Boundary scattering in the principal chiral model

An introduction to our recent work (hep-th/0104212) on the (G ×G-invariant) principal chiral model with boundary is presented. We found that both classically integrable boundary conditions and quantum boundary is presentedS-matrices were classified by the symmetric spacesG/H. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.     

On topological excitations with isovector topological charges

It is shown that degenerate systems with order parameter ψ taking on values in compact homogeneous subspaces T _G or G/T _G (where G is a simple compact group and T _G is its maximum commutative subgroup) possess a rich collection of topological excitations (vortices or, correspondingly, instantons) with isovector topological charges. The corresponding homotopy groups are found for all G. The possibility of a topological interpretation of the quantum numbers of the groups and particles is discussed. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 10, 758–762 (25 May 1996)     

On braided poisson and quantum inhomogeneous groups

The well known incompatibility between inhomogeneous quantum groups and the standardq-deformation is shown to disappear (at least in certain cases) when admitting the quantum group to be braided. Braided quantumISO(p, N - p) containingSO _q (p, N - p) with |q|=1 are constructed forN=2p, 2p + 1, 2p + 2. Their Poisson analogues (obtained first) are presented as an introduction to the quantum case. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

Topological interpretation of quantum numbers

The paper shows how vector topological charges for topological excitations in nonlinear σ-models on compact one-dimensional spaces T _G and G/T _G can be defined (here G is a simple compact Lie group and T _G is its maximal commutative subgroup). Explicit solutions, their energies and interactions between different topological charges have been obtained. A possibility of topological interpretation of quantum numbers of groups and particles is discussed. Zh. éksp. Teor. Fiz. 116, 1131–1147 (October 1999)     

Quantum Double Actions on Operator Algebras and Orbifold Quantum Field Theories

Starting from a local quantum field theory with an unbroken compact symmetry group G in 1+1-dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group G are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group G the extended theory is acted upon in a completely canonical way by the quantum double D(G) and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which is expected to hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitrary locally compact groups and our methods are adapted to chiral theories on the circle. Received: 4 September 1996 / Accepted: 6 May 1997     

Automorphisms and symmetries of quantum logics

Given an amalgam of groups then every quantum logicQ ₀ = (L ₀,M ₀) (L ₀ is aσ-orthomodular poset,M ₀ is a full set of states on it) satisfying some reasonable conditions can be embedded in a quantum logicQ = (L, M), in which (1) all the automorphisms ofL form a group ∼-G ₁, (2) all the automorphisms ofM form a group ∼-G ₂, and (3) all the symmetries ofQ form a group ∼-G ₀. The quantum logic of all closed subspaces of a Hilbert spaceH and all its measures satisfies the conditions required fromQ ₀; hence, enlarging it, one can obtain “anything.”     

Quantum Invariant Measures

We derive an explicit expression for the Haar integral on the quantized algebra of regular functions ℂ _q [K] on the compact real form K of an arbitrary simply connected complex simple algebraic group G. This is done in terms of the irreducible ✶-representations of the Hopf ✶-algebra ℂ _q [K]. Quantum analogs of the measures on the symplectic leaves of the standard Poisson structure on K which are (almost) invariant under the dressing action of the dual Poisson algebraic group K ^✶ are also obtained. They are related to the notion of quantum traces for representations of Hopf algebras. As an application we define and compute explicitly quantum analogs of Harish-Chandra c-functions associated to the elements of the Weyl group of G. Received: 26 January 2001 / Accepted: 31 May 2001     

Quantum Vertex Representations via Finite Groups¶and the McKay Correspondence

We establish a q-analog of our recent work on vertex representations and the McKay correspondence. For each finite group Γ we construct a Fock space and associated vertex operators in terms of wreath products of $Γ×ℂ^× and the symmetric groups. An important special case is obtained when Γ is a finite subgroup of SU ₂, where our construction yields a group theoretic realization of the representations of the quantum affine and quantum toroidal algebras of ADE type. Received: 17 August 1999 / Accepted: 2 December 1999     

Mean Field Magnetic Phase Diagrams for the Two Dimensional <Emphasis Type="Italic">t</Emphasis> — <Emphasis Type="Italic">t</Emphasis>′ — <Emphasis Type="Italic">U</Emphasis> Hubbard Model

We study the ground state phase diagram of the two dimensional t — t′ — U Hubbard model concentrating on the competition between antiferro-, ferro-, and paramagnetism. It is known that unrestricted Hartree–Fock- and quantum Monte Carlo calculations for this model predict inhomogeneous states in large regions of the parameter space. Standard mean field theory, i.e., Hartree–Fock theory restricted to homogeneous states, fails to produce such inhomogeneous phases. We show that a generalization of the mean field method to the grand canonical ensemble circumvents this problem and predicts inhomogeneous states, represented by mixtures of homogeneous states, in large regions of the parameter space. We present phase diagrams which differ considerably from previous mean field results but are consistent with, and extend, results obtained with more sophisticated methods. PACS: 71.10.Fd, 05.70.Fh, 75.50.Ee     

