Topological Quantum Field Theory for Abelian Topological Phases and Loop Braiding Statistics in -Dimensions
Topological qauntum field theory(TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In D, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasi-particle 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 D and it has been shown that loop like 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 loop like excitation in D. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.
- I Introduction
- II General considerations
III gauge theory
- III.1 A simple example
- III.2 General theory
- IV -type gauge theory
- V Discussions and conclusions
- A A review of Abelian Chern-Simons particle Braiding Statistics
- B Computation of the effective action
- C A microscopic derivation for the gauge transformation of type theory
- D Equivalent actions
Gapped phases of quantum matter are naturally described by topological quantum field theories (TQFT) at low energy and long distance. For example, Abelian and non-Abelian Chern-Simons theories in spacetime are believed to capture the topological properties of fractional quantum Hall states Wen (2004); Nayak et al. (2008), and gauge theories(which can be described by mutual Chern-Simons theory) have been proposed to describe some quantum spin liquids ( spin liquids). Essentially these topological gauge theories encode nontrivial quantum statistics of low-energy excitations in a gapped phase. In two dimensions, low-energy excitations consist of localized quasiparticles, and their exchange and braiding statistics have been well understood.
In spacetime, which is the dimension of the physical world, low-energy excitations are dramatically different: besides point-like particles, there can be loop-like excitations. A familar example is vortex lines in type-II superconductors. It is well-known that in three dimensions point-like particles can only have bosonic or fermionic exchange statistics, and no nontrivial mutual braiding statistics. On the other hand, there can be nontrivial braiding statistics between particles and loops, e.g. in discrete Abelian gauge theories. Recently a new kind of braiding statistics between loop-like excitations, involving two loops linked to a third one, was discovered in Dijkgraaf-Witten gauge theories Dijkgraaf and Witten (1990); Wang and Levin (2014, 2015); Jiang et al. (2014); Wang and Wen (2015); Wan et al. (2015); Moradi and Wen (2015); Jian and Qi (2014); Bi et al. (2014); Walker and Wang (2012); von Keyserlingk et al. (2013); von Keyserlingk and Burnell (2015); Putrov et al. (2017).
Another impetus for interest in -dimensional topological gauge theories comes from the study of symmetry-protected topological(SPT) phases Chen et al. (2013). These are short-range entangled gapped phases, which in the absence of any symmetries are continuously connected to a trivial product state, but with certain symmetry they become topologically distinct. When is unitary, one can understand the SPT phases by promoting to a local gauge symmetry Levin and Gu (2012). Once the matter fields (i.e. SPT) are integrated out, one obtains a topological gauge theory at low energy. The nontriviality of SPT phases then manifests through the nontrivial braiding statistics of gauge fluxes in the gauged theory. This approach has been shown to correctly characterize all SPT phases with finite, Abelian unitary symmetries in both two and three dimensions Cheng and Gu (2014); Wang and Levin (2014, 2015); Cheng et al. (2018).
Moreover, TQFTs also provide us a powerful tool to understand topological phase transitions. In fact, the continuum quantum fields in a TQFT should be regarded as emergent collective degrees of freedom in the vincinity of topological quantum phase transition, and the TQFT captures the topological Berry phase term induced by these collective degrees of freedom. Given the important roles played by field theories in the study of two-dimensional topological phases and their phase transitions, it is desired to have a similar systematic construction of field theories in three dimensions.
In this paper we introduce TQFTs that describe nontrivial loop braiding statistics in Abelian topological phases. The theories that we consider are all gauge theories, which naturally have non-local observables (Wilson loops and surfaces). Since we are interested in , the gauge theories involve both -form and -form gauge fields. We write down all possible Schwartz-type topological field theories that describe Abelian excitations. Namely, we require the action to be invariant under smooth diffeomorphisms, so they should be built out of the differential forms with wedge product. A similar approach was taken in LABEL:Wang_unpub to write down response theories for SPT phases, where the gauge fields are treated as background fields. In contrast, we are interested in truly dynamical gauge theories. An important point is that the action needs to have gauge invariance, in a manner whose precise meaning will be specified below. This requirement severely restricts possible terms that can appear in the action. We extract particle and loop braiding statistics for some of these topological gauge theories which result in Abelian statistics. We hope this work will stimulate future theoretical studies on general non-Abelian topological phases and topological phase transitions.
