General theory of topological phases of matter with symmetry

Many major discoveries in the past 40 years in condensed matter physics centered on the experimental or theoretical discoveries of quantized topological invariants that can distinguish quantum phases of matter. For example, the Chern number, which sets the quantized Hall conductivity in integer and fractional quantum Hall systems, or various Z/2 invariants that can distinguish topological insulators and superconductors.  

This raises the question: How do we understand the complete set of possible topological invariants that can describe a gapped quantum phase of matter? What theory should we use to systematically characterize such systems? This question has been answered in the context of free fermion systems, where interactions are completely ignored, through the development of topological band theory. But how do we understand systems in the presence of interactions? Over the past decade, my collaborators and I have provided an answer to this question for topological phases of matter in two spatial dimensions. 

In 2014, my collaborators and I wrote down a precise set of algebraic data and consistency conditions, which define a G-crossed braided tensor category, and which provides the basic mathematical framework to characterize and classify (2+1)D topological phases of matter:

• M. Barkeshli, P. Bonderson, M. Cheng, Z. Wang, Symmetry Fractionalization, Defects, and Gauging of Topological Phases, arXiv:1410.4540, Phys. Rev. B 100, 115147 (2019)

Just like group theory is used to characterize crystals based on their space group symmetries, G-crossed braided tensor categories can be used to characterize topological phases of matter with symmetry in two spatial dimensions.  

Our original theory was developed for bosonic topological phases of matter; that is, phases of matter where the constituent degrees of freedom are bosons. Recently, we have understood how to generalize the theory to describe fermionic topological phases as well:

• D. Bulmash, M. Barkeshli, Fermionic symmetry fractionalization in (2+1)D, arXiv:2109.10913

• M. Barkeshli, Y.-A. Chen, P.-S. Hsin, N. Manjunath, Classification of (2+1)D invertible fermionic topological phases with symmetry, arXiv:2109.11039

In particular, invertible topological phases of matter, like integer quantum Hall states and topological insulators, are a special class of topological phases of matter that do not contain any topologically non-trivial excitations (that is, no anyons).  40 years since the experimental discovery of the integer quantum Hall effect, we finally understand how to systematically characterize and classify symmetric fermionic invertible topological phases of matter, by simply solving a set of consistency equations. 

Anomalies of topological phases of matter and 4-manifold invariants

One of the most important phenomena in quantum field theory is that of a quantum anomaly ('t Hooft anomaly to be specific). The reason is that anomalies, along with symmetry, are two properties of a quantum system that remain invariant no matter what energy or length scale we study the system at. The technical statement is that anomalies are invariant under renormalization group flow. For continuous symmetries, anomalies can often be understood using perturbation theory. However for discrete symmetries, anomalies necessarily require non-perturbative techniques. 

In contemporary physics, anomalies in d dimensions are understood to be equivalent invertible topological phases of matter in d+1 dimensions. The reason is that a system in d dimensions with an anomaly cannot exist on its own purely in d dimensions, but can exist at the boundary of a d+1 dimensional invertible topological phase of matter. Therefore, given a description of a d-dimensional quantum field theory, computing the anomaly means finding a way to extract a d+1 dimensional invertible topological phase. 

One important reason that we are interested in anomalies in condensed matter physics is that in the last few years we understood that the celebrated Lieb-Schultz-Mattis theorem (which goes back to the 1960s), is closely related to the physics of anomalies. In particular, a condensed matter system with a non-trivial LSM constraint has a mixed anomaly between translation symmetry and the on-site symmetry. This was first observed in a paper I wrote with my collaborators, where we used this insight to significantly strengthen the LSM theorem and its higher-dimensional generalizations:

• M. Cheng, M. Zaletel, M. Barkeshli, A. Vishwanath, P. Bonderson, Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface, arXiv:1511.02263, Phys. Rev. X 6, 041068 (2016)

In the past few years my collaborators and I have been pioneering techniques to systematically compute anomalies for general symmetry groups in (2+1)D topological quantum field theories:

• M. Barkeshli, P. Bonderson, M. Cheng, Z. Wang, Symmetry Fractionalization, Defects, and Gauging of Topological Phases, arXiv:1410.4540, 

