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Register a new userVacuum energy for generalised Dirac combs at $T = 0$
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Added by
Jose M Munoz-Castaneda
Publication date
2018
fields
Physics
and research's language is
English
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The quantum vacuum energy for a hybrid comb of Dirac $delta$-$delta$ potentials is computed using the energy of the single $delta$-$delta$ potential over the real line that makes up the comb. The zeta function of a comb periodic potential is the continuous sum of zeta functions over the dual primitive cell of Bloch quasi-momenta. The result obtained for the quantum vacuum energy is non-perturbative in the sense that the energy function is not analytical for small couplings
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A new nonlinear integral equation (NLIE) describing the thermodynamics of the Heisenberg spin chain is derived based on the t-W relation of the quantum transfer matrices. The free energy of the system in a magnetic field is thus obtained by solving the NLIE. This method can be generalized to other lattice quantum integrable models. Taking the SU(3)-invariant quantum spin chain as an example, we construct the corresponding NLIEs and compute the free energy. The present results coincide exactly with those obtained via other methods previously.
A good understanding of conformal field theory (CFT) at c=0 is vital to the physics of disordered systems, as well as geometrical problems such as polymers and percolation. Steady progress has shown that these CFTs should be logarithmic, with indecomposable operator product expansions, and indecomposable representations of the Virasoro algebra. In one of the earliest papers on the subject, V. Gurarie introduced a single parameter b to quantify this indecomposability in terms of the logarithmic partner t of the stress energy tensor T. He and A. Ludwig conjectured further that b=-5/8 for polymers and b=5/6 for percolation. While a lot of physics may be hidden behind this parameter - which has also given rise to a lot of discussions - it had remained very elusive up to now, due to the lack of available methods to measure it experimentally or numerically, in contrast say with the central charge. We show in this paper how to overcome the many difficulties in trying to measure b. This requires control of a lattice scalar product, lattice Jordan cells, together with a precise construction of the state L_{-2}|0>. The final result is that b=5/6 for polymers. For percolation, we find that b=-5/8 within an XXZ or supersymmetric representation. In the geometrical representation, we do not find a Jordan cell for L_0 at level two (finite-size Hamiltonian and transfer matrices are fully diagonalizable), so there is no b in this case.
Motivated by various problems in physics and applied mathematics, we look for constraints and properties of real Fourier-positive functions, i.e. with positive Fourier transforms. Properties of the Dirac comb distribution and of its tensor products in higher dimensions lead to Poisson resummation, allowing for a useful approximation formula of a Fourier transform in terms of a limited number of terms. A connection with the Bochner theorem on positive definiteness of Fourier-positive functions is discussed. As a practical application, we find simple and rapid analytic algorithms for checking Fourier-positivity in 1- and (radial) 2-dimensions among a large variety of real positive functions. This may provide a step towards a classification of positive positive-definite functions.
Recently we introduced a new technique for computing the average free energy of a system with quenched randomness. The basic tool of this technique is a distributional zeta-function. The distributional zeta-function is a complex function whose derivative at the origin yields the average free energy of the system as the sum of two contributions: the first one is a series in which all the integer moments of the partition function of the model contribute; the second one, which can not be written as a series of the integer moments, can be made as small as desired. In this paper we present a mathematical rigorous proof that the average free energy of one disordered $lambdavarphi^{4}$ model defined in a zero-dimensional space can be obtained using the distributional zeta-function technique. We obtain an analytic expression for the average free energy of the model.
A general, multi-component Eulerian fluid theory is a set of nonlinear, hyperbolic partial differential equations. However, if the fluid is to be the large-scale description of a short-range many-body system, further constraints arise on the structure of these equations. Here we derive one such constraint, pertaining to the free energy fluxes. The free energy fluxes generate expectation values of currents, akin to the specific free energy generating conserved densities. They fix the equations of state and the Euler-scale hydrodynamics, and are simply related to the entropy currents. Using the Kubo-Martin-Schwinger relations associated to many conserved quantities, in quantum and classical systems, we show that the associated free energy fluxes are perpendicular to the vector of inverse temperatures characterising the state. This implies that all entropy currents can be expressed as averages of local observables. In few-component fluids, it implies that the averages of currents follow from the specific free energy alone, without the use of Galilean or relativistic invariance. In integrable models, in implies that the thermodynamic Bethe ansatz must satisfy a unitarity condition. The relation also guarantees physical consistency of the Euler hydrodynamics in spatially-inhomogeneous, macroscopic external fields, as it implies conservation of entropy, and the local-density approximated Gibbs form of stationarity states. The main result on free energy fluxes is based on general properties such as clustering, and we show that it is mathematically rigorous in quantum spin chains.
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