Kitaev materials are promising for realizing exotic quantum spin liquid phases, such as a non-Abelian chiral spin liquid. Motivated by recent experiments in these materials, we theoretically study the novel field-induced quantum phase transitions from the non-Abelian chiral spin liquid to the symmetry-broken zigzag phase and to the trivial polarized state. Utilizing the recently developed dualities of gauge theories, we find these transitions can be described by critical bosons or gapless fermions coupled to emergent non-Abelian gauge fields, and the critical theories are of the type of a QCD
3-Chern-Simons theory. We propose that all these exotic quantum phase transitions can potentially be direct and continuous in Kitaev materials, and we present sound evidence for this proposal. Therefore, besides being systems with intriguing quantum magnetism, Kitaev materials may also serve as table-top experimental platforms to study the interesting dynamics of emergent strongly interacting quarks and gluons in
2+1 dimensions.
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 find a series of non-Abelian topological phases that are separated from the deconfined phase of
ZN gauge theory by a continuous quantum phase transition. These non-Abelian states, which we refer to as the ``twisted''
ZN states, are described by a recently studied $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons (CS) field theory. The $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ CS theory provides a way of gauging the global
Z2 electric-magnetic symmetry of the Abelian
ZN phases, yielding the twisted
ZN states. We introduce a parton construction to describe the Abelian
ZN phases in terms of integer quantum Hall states, which then allows us to obtain the non-Abelian states from a theory of
Z2 fractionalization. The non-Abelian twisted
ZN states do not have topologically protected gapless edge modes and, for
N>2, break time-reversal symmetry.
Topological quantum field theory (TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In
2+1d, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasiparticle braiding statistics, chiral central charge, and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in
3+1d and it has been shown that looplike excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and looplike excitation in
3+1d. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.
Topological phase transitions involving intrinsic topological orders are usually characterized by qualitative changes of ground state quantum entanglement, which cannot be described by conventional mean-field theories with local order parameters. Here, we apply the lattice Chern-Simons theory to study frustrated quantum magnets and show that the conventional concepts, such as the order parameter and symmetry breaking, can still play a crucial role in certain topological phase transitions. The lattice Chern-Simons representation establishes a nonlocal mapping from quantum spin models to interacting spinless Dirac fermions. We show that breaking certain emergent symmetries of the fermionic theory could provide a unified approach to describing both magnetic and topological orders, as well as the topological phase transitions between them. We apply this method to the perturbed spin-1/2 ${J}_{1}\ensuremath{-}{J}_{2}$ XY model on the honeycomb lattice and predict a nonuniform chiral spin liquid ground state in the strong frustration region. This is further verified by our high-precision tensor network calculations. These results suggest that the lattice Chern-Simons theory can simplify the complicated topological phase transitions to effective mean-field theories in terms of fermionic degrees of freedom, which lead to different understandings that help to understand the frustrated quantum magnets.