nLab topological order

This entry is about topological orders of materials in condensed matter physics. For topological orders of directed acyclic graphs in graph theory, see linear extension of a partial order.

---

Idea

In solid state physics, by topological order (Wen 89, Wen & Niu 90, Wen 91, 93, 95, Gu & Wen 09, p. 2, Chen, Gu & Wen 10) one refers to a phenomenon that may (but need not) be exhibited by quantum materials that are in a topological phase of matter. Hence there is an implication

$$\text{topological order} \;\;\;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\;\;\; \text{topological phase} \,,$$

but not the other way around.

Specifically, the ground state of a topologically ordered material is rich in structure (besides being topologically stable), which need not be the case for a general topological phase. In particular a topologically ordered ground state is “degenerate” (a standard but somewhat unfortunate jargon in this context: it really refers to the energy eigenvalues being degenerate) in that there is a $\geq 2$-dimensional Hilbert space of quantum states that all have the same lowest energy (within experimentally relevant approximation).

Moreover, for this to qualify as topologically ordered, one typically demands that in its degenerate ground state the system may exhibit “anyonic defects”. A popular more succinct way of making this somewhat more precise is to say that the dynamics of (the electrons in) a topologically ordered material, when restricted to the energy=0 ground states, is a topological field theory equal or akin to a Chern-Simons theory with Wilson line-insertions: these Wilson lines being the worldlines of the anyon-defects.

In short then, topological order is meant to be that aspect of topological phases of matter which is related to the existence of anyons in the material, in one way or other. Via this relation, topological order is closely related to considerations in topological quantum computation.

Via degenerate anyonic ground states

The definition is traditionally a little vague, but the hallmark of a topological order is meant to be the presence, in a topological phase, of some or preferably all of the following phenomena (Gu & Wen 09, p. 2, called the “modern day philosopher’s stone” in Tsvelik 2014b):

1. degenerate ground state (Wen 89, p. 4, Wen 95, Sec. 1.1);

2. Berry connection with non-abelian holonomy;

3. anyon excitations/defects whose wavefunctions constitute braid group representations (“fractional statistics”).

The original articles (Wen 89, Wen & Niu 90, Wen 91, 93, 95) proposed to declare that topological phases with distinct ground state degeneracy exhibit distinct topological order. The demand that also the Berry connection should be non-abelian for there to be a topological order seems to appear first in Wen 91 review (then also Gu & Wen 09, p. 2). Or maybe the claim is rather that two distinct topological orders may have the same ground state degeneracy but be distinguished by their Berry connections.

Remark

(relation between these conditions)
By “degenerate ground state” one just means that the sub-Hilbert space of quantum states (for crystalline materials these are the quantum states of electrons in the crystal lattice of atomic nuclei) of those that have the lowest possible energy, has dimension $\geq 2$.

(It has been proposed that the presence of “short-range entanglement” in quantum materials implies that the ground state Hilbert space is 1-dimensional. In this sense, the condition that the ground state be degenerate implies that there is no “short-range entanglement” if there is “topological order”.)

Since (by Bloch-Floquet theory) the quantum states of these electrons form the space of sections of the Bloch bundle, and since (by assumption of a topological phase) the ground state involves sections only of the gapped sub-bundle which is called the valence bundle, this is closely related to the valence bundle having rank $\geq 2$.

(In making these statement there is some tacit switching between single-electron theory and its second quantization involved, sorting out of which would be necessary for being more precise about these matters.)

But since the Berry connection is a connection on the valence bundle, it can have non-abelian holonomy (namely in U(n) for $n \geq 2$) only if the bundle’s rank is $\geq 2$.

Similarly, the presence of anyons broadly means that there are families of adiabatic defomrations of the material which have the effect of transforming the ground state by linear operators forming a braid group representation. This is most interesting when the representation is non-abelian, which again requires that the Hilbert space of ground states that it is represented on has dimension $\geq 2$.

