We propose and prove a family of generalized Lieb-Schultz-Mattis~(LSM) theorems for symmetry protected topological~(SPT) phases on boson/spin models in any dimensions. The ``conventional'' LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin-1/2 particles per unit cell, forbids a symmetric short-range-entangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a non-trivial SPT phase with anomalous boundary excitations. Depending on models, they can be either strong or ``higher-order'' crystalline SPT phases, characterized by non-trivial surface/hinge/corner states. Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory. We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.
Lieb-Schultz-Mattis (LSM) theorem and its various generalizations provide powerful guidance towards the search for novel phases of matter. Here, the authors propose and prove a generalized LSM theorem suitable for 2+1D lattice models of interacting bosons or spins, with both magnetic flux and fractional spin in the unit cell. One outcome of this theorem is that, under certain conditions, the gapped ground states preserving all symmetries must be a nontrivial symmetry-protected topological (SPT) phase. Such symmetry-enforced SPTs display a dyonic character in that they associate charge with symmetry flux, which is demonstrated using quasiexactly solvable models constructed by decorating quantum dimer models.
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.
Inspired by the Chern-Simons effective theory description of symmetry protected topological (SPT) phases in two dimensions, we present a projective construction for many-body wave functions of SPT phases. Using this projective construction, we can systematically write down trial wave functions of SPT phases on a lattice. An explicit example of SPT phase with U(1) symmetry is constructed for two types of bosons with filling ${\ensuremath{\nu}}_{{b}_{1}}={\ensuremath{\nu}}_{{b}_{2}}=\frac{1}{2}$ per site on a square lattice. We study continuous phase transitions between different U(1)-SPT phases based on projective construction. The effective theory around the critical point is an emergent
QED3 with fermion number
Nf=2. Such a continuous phase transition, however, needs fine tuning, and in general there are intermediate phases between different U(1)-SPT phases. We show that such an intermediate phase has the same response as an anyon superconductor, and hence dub it ``anyon superfluid.'' A schematic phase diagram of interacting bosons with U(1) symmetry is depicted.
As condensed matter theorists, we always try to seek new quantum phases of matter that are not possible in classical physics. In this dissertation, I discussed a new type of quantum disordered phases known as symmetry protected topological (SPT) phases, which is a generalization of the topological insulator to interacting fermion or boson/spin systems with various symmetries. In the first part of this thesis, a nonlinear σ-model (NLσM) field theory is introduced as a powerful tool to describe the properties of the bosonic SPT phases. Secondly, we want to answer the question of how to detect the SPT states from their bulk properties. Introducing gauge fields was shown to be an effective theoretical tool to study bulk properties of SPT phases. Furthermore, we investigated anyon and loop statistics of gauged SPT states in the framework of NLσM. We also designed a new numerical probe, so-called strange correlator, which can distinguish SPT states from trivial states based on the bulk wavefunction on a closed manifold. Thirdly, several aspects of surface states of SPT phases are discussed. 1. A surface phase transition of 3d topological insulator is studied through a new controlled expansion method with the help of the recently discovered fermion-vortex duality. 2. A new strongly interacting conformal field theory on the surface of 3d bosonic SPT state is also found by a controlled renormalization group calculation. 3. we made a connection between the surface of SPT phase and the Lieb-Schultz-Mattis (LSM) theorem, which enables us to identity the SU(N) and SO(N) spin systems that permit a featureless ground state in 2d and 3d. Finally, we proposed the first experimental realization of bosonic SPT state in dimension higher than 1. We established a general relation between interacting multi-layer fermionic SPTs and bosonic SPT with the same symmetry, which motivates a proposal of realizing 2+1d bosonic SPT phase in bilayer graphene system.
We introduce a lattice field theory that describes the transition between a superfluid (SF) and a bosonic topological Mott Insulator (tMI) -- a
U(1) symmetry protected topological phase labeled by an integer level
k and possessing an even integer
2khe2 quantized Hall conductance. Our model differs from the usual
2+1d XY model by a topological term that vanishes on closed manifolds and in the absence of an applied gauge field, which implies that the critical exponents of the SF-tMI transition are identical to those of the well-studied
2+1d XY transition. Our formalism predicts a "level-shift" symmetry: in the absence of an applied gauge field, the bulk correlation functions of all local operators are identical for models differing by the topological term. %, for example, near the SF-MI and SF-tMI transitions, and hence the extremely well-studied critical exponents of the
2+1d XY model apply to the SF-tMI transition. In the presence of a background gauge field, the topological term leads to a quantized Hall response in the tMI phase, and we argue that this quantized Hall effect persists in the vicinity of the phase transition into the SF phase. Our formalism paves the way for other exact lattice descriptions of symmetry-protected-topological (SPT) phases, and mappings of critical exponents between transitions from symmetry-breaking to trivial states and transitions from symmetry-breaking to SPT states. A similar "level shift" symmetry should appear between all group cohomology SPT states protected by the same symmetry.