Ii General considerations
Let us first discuss some general aspects of three-dimensional topologically ordered states.
ii.1 Excitations in 3D topological orders
We list our physical assumptions of the general structures of topological excitations in 3D topological orders (TOs):
3D gapped topological phases can support two kinds of excitations: quasiparticles and quasi-strings. In the absence of boundary, quasi-strings always form closed loops. We assume that there are a finite number of topologically distinct types of quasiparticles and quasi-stringsLan et al. (2017).
For each type of quasi-string, one can create a single loop of this type out of the vacuum by a membrane operator. In other words, this single loop can be continuously shrinked to a local excitation. We say these are “neutral” loop excitations. On the other hand, if it shrinks to a topologically nontrivial quasi-particle, we say it is “charged”.
One can obviously define fusion of quasiparticles, as well as fusion of (unlinked) neutral loops. Thus the set of quasiparticles form a unitary fusion category. In fact, they can further be endowed with braiding. However, because of the dimensionality, the braiding must be symmetric. This strongly constraints the structure of quasiparticles: the fusion category must be the category of irreducible linear representations of some finite group , denoted by Deligne (2002). They can have bosonic or fermionic exchange statistics.
There should be a generalized notion of braiding non-degeneracy in three dimensions. More concretely, there must be braiding processes that allow one to distinguish different types of quasiparticles from each other. Since braiding between quasiparticles are trivial, one has to use the braiding between quasiparticles and loops. In this regard, we only need unlinked single loops. It is then reasonable to postulate that one should be able to distinguish all types of quasiparticles by the braiding between quasiparticles and single neutral loops. Furthermore, such particle-loop braiding must be consistent with the fusion rules of quasiparticles: for a fixed type of loop excitation , denote the braiding between and a quasiparticle of type by . Then
Therefore, defines a character on the category. It is easy to see that such characters are nothing but the characters of the representations. Since characters are class functions, we have seen that each type of quasi-strings must correspond to a conjugacy class of , uniquely.
Braiding statistics between quasi-strings can be very complicated, since quasi-strings may be knotted and/or linked. It was proposedWang and Levin (2014); Jiang et al. (2014) that the most fundamental braiding process of quasi-string braiding involves three loops (Fig. 1): loop is braided around loop , while both are linked to a third loop . Simple two-loop process cannot capture the essence of 3D topological orders (TOs), and many complicated processes can be decomposed to a sequence of three-loop processes. So far, all known 3D TOs can be characterized by the three-loop braiding statistics. Nevertheless, whether the three-loop braiding statistics is complete for 3D TOs remains an open question.
Note that in this discussion, we assume both quasiparticles and quasistrings are free to move in space, and exclude the fracton topological order with immobile excitations Haah (2011); Vijay et al. (2015, 2016).
ii.2 Topological gauge theories in d
We aim to study topological gauge theories, with -form and -form gauge fields Kapustin and Thorngren (2013), to describe TOs in 3+1 dimensions. This is natural since -form gauge fields minimally couple to worldlines of particles and -form gauge fields couple to worldsheets of strings. We will only consider U gauge fields for simplicity.
To begin with, we enumerate all possible types of topological terms (dropping the indices for components of the gauge fields):
where is 1-form and is 2-form. is the familiar BF term, describing the simplest discrete gauge theories. It is natural to include BF terms in the TQFT from the onset, for the following reason: to describe a discrete (i.e. ) gauge theory in a continuum formalism, we can start from a gauge field and add charge- bosonic matter fields. The Higgs phase effectively realizes a gauge theory. By performing a standard duality transformation, this Higgs theory can be rewritten as a topological BF theory.
We now consider the other topological terms. Conventionally, we require that the Lagrangian is invariant (up to boundary terms) under the following gauge transformations:
Here is a -valued function (mod ) and is a -form. This gauge-invariance condition then excludes all the other terms except the BF term. In order to describe more exotic statistical properties, it is necessary to generalize the notion of gauge transformations. For example, when gauge transforming , we should also allow to transform:
Here is a -form built out of and , such that . Similarly, when gauge transforming by we allow to be shifted by :
Here is constant.
Therefore, our first working assumption is that the allowed topological terms are those that can satisfy generalized gauge transformations Eq. (8) and Eq. (9) with appropriate choices of and . This is different from the approach taken in LABEL:Wang_unpub, where the topological terms are introduced as responses of SPT phases to non-dynamical symmetry gauge fields, and a flat connection condition is imposed to recover gauge invariance.