However, there are also non-trivial “abelian anyons”, namely braid group representations which are just complex 1-dimensional, and – even if not as interesting as their non-abelian cousins – these are counted as perfectly valid instances of the concept of anyons (in fact it was first a struggle and then a breakthrough to detect abelian anyons in experiment). Some authors, especially those using the language of fusion categories to speak about anyons, require of a “topological order” only that it contains anyons and possibly just abelian anyons, in which case it is maybe not so clear whether the ground state needs to be degenerate and/or the Berry connection be non-abelian.

Via topological entanglement entropy

It is now thought that some/all of the above characteristics may be captured quantum information theoretically in terms of the entanglement entropy of the ground state:

The non-degeneracy of the ground state is related to absence of “short-range entanglement”; and the existence of anyon excitations is related to the presence of “long-range entanglement” (Chen, Gu & Wen 10, Sec. 5) witnessed by a non-vanishing topological entanglement entropy (Kitaev & Preskill 2006, Levin & Wen 2006).

Often one sees topological order being related also to (1) strong interaction and/or (2) strong quantum correlation (Wen 91 review) between the electrons (while the classical electron band theory that gives rise to the widely-accepted K-theory classification of topological phases of matter assumes that it is sensible to neglect the interaction of electrons between each other and only retain their interaction to an effective Coulomb background field).

Beware that the use of the term “correlation” in the context of topological order (cf. Wen 91 review) is always meant as “quantum correlation” and specifically as “non-classical quantum correlation” and as such used as a synonym for quantum entanglement (cf. ZCZW 19, §1.5 and generally Luo & Luo 03, p. 3). In contrast, long-range classical correlation is indicative of non-topological Landau theory-phases and hence the opposite of what is relevant here.

Together with the entanglement-theoretic characterization just mentioned, the logic here seems to be the following sequence of schematic implications:

(diagram adapted from SS22)

The first steps in this sequence is intuitively plausible and widely expected to hold (eg. Lu & Viyah 2022, p. 1) but not currently derivable from first principles (Zaanen, Liu, Sun & Schalm 2015, p. 527):

If the Coulomb interaction between the electrons – which by itself is certainly strong and long-range – cannot be neglected (hence if the averaging- or screening-effects that make electron band theory work do not apply) then this strong interaction makes the electrons in the ground state be correlated with each other, one way or other, across non-negligible distances; and quantum mechanically this leads to the ground state’s entanglement entropy having long-range contributions, which, essentially by definition, means that it has a constant contribution by the topological entanglement entropy.

Examples

under construction

• quantum Hall effect

• non-Abelian quantum Hall state (see Wikipedia)

• chiral spin liquid? (see Wikipedia)

• Z2 spin liquid? (see Wikipedia)

Related concepts

• topological state of matter,

  • quantum Hall effect

• TQFT, Chern-Simons theory

• string-net model

• quantum computing

• modular tensor category

• entanglement

Literature

Original articles

The proposal that ground state degeneracy is a signature of “topological order”:

• Xiao-Gang Wen, Vacuum degeneracy of chiral spin states in compactified space, Phys. Rev. B 40 (1989) 7387(R) $[$doi:10.1103/PhysRevB.40.7387$]$

• 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 (1990) 9377 $[$doi:10.1103/PhysRevB.41.9377$]$

• Xiao-Gang Wen, Non-Abelian statistics in the fractional quantum Hall states, Phys. Rev. Lett. 66 (1991) 802 [doi:10.1103/PhysRevLett.66.802, pdf]

• Xiao-Gang Wen, Topological order and edge structure of $\nu = 1/2$ quantum Hall state, Phys. Rev. Lett. 70 (1993) 355 $[$doi:10.1103/PhysRevLett.70.355$]$

• Xiao-Gang Wen, Topological orders and Edge excitations in FQH states, Advances in Physics 44 (1995) 405 $[$arXiv:cond-mat/9506066v2, doi:10.1080/00018739500101566$]$

The additional requirement that the Berry connection be non-abelian (and/or the presence of anyons):

• Zheng-Cheng Gu, Xiao-Gang Wen, p. 2 of: Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order, Phys. Rev. B 80 155131 (2009) $[$arXiv:0903.1069, doi:10.1103/PhysRevB.80.155131$]$