Phys. Rev. B 100, 115147 (2019)

• M. Barkeshli, P. Bonderson, M. Cheng, C.-M. Jian, K. Walker, Reflection and time reversal symmetry enriched  topological phases of matter: path integrals, non-orientable manifolds, and anomalies, arXiv:1612.07792, Communications in Mathematical Physics volume 374, pages1021–1124 (2020)

• M. Barkeshli and M. Cheng, Time-reversal and spatial reflection symmetry localization anomalies in (2+1)D topological phases of matter, arXiv:1706.09464, 

Phys. Rev. B 98, 115129 (2018)

• M. Barkeshli and M. Cheng, Relative anomalies in (2+1)D symmetry enriched topological states, arXiv:1906.10691, 

SciPost Phys. 8, 028 (2020)

• D. Bulmash and M. Barkeshli, Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions, arXiv:2003.11553, 

Phys. Rev. Research 2, 043033 (2020)

• S. Tata, R. Kobayashi, D. Bulmash, and M. Barkeshli, Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTs, arXiv:2104.14567  

• D. Bulmash, M. Barkeshli, Anomaly cascade in (2+1)D fermionic topological phases, arXiv:2109.10922

• R. Kobayashi and M. Barkeshli, (3+1)D path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)D topological phases, arXiv:2111.14827

One of the interesting byproducts of our above results is that we are naturally led to some new constructions of topological invariants of 4-manifolds defined on principal G bundles, with or without spin structures. In particular, in Tata et. al. , we even found explicit topologically invariant combinatorial state sums that can distinguish 4-manifolds that are homeomorphic but not diffeomorphic (e.g. real vs. fake RP4). In Kobayashi et. al, we found new combinatorial state sums for invariants of topologically non-trivial, curved U(1) bundles over 4-manifolds, including spin^c structures. 

Classifying crystalline topological states and quantized crystalline response theories

My student Naren Manjunath and I have been using these theories to provide systematic classifications of topological phases, like fractional Chern insulators, in the presence of crystalline space group symmetries. Our work has led to predictions of several new kinds of topological invariants and fractional quantized responses:

• Naren Manjunath and Maissam Barkeshli, Classification of fractional quantum Hall states with spatial symmetries, arXiv:2012.11603

• Naren Manjunath and Maissam Barkeshli, Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry, arXiv:2005.10265, Phys. Rev. Research 3, 013040 (2021)

We are currently studying the predictions of our theory through numerical calculations of concrete microscopic models. 

Extracting topological invariants from many-body wave functions

Suppose we are given a ground state wave function of an interacting many-body topological phase of matter. Is it possible to extract all possible topological invariants that fully characterize the phase of matter, entirely from this ground state wave function? More specifically, suppose we have access to the equal-time correlators of a ground state wave function defined on a simply connected patch of space. Can we, without any knowledge of the underlying Hamiltonian, extract all of the topological invariants of the phase? 

I have recently been tackling this research direction. We have so far written two papers:

• H. Dehghani, Z.-P. Cian, M. Hafezi, M. Barkeshli, Extraction of many-body Chern number from a single wave function, arXiv:2005.13677, 

Phys. Rev. B 103, 075102 (2021)

• Z.-P. Cian, H. Dehghani, A. Elben, B. Vermersch, G. Zhu, M. Barkeshli, P. Zoller, M. Hafezi, Many-body Chern number from statistical correlations of randomized measurements, arxiv:2005.13543, Phys. Rev. Lett. 126, 050501 (2021)

In the first paper, we showed how the many-body Chern number can be extracted from a single ground state wave function. In the second paper, we then designed a protocol to extract the Chern number through the statistical correlations of randomized measurements of a quantum state, without requiring any knowledge of any Hamiltonian. 

