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Register a new userTensor RG approach to high-temperature fixed point
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We study a renormalization group (RG) map for tensor networks that include two-dimensional lattice spin systems such as the Ising model. Numerical studies of such RG maps have been quite successful at reproducing the known critical behavior. In those numerical studies the RG map must be truncated to keep the dimension of the legs of the tensors bounded. Our tensors act on an infinite-dimensional Hilbert space, and our RG map does not involve any truncations. Our RG map has a trivial fixed point which represents the high-temperature fixed point. We prove that if we start with a tensor that is close to this fixed point tensor, then the iterates of the RG map converge in the Hilbert-Schmidt norm to the fixed point tensor. It is important to emphasize that this statement is not true for the simplest tensor network RG map in which one simply contracts four copies of the tensor to define the renormalized tensor. The linearization of this simple RG map about the fixed point is not a contraction due to the presence of so-called CDL tensors. Our work provides a first step towards the important problem of the rigorous study of RG maps for tensor networks in a neighborhood of the critical point.
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We discuss the decay of unstable states into a quasicontinuum using models of the effective Hamiltonian type. The goal is to show that exponential decay and the golden rule are exact in a suitable scaling limit, and that there is an associated renormalization group (RG) with these properties as a fixed point. The method is inspired by a limit theorem for infinitely divisible distributions in probability theory, where there is a RG with a Cauchy distribution, i.e. a Lorentz line shape, as a fixed point. Our method of solving for the spectrum is well known; it does not involve a perturbation expansion in the interaction, and needs no assumption of a weak interaction. We use random matrices for the interaction, and show that the ensemble fluctuations vanish in the scaling limit. Thus the limit is the same for every model in the ensemble with probability one.
In search of non-trivial field theories in high dimensions, we study further the tensor representation of the $O(N)$-symmetric $phi^4$ field theory introduced by Herbut and Janssen (Phys. Rev. D. 93, 085005 (2016)), by using four-loop perturbation theory in two cubic interaction coupling constants near six dimensions. For infinitesimal values of the parameter $epsilon=(6-d)/2$ we find infrared-stable fixed point with two relevant quadratic operators for $N$ within the conformal windows $1<N<2.653$ and $2.999<N<4$, and compute critical exponents at this fixed point to the order $epsilon^4$. Taking the four-loop beta-functions at their face value we determine the higher-order corrections to the edges of the above conformal windows at finite $epsilon$, to find both intervals to shrink to zero above $epsilonapprox 0.15$. The disappearance of the conformal windows with the increase of $epsilon$ is due to the collision of the Wilson-Fisher $mathcal{O}(epsilon)$ infrared fixed point with the $mathcal{O}(1)$ mixed-stable fixed point that appears at two and persists at higher loops. The latter may be understood as a Banks-Zaks type fixed point that becomes weakly coupled near the right edge of either conformal window. The consequences and issues raised by such an evolution of the flow with dimension are discussed. It is also shown both within the perturbation theory and exactly that the tensor representation at $N=3$ and right at the $mathcal{O}(epsilon)$ infrared-stable fixed point exhibits an emergent $U(3)$ symmetry. A role of this enlarged symmetry in possible protection of the infrared fixed point at $N=3$ is noted.
We introduce a systematic mathematical language for describing fixed point models and apply it to the study to topological phases of matter. The framework established is reminiscent to that of state-sum models and lattice topological quantum field theories, but is formalized and unified in terms of tensor networks. In contrast to existing tensor network ansatzes for the study of ground states of topologically ordered phases, the tensor networks in our formalism directly represent discrete path integrals in Euclidean space-time. This language is more immediately related to the Hamiltonian defining the model than other approaches, via a Trotterization of the respective imaginary time evolution. We illustrate our formalism at hand of simple examples, and demonstrate its full power by expressing known families of models in 2+1 dimensions in their most general form, namely string-net models and Kitaev quantum doubles based on weak Hopf algebras. To elucidate the versatility of our formalism, we also show how fermionic phases of matter can be described and provide a framework for topological fixed point models in 3+1 dimensions.
The electrocaloric effect (ECE), i.e., the reversible temperature change due to the adiabatic variation of the electric field, is of great interest due to its potential technological applications. Based on entropy arguments, we present a new framework to attain giant ECE. Our findings are fourfold: $i$) we employ the recently-proposed electric Gruneisen parameter $Gamma_E$ to quantify the ECE and discuss its advantages over the existing so-called electrocaloric strength; $ii$) prediction of giant caloric effects $close$ to $any$ critical end point; $iii$) proposal of potential key-ingredients to enhance the ECE; $iv$) demonstration of $Gamma_E$ as a proper parameter to probe quantum ferroelectricity in connection with the celebrated Barretts formula. Our findings enable us to interpret the recently-reported large ECE at room-temperature in oxide multilayer capacitors [Nature 575, 468 (2019)], paving thus the way for new venues in the field.
We consider $N$ particles in the plane influenced by a general external potential that are subject to the Coulomb interaction in two dimensions at inverse temperature $beta$. At large temperature, when scaling $beta=2c/N$ with some fixed constant $c>0$, in the large-$N$ limit we observe a crossover from Ginibres circular law or its generalization to the density of non-interacting particles at $beta=0$. Using several different methods we derive a partial differential equation of generalized Liouville type for the crossover density. For radially symmetric potentials we present some asymptotic results and give examples for the numerical solution of the crossover density. These findings generalise previous results when the interacting particles are confined to the real line. In that situation we derive an integral equation for the resolvent valid for a general potential and present the analytic solution for the density in case of a Gaussian plus logarithmic potential.
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