For simplicity, let us consider a BF theory with the other topological terms all of the same type in our list. For all four types of topological terms, one can indeed find generalized gauge transformations to make sense of the topological gauge theories. We will focus on and terms in the following sections. It has been proposed that type terms are responsible for three-loop braiding statistics Kapustin and Thorngren (2014); Ye and Gu (2016); Wang et al. (2015); Chen et al. (2016); Tiwari et al. (2017), and we will derive this result explicitly. We also found that type terms can alter the exchange statistics of point-like excitations (i.e. from bosonic to fermionic). The SPT response theory indicates that the type terms actually describe non-Abelian three-loop braiding statistics Gu et al. (2016). Recently, it has also been conjectured that the type terms are related to non-Abelian particle-loop braiding statisticsChan et al. (2018).
Iii gauge theory
Let us start with TQFT containing a cubic term . We will show that such TQFTs can describe the three-loop braiding statistics.
iii.1 A simple example
To begin with, let us consider the following gauge theory with three gauge fields () corresponding to gauge group :
This Lagrangian is an example of the general theory Eq. (31) below with and otherwise.
The Lagrangian Eq. (10) is gauge invariant (up to total derivative) under the following gauge transformations:
Here, we have again defined and . And and () are 0-form and 1-form gauge transformation parameters respectively.
To quantize the theory Eq. (10), we first integrate over and , which impose the flat connection condition for and on spacial manifold. Then we can use the standard canonical quantization procedure to quantize the theory Eq. (10), where only the terms with differential along time direction remain.
iii.1.1 Quantization and periodicity
Since is quantized as
for arbitrary closed surface before and after the gauge transformation Eq. (11), we have the quantization of : and . Therefore the final quantization is , where is the least common multiplier of and .
It will be shown later that and should be identified, for they give the same loop braiding statistics. Combined with the quantization of as an integer multiple of , we see that ().
iii.1.2 Membrane operators
Observables in gauge theory are gauge invariant Wilson operators. In our theory, the gauge invariant Wilson loops are
where is a closed curve. They are invariant under gauge transformation Eq. (11).
However, the usual Wilson surface operator for a closed surface in spacetime is not gauge invariant. Therefore, we modify its definition to be
where is a volume such that . One can check that the new Wilson surface operators are indeed invariant under gauge transformation Eq. (11). In the canonical quantization, however, we only consider the Wilson surface operators in three dimensional spacial manifold. Since is a flat connection in space in the canonical quantization procedure, we can drop the term in the above definition and have a simpler expression for spacial Wilson surface operators:
where is a closed surface in three dimensional space.
iii.1.3 Canonical quantization
We can do canonical quantization of the theory Eq. (10). By definition, the canonical momentum for is
Using the canonical quantization conditions for and , one can show the commutation relations for ’s and ’s are
The non-commutativity between ’s come from the requirement of .
iii.1.4 Three-loop braiding
The commutation relations of ’s and ’s contain the information of braiding statistics of point-like and loop-like excitations associated with the Wilson line and surface operators.
Consider a line and a surface intersecting transversely once. Using the commutation relation , one can show that
Therefore, we can obtain the group commutator of Wilson line and surface operators:
This is the well-known result that braiding a species charge around a fundamental flux line of gauge theory of species gives a statistics phase . In the following, we will show there is non-trivial three-loop braiding statistics for the theory Eq. (10).
The Berry phase accumulated in the process corresponds to the three-loop braiding, where two loops with unit and fluxes are linked to a base loop with unit flux, can be calculated as Wang and Levin (2014); Jiang et al. (2014); Yoshida (2017)
where we have chosen the three surfaces to be and plane. The basic idea of the process is that first create a base loop , and do a full braiding of two other loops linked to , then annihilate the base loop and do a full braiding of the two other loops. Using commutation relations of ’s and ’s, one can directly show that the only non-trivial three-loop braiding phase factors are
The topological invariant for the three-loop braiding is Wang and Levin (2015). So the nontrivial ones are
From the above expressions, we see that and give the same topological invariants. Therefore we identify these two values of and have ().
