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Symmetry Principle for Topologically Ordered Phases
Yuji Hirono
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DOI: 10.22661/AAPPSBL.2019.29.2.45

Symmetry Principle for Topologically Ordered Phases



Determining phase structures has been one of the fundamental problems in quantum field theories. Different phases are classified according to the symmetry: this is the basic principle of Landau-Ginzburg theory based on local order parameter fields. It has been realized that this theory fails to detect the so-called topological order, which exhibits robust ground-state degeneracy depending on the spacetime topology, fractionalized excitations, and long-range entanglement. In this article, we review how to characterize and describe topologically ordered states. Properties of such states are captured by topological field theories. The appearance of topological order can be understood as a consequence of an emergence of so-called higher-form symmetry and its subsequent spontaneous breaking. In this article, we review how to characterize and describe topologically ordered physical systems. We also review our recent work on the possibility of topological order in dense quantum chromodynamic (QCD) matter. We show that the low-energy effective theory of a color superconductor can be written as a topological BF theory coupled with massless Nambu-Goldstone bosons. We discuss physical consequences of the effective theory and their implications.


In many-body physics, determination of the phase diagram is one of the most basic problems. According to the Ginzburg-Landau paradigm, the phases of matter can be classified in terms of symmetry breaking patterns. In this theory, there is a local order parameter field and the symmetry is determined from the value of the field. In recent years, it has been recognized that the Ginzburg-Landau paradigm is not detailed enough to classify phases. There is another type of order which cannot be detected by the local order parameter, called topological order [1, 2]. Hallmark features of topological order are the existence of fractionalized excitations and robust ground-state degeneracy that depends on the spacetime topology. S-wave superconductors [3] and fractional quantum Hall systems are examples of the physical systems that exhibit topological order. Identifying the nature of topological order of physical states is important in determining the phase diagram of matter, because the states with different topological order cannot be connected without a phase transition.

Although topological order cannot be captured by the Ginzburg-Landau theory, there is a symmetry principle for the detection of topological order. It has been proposed that topological order can be classified and characterized by emergent higher-form symmetries [4, 5]. Low-energy behavior of states with topological order are described by topological field theories with emergent discrete higher-form symmetries. For example, s-wave superconductors are described by BF theory at level 2. Furthermore, when the system is topologically ordered, the discrete symmetries are spontaneously broken.

In this article, we will first give an introduction to what topological order is and how such a system is described, taking s-wave superconductors in 2+1 dimensions as an example. We will explain how the fractional statistics and degeneracy of ground states are encoded in the effective topological field theory at low energies. We will also review what a higher-form symmetry is, and it will be shown that BF theory has two ℤ2 1-form symmetries. Furthermore, we will review our recent attempt to use those concepts in constraining the phase diagram of dense QCD matter. We will derive a dual gauge theory for color superconductors and discuss the topological properties.

Topological order in s-wave superconductors

Let us demonstrate how the features specific to topological order arise, taking an s-wave superconductor in 2+1 dimensions as an example. We first derive the topological field theory for s-wave conductors via a duality transformation. We start with the Abelian Higgs model,

where 𝜙 is the phase-part of the Cooper pair field, a is a U(1) gauge field, d is the exterior derivative, and ★ is the Hodge dual operation. Here, the amplitude fluctuations of the Higgs field are neglected because they are heavy. The parameter k ∈ ℤ is 2, which is the charge of the Cooper pair. We introduce a 2-form field h through the Hubbard-Stratonovich transformation,

The equation of motion (EOM) from varying 𝜙 reads dh = 0. We can solve this explicitly by taking h = db, where b is a 1-form gauge field. Then we have

Noting that db = 0 from the EOM for the gauge potential a, we can further drop the kinetic term for b. The resulting action is that of BF theory at level k,

BF theory is an example of a topological field theory. There is no bulk degree of freedom. The field strengths of the two gauge fields both vanish, da = 0 and db = 0. Still, the theory is not trivial and captures the essence of topologically ordered states, as we shall see soon. Gauge-invariant observables of this theory are the Wilson loop operator and vortex operator,

where C1 and C2 are closed paths in 2 + 1D spacetime dimensions.

