The zero-temperature phase diagram of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N)$ antiferromagnetic Heisenberg spin chain is investigated by means of complementary field theory and numerical approaches for general
N. A fully gapped
SU(N) valence bond solid made of
N sites is formed above a critical value of
J2/J1 for all
N. We find that the extension of this
N-merized phase for larger values of
J2 strongly depends on the parity of
N. For even
N, the phase smoothly interpolates to the large
J2 regime where the model can be viewed as a zigzag
SU(N) two-leg spin ladder. The phase exhibits both a
N-merized ground state and incommensurate spin-spin correlations. In stark contrast to the even case, we show that the
N-merized phase with odd
N has only a finite extent with no incommensuration. A gapless phase in the
SU(N)1 universality class is stabilized for larger
J2 that stems from the existence of a massless renormalization group flow from
SU(N)2 to
SU(N)1 conformal field theories when
N is odd.
According to field theory computations, the zero-temperature phase diagram of the J1-J2 SU(N) spin chains depends on the parity of N. While a transition from critical to a gapped SU(N) valence bond solid is always observed at small J2s, we find that for odd N a gapless phase reopen in the large J2 limit. This phase is in the SU(N)1 universality class due to the existence of a massless renormalization group flow from SU(N)2 to SU(N)1 conformal field theories.In this talk, after presenting our RG analysis, I will discuss the strengths and limits of (i)DMRG applied to SU(N) spin chains, which allowed us to confirm (some of) the field theory predictions, and also revealed the existence of a commensurate to incommensurate transition within the valence bond phase for even N.
We study the phase diagram of isotropic spin-1 models in the vicinity of the Uimin-Lai-Sutherland (ULS) model. This is done with the help of a level-one SU(3) Wess-Zumino-Witten model with certain marginal perturbations. We find that the renormalization-group flow has infrared stable and unstable trajectories divided by a critical line on which the ULS model is located. The infrared unstable trajectory produced by a marginally relevant perturbation generates an exponential mass gap for the Haldane phase, and thus the universality class of the transition from the massless phase to the Haldane phase at the ULS point is identified with the Berezinski\ifmmode \breve{}\else \u{}\fi{}\ifmmode \imath \else \i \fi{}-Kosterlitz-Thouless type. Our results support recent numerical studies by F\'ath and S\'olyom. In the massless phase, we calculate logarithmic finite-size corrections of the energy for the SU(\ensuremath{\nu})-symmetric and asymmetric models in the massless phase.
In this paper we study the ground-state properties of a ladder Hamiltonian with chiral
SU(2)-invariant spin interactions, a possible first step toward the construction of truly two-dimensional nontrivial systems with chiral properties starting from quasi-one-dimensional ones. Our analysis uses a recent implementation by us of
SU(2) symmetry in tensor network algorithms, specifically for infinite density matrix renormalization group. After a preliminary analysis with Kadanoff coarse graining and exact diagonalization for a small-size system, we discuss its bosonization and recap the continuum limit of the model to show that it corresponds to a conformal field theory, in agreement with our numerical findings. In particular, the scaling of the entanglement entropy as well as finite-entanglement scaling data show that the ground-state properties match those of the universality class of a
c=1 conformal field theory (CFT) in
(1+1) dimensions. We also study the algebraic decay of spin-spin and dimer-dimer correlation functions, as well as the algebraic convergence of the ground-state energy with the bond dimension, and the entanglement spectrum of half an infinite chain. Our results for the entanglement spectrum are remarkably similar to those of the spin-
21 Heisenberg chain, which we take as a strong indication that both systems are described by the same CFT at low energies, i.e., an
SU(2)1 Wess-Zumino-Witten theory. Moreover, we explain in detail how to construct matrix product operators for
SU(2)-invariant three-spin interactions, something that had not been addressed with sufficient depth in the literature.
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.
We show how the low energy properties of the two-leg XXZ spin-1/2 ladders with general anisotropy parameter Δ on closed geometries can be accounted for in the framework of the m -reduction procedure developed previously (Cristofano et al 2000 Mod. Phys. Lett. A 15 547; Cristofano et al 2000 Mod. Phys. Lett. A 15 1679; Cristofano et al 2002 Nucl. Phys. B 641 547; Cristofano et al 2004 J. High Energy Phys. JHEP06(2004) 056). In the limit of quasi-decoupled chains, a conformal field theory (CFT) with central charge c = 2 is derived and its ability to describe the model with different boundary conditions is shown. Special emphasis is given to the Mobius boundary conditions which generate a topological defect corresponding to non-trivial single-spinon excitations. Then, in the case of the two-leg XXX ladders we discuss in detail the role of various perturbations in determining the renormalization group flow starting from the ultraviolet (UV) critical point with c = 2.