a property expected to pave the way toward classifying the model's equilibrium (KMS) states, extending recent results for the planar case.