Higher-form symmetries and the characterization of topological order

It has been realized that topological order can be characterized by the emergence of discrete higher-form symmetry and its subsequent spontaneous breaking. A higher-form symmetry [5], or a generalized global symmetry, is a generalization of the concept of symmetry. Let us briefly explain what it is. Recall that, in the case of an ordinary symmetry in quantum field theories, the charged objects are local fields. For example, a U(1) symmetry transformation rotates the phase of a local operator as

In the case of a higher-form symmetry, corresponding charged objects are extended objects, like Wilson loops. A 1-form transformation acts on a Wilson loop as

Many of the concepts that appear in the discussion of symmetry can be generalized as well. For example, in the presence of a continuous higher form symmetry, there exists a corresponding conserved charge. When a continuous higher form symmetry is spontaneously broken, there appears a Nambu-Goldstone (NG) mode. For example, Maxwell theory without matter is invariant under two continuous 1-form symmetries (electric and magnetic). The corresponding conserved charges are the electric flux and the magnetic flux. Photons are understood as a NG mode associated with the spontaneous breaking of a continuous 1-form symmetry.

In the case of BF theory for superconductors, there is an emergent ℤk 1-form symmetry. The charged object under this symmetry is the Wilson loop operator,

In terms of the gauge field, this transformation is written as

where is a flat (meaning d位=0) 1-form connection with property C ∈ 2𝜋ℤ for a closed loop C. The action is also invariant under

which corresponds to the discrete phase rotation of the vortex operator,

In the superconducting phase, the two 1-form symmetries are spontaneously broken, by which we mean that

for a large loop C. In this case, those operators are topological, meaning that correlation functions only depend on the topology of the world lines and the shapes of those lines are irrelevant as long as the topology is unchanged.

Fractional braiding and ground state degeneracy

Using the effective action, we can evaluate the following correlation function as

where link(C1,C2) is the linking number of the world lines C1 and C2 in 2+1 dimensions. Nonzero linking number only appears when the world lines are linked (see Fig. 1 for an example). When k≠1, this correlation function indicates that vortices and particles obey fractionalized statistics.


Fig. 1: Example of linked world lines.

Now, let us show the ground state degeneracy. To do this, it is convenient to use the Hamiltonian formalism. The Wilson loop and vortex operators satisfy


where C1 and C2 are now in the 2D spatial slice and I(C1,C2 ) is the intersection number of the two lines. This relation entails that W(C) induces the discrete phase rotation of V(C), and vice versa. The physical meaning of the actions of those operators is as follows: W(C) measures magnetic flux and creates an electric flux, and V(C) measures electric flux and creates a magnetic flux. When there is a "hole" in space, we can create an electric or magnetic flux without changing energy. This is the origin of the degeneracy. Accordingly, the number of degenerate ground states depends on the number of holes.


Fig. 2: Paths of Wilson loops and vortex operators on a torus.

Suppose the space is 2-dimensional torus. Then there are two directions along which a closed loop can wind, which we denote by x and y. Correspondingly, we have Wilson loop operators Wx, Wy and vortex operators Vx, Vy (see Fig.2). The operator algebra (1) now reads

where we have taken k = 2 and other combinations are commutative. Those operators commute with Hamiltonian, [Wi, H] = 0 and [Vi, H] = 0.

Let us denote a ground state by |惟>. We can take this state as an eigenstate of Wx,

since Wx commutes with the Hamiltonian. Wx counts the number of magnetic fluxes mod 2. We define a state by operating Vy on this,

This state has the same energy as |惟>, because Vy commutes with the Hamiltonian, and is threaded with a flux (See Fig. 3). We can show that <惟|惟'> = 0 as

Thus, |惟'> is a different state from |惟>, meaning that the ground states are degenerate. In this case, there are 4 ground states in total.


