Topological non-linear σ-model, higher gauge theory, and a systematic construction of 3+1D topological orders for boson systems

October 14, 2020

A summary of arXiv:1808.09394  DOI: 10.1103/PhysRevB.100.045105

Topological order describes a new kind of quantum phases of matter at zero temperature. This paper describes a systematic way to construct models to realize various topological order in bosonic systems. In three spacial dimensions, we argue that our construction can even realize all the possible topological order in bosonic systems.  This construction deepen our systematic understanding of topological order, and the concrete models help us to study various novel experimental phenomena arisen from the topological order.

Topological phase transition is not topological phase transition

Let us consider a quantum phase transition (a phase transition at zero temperature) from an ordered phase to a disordered phase, driven by the quantum fluctuations of the order parameter. We like to ask if the disordered phase has topological order or not.

The importance of the topological defects in fluctuations that drive phase transitions have been emphasized by Kosterlitz and Thouless, which shared 2016 Nobel prize ``for theoretical discoveries of topological phase transitions and topological phases of matter''.

In this paper, we show that the phase transitions driven by fluctuations with all possible topological defects produce disordered states that have no topological order, and correspond to non-topological phase transitions. On the other hand, transitions driven by fluctuations without any topological defects usually produce disordered states that have non-trivial topological orders, and correspond to topological phase transitions.  Thus topological phase transition (driven by topological defects) is not topological phase transition (that changes topological orders)

In particular, the topological order in the disordered state is determined by to topology of the space of the fluctuating order parameter, which is called the target space.  If only the first homotopy group of the target space is non-trivial, then the topological order is described by a gauge theory with the gauge group given by the first homotopy group.  If only the first and second homotopy groups are non-trivial, then the topological order is described by a 2-gauge theory, a special higher gauge theory.

Models that realize all 3+1D bosonic topological orders

The fluctuations of the order parameter may be twisted by a topological term, which can give rise to more general topological order for the disordered state.  As a result, we find that the resulting topological order is characterized by a pair: the target space and a cocycle on the target space.  With this more general result, we find that all 3+1D topological order for bosonic systems can be realized by the disordered state, whose target space has only non-trivial first and second homotopy groups, and the second homotopy group is only Z2 or trivial.  In other words, this paper gives rise to a systematic construction of bosonic models that can realize all 3+1D topological orders, which is an important result.

Only 10 years ago, systemic and classifying understanding of strongly correlated systems appeared to an impossible task.  At that time, the only systemic understanding is Landau symmetry breaking theory.  Since then, we have obtained systemic and classifying understanding of strongly correlated 1+1D gapped phases at zero temperature.  This paper presents a systemic and classifying understanding of strongly correlated 3+1D gapped liquid phases at zero temperature. It is an important mile stone.

1) It is known that some topological orders are described by gauge theory with finite gauge group.  Knowing the theoretical existence of 2-gauge theory, one may wonder, do we have condensed matter systems on lattice that can produce topological orders described by  2-gauge theory. This paper provide a systematics answer to this question.  We constructed exactly soluble bosonic systems on lattice to realize all the higher gauge theories (including all the gauge theories and 2-gauge theories).

2) We find that many higher gauge theories are equivalent and describe the same topological order. We identified a small subset of 2-gauge theories and argue that the subset can realize a large class of topological order in 3+1D bosonic systems.  The rest of 3+1D topological order can be realized by topological nonlinear sigma models.  This result allows us to classify all topological order in 3+1D via concrete models. Using those models, we can study universal experimental properties of all 3+1D topological orders.