Twist defects and topological domain walls in topologically ordered states

• M. Barkeshli and X.-G. Wen, [U(1) x U(1)] \rtimes Z2 Chern-Simons Theory and Z4 Parafermion Fractional Quantum Hall States, arXiv:0909.4882

Physical Review B 81, 045323 (2010)

• M. Barkeshli and Xiao-Liang Qi, Topological Nematic States and Non-Abelian Lattice Dislocations, arXiv:1112.3311, 

Phys. Rev. X 2, 031013 (2012)

• M. Barkeshli, C.M. Jian, and X.-L. Qi, Twist defects and projective non-abelian statistics, arXiv:1208.4834 , 

Phys. Rev. B 87, 045130 (2013) 

• M. Barkeshli, C.M. Jian, and X.-L. Qi, Theory of defects in Abelian topological states, arXiv:1305.7203, 

Phys. Rev. B 88, 235103 (2013)

• M. Barkeshli, C.M. Jian, and X.-L. Qi, Classification of Topological Defects in Abelian Topological States, arXiv:1304.7579,

       Phys. Rev. B 88, 241103(R) (2013)

• M. Barkeshli and X.-L. Qi, Synthetic Topological Qubits in Conventional Bilayer Quantum Hall Systems, arXiv:1302.2673,

Phys. Rev. X 4, 041035 (2014)

• M. Barkeshli, H.-C. Jiang, R. Thomale, X.-L. Qi, Generalized Kitaev Models and Slave Genons, arXiv:1405.1780,

 Phys. Rev. Lett. 114, 026401 (2015)

• A. Vaezi, M. Barkeshli, Fibonacci Anyons From Abelian Bilayer Quantum Hall States, arXiv:1403.3383, 

Phys. Rev. Lett. 113, 236804 (2014)

• M. Barkeshli, Erez Berg, Steven Kivelson, Coherent Transmutation of Electrons into Fractionalized Anyons, arXiv:1402.6321, 

Science, 346 6210 (2014)

• M. Barkeshli, Yuval Oreg, X.-L. Qi, Experimental Proposal to Detect Topological Ground State Degeneracy, arXiv:1401.3750

• M. Barkeshli, P. Bonderson, M. Cheng, Z. Wang, Symmetry Fractionalization, Defects, and Gauging of Topological Phases, arXiv:1410.4540, Phys. Rev. B 100, 115147 (2019)

• M. Barkeshli, C. Nayak, Superconductivity Induced Topological Phase Transition at the Edge of Even Denominator Fractional Quantum Hall States, arXiv:1507.06305

• M. Barkeshli, Charge 2e/3 superconductivity and topological degeneracies without localized zero modes in bilayer fractional quantum Hall states, arXiv:1604.00683, 

Phys. Rev. Lett. 117, 096803 (2016) 

• J. Cano, M. Cheng, M. Barkeshli, D. J. Clarke, C. Nayak, Chirality-Protected Majorana Zero Modes at the Gapless Edge of Abelian Quantum Hall States, arXiv:1505.07825, Phys. Rev. B 92, 195152 (2015) 

Quantum criticality

• Yahya Alavirad and M. Barkeshli, Anomalies and unnatural stability of multi-component Luttinger liquids in Zn x Zn spin chains, arXiv:1910.00589, Phys. Rev. B 104, 045151 (2021)

• M. Barkeshli, N.Y. Yao, C.R. Laumann, Continuous Preparation of a Fractional Chern Insulator, arXiv:1407.7034, 

Phys. Rev. Lett. 115, 026802 (2015) 

• M. Barkeshli, Transitions Between Chiral Spin Liquids and Z2 Spin Liquids, arXiv:1307.8194

• M. Barkeshli and John McGreevy, Continuous transitions between composite Fermi liquid and Landau Fermi liquid: a route to fractionalized Mott insulators, arXiv:1206.6530, 

Phys. Rev. B 86, 075136 (2012) 

• M. Barkeshli and John McGreevy, Continuous transition between fractional quantum Hall and superfluid states, arXiv:1201.4393, 

Phys. Rev. B 89, 235116 (2014)

• M. Barkeshli and X.-G. Wen, Anyon Condensation and Topological Phase Transitions in Non-Abelian States, arXiv:1007.2030,

Phys. Rev. Lett. 105, 216804 (2010)

•  M. Barkeshli and X.-G. Wen, Phase transitions in Zn gauge theory and twisted Zn topological phases, arXiv:1012.2417

Phys. Rev. B 86, 085114 (2012)

• M. Barkeshli and X.-G. Wen, Bilayer quantum Hall phase transitions and the orbifold non-Abelian fractional quantum Hall states,  arXiv:1010.4270