Fig. 3: Pictorial expression of |惟'>. The dotted loop indicates a magnetic flux.


Let us apply the symmetry principle of topological order to the phase diagram of QCD matter. QCD is the theory of strong interaction and describes the properties of quarks and gluons. It is a non-Abelian gauge theory, and shows a rich variety of phenomena at finite densities and temperatures [6, 7]. We here consider dense QCD matter, that is realized inside neutron stars or low-energy heavy-ion collisions.

At low temperature, below several trillion degrees, quarks and gluons are confined in hadrons such as protons and neutrons. When hadrons are packed ever more densely, then at some point, baryons start to form pairs and Bose-Einstein condensation takes place: this is nucleon superfluidity. There is observational evidence of the formation of nucleon superfluidity [8]. The pair-formation and annihilation processes lead to an enhanced cooling through neutrino emission, which explains the observed rapid cooling.

On the other hand, if we consider the limit of asymptotically large baryon densities, there are large Fermi surfaces of quarks. As a result, the typical momentum exchanged in the collisions of quarks is large, meaning that weak-coupling calculations are reliable thanks to the asymptotic freedom of QCD. The interactions between quarks are mediated by gluons and there is an attractive channel. The existence of a well-defined Fermi surface and an attractive interaction are the sufficient condition for the Cooper instability to occur. As a result, two quarks form a pair and Bose-Einstein condensation takes place. Thus, QCD matter at very high density is naturally a color superconductor.

Schaefer and Wilczek claimed that those two phases, a nucleon superfluid and a color superconductor, can be smoothly connected without a phase transition [9]. Despite apparent differences, they argued that the two phases are essentially indistinguishable. This is called the quark-hadron continuity scenario. The observation is based on the fact that the two phases have the same symmetry, in the spirit of Ginzburg-Landau-type classification of phases using local order parameters. However, as we have discussed already, the Ginzburg-Landau theory misses the possibility of topological order. Below we will examine this possibility and obtain some constraints on possible phase structures.

Dual effective description for color superconductors

We here consider three light flavors of quarks. The order parameter of color superconductivity is the diquark condensate 桅,

where C ≡ i纬02 is the charge-conjugation matrix, and , , 路路路 and i, j,路路路 are indices for color and flavor, respectively. This field is in the anti-fundamental representation of both color SU(3)c and flavor SU(3)f. The pairing pattern is s-wave and has positive parity. The gauged Ginzburg-Landau Lagrangian can be written as

Here, aSU(3) represents the SU(3)c gauge field, and G is the field strength. The potential Veff is taken to have the minimum at 桅桅 = Δ21. As a result, the global symmetry is spontaneously broken as

At the mean-field level, we can choose the gauge so that

In terms of this gauge-fixed field, the symmetry breaking pattern looks like

A ground state is invariant under simultaneous rotations of color and flavor groups, which is why this state is called the color-flavor locked (CFL) phase. Because of the diquark condensation, all the gluons acquire masses via the Higgs mechanism.

Let us consider the low-energy theory of the CFL phase. We here give degenerate masses for all flavors of quarks, and then the pions are massive and can be dropped. The massless degree of freedom in this phase is the NG boson associated with the breaking of U(1) symmetry (we call them phonons). We choose the gauge where 桅 takes the diagonal form,

where 𝜙i are phase variables. In the CFL phase, there are topologically stable vortices when the phases 𝜙i have nontrivial windings. The superfluid circulation of a minimal energy vortex is fractional, 1/3 [11]. Those vortices have various interesting properties, see [12] for a review.

