Xiao-Gang Wen have been working on theoretical condensed matter physics, in particular, in the area of strongly correlated quantum matter. Those work have lead to new concepts, new quantum phases, and new theoretical formalisms:

#### The introduced concepts:

- Topological order [1] (Systems without any symmetry can form different phases)
- Topological degeneracy [2] (Robust against any local perturbations)
- Projective symmetry group [3] (A “fractionalized representation” generalizing fractionalized charge)
- Topological entanglement entropy [4]
- Long-range entanglement [5]
- Symmetry protected trivial (SPT) order, also known as symmetry protected topological order [6, 7, 8] (Trivial product states can form different phases in the presence of symmetry even without spontaneous symmetry breaking)
- Symmetry enriched topological order [3, 9, 10, 5, 11] (The same topological order can form different phases in the presence of symmetry without spontaneous symmetry breaking)
- Group super-cohomology [12] (A twisted group cohomology)
- Algebraic higher symmetry [13, 14] (A generalization of symmetry and higher symmetry)
- Categorical symmetry [13, 14] (Another name for non-invertible gravitational anomaly, providing a holographic way to describe symmetry, higher symmetry, and algebraic higher symmetry)

#### The constructed quantum states of matter:

- Chiral spin liquids [15] (Topological order with emergent semions)
- Chiral Luttinger liquids [16, 17] (on edge of fractional quantum Hall states)
- Z
_{2}spin liquid [18] (with Z_{2}topological order, also known as toric code) - Non-Abelian quantum Hall states [19] (Topological order with non-Abelian statistics)
- string-net condensation 20, 21 (A mechanism to produce Abelian and non-Abelian topological order)
- New topological insulators/superconductors and SPT phases for interacting bosons/fermions [7, 8, 12]

#### The developed theoretical formalisms:

- K-matrix classification of 2+1D Abelian topological order [23, 24]
- Projective construction for non-Abelian states [25]
- Emergence and unification of Fermi statistics and gauge interaction in any dimensions from qubits [20], 22](A unification of matter and information. A concrete realization of
*it from qubit*,*matter from qunatum information*) - Unitary fusion category classification of 2+1D topological orders with gappable edge [21, 26]
- Projective-symmetry-group theory for symmetry enriched topological orders [3, 27, 10, 5]
- Gauge theory for high Tc superconductors [28, 29]
- Pattern-of-zeros theory of non-Abelian fractional quantum Hall states [30, 31]
- Classification of 1+1D gapped quantum phases [32, 33]
- Classification of 1+2D gapped quantum phases with finite symmetry for bosonic/fermionic systems [39, 40]
- Classification and realizations of 3+1D topological orders for bosonic systems [34, 35, 36]
- Group cohomology theory of SPT orders [7, 8]
- Classify perturbative and gloal gauge anomalies using group cohomology theory [37]
- Classify perturbative/global gravitational anomalies using topolo. orders in one higher dimension [26]
- Use SPT approach to solve the chiral fermion problem (i.e. putting the standard model on 3+1D lattice)[38]

### References

[1] Xiao-Gang Wen, “Topological orders in rigid states”, Int. J. Mod. Phys. B 04 239 (1990).

- The first detailed discussion of topological order. Proposed a conjecture: the non-Abelian Berry’s phase of degenerate ground states and the induced mapping class group representations completely characterizes(i.e. define) topological orders macroscopically.

[2] Xiao-Gang Wen, Q. Niu, “Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces”, Phys. Rev. B 41 9377 (1990).

- Showed that the ground state degeneracy of fractional quantum Hall states is robust against any local perturbations, which led to the concept of topological degeneracy. Topological degeneracy can be used as qubits to implement robust quantum memories and fault-tolerant topological quantum computations.

[3] Xiao-Gang Wen, “Quantum orders and symmetric spin liquids”, Phys. Rev. B 65 165113 (2002). cond-mat/0107071

- Introduced projective symmetry group to describe symmetry enriched topological orders. Projective symmetry group describes the symmetry of the topological excitations, which can be viewed as a “fractionalized” symmetry of the parent system.

[4] Michael Levin, Xiao-Gang Wen, “Detecting Topological Order in a Ground State Wave Function”, Phys. Rev. Lett. 96 110405 (2006). cond-mat/0510613

- Used the universal (topological) entanglement entropy to detect topological orders, which revealed a connection of topological order with quantum entanglement.

[5] Xie Chen, Zheng-Cheng Gu, Xiao-Gang Wen, “Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order”, Phys. Rev. B 82 155138 (2010). arXiv:1004.3835

- Introduced the notion of long-range quantum entanglement via local unitary transformation. Long-range quantum entanglement is the essence of topological order and quantum order. Introduced the notion of symmetry enriched topological order as long-range entangled states with symmetry.

