Asymptotic Freedom and Large Spin Antiferromagnetic Chains

Building on the mapping of large-S spin chains onto the O(3) nonlinear sigma model with coupling constant 2/S, and on general properties of that model (asymptotic freedom, implying that perturbation theory is valid at high energy, and Elitzur's conjecture that rotationally invariant quantities are infrared finite in perturbation theory), we use the Holstein-Primakoff representation to derive analytic expressions for the equal-time and dynamical spin-spin correlations valid at distances smaller than S-1 exp(pi S) or at energies larger than JS(2) exp(-pi S), where J is the Heisenberg exchange coupling. This is supported by comparing the static correlations with quantum Monte Carlo simulations for S = 5/2.

Asymptotic freedom, Haldane gap and edge states of SU(N) spin chains

Samuel Gozel

In this thesis we study various one-dimensional quantum spin systems with SU(2) and SU(N) symmetry. We investigate the short-distance behavior of the SU(2) Heisenberg model in the limit of large spin and show that there exists an extended regime where perturbation theory, in the form of spin-wave theory, can be successfully applied. The reason for this perturbative regime stems from the asymptotic freedom of the nonlinear sigma model onto which the Heisenberg model can be mapped. When considering observables which respect the rotational invariance of the model, we observe a cancellation of infrared divergences in the perturbative expansions, leading to a meaningful description of correlation functions. We then turn to the study of SU(N) models. Building on the representation theory of the SU(N) group and on the matrix product state (MPS) formalism, we introduce a generic method to construct Affleck-Kennedy-Lieb-Tasaki (AKLT) models having edge states described by any self-conjugate irreducible representation (irrep) of SU(N). A simple example is given by a spin-1 AKLT model having spin-1 edge states. The phase transition between this model and the original AKLT model is shown to be continuous and to correspond to a topological phase transition described by the SU(2) level-1 Wess-Zumino-Witten conformal field theory universality class. In addition we study an SU(3) AKLT model for the 3-box symmetric irrep at each site. We demonstrate that the edge states are adjoint edge irreps, we extract its correlation length and provide a useful construction as an optimal MPS. Finally, we develop a density matrix renormalization group algorithm based on standard Young tableaus and subduction coefficients to make full use of the non-abelian symmetry and to investigate the SU(3) Heisenberg model with 3-box symmetric irrep at each site. We show that the model has a finite gap above the singlet ground state, in agreement with an extension of the Haldane conjecture to SU(3) chains in the fully symmetric irreps. We also argue that there are five branches of elementary excitations living in four different irreps, each of which is gapped.

EPFL2020

A new six-dimensional analytical potential up to chemically significant energies for the electronic ground state of hydrogen peroxide

Thomas Rizzo

We report calculations of the electronic ground state potential energy surface (PES) of hydrogen peroxide covering, in an almost global fashion, all six internal degrees of freedom by two different ab initio techniques. Density functional theory (DFT) calculations using the Becke 3 parameter Lee-Yang-Parr (B3LYP) hybrid functional and multiconfigurational second order perturbation theory (CASPT2) calculations, both using large basis sets, are performed for a wide range of geometries (8145 DFT and 5310 CASPT2 single-point energies). We use a combined data set of mostly DFT with additional CASPT2 ab initio points and the complete CASPT2 surface to fit a total of four different 6D analytical representations. The resulting potentials contain 70-76 freely adjusted parameters and represent the ground state PES up to 40000 cm(-1) above the equilibrium energy with a standard deviation of 100-107 cm(-1) without any important artifacts. One of the model surfaces is further empirically refined to match the bond dissociation energy D-0 for HOOH --> 2OH. The potentials are designed for energy regions accessible by vibrational fundamental and overtone spectroscopy including the dissociation channel into hydroxyl radicals. Characteristic properties of the model surfaces are investigated by means of stationary point analyses, torsional barrier heights, harmonic frequencies, low-dimensional cuts and minimum energy paths for dissociation. Overall good agreement with high-level ab initio calculations, especially for the CASPT2 based potentials, is achieved. The drastic change in geometry at intermediate O-O distances, which reflects the transition from covalent to hydrogen bonding, is reproduced quantitatively. We calculate fully 6D anharmonic zero point energies and ground state torsional splittings with the diffusion quantum Monte Carlo method in perfect agreement, within statistical error bars, with experiment for the CASPT2 based potentials. Variational vibrational calculations in the (4+2)D adiabatic approximation yield energy levels and torsional splittings from the ground state up to predissociative states, satisfactorily reproducing the experimental transition wavenumbers. (C) 1999 American Institute of Physics. [S0021-9606(99)30205-1].

1999

A new six-dimensional analytical potential up to chemically significant energies for the electronic ground state of hydrogen peroxide

Thomas Rizzo

1999

Asymptotic freedom, Haldane gap and edge states of SU(N) spin chains

Samuel Gozel

EPFL2020