Acknowledgement

This is the lecture notes taken from the 2020 lectures on topological order [https://www.koushare.com/live/topology], the references are important for future study.

孔良：Infinite more is more different

Try to give a categorical unification of many notions in quantum many body problem, quantum gravity, quantum entanglement. The “inifity” is essentially different from finite, where new mathematical structures emerge. The nature of strong interacting systems in quantum many body also adopt this feature.

This talk(rather than a lecture) is inviting, a more detailed version can be found in [https://arxiv.org/abs/2205.05565]

王晨杰：Quantum anomaly and bulk-boundary correspondence

The anomaly in QFT is a powerful tool in CMP. The anomalous topological order(surface) must be attached to some topological nontrivial topological order(SPT/SET). The anomaly is classfied into several types: 't Hooft, Gravitational, gauge-gravitational mixed, invertible/non-invertible(w/o GSD,w/o anyon)

't Hooft anomaly basicly means the global symmetry can’t be gauged, turning on the probing(external) gauge field, we will obtain anomalous gauge response. This anomaly means the theory can’t have a lattice regularization(for example the Nilsen-Ninomiya theorem/fermion doubling). The anomalous theory can’t exist alone, but being the boudanry of some bulk topological order which is topologically nontrivial. Thus the whole space is anomaly free, and the bulk theory is protected/enriched by the corresponding theory.

Gravitational anomaly is related to the topology, which is manifested by coupling to a external gravitational field(metric field), i.e, curving the spacetime. This is also related to the chiral central charge, which is manifested when the CFT description is set up.

The effective field theory of T.O is a TQFT, coupling to the external gauge field(gauge response), we can study its excitation(anyon), the excitation spectrum depend on symmetry conditions(parameter) thus symmetry enriched. For example, the toric code and double semion can be described in terms of double layer Chern-Simon(non-chiral T.O,FQHE is chiral T.O.).

顾正澄：Topological order and ground state wave function

Viewed from ground state, Topological order phase is a gapped quantum phase that can’t be deformed(using generalized local unitary transformation) into a tensor product state. This also means these phases have long-range entanglement(but don’t have long-range order,all correlation functions decaying). With symmetry, it’s a SET state. There’s related SPT state, which have no intrinsic T.O, but can’t be deformed into a tensor product state(independent atomic insulators) perserving symmetry.

The construction of specific ground state for 2D T.O is using lattice models(this can’t be easily done for chiral T.O,like FQH)(From field theoretical point, this is constructing lattice gauge theory of TQFT), the construction will involve the wavefunction renormalization operations(related to retriangulation of lattice,etc) since ar RG fixed point, only the topology matters, then there will be consistency conditions for these operations together with gauge choices. These then defines possible data of a G.S wavefunction at least for bosonic case(having extra saturate condition). These data(solutions) define a unitary tensor category.

This view is related to the excitation view, since the algebra of anyons(anyon diagramatics or TQFT/CFT algebra) and the algebra of the Ribbon operators also define a unitary tensor category, possibly with more datas.

Models include Toric code, double semion(related in cohomology), quantum double, twisted quantum double(Dijkgraaf-Witten), string-net(Levin-Wen,Tuaev-Viro), etc. The construction of fermionic topological order lattice models are rather hard, which can be studied using anomaly.

GS wavefunction can also be constructed using TQFT/CFT, for example the Laughlin wavefunction(chiral/holomorphic) of FQHE, and Moore-Reed, etc.

Gu: Introduction to SPT phases

The definition of SPT phases: they are gapped phases with short-range order, short-range entangled(only 1-dim support space) quantum phases. SPT phases have unique ground state over closed manifold, the bulk is topological trivial, but the edge is topological nontrivial. The boundary is anomalous, may be degenerate, gappless, etc. These boundary states are protected by symmetry. This means using symmetric generalized local unitary transformations we can’t deform the ground state wavefunction to a tensor product state.

The classification of these states is similar to the method of classifying intrinsic topological order(like Toric code, double semion). This can be done by constructing all possible fixed point ground state wavefucntion. SPT is simpler since the ground state wavefucntional is unique and thus the 1-dimensional.

The general idea is, by discretizing the space, we obtain some branching lattice, and we put the D.O.F(labeled by protecting group elements) on the vertices(this constructinon is dual to the constructions of topological order). Since we are considering topological phases, the equivalence class is determined by SLU, manifested in lattice models, we are actually doing the retriangulation, in view of field theory or Hamiltonian, we are doing the renormalization group flow, by perturbing the theory. The final result should be a fixed point wavefunction(al)! This means the theory is topological and is independent of the triangulation, thus there will be consistency conditions. The retriangulation is furnished by fundamental moves, these fundamental moves will give rise to phases labeled by cochains. The consistency condition then implies these cochains satisying the cocycle equation, thus they are cocycles. Physically, the local U(1) phase ambiguity of quantum states exactly give rise to coboundary transformations and may simplify the procedure of solving the cocycles. Then the classification of these lattice models are determined by the group cohomology.

Since these wavefunctions are at the fixed point, the CFT description may be useful.

These group cohomology actually label the boundary states, for example the AKLT model(equivalent to the Haldane phase) the nontrivial state correspond to non trivial projective representation(thus nontrivial elements in cohomology group).

These wavefucntions then give rise to projective Hamiltonian which is exactly solvable. And include both interacting and noninteracting cases, at least for bosonic case.

Another way to study the physics of these topological phases is by gauging the global symmetry(coupling to external gauge fields, anomaly detecting) study the excitations! This is also the case of topological orders. Especially we can consider the effective field theory of these SPT phases, which is related to generalization of Chern-Simons theory.

There’s duality between topological order and SPT phases, for example the T.C/double-semion vs parameter phases(two of them) of Ising model(Z2). The data of group cohomology actually can define a MTC thus this is not very different from the construction of wavefunction of topological order and the anyon excitation algebra.

The fermionic case is more complicated, since the lattice construction is not always well defined(anomaly) the possible approach may be the super group cohomology. Involving the input of fermionic entities, this is like the supersymmetry, where the manifold is punctured with fermions and the mathematical structure is related to differential geometry.

For free fermions the SPT phases is classified by K theory!

叶鹏： Effective topological field theory of topological phases of matter with global symmetry.

In strong correlated systems, the Hamiltonian is too complex to solve and to understand, we have to seek for effective description of the low-energy physics, this need inspired guess and some techniques to sort out the real relevant D.O.F and their effective field theory. For topological phases of matter, these will be all kinds of topological field theories, to extract the physics we have to gauging them to probe their gauge response.

SPT phases are topological trivial TQFT with further global symmetry included, the ground state can be shown to be unique. While SET phases are nontrivial topological TQFT with further global symmetry, with ground state shown to be degenerate and the excitations(gauge responses) are anyons with braiding statistics and fractional gauge charges.

The talk start with an example of t-J model, where the t is for hopping, which is weak correlated and the J is for superexchange effect between spins, which is strong correlated. The low-energy regime(low energy subspace, to construct these theories we need to project into these spaces, this is down by including further constraints, usually using Lagrange multiplier after some deformation/procedure) implies further constraints, known as non-doubly occupancy condition, this is the large-U limit of the Hubbard model. To extract the effective D.O.F, we use the technique known as slave-particle, this is one type of projective constructions.

The basic idea is to include axulliary fields(particles) to enlarge the Hilbert space, introducing redundant gauge symmetry and D.O.F then projecting back to the physical Hilbert space(states) using constraints. These constraints then can be included into the meanfield theory of these axulliary particles. Usually these constraints means the axulliary particles are confined to be a physical particle. These axulliary particles are understood as originally confined by the redundant gauge interactions and became the relevant D.O.F in the effctive low-energy regime, producing the physically observed responses(coupling to external probing gauge field,usually EM field or Spin field).

The Lagrange multiplier here usually turn out to be the gauge field of the redundant gauge symmetry. We can study their physics in two ways, first integrating out the redundant gauge field(Lagrange multiplier) to obtain a effective theory of axulliary particles, further integrate out the external field to obtain response and compare to experiments. We can also integrate out the axulliary particles first to see if the redundant gauge fields are well-behaved, since their Maxwell(or Y-M,more explicitly dynamical terms) term is not included, the redundant gauge coupling is actually infinite(confining phase).

Usually to make sure the gauge coupling is not infinite, we have to gap these redundant gauge fields, this usually lead to the meanfield theory of axulliary particles being certain TQFT that will produce topological masses(like Chern-Simons), thus these terms dominate and we can safely ignore the dynamic terms.

Another example is the v=1/3 Laughlin FQHE state. Whose effective excitations are partons, these effective field theoretical description help us to understand the wavefunctions.

For the construction of TQFT theories of SPT, see the slices and the original papers, the talk is really full of details and not easy to follow. Some features should be noticed:

1. the topological terms are different in different dimensions(BF term and generalized Wen-Zee term, etc) and there are twistings terms to include symmetry. This is actually similar to Dijkraaf-Witten TQFT, though the talker implies there are SPT phases beyond D-W TQFT, which is also beyond the cohomology classification(releted to the construction of NLSMs and wavefunctions). This is interesting.
2. The coeffeicents(matrix) of the topological terms include the informations of the excitations, for example bosonic/fermionic, GSD, chiral/non-chiral(chiral central charge), excitation(anyon) statistics(by gauging SPT, we actually obtain topological order phases). The classification is obtained physically by constraints on the coefficents like Dirac quantization condition(related to large gauge transformations) or symmetry constraints. These quantization can also be obtained from duality, which usually describe some superfluid/superconductors.
3. In a given TQFT gauge theory, the phases are really complex, we have to be careful. The topologcial trivial phases(some special level) involve trivial braiding statistics, thus are starting point of the study of SPT(no T.O.)

The main refecences:

1. Topological quantum field theory of three-dimensional bosonic abelian-symmetry-protected topological phases_Ye,Gu16
2. Generalized Wen-Zee terms_Han,Wang,PY19
3. Gauge theory of strongly-correlated symmetric topological phases_PY20