Apart from the three-loop braiding calculated above, there are also processes of three-loop half-braidings when two loops are linked to . We can first create a base loop by . Since the full braiding process is given by in Eq. (26), we can do a half-braiding by , which do not move the two loops back to its original places. Therefore, the three-loop half-braiding phase can be calculated by
Using the canonical commutation relations and the fact is a state without any flux loops (=0), one can show directly that all half-braiding phases are for this particular theory Eq. (10).
iii.2 General theory
Now let us consider the most general partition function of the type TQFT:
where the Lagrangian is given by
Here all the repeated indexes are summed over automatically. Although we can choose without loss of generality, we would not to impose this condition for the coefficients in the following discussions.
Naively the theory is not invariant under the gauge transformation of . To recover gauge invariance, we need to let the gauge transformation also acts on :
where the gauge parameters are quantized as and on closed line and surface . It is easy to check that the theory is indeed gauge-invariant with this definition of gauge transformations. We notice that one can come up with different gauge transformations to make the action gauge-invariant, however our choice in Eq. (32) is motivated and justified by a microscopic derivation of the action (31) (with non-compact fields). We note that the gauge transformation define here is different from the one defined in Ref. Chen et al., 2016. In Appendix B, we will provide microscopic derivation of such a twisted gauge transformation.
iii.2.1 Quantization and Periodicity
Quantization of requires and , so must be an integer multiple of . We will show later that the theories and have the same braiding invariants. Combine this identification with the previous quantization, we have ().
iii.2.2 Membrane Operators
We now compute the physical observables in the theory. Due to the cubic form of (31), we are no longer able to integrate out the gauge fields exactly to obtain an effective action of matter fields. Therefore we proceed with canonical quantization.
First let us define gauge-invariant physical observables. Wilson loops take the conventional form
Here is any closed curve. For later use, we will also define
The gauge invariant Wilson surface operators for a closed surface in spacetime are
where is a volume such that . One can check that the Wilson surface operators, with an additional Chern-Simons density term compared to the usual definition, are invariant under the gauge transformation Eq. (32). In the canonical quantization, we only need to consider the Wilson surface operators in three dimensional spacial manifold. After integrating out the Lagrangian multiples and obtaining the flat connections in space, we can drop the terms in the above definition and have a simpler expression for spacial Wilson surface operators:
iii.2.3 Canonical quantization and Membrane Algebra
To carry out canonical quantization, the four manifold has to be where corresponds to the time direction. Again, the time components are all Lagrange multipliers and just enforce the constraint that when there are no external sources, and the Hilbert spaces are flat connections of and modulo gauge transformations.
By definition, the canonical momentum for is
From the canonical quantization conditions
we obtain the commutation relations between fields ’s and ’s:
If we consider the commutation relation , we need only to antisymmetrize the indices and , and for the last equation:
This equation turns out to be related to the three-loop braiding statistics. We also note that only the antisymmetric part of the first two indices of appears in the above commutation relation.
For closed line and closed surface intersecting transversely, we have the commutation relation between the Wilson loop and surface operators:
is the signed intersection number of and .  is the tangent (normal) direction of () at point . If and intersect non-transversely, then the commutation relation is zero due to Eq. (41b).
By using the Baker-Campbell-Hausdorff formula, we can obtain the group commutator of Wilson line and surface operators:
This is the well-known result that braiding a species charge around a fundamental flux line of gauge theory of species gives a statistics phase .
iii.2.4 Three loop braiding
Now let us move to the statistics of loops. We assume two closed surfaces and embedded in intersect transversely. The intersection is then a collection of (directed) closed lines. The direction of the line is given by locally, where is the local normal direction of the surface . Using the convention and the commutation relations Eq. (42), one can show straightforwardly that
The line integral in on the right hand side is along the direction defined above.
Note that does not intersect transversely with neither or . Therefore, commutes with both and . And by using the Baker-Campbell-Hausdorff formula again, we have
In order to reveal the nontrivial statistical properties, we consider whose dimensions we refer to as . We will use to denote the three spatial directions. The nontrivial braiding statistics of particle and loop excitations manifests in the algebra of Wilson operators defined on non-contractible cycles and surfaces (i.e. nontrivial cohomology classes in and ). By definition, such surfaces are not the boundary of any open volume. As illustrated above, we can use alternative definitions of Wilson surface operators purely on the surface. And the commutation relations, hence the braiding statistics, are all the same for these operators. For , we denote the non-contractible cycles by and the three nontrivial surfaces by .