A very strong difference property for semisimple compact connected lie groups

Let G be a topological group. For a function f: G → ℝ and h ∈ G, the difference function Δ _h f is defined by the rule Δ _h f(x) = f(xh) − f(x) (x ∈ G). A function H: G → ℝ is said to be additive if it satisfies the Cauchy functional equation H(x + y) = H(x) + H(y) for every x, y ∈ G. A class F of real-valued functions defined on G is said to have the difference property if, for every function f: G → ℝ satisfying Δ _h f ∈ F for each h ∈ G, there is an additive function H such that f − H ∈ F. Erdős’ conjecture claiming that the class of continuous functions on ℝ has the difference property was proved by N. G. de Bruijn; later on, F. W. Carroll and F. S. Koehl obtained a similar result for compact Abelian groups and, under the additional assumption that the other one-sided difference function ∇ _h f defined by ∇ _h f(x) = f(xh) − f(x) (x ∈ G, h ∈ G) is measurable for any h ∈ G, also for noncommutative compact metric groups. In the present paper, we consider a narrower class of groups, namely, the family of semisimple compact connected Lie groups. It turns out that these groups admit a significantly stronger difference property. Namely, if a function f: G → ℝ on a semisimple compact connected Lie group has continuous difference functions Δ _h f for any h ∈ G (without the additional assumption concerning the measurability of the functions of the form ∇ _h f), then f is automatically continuous, and no nontrivial additive function of the form H is needed. Some applications are indicated, including difference theorems for homogeneous spaces of compact connected Lie groups.     

Glauber Dynamics for the Quantum Ising Model in a Transverse Field on a Regular Tree

Motivated by a recent use of Glauber dynamics for Monte Carlo simulations of path integral representation of quantum spin models (Krzakala et al. in Phys. Rev. B 78(13):134428, 2008), we analyse a natural Glauber dynamics for the quantum Ising model with a transverse field on a finite graph G. We establish strict monotonicity properties of the equilibrium distribution and we extend (and improve) the censoring inequality of Peres and Winkler to the quantum setting. Then we consider the case when G is a regular b-ary tree and prove the same fast mixing results established in Martinelli et al. (Commun. Math. Phys. 250(2):301–334, 2004) for the classical Ising model. Our main tool is an inductive relation between conditional marginals (known as the “cavity equation”) together with sharp bounds on the operator norm of the derivative at the stable fixed point. It is here that the main difference between the quantum and the classical case appear, as the cavity equation is formulated here in an infinite dimensional vector space, whereas in the classical case marginals belong to a one-dimensional space.     

Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials

We construct noncommutative “Riemannian manifold” structures on dual quasitriangular Hopf algebras such as ℂ _q [SU ₂] with its standard bicovariant differential calculus, using the quantum frame bundle approach introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the n-bein by the Maurer–Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras ℂ[G] with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group ℂ[S ₃] is worked out in full detail and a unique torsion free and cotorsion free or “Levi–Civita” connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as S ₃. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form. Received: 22 June 2000 / Accepted: 26 August 2001     

Gauge transformations and quasitriangularity

Natural conditions on a Poisson/quantum group G to implement Poisson/quantum gauge transformations on the lattice are investigated. In addition to our previous result that transformations on one lattice link require G to be coboundary, it is shown that for a sequence of links one needs a quasitriangular G.     

Quantum deformation of lorentz group

A one parameter quantum deformationS _μ L(2,ℂ) ofSL(2,ℂ) is introduced and investigated. An analog of the Iwasawa decomposition is proved. The compact part of this decomposition coincides withS _μ U(2), whereas the solvable part is identified as a Pontryagin dual ofS _μ U(2). It shows thatS _μ L(2,ℂ) is the result of the dual version of Drinfeld's double group construction applied toS _μ U(2). The same construction applied to any compact quantum groupG _c is discussed in detail. In particular the explicit formulae for the Haar measures on the Pontryagin dualG _d ofG _c and on the double groupG are given. We show that there exists remarkable 1-1 correspondence between representations ofG and bicovariant bimodules (“tensor bundles”) overG _c . The theory of smooth representations ofS _μ L(2,ℂ) is the same as that ofSL(2,ℂ) (Clebsh-Gordon coefficients are however modified). The corresponding “tame” bicovariant bimodules onS _μ U(2) are classified. An application to 4D ₊ differential calculus is presented. The nonsmooth case is also discussed.     

The Matrix Algebras of Continuous Groups with Antilinear Operations

The corepresentation theory of continuous groups is presented without the assumption that the subgroup G of the group with antilinear operations is unitary. Continuous groups of the form: G+a ₀ G are defined, where G denotes a linear Lie group and a ₀ denotes an antilinear operation which fulfils the condition a²₀=±1a^{2}_{0}=\pm1. The matrix algebras connected with the groups G+a ₀ G are defined. The structural constants of these algebras fulfill the conditions following from the Jacobi identities. Applications are presented to the groups G=SU(d), d=1,2,… , for a ₀=K, the complex conjugation operation, and to the group SL(2,C) for a ₀=K or Θ, the time-reversal operation.     

Quantum corrections to the conductance of n-GaAs films in a strong magnetic field

The conductance of doped n-GaAs films is studied experimentally as a function of magnetic field and temperature in strong magnetic fields right up to the quantum limit (ħω_c = E _F). The Hall conductance G _xy is virtually independent of temperature T until the transverse conductance G _xx is quite large compared with e ²/h. In strong fields, when G _xx becomes comparable to e ²/h, G _xy starts to depend on T. The difference between the conductances G _xx at the two temperatures 4.2 and 0.35 K depends only weakly on the magnetic field H over a wide range of magnetic fields, while the conductances G _xx themselves vary strongly. The results can be explained by quantum corrections to the conductance as a result of the electron-electron interaction in the diffusion channel. The possibility of quantization of the Hall conductance as a result of the electron-electron interaction is discussed. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 3, 201–206 (10 February 1998)