[6] Zheng-Cheng Gu, Xiao-Gang Wen, “Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order”, Phys. Rev. B 80 155131 (2009). arXiv:0903.1069

- Introduced the notion of symmtery protected trivial (SPT) order, to describe distinct short-range entangled phases without spontaneous symmetry breaking.

[7] Xie Chen, Zheng-Xin Liu, Xiao-Gang Wen, “Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations”, Phys. Rev. B 84 235141 (2011). arXiv:1106.4752

- Discovered the first example of 2+1D SPT order protected by Z_{2}on-site symmetry.

[8] Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, Xiao-Gang Wen, “Symmetry protected topological orders and the group cohomology of their symmetry group”, Phys. Rev. B 87 155114 (2013). arXiv:1106.4772

- Developed a systematic theory of bosonic SPT order in any dimensions, in terms of group cohomology theory. This led to the discovery of bosonic topological insulators/superconductors, and many other SPT phases.

[9] Su-Peng Kou, Michael Levin, Xiao-Gang Wen, “Mutual Chern-Simons theory for Z2 topological order”, Phys. Rev. B 78 155134 (2008). arXiv:0803.2300

[10] Su-Peng Kou, Xiao-Gang Wen, “Translation-symmetry-protected topological orders in quantum spin systems”, Phys. Rev. B 80 224406 (2009). arXiv:0907.4537

[11] Tian Lan, Liang Kong, Xiao-Gang Wen, “Classification of (2+1)-dimensional topological order and symmetry-protected topological order for bosonic and fermionic systems with on-site symmetries”, Phys. Rev. B 95 235140 (2017). arXiv:1602.05946

- Gapped quantum liquids (GQL) include both topologically orders (with long range entanglement) and SPT orders (with short range entanglement). We proposed that 2+1D bosonic/fermionic GQLs with a finite on-site symmetry are classified by non-degenerate unitary braided fusion categories over a symmetric fusion category E, together with their modular extensions and total chiral central charges. The symmetric fusion category E encodes the on-site symmetry and the underlying bosonic/fermionic nature of the lattice models.

[12] Zheng-Cheng Gu, Xiao-Gang Wen, “Symmetry-protected topological orders for interacting fermions – fermionic topological non-linear sigma-models and a group super-cohomology theory”, Phys. Rev. B 90 115141 (2014). arXiv:1201.2648

- Introduced a group super-cohomology theory to describe fermionic SPT order. This led to the discovery of new topological insulators/superconductors for interacting fermions, and many new fermionic SPT phases.

[13] Wenjie Ji, Xiao-Gang Wen, “Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions”, (2019). arXiv:1912.13492

- Discovered that one of the strongly correlated gapless states at the critical point of Landau symmetry breaking transition has an unbroken dual algebraic (n - 1)-symmetry G^{(n-1)}in n-dimensional space, in addition to the usual unbroken symmetry G. Symmetry and dual algebraic higher symmetry together form the larger categorical symmetry. We propose the emergence of categorical symmetry to be a general and essential feature of a critical point, and give examples to show that emergent maximal categorical symmetry determines the low energy dynamics of strongly correlated critical points. We also propose to use the categorical symmetry of a critical point to determine the AdS/CFT dual of the critical point.

[14] Liang Kong, Tian Lan, Xiao-Gang Wen, Zhi-Hao Zhang, Hao Zheng, “Algebraic higher symmetry and categorical symmetry – a holographic and entanglement view of symmetry ”, (2020). arXiv:2005.14178

- Generalized higher symmetry to algebraic higher symmetry which is beyond higher group. Found that non-invertible gravitational anomaly (i.e. topological order in one higher dimension) is a generaic and universal way to describe symmetries, higher symmetries, and the more general algebraic higher symmetries. To stress its connection of symmetry, non-invertible gravitational anomaly is referred to as categorical symmetry. Such a holographic point of view for symmetry led to a classification of SPT and SET orders with symmetry, higher symmetry, or algebraic higher symmetry, in any dimensions.

[15] Xiao-Gang Wen, Frank Wilczek, A. Zee, “Chiral spin states and superconductivity”, Phys. Rev. B 39 11413 (1989).

- Showed that a spin liquid that breaks time-reversal symmetry (chiral spin liquid) are described by Chern-Simons theory with emergent semions. This led to the discovery of topological order.

[16] Xiao-Gang Wen, “Gapless boundary excitations in the quantum Hall states and in the chiral spin states”, Phys. Rev. B 43 11025 (1991).

- Developed chiral Luttinger liquid theory to describe gapless edge states of 2+1D chiral topological orders (such as FQH states and chiral spin states).

[17] Xiao-Gang Wen, “Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall states”, Phys. Rev. B 41 12838 (1990).