Suggestion that topological order goes along with long-range entanglement:

• Xie Chen, Zheng-Cheng Gu, Xiao-Gang Wen, Section V of: Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order Phys. Rev. B 82 155138 (2010) $[$arXiv:1004.3835, doi:10.1103/PhysRevB.82.155138$]$

Review

• Xiao-Gang Wen, Topological orders and Chern-Simons theory in strongly correlated quantum liquid, International Journal of Modern Physics B 05 10 (1991) 1641-1648 $[$doi:10.1142/S0217979291001541$]$

Textbook accounts:

• Bei Zeng, Xie Chen, Duan-Lu Zhou, Xiao-Gang Wen:

  Part III of: Quantum Information Meets Quantum Matter – From Quantum Entanglement to Topological Phases of Many-Body Systems, Quantum Science and Technology (QST), Springer (2019) $[$arXiv:1508.02595, doi:10.1007/978-1-4939-9084-9$]$

• Tudor D. Stanescu, Section 6.2 of: Introduction to Topological Quantum Matter & Quantum Computation, CRC Press 2020 (ISBN:9780367574116)

Further review:

• Chetan Nayak, Steven H. Simon, Ady Stern, M. Freedman, Sankar Das Sarma, Non-Abelian anyons and topological quantum computation, Rev Mod Phys 80:3 (Aug 2008) 1083–1159 MR2009g:81041 doi

• Xiao-Gang Wen, An introduction of topological order 2009 (pdf slides, article)

• Michel Fruchart, David Carpentier, An Introduction to Topological Insulators, Comptes Rendus Physique 14 (2013) 779-815 (arXiv:1310.0255)

• Philip Ball, Making the world from topological order, National Science Review 6 2 (2019) 227–230 $[$doi:10.1093/nsr/nwy116$]$

  (an interview with Xiao-Gang Wen)

See also:

• Wikipedia: topological order

Early discovery articles

• Xiao-Gang Wen, Vacuum Degeneracy of Chiral Spin State in Compactified Spaces, Phys. Rev. B, 40, 7387 (1989).

• Xiao-Gang Wen, Topological Orders in Rigid States, Int. J. Mod. Phys. B4, 239 (1990)

• Xiao-Gang Wen and Qian Niu, Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces, Phys. Rev. B41, 9377 (1990)

• E. Keski-Vakkuri and Xiao-Gang Wen, Ground state structure of hierarchical QH states on torus and modular transformation Int. J. Mod. Phys. B7, 4227 (1993).

Topologically ordered materials

On semimetals with degenerate ground states:

• Huaiming Guo, Yongfei Jia, Interaction-driven phases in a Dirac Semimetal: Exact Diagonalization Results, J. Phys.: Condens. Matter 26 475601 (2014) [arXiv:1402.4274, doi:10.1088/0953-8984/26/47/475601]

Other articles

On detection of topological order by observing modular transformations on the ground state:

• Zhuan Li, Roger S. K. Mong, Detecting topological order from modular transformations of ground states on the torus, Phys. Rev. B 106 (2022) 235115 [doi:10.1103/PhysRevB.106.235115, arXiv:2203.04329]

See also:

• Davide Gaiotto, Anton Kapustin, Spin TQFTs and fermionic phases of matter, arxiv/1505.05856

• Nicholas Read, Subir Sachdev, Large-$N$ expansion for frustrated quantum antiferromagnets, Phys. Rev. Lett. 66 1773 (1991) (on $Z_2$ topological order)

• Xiao-Gang Wen, Mean Field Theory of Spin Liquid States with Finite Energy Gap and Topological orders, Phys. Rev. B 44 2664 (1991). (on $Z_2$ topological order)

• Xiao-Gang Wen, Non-Abelian Statistics in the FQH states

  Phys. Rev. Lett. 66, 802 (1991).

• Moore, Gregory; Read, Nicholas. Nonabelions in the fractional quantum hall effect Nuclear Physics B 360 (2–3): 362 (1991).