Quantum error correction and fault-tolerant quantum computation 

• A. Lavasani, G. Zhu, M. Barkeshli, Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes, arXiv:1901.11029, 

Quantum 3, 180 (2019)

• G. Zhu, A. Lavasani, M. Barkeshli, Instantaneous braids and Dehn twists in topologically ordered states, arXiv:1806.06078

Phys. Rev. B 102, 075105 (2020)

• G. Zhu, A. Lavasani, M. Barkeshli, Universal Logical Gates on Topologically Encoded Qubits via Constant-Depth Unitary Circuits, arXiv:1806.02358 

Phys. Rev. Lett. 125, 050502 (2020)

JQI News article on our work above: https://jqi.umd.edu/news/quantum-computers-do-instantaneous-twist

• A. Lavasani, M. Barkeshli, Low overhead Clifford gates from joint measurements in surface, color, and hyperbolic codes, arXiv:1804.04144, 

Phys. Rev. A 98, 052319 (2018)

• G. Zhu, M. Hafezi, M. Barkeshli, Quantum Origami: Transversal Gates for Quantum Computation and Measurement of Topological Order, arXiv:1711.05752, 

Phys. Rev. Research 2, 013285 (2020)

• Maissam Barkeshli and Michael Freedman, Modular transformations through sequences of topological charge projections, arXiv:1602.01093, 

Phys. Rev. B 94, 165108 (2016)

• M. Barkeshli, J.D. Sau, Physical Architecture for a Universal Topological Quantum Computer based on a Network of Majorana Nanowires, arXiv:1509.07135 

• M. Barkeshli and X.-L. Qi, Synthetic Topological Qubits in Conventional Bilayer Quantum Hall Systems, arXiv:1302.2673,

Phys. Rev. X 4, 041035 (2014)

Entanglement transitions in monitored random quantum circuits 

• Ali Lavasani, Yahya Alavirad, Maissam Barkeshli, Topological order and criticality in (2+1)D monitored random quantum

 circuits, arXiv:2011.06595,  Phys. Rev. Lett.  127, 235701 (2021)

• Ali Lavasani, Yahya Alavirad, Maissam Barkeshli, Measurement-induced topological entanglement transitions in symmetric random quantum circuits, arXiv:2004.07243, 

Nature Physics 17, 342 - 347 (2021)

A "News and Views" article in Nature Physics on our work:

Brayden Ware and Romain Vasseur, Measurements make the phase, Nature Physics 17 298-299 (2021)

Quantum spin liquids 

• M. Barkeshli, Erez Berg, Steven Kivelson, Coherent Transmutation of Electrons into Fractionalized Anyons, arXiv:1402.6321, 

Science, 346 6210 (2014)

• M. Barkeshli, H.-C. Jiang, R. Thomale, X.-L. Qi, Generalized Kitaev Models and Slave Genons, arXiv:1405.1780, Phys. Rev. Lett. 114, 026401 (2015)

• M. Barkeshli, Transitions Between Chiral Spin Liquids and Z2 Spin Liquids, arXiv:1307.8194

• M. Barkeshli, Hong Yao, and Steven A. Kivelson, Gapless Spin Liquids: Stability and Possible Experimental Relevance, arXiv:1208.3869, 

Phys. Rev. B 87, 140402(R) (2013)

• A.C. Potter, M. Barkeshli, J. McGreevy, and T. Senthil, Quantum spin liquids and the metal-insulator transition in doped semiconductors, arXiv:1204.1342, Phys. Rev. Lett. 109, 077205 (2012)

Fractons

• D. Bulmash and M. Barkeshli, Gauging fractons: immobile non-Abelian quasiparticles, fractals, and position-dependent degeneracies, arXiv:1905.05771, 

Phys. Rev. B 100, 155146 (2019)

• D. Bulmash, M. Barkeshli, Generalized U(1) Gauge Field Theories and Fractal Dynamics, arXiv:1806.01855

• D. Bulmash, M. Barkeshli, The Higgs Mechanism in Higher-Rank Symmetric U(1) Gauge Theories,

arXiv:1802.10099, Phys. Rev. B 97, 235112 (2018)