According to Ref. Wang and Levin, 2014; Jiang et al., 2014; Yoshida, 2017, the Berry phase accumulated in the process corresponds to the three-loop braiding, where two loops with unit and fluxes are linked to a base loop with unit flux, is
where we have used the fact .
Since is an integer multiple of , we can parametrize it to be where is an integer and is the least common multiple of and . The topological invariant for the three-loop braiding is then given by
This is consistent with the results in Ref. Wang and Levin, 2015.
Similar to the three-loop braiding phase factor , the three-loop half-braiding phase factor can also be obtained. Naively, we have , since twice the half-braiding is a full-braiding. But there is an ambiguity of in , for both and are defined modulo .
Let us calculate in detail the three-loop half-braiding statistics from the canonical quantization. Similar to the expression Eq. (50), the process of three-loop half-braiding can be written as (see the discussions above Eq. (29))
Using the Baker-Campbell-Hausdorff formula, we have
where is a real number by using the commutation relations of ’s and ’s. The three-loop half-braiding phase factor is then
which is exactly the naive result: half of the full-braiding result Eq. (50) or Eq. (51). Note that the term in Eq. (53) contributes to the half-braiding phase factor as terms like , which is trivial for is the ground state without any flux, i.e., .
From these braiding invariants, we see that and would give the same results. Combined with the quantization of , we have the result () claimed above.
Iv -type gauge theory
The other Abelian family of TQFT is given by the following action:
The single-component version of the gauge theory (55) was first introduced in LABEL:HorowitzCMP, and its relevance to 3D topological phases of matter has been increasingly appreciated in recent years Walker and Wang (2012); von Keyserlingk et al. (2013); von Keyserlingk and Burnell (2015); Ye and Gu (2015). Naively, the action is not invariant under the gauge transformations of . As observed in LABEL:HorowitzCMP, gauge invariance can be achieved with the following generalized gauge transformations
Notice that with this definition, is no longer gauge-invariant. Instead, can be used to construct a Maxwell-type kinetic term (in addition to the Maxwell term for built from the -form curvature tensor of ). The above type TQFT can be formally rewritten as a type term.(We note that the type term is a total derivative which is dropped away here.)
Let us check the gauge invariance explicitly, which will also lead to a quantization condition for . For now let us assume that the theory is defined on a closed -manifold. The variation of the action under the -form gauge transformation becomes
The first term is a total derivative. Integral of the second term is quantized: 111This is the intersection form on the -manifold.
For the action to be gauge-invariant on any space-time manifold, the second term must an integral multiple of which requires and to be even integers. Therefore we find a quantization condition
Notice, however, that on a spin manifold, the (58) quantizes to an even integer. So can be any integer if we are considering fermionic theories which can only be defined on spin manifolds. We will see that if any of is odd, the theory indeed admits transparent fermionic excitations.
On the other hand, we notice that because is compact, in order to keep the periodicity in (56), should be an integer. Similarly, is also an integer. So is a multiple of where lcm means the least common multiple. We will write
The only constraint then is that is even.
We now compute the physical observables in the quantum theory. To motivate, let us couple the gauge fields to sources:
First we need to make sure that the coupling term is gauge-invariant. Invariance under -form gauge transformation gives the usual conservation law: . However, under the -form gauge transformation
So we must impose a different conservation law
The physical interpretation is that point-like excitations are the end of string-like excitations. If , all strings are closed loops.
A completely equivalent viewpoint is to consider the expectation values of gauge-invariant operators, which are Wilson loops and surfaces. The Wilson surface operators are defined as
Here is a closed surface. One might attempt to construct Wilson loop operators as
However, it is no invariant under -form gauge transformations. In order to restore gauge invariance, we have to attach to a surface such that and define
It is easy to see that evaluating the expectation values of Wilson loop/surface operators using path integrals is the same as computing the path integral in the presence of sources.
We can integrate out the gauge fields to obtain an effective action of the source fields. Since the action is Gaussian, let us write down the equations of motion first:
In the Lorentz gauge , we find
Particle current is defined by the worldlines:
is the worldline of the particle carrying charges of the gauge field .
The string current needs some care Lechner and Marchetti (2000). As we have noted, the gauge structure of the theory requires that each worldline bounds a (open) worldsheet, the choice of which is not unique. We will use the following worldsheet: