Simplifying the spectral analysis of the volume operator

The volume operator plays a central role in both the kinematics and dynamics of canonical approaches to quantum gravity which are based on algebras of generalized Wilson loops. We introduce a method for simplifying its spectral analysis, for quantum states that can be realized on a cubic three-dimensional lattice. This involves a decomposition of Hilbert space into sectors transforming according to the irreducible representations of a subgroup of the cubic group. As an application, we determine the complete spectrum for a class of states with six-valent intersections.Comment: 19 pages, TeX, to be published in Nucl. Phys.

Low temperature specific heat and possible gap to magnetic excitations in the Heisenberg pyrochlore antiferromagnet Gd2Sn207

The Gd2Sn2O7 pyrochlore Heisenberg antiferromagnet displays a phase transition to a four sublattice Neel ordered state at a temperature near 1 K. Despite the seemingly conventional nature of the ordered state, the specific heat has been found to be described in the temperature range 350-800 mK by an anomalous T-squared power law. A similar temperature dependence has also been reported for Gd2Ti2O7, another pyrochlore Heisenberg material. Such anomalous T-squared behavior in Cv has been argued to be correlated to an unusual energy-dependence of the density of states which also seemingly manifests itself in low-temperature spin fluctuations found in muon spin relaxation experiments. In this paper, we report calculations of Cv that consider spin wave like excitations out of the Neel order observed in Gd2Sn2O7 and argue that the parametric T-squared behavior does not reflect the true low-energy excitations of Gd2Sn2O7. Rather, we find that the low-energy excitations of this material are antiferromagnetic magnons gapped by single-ion and dipolar anisotropy effects, and that the lowest temperature of 350 mK considered in previous specific heat measurements accidentally happens to coincide with a crossover temperature below which magnons become thermally activated and Cv takes an exponential form. We argue that further specific heat measurements that extend down to at least 100 mK are required in order to ascribe an unconventional description of magnetic excitations out of the ground state of Gd2Sn2O7 or to invalidate the standard picture of gapped excitations proposed herein.Comment: 12 pages, 13 figures; shortened introduction and added 1 figur

A double-sum Kronecker-type identity

We prove a double-sum analog of an identity known to Kronecker and then express it in terms of functions studied by Appell and Kronecker's student Lerch, in so doing we show that the double-sum analog is of mixed mock modular form. We also give related symmetric generalizations.Comment: Major revisions. Identities (1.10) and (1.11) are ne

Imposing det E > 0 in discrete quantum gravity

We point out that the inequality det E > 0 distinguishes the kinematical phase space of canonical connection gravity from that of a gauge field theory, and characterize the eigenvectors with positive, negative and zero-eigenvalue of the corresponding quantum operator in a lattice-discretized version of the theory. The diagonalization of the operator det E is simplified by classifying its eigenvectors according to the irreducible representations of the octagonal group.Comment: 10 pages, plain Te

The square-kagome quantum Heisenberg antiferromagnet at high magnetic fields: The localized-magnon paradigm and beyond

We consider the spin-1/2 antiferromagnetic Heisenberg model on the two-dimensional square-kagome lattice with almost dispersionless lowest magnon band. For a general exchange coupling geometry we elaborate low-energy effective Hamiltonians which emerge at high magnetic fields. The effective model to describe the low-energy degrees of freedom of the initial frustrated quantum spin model is the (unfrustrated) square-lattice spin-1/2 $XXZ$ model in a $z$-aligned magnetic field. For the effective model we perform quantum Monte Carlo simulations to discuss the low-temperature properties of the square-kagome quantum Heisenberg antiferromagnet at high magnetic fields. We pay special attention to a magnetic-field driven Berezinskii-Kosterlitz-Thouless phase transition which occurs at low temperatures.Comment: 6 figure

Quantum groups, Yang-Baxter maps and quasi-determinants

For any quasi-triangular Hopf algebra, there exists the universal R-matrix, which satisfies the Yang-Baxter equation. It is known that the adjoint action of the universal R-matrix on the elements of the tensor square of the algebra constitutes a quantum Yang-Baxter map, which satisfies the set-theoretic Yang-Baxter equation. The map has a zero curvature representation among L-operators defined as images of the universal R-matrix. We find that the zero curvature representation can be solved by the Gauss decomposition of a product of L-operators. Thereby obtained a quasi-determinant expression of the quantum Yang-Baxter map associated with the quantum algebra $U_{q}(gl(n))$. Moreover, the map is identified with products of quasi-Pl\"{u}cker coordinates over a matrix composed of the L-operators. We also consider the quasi-classical limit, where the underlying quantum algebra reduces to a Poisson algebra. The quasi-determinant expression of the quantum Yang-Baxter map reduces to ratios of determinants, which give a new expression of a classical Yang-Baxter map.Comment: 46 page

Coupling coefficients of SO(n) and integrals over triplets of Jacobi and Gegenbauer polynomials

The expressions of the coupling coefficients (3j-symbols) for the most degenerate (symmetric) representations of the orthogonal groups SO(n) in a canonical basis (with SO(n) restricted to SO(n-1)) and different semicanonical or tree bases [with SO(n) restricted to SO(n'})\times SO(n''), n'+n''=n] are considered, respectively, in context of the integrals involving triplets of the Gegenbauer and the Jacobi polynomials. Since the directly derived triple-hypergeometric series do not reveal the apparent triangle conditions of the 3j-symbols, they are rearranged, using their relation with the semistretched isofactors of the second kind for the complementary chain Sp(4)\supset SU(2)\times SU(2) and analogy with the stretched 9j coefficients of SU(2), into formulae with more rich limits for summation intervals and obvious triangle conditions. The isofactors of class-one representations of the orthogonal groups or class-two representations of the unitary groups (and, of course, the related integrals involving triplets of the Gegenbauer and the Jacobi polynomials) turn into the double sums in the cases of the canonical SO(n)\supset SO(n-1) or U(n)\supset U(n-1) and semicanonical SO(n)\supset SO(n-2)\times SO(2) chains, as well as into the_4F_3(1) series under more specific conditions. Some ambiguities of the phase choice of the complementary group approach are adjusted, as well as the problems with alternating sign parameter of SO(2) representations in the SO(3)\supset SO(2) and SO(n)\supset SO(n-2)\times SO(2) chains.Comment: 26 pages, corrections of (3.6c) and (3.12); elementary proof of (3.2e) is adde

Integrable properties of sigma-models with non-symmetric target spaces

It is well-known that sigma-models with symmetric target spaces are classically integrable. At the example of the model with target space the flag manifold U(3)/U(1)^3 -- a non-symmetric space -- we show that the introduction of torsion allows to cast the equations of motion in the form of a zero-curvature condition for a one-parametric family of connections, which can be a sign of integrability of the theory. We also elaborate on geometric aspects of the proposed model.Comment: 8 pages, 1 figur

Second order dissipative fluid dynamics from kinetic theory

We derive the equations of second order dissipative fluid dynamics from the relativistic Boltzmann equation following the method of W. Israel and J. M. Stewart [1]. We present a frame independent calculation of all first- and second-order terms and their coefficients using a linearised collision integral. Therefore, we restore all terms that were previously neglected in the original papers of W. Israel and J. M. Stewart