The relation between quantum spin chains and conformal field theories is reexamined. Using a generalized Hubbard model representation it is argued that the critical theory for generic half-odd-integer spin antiferromagnets is the Wess-Zumino-Witten model (WZW model) with topological coupling, k=1, whereas generic integer spin antiferromagnets have an energy gap. The higher-k WZW models (which describe integrable higher spin models) are multicritical points in the space of all spin Hamiltonians. The k=1 WZW model represents a stable fixed point for many theories including WZW models of arbitrary odd k with relevant operators added, generalized Hubbard or Thirring models with an odd number of colors and the O(3) \ensuremath{\sigma} model at topological angle \ensuremath{\theta}=\ensuremath{\pi}.
We study topological phases of interacting systems in two spatial dimensions in the absence of topological order (i.e., with a unique ground state on closed manifolds and no fractional excitations). These are the closest interacting analogs of integer quantum Hall states, topological insulators, and superconductors. We adapt the well-known Chern-Simons
K-matrix description of quantum Hall states to classify such ``integer'' topological phases. Our main result is a general formalism that incorporates symmetries into the
K-matrix description. Remarkably, this simple analysis yields the same list of topological phases as a recent group cohomology classification, and in addition provides field theories and explicit edge theories for all these phases. The bosonic topological phases, which only appear in the presence of interactions and which remain well defined in the presence of disorder, include (i) bosonic insulators with a Hall conductance quantized to even integers, (ii) a bosonic analog of quantum spin Hall insulators, and (iii) a bosonic analog of a chiral topological superconductor, whose
K matrix is the Cartan matrix of Lie group
E8. We also discuss interacting fermion systems where symmetries are realized in a projective fashion, where we find the present formalism can handle a wider range of symmetries than a recent group super-cohomology classification. Lastly, we construct microscopic models of these phases from coupled one-dimensional systems.
Motivated by recent numerical studies where various
SU(2) Chern-Simons-matter theories emerge, we analytically study topological phases and quantum criticality in two-dimensional systems described by such theories. First, we classify
SU(2)k topological orders in all lattice spin systems with a
p4×SO(3) symmetry, where
k is an arbitrary nonzero integer. We find that for each odd
k, the topological order can emerge in systems with an arbitrary Lieb-Schultz-Mattis (LSM) anomaly, and the symmetry cannot permute anyons. If the system has a nontrivial (respectively, trivial) LSM anomaly, then there is exactly one (respectively, nine) symmetry-enriched topological (SET) phases. On the other hand,
SU(2)k topological order with any even
k can only emerge in systems with a trivial LSM anomaly. If
k∈/{6,10,14,⋯}, the symmetry cannot permute anyons, and there are 16 SET phases. If
k∈{6,10,14,⋯}, there are 4 different ways how the symmetry can permute anyons, and there are 64 SET phases. Next, we analyze the
SU(2)k Chern-Simons theories coupled to
Nf flavors of gapless matter fields that can be either bosonic or fermionic. For both types of theories, we consider a joint large-
Nf and large-
k limit with
Nf/k fixed, and compute the scaling dimensions of the bilinear operators of the bosons or fermions to the order of
1/Nf. These results sharpen our understanding of these emergent exotic topological phases and quantum criticality, and provide useful guidance to explore them further.
We compare the excitation spectra in the presence of a magnetic field of a number of integrable (exactly solvable) and nonintegrable quantum spin chains of various spin value s. The archetypal Bethe-ansatz integrable model is the s= 1/2 Heisenberg antiferromagnet (HB AFM). The excitation spectra are characterized by a soft mode which tracks across the Brillouin zone as the field increases to its saturation value. A class of Bethe-ansatz integrable models with SU(2) symmetry and the general spin s display excitation spectra qualitatively similar to the spin- 1/2 model above, for all s. A second class of Bethe-ansatz integrable models has SU(n) symmetry, where n=2s+1. Like the SU(2) integrable chains, these models have gapless excitation spectra, but the basic Brillouin zone changes from k=±2π/(2s+1)a. Studies show that periodicity of the SU(3) member of the class changes (increases) as the field increases to saturation. For both classes of integrable models, there is a single type of excitation pattern which is generically similar for all s. In the case of the other models, on the other hand, numerical studies show that the excitations divide into at least two distinct classes. In the case of the s=1 HB AFM, at high fields (corresponding to SzT=N,N−1, . . .,N/2) the excitations map approximately onto the complete set of excitations for s= 1/2 , whereas at low fields (SzT=N/2,N/2−1,. . .,0) the excitations have notable classical character. In the case of the s=1 model with pure biquadratic exchange, one set of excitations, corresponding to SzT even (SzT=N,N−2,. . .,2,0), again shows an approximate mapping to the complete excitation set for s= 1/2 . The second class of excitations, corresponding to SzT odd, are very different. They are symmetric about k=±π/2a for all SzT, i.e., correspond to a basic Brillouin zone of ±π/2a.
Effective field theory is a very useful technique for understanding quantum many body systems. We use this approach to study a certain class of critical quantum spin-1 chains and symmetry protected topological phases of matter in two spatial dimensions. In the first context we consider the SU(3) symmetric spin-1 chain with additional interactions. We use bosonization to demonstrate that the effective field theories describing the critical behaviour of these spin chains can be mapped to free compact boson conformal field theories (CFTs) with central charge c=2. We also describe how some predictions from field theory can be verified in numerical calculations using exact diagonalization (ED) and the density matrix renormalization group (DMRG) algorithm. In particular, the bosonization method gives a formula for the evolution of four Tomonaga-Luttinger liquid (TLL) parameters as a function of the lattice parameters. Using the analytic formulae for the scaling dimensions in terms of the TLL parameters and matching of lowest scaling dimensions, we numerically calculate these field-theoretic parameters and confirm that their evolution agrees with the prediction using bosonization. We also tackle aspects of the challenging problem of understanding interacting topological phases. In this context, we aim to understand the effects if interactions in certain classes of symmetry protected topological (SPT) phases of matter. We consider non-chiral SPT phases in two spatial dimensions protected by a discrete symmetry such as Z_K or Z_K x Z_K symmetry. We argue that modular invariance/noninvariance of the partition function of the one-dimensional edge theory can be used to diagnose whether, by adding a suitable potential, the edge theory can be gapped or not without breaking the symmetry. By taking bosonic phases described by Chern-Simons K-matrix theories and fermionic phases relevant to topological superconductors as examples, we demonstrate explicitly that when modular invariance is achieved, we can construct an interaction potential that is consistent with the symmetry and can completely gap out the edge. We also briefly discuss preliminary results of a numerical approach for simulating the 3 dimensional Landau Level problem. This includes a brief description of a potential application of variational Monte Carlo to spin-orbit coupled ab initio systems.
We systematically study the phase diagram of
S=2 spin chain by means of density-matrix renormalization group and exact diagonalization. We confirm the presence of a dimer phase in the Affleck-Kennedy-Lieb-Tasaki (AKLT) and Scalapino-Zhang-Hanke (SZH) model and find that the whole phase boundary between dimer and SZH phases, including the multicritical point, is a critical line with central charge
c=5/2. Finally, we propose and confirm that this line corresponds to
SO(5)1 Wess-Zumino-Witten conformal field theory.