- Predicted power-law tunneling I-V curve between FQH edges, which was observed by F. P. Milliken, C. P. Umbach and R. A. Webb; Solid State Comm. 97, 309, 1995; A. M. Chang, L. N. Pfeiffer, and K. W. West; Phys. Rev. Lett. 77, 2538, 1996

[18] Xiao-Gang Wen, “Mean-field theory of spin-liquid states with finite energy gap and topological orders”, Phys. Rev. B 44 2664 (1991).

- Discovered a Z_{2}spin liquid that realize the simplest Z_{2}topological order, which is described by an emergent Z_{2}gauge theory at low energies. Z_{2}topological order is also known as toric code.

[19] Xiao-Gang Wen, “Non-Abelian Statistics in the FQH states”, Phys. Rev. Lett. 66 802 (1991).

- One of the first papers that constructed non-Abelian FQH states with non-Abelian Statistics. The low energy effective theories are described by SU(N)_{k}Chern-Simons theories for any N and k.

[20] Michael Levin, Xiao-Gang Wen, “Fermions, strings, and gauge fields in lattice spin models”, Phys. Rev. B 67 245316 (2003). cond-mat/0302460

- Introduced statistics hopping algebra to show the emergence of fermions from bosonic lattice models in any dimensions, if the bosonic models have a “string condensation” (now known as a long-range entanglement). Showed that fermions always emerge together with gauge theory and emergent fermions always carry a non-trivial gauge charge. This led to the point of view that gauge theory and Fermi statistics are reflections of long-range entanglement.

[21] Michael A. Levin, Xiao-Gang Wen, “String-net condensation: A physical mechanism for topological phases”, Phys. Rev. B 71 045110 (2005). cond-mat/0404617

- Introduced the notion of string-net condensation, which led to a description and classification of 2+1D topological orders with gappable edge, in terms of spherical fusion category. It is a mechanism to make the ends of condensed string to have Fermi, fractional, or non-Abelian statistics.

[22] Michael Levin, Xiao-Gang Wen, “Colloquium: Photons and electrons as emergent phenomena”, Rev. Mod. Phys. 77 871 (2005). cond-mat/0407140

- String-net condensation provides a way to unify light and electrons, or more precisely, to unify gauge interaction and Fermi statistics.

[23] B. Blok, Xiao-Gang Wen, “Effective theories of Fractional Quantum Hall Effect: Hierarchical Construction”, Phys. Rev. B 42 8145 (1990).

- Used K-matrix (integer symmetric matrix with even or odd diagonals) to systematically describe Abelian FQH states.

[24] Xiao-Gang Wen, A. Zee, “Classification of Abelian quantum Hall states and matrix formulation of topological fluids”, Phys. Rev. B 46 2290 (1992).

- Used K-matrix to classify Abelian topological orders for bosons and fermions.

[25] Xiao-Gang Wen, “Projective construction of non-Abelian quantum Hall liquids”, Phys. Rev. B 60 8827 (1999). cond-mat/9811111

[26] Liang Kong, Xiao-Gang Wen, “Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions”, (2014). arXiv:1405.5858

- Developed a theory for topological orders and gravitational anomalies in any dimensions, using higher category theory. Proposed a holographic principle of topological order: boundary uniquely determines the bulk, which in turn implies that topological orders in different dimensions form a chain complex of commutative moniods (similar to differential forms). Many properties of topological orders are discussed.

[27] Xiao-Gang Wen, “Quantum order from string-net condensations and the origin of light and massless fermions”, Phys. Rev. D 68 065003 (2003). hep-th/0302201

- Constructed a lattice spin model that has emergent U(1)×SU(2)×SU(3) gauge theory coupling non-chirally to 4 families of massless fermions. The model also demonstrated how projective symmetry group protects the masslessness of the emergent fermions.

[28] Patrick A. Lee, Naoto Nagaosa, Xiao-Gang Wen, “Doping a Mott insulator: physics of high-temperature superconductivity”, Rev. Mod. Phys. 78 17 (2006). cond-mat/0410445

- A review of the SU(2) slave-boson theory of high temperature superconductors.

[29] Tiago C. Ribeiro, Xiao-Gang Wen, “New mean field theory of the tt’t”J model applied to the high-Tc superconductors”, Phys. Rev. Lett. 95 057001 (2005). cond-mat/0410750

- A dopon theory (a fully fermionic mean field theory) of high Tc superconductor. The doping is described by electrons or hole-like electrons, coupled to d-wave-paired (staggered-flux) phase of emergent fermions (the spinons) that describe the spin dynamics. A change from d-wave-paired phase to Fermi liquid phase for spinons in spin part can cause a change from Metal* ((small Fermi surface) to Metal (large Fermi surface)

[30] Xiao-Gang Wen, Zhenghan Wang, “Classification of symmetric polynomials of infinite variables: Construction of Abelian and non-Abelian quantum Hall states”, Phys. Rev. B 77 235108 (2008). arXiv:0801.3291

- Introduced pattern-of-zeros theory to systematically describe non-Abelian fractional quantum Hall states.