• Xiao-Gang Wen and Yong-Shi Wu, Chiral operator product algebra hidden in certain FQH states

  Nucl. Phys. B419, 455 (1994).

• Alexei Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303:1, January 2003; Anyons in an exactly solved model and beyond, Annals of Physics 321:1, January 2006

• Michael Levin, Xiao-Gang Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B, 71, 045110 (2005).

• A. Kitaev, C. Laumann, Topological phases and quantum computation, arXiv/0904.2771

• Alexei Kitaev, John Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404 (2006)

• Levin M. and Wen X-G., Detecting topological order in a ground state wave function, Phys. Rev. Letts.,96(11), 110405, (2006)

• Jan Carl Budich, Björn Trauzettel, From the adiabatic theorem of quantum mechanics to topological states of matter, physica status solidi (RRL) 7, 109 (2013) arXiv:1210.6672

• Kumar S. Gupta, Amilcar Queiroz, Anomalies and renormalization of impure states in quantum theories, arxiv/1306.5570

• Frank Wilczek, Anthony Zee, Appearance of gauge structure in simple dynamical systems, Physical Review Letters 52 24 (1984) 2111 $[$doi:10.1103/PhysRevLett.52.2111$]$

• Amit Jamadagni, Hendrik Weimer, An Operational Definition of Topological Order (arXiv:2005.06501)

• Nathanan Tantivasadakarn, Ashvin Vishwanath, Ruben Verresen, A hierarchy of topological order from finite-depth unitaries, measurement and feedforward [arXiv:2209.06202]

Discussion via twisted equivariant K-theory:

• Hisham Sati, Urs Schreiber: Anyonic topological order in TED K-theory [arXiv:2206.13563]

Topological entanglement entropy

General

Identification of a contribution to entanglement entropy at absolute zero which is independent of the subsystem‘s size (“topological entanglement entropy”, “long-range entanglement”), reflecting topological order and proportional to the total quantum dimension of anyon excitations:

• Alexei Kitaev, John Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 (arXiv:hep-th/0510092)

• Michael Levin, Xiao-Gang Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett., 96, 110405 (2006) $[$arXiv:cond-mat/0510613, doi:10.1103/PhysRevLett.96.110405$]$

  (in view of string-net models)

Review:

• Shunsuke Furukawa, Entanglement Entropy in Conventional and Topological Orders, talk at Topological Aspects of Solid State Physics 2008 (pdf, pdf)

• Tarun Grover, Entanglement entropy and strongly correlated topological matter, Modern Physics Letters A 28 05 (2013) 1330001 $[$doi:10.1142/S0217732313300012$]$

• Bei Zeng, Xie Chen, Duan-Lu Zhou, Xiao-Gang Wen:

  Sec. 5 of: Quantum Information Meets Quantum Matter – From Quantum Entanglement to Topological Phases of Many-Body Systems, Quantum Science and Technology (QST), Springer (2019) $[$arXiv:1508.02595, doi:10.1007/978-1-4939-9084-9$]$

In terms of Renyi entropy (it’s independent of the Renyi entropy parameter):

• Ulrich Schollwöck, (Almost) 25 Years of DMRG - What Is It About? (pdf)

and in the example of Chern-Simons theory:

• Aditya Dwivedi, Siddharth Dwivedi, Bhabani Prasad Mandal, Pichai Ramadevi, Vivek Kumar Singh, Topological entanglement and hyperbolic volume, J. High Energy Phys. 21021 172 (2021) 172 [arXiv:2106.03396, doi:10.1007/JHEP10(2021)172]

Discussion in the dimer model:

• Shunsuke Furukawa, Gregoire Misguich, Topological Entanglement Entropy in the Quantum Dimer Model on the Triangular Lattice, Phys. Rev. B 75, 214407 (2007) (arXiv:cond-mat/0612227)

Discussion via holographic entanglement entropy:

• Ari Pakman, Andrei Parnachev, Topological Entanglement Entropy and Holography, JHEP 0807: 097 (2008) (arXiv:0805.1891)

• Andrei Parnachev, Napat Poovuttikul, Topological Entanglement Entropy, Ground State Degeneracy and Holography, Journal of High Energy Physics volume 2015, Article number: 92 (2015) (arXiv:1504.08244)

See also:

• Tatsuma Nishioka, Tadashi Takayanagi, Yusuke Taki, Topological pseudo entropy (arXiv:2107.01797)

Relation to strong interaction

Relation of long-range entanglement to strong interaction:

• Jan Zaanen, Yan Liu, Ya-Wen Sun, Koenraad Schalm, Holographic Duality in Condensed Matter Physics, Cambridge University Press 2015 $[$doi:10.1017/CBO9781139942492$]$

  In a way it appears obvious that the strongly interacting bosonic quantum critical state is subject to long-range entanglement. Nonetheless, the status of this claim is conjectural.

  It is at present impossible to arrive at more solid conclusions that are based on rigorous mathematical procedures. It does illustrate emphatically the central challenge in the pursuit of field-theoretical quantum information: there are as yet not general measures available to precisely enumerate the meaning of long-range entanglement in such seriously quantum field-theoretical systems. $[$p. 527$]$

• Tsung-Cheng Lu, Sagar Vijay, Characterizing Long-Range Entanglement in a Mixed State Through an Emergent Order on the Entangling Surface $[$arXiv:2201.07792$]$

  strongly interacting quantum phases of matter at zero temperature can exhibit universal patterns of long-range entanglement

Characterizing topological order

On characterizing anyon braiding / modular transformations on topologically ordered ground states by analysis of (topological) entanglement entropy of subregions:

• Yi Zhang, Tarun Grover, Ari M. Turner, Masaki Oshikawa, Ashvin Vishwanath, Quasiparticle statistics and braiding from ground-state entanglement, Phys. Rev. B 85 (2012) 235151 $[$doi:10.1103/PhysRevB.85.235151$]$

• Yi Zhang, Tarun Grover, Ashvin Vishwanath, General procedure for determining braiding and statistics of anyons using entanglement interferometry, Phys. Rev. B 91 (2015) 035127 $[$arXiv:1412.0677, doi:10.1103/PhysRevB.91.035127$]$

• Zhuan Li, Roger S. K. Mong, Detecting topological order from modular transformations of ground states on the torus, Phys. Rev. B 106 (2022) 235115 [doi:10.1103/PhysRevB.106.235115, arXiv:2203.04329]

Simulation and experiment

Experimental observation:

• A. Hamma, W. Zhang, S. Haas, and D. A. Lidar, Entanglement, fidelity, and topological entropy in a quantum phase transition to topological order, Phys. Rev. B 77, 155111 (2008) (doi:10.1103/PhysRevB.77.155111, arXiv:0705.0026)

• Hong-Chen Jiang, Zhenghan Wang, Leon Balents, Identifying Topological Order by Entanglement Entropy, Nature Physics 8 902-905 (2012) $[$arXiv:1205.4289$]$

Detection of long-range entanglement entropy in quantum simulations on quantum computers:

• Realizing topologically ordered states on a quantum processor, Science 374 6572 (2021) 1237-1241 $[$doi:10.1126/science.abi8378$]$

• Probing topological spin liquids on a programmable quantum simulator, Science 374 6572 (2021) 1242-1247 $[$doi:10.1126/science.abi8794$]$

exposition in:

• Isabelle Dumé, Long-range quantum entanglement measured at last, PhysicsWorld (Jan 2022)

Anyonic topological order in terms of braided fusion categories

Claim and status

In condensed matter theory it is folklore that species of anyonic topological order correspond to braided unitary fusion categories/modular tensor categories.