With this gauge fixing, the gauged GL Lagrangian reduces to

where the kinetic term for the gauge fields are dropped because they are heavy. Taking an Abelian duality as in the case of s-wave superconductor, we obtain a dual effective gauge theory,

where bi = (bi)渭谓 dx dx are 2-form gauge fields (bi is dual of 𝜙i). Here, the matrix K is given by

This theory describes the massless phonon fields as well as the nontrivial winding between the vortices and Wilson loops. The observable Wilson lines are generated by

The vortices are represented by the following vortex operators,

where S is a 2-dimensional world sheet of a vortex. We can evaluate the correlators of Wilson loops and vortex operators using the dual action, and show that

where link(C, S) ∈ ℤ is the linking number of C and S (note that 1-dimensional loop and 2-dimensional closed surface can make a link in 3+1 dimensions). The division on the left-hand side is necessary to extract the information of linking, because the vortex operators are not topological because of their coupling to massless phonons. Here, the matrix K+Ai is the so-called Moore-Penrose inverse of K, which is a generalization of the matrix inverse. It is defined as a matrix satisfying


It always exists and is unique for any kind of matrix. Explicitly, it is given by

Using this, the braiding phase is determined by the elements of the following matrix,

Therefore, we find that there is a ℤ3 fractional statistics between vortices and Wilson loops. This reproduces the observation made in [13] as a special case. The fractional statistics of vortices and Wilson loops appears as a consequence of an emergent ℤ3 2-form symmetry in the effective theory,

where is a flat 2-form U(1) connection that satisfies S ∈ 2𝜋ℤ where S is a 2-dimensional subspace with no boundary. Indeed, the variation of the action under this transformation is

and this transformation is a symmetry of the low-energy theory.

Unlike the case of an s-wave superconductor, there is no discrete 1-form symmetry for aA. Furthermore, the ℤ3 2-form symmetry is unbroken. The order parameter for this symmetry is the expectation value of vortex operators. When they obey the perimeter law, the symmetry is spontaneously broken. However, vortices couple to massless phonons, which leads to the logarithmic confining potential between vortices and antivortices. As a result, the vortex operator does not obey the perimeter law and the order parameter vanishes. Thus, this symmetry is not spontaneously broken. Consequently, the resulting ground state does not exhibit topological order, unlike the case of s-wave superconductors.

Let us finally discuss the implication of the analysis above to the quark-hadron continuity scenario. Although the CFL phase exhibit fractional braiding between vortices and colored particles, we have learned that the CFL phase is not a topologically ordered state. Here, vortices are not "topological" operators, meaning that their shapes affect the correlation functions because of the coupling to massless phonons. On the other hand, in the nucleon superfluid phase, the degrees of freedom are hadrons and all the gauge fields are confined. Assuming that this phase is not topologically ordered, the topological properties of the CFL phase and nucleon superfluid phase are the same. Therefore, those two phases can be connected and the quark-hadron continuity scenario is still alive.


In this article, we reviewed the basic aspects of topological order. We explained that the low-energy physics of a topologically ordered state is captured by topological field theories. The appearance of topological order can be understood as an emergence of discrete higher-form symmetries and their spontaneous breaking. Those symmetry principles can be used to diagnose the nature of the ordering in a phase. Such analysis also gives nontrivial constraints on the structure of phase diagrams, because at least one phase transition is necessary between states with different topological order.

We then discussed the application of those concepts to dense QCD matter. We have derived a dual gauge theory for the CFL phase of color superconductivity, from which we have shown the fractional braiding statistics of vortices and colored particles. However, the discrete higher-form symmetry is unbroken in this phase and it is not a topologically ordered state, hence this phase can be still smoothly connected to nucleon superfluid phase without a phase transition. For details on this work, see [14].

Acknowledgements: The author thanks Yuya Tanizaki for helpful discussions and collaborations that led to the work discussed here, and Yoshimasa Hidaka for insightful discussions.


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Yuji Hirono is a Junior Research Group Leader at the Asia Pacific Center for Theoretical Physics (APCTP). After obtaining his PhD. at the University of Tokyo, he worked as a JSPS postdoc at Stony Brook University and then as a research associate at Brookhaven National Laboratory. His main research field is theoretical nuclear physics.

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