[31] Xiao-Gang Wen, Zhenghan Wang, “Topological properties of Abelian and non-Abelian quantum Hall states from the pattern of zeros”, Phys. Rev. B 78 155109 (2008). arXiv:0803.1016

- Introduced pattern-of-zeros theory to systematically describe non-Abelian fractional quantum Hall states.

[32] Xie Chen, Zheng-Cheng Gu, Xiao-Gang Wen, “Classification of Gapped Symmetric Phases in 1D Spin Systems”, Phys. Rev. B 83 035107 (2011). arXiv:1008.3745

- A classification of all gapped 1+1D quantum phases.

[33] Xie Chen, Zheng-Cheng Gu, Xiao-Gang Wen, “Complete classification of 1D gapped quantum phases in interacting spin systems”, Phys. Rev. B 84 235128 (2011). arXiv:1103.3323

- A classification of all gapped 1+1D quantum phases.

[34] Tian Lan, Liang Kong, Xiao-Gang Wen, “Classification of (3+1)D bosonic topological orders: the case when pointlike excitations are all bosons”, Phys. Rev. X 8 021074 (2018). arXiv:1704.04221

- Used pointed fusion 2-categories with trivial 1-morphism to classify 3+1D bosonic topological orders, that have only emergent bosonic point-like excitations. This allowed us to show that those 3+1D bosonic topological orders are all realized by Dijkgraaf-Witten models.

[35] Tian Lan, Xiao-Gang Wen, “Classification of 3+1D bosonic topological orders (II): the case when some pointlike excitations are fermions”, Phys. Rev. X 9 021005 (2019). arXiv:1801.08530

- Proposed a classification of 3+1D bosonic topological orders, that have emergent fermionic point-like excitations. Those topological orders can be devided into to two classes: EF1 where triple-string junctions can have no Majorana zero modes, and EF2 where some triple-string junctions must have Majorana zero modes. The EF1 topological orders are classified by pointed fusion 2-categories with a Z_{2}pointed 1-morphism. The EF2 topological orders are are classified by pointed fusion 2-categories with a Z_{2}pointed 1-morphism and a quantum dimension 2^{1∕2}1-morphism.

[36] Chenchang Zhu, Tian Lan, Xiao-Gang Wen, “Topological nonlinear σ-model, higher gauge theory, and a systematic construction of 3+1D topological orders for boson systems”, Phys. Rev. B 100 045105 (2019). arXiv:1808.09394

- Found lattice realization of all 3+1D bosonic topological orders. In particular, all the EF1 topological orders are realized by 2-gauge theories with π_{1}= a finite group and π_{2}= Z_{2}. This implies that a 3+1D discrete n-gauge theories is either dual to an 1-gauge theories (the Dijkgraaf-Witten models) or a 2-gauge theories of the above form.

[37] Xiao-Gang Wen, “Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders”, Phys. Rev. D 88 045013 (2013). arXiv:1303.1803

- A classification of perturbative and global symmetry/gravitational anomalies. Roughly, symmetry (t’ Hooft) anomalies = SPT orders in one higher dimension. Gravitational anomalies = topological orders in one higher dimension.

[38] Xiao-Gang Wen, “A Lattice Non-Perturbative Definition of an SO(10) Chiral Gauge Theory and Its Induced Standard Model”, Chinese Phys. Lett. 30 111101 (2013). arXiv:1305.1045

- Used the SPT approach to solve the long standing chiral fermion problem. In particular, found a 3+1D lattice realization of the chiral SO(10) grand unified theory (and the associated standard model). Pointed out that gapless field theory with certain global anomalies can be realized by lattice model in the same dimension.

[39] Tian Lan, Liang Kong, Xiao-Gang Wen, “Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries”, Phys. Rev. B. 94 155133 (2016). arXiv:1507.04673:

- Classification of 2+1D fermionic topological order via braided fusion catgeories over sVec and their modular extensions.

[40] Tian Lan, Liang Kong, Xiao-Gang Wen, “Classification of (2+1)-dimensional topological order and symmetry-protected topological order for bosonic and fermionic systems with on-site symmetries”, Phys. Rev. B. 95 235104 (2017). arXiv:1602.05946:

- Gapped quantum liquids (GQL) include both topologically orders (with long range entanglement) and SPT orders (with short range entanglement). We proposed that 2+1D bosonic/fermionic GQLs with a finite on-site symmetry are classified by non-degenerate unitary braided fusion categories over a symmetric fusion category $E$, together with their modular extensions and total chiral central charges. The symmetric fusion category $E$ encodes the on-site symmetry and the underlying bosonic/fermionic nature of the lattice models.