The origin of the claim is:

• Alexei Kitaev, Section 8 and Appendix E of: Anyons in an exactly solved model and beyond, Annals of Physics 321 1 (2006) 2-111 $[$doi:10.1016/j.aop.2005.10.005$]$

Early accounts re-stating this claim (without attribution):

• Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, Sankar Das Sarma, pp. 28 of: Non-Abelian Anyons and Topological Quantum Computation, Rev. Mod. Phys. 80 1083 (2008) $[$arXiv:0707.1888, doi:10.1103/RevModPhys.80.1083$]$

• Zhenghan Wang, Section 6.3 of: Topological Quantum Computation, CBMS Regional Conference Series in Mathematics 112, AMS (2010) $[$ISBN-13: 978-0-8218-4930-9, pdf$]$

Further discussion (mostly review and mostly without attribution):

• Simon Burton, A Short Guide to Anyons and Modular Functors $[$arXiv:1610.05384$]$

  (this one stands out as still attributing the claim to Kitaev (2006), Appendix E)

• Eric C. Rowell, Zhenghan Wang, Mathematics of Topological Quantum Computing, Bull. Amer. Math. Soc. 55 (2018), 183-238 (arXiv:1705.06206, doi:10.1090/bull/1605)

• Tian Lan, A Classification of (2+1)D Topological Phases with Symmetries $[$arXiv:1801.01210$]$

• From categories to anyons: a travelogue $[$arXiv:1811.06670$]$

• Colleen Delaney, A categorical perspective on symmetry, topological order, and quantum information (2019) $[$pdf, uc:5z384290$]$

• Colleen Delaney, Lecture notes on modular tensor categories and braid group representations (2019) $[$pdf, pdf$]$

• Liang Wang, Zhenghan Wang, In and around Abelian anyon models, J. Phys. A: Math. Theor. 53 505203 (2020) $[$doi:10.1088/1751-8121/abc6c0$]$

• Parsa Bonderson, Measuring Topological Order, Phys. Rev. Research 3, 033110 (2021) $[$arXiv:2102.05677, doi:10.1103/PhysRevResearch.3.033110$]$

• Zhuan Li, Roger S.K. Mong, Detecting topological order from modular transformations of ground states on the torus $[$arXiv:2203.04329$]$

• Eric C. Rowell, Braids, Motions and Topological Quantum Computing $[$arXiv:2208.11762$]$

• Sachin Valera, A Quick Introduction to the Algebraic Theory of Anyons, talk at CQTS Initial Researcher Meeting (Sep 2022) $[$pdf$]$

• Willie Aboumrad, Quantum computing with anyons: an F-matrix and braid calculator $[$arXiv:2212.00831$]$

Emphasis that the expected description of anyons by braided fusion categories had remained folklore, together with a list of minimal assumptions that would need to be shown:

• Sachin J. Valera, Fusion Structure from Exchange Symmetry in (2+1)-Dimensions, Annals of Physics 429 (2021) $[$

  arXiv:2004.06282, doi:10.1016/j.aop.2021.168471$]$

An argument that the statement at least for SU(2)-anyons does follow from an enhancement of the K-theory classification of topological phases of matter to interacting topological order:

• Hisham Sati, Urs Schreiber, Anyonic topological order in TED K-theory, Rev. Math. Phys. (20223) $[$arXiv:2206.13563, doi:10.1142/S0129055X23500010$]$

Further discussion

Relation to ZX-calculus:

• Fatimah Rita Ahmadi, Aleks Kissinger, Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics $[$arXiv:2211.03855$]$

On detection of topological order by observing modular transformations on the ground state:

• Zhuan Li, Roger S. K. Mong, Detecting topological order from modular transformations of ground states on the torus, Phys. Rev. B 106 (2022) 235115 $[$doi:10.1103/PhysRevB.106.235115, arXiv:2203.04329$]$

See also:

• Liang Kong, Topological Wick Rotation and Holographic Dualities, talk at CQTS (Oct 2022) $[$pdf$]$

Further resources

Research groups:

• Univ. of Maryland, joint quantum institute

• Topology and correlation in condensed matter (Germany)

• Center for topological matter (Korea)

• Microsoft Research Station Q (KITP, Santa Barbara)

Conference and seminar cycles:

• seminar in Koeln Topological states of matter

• Topological Phases of Matter: Simons Center, June 10-14, 2013, videos available

• CECAM 2013, Topological Phases in Condensed Matter and Cold Atom Systems: towards quantum computations description