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.
Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.
General theory:
In metamaterials:
topological phononics (sound waves?)
For quantum computation:
basics
Examples
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
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.
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):
degenerate ground state (Wen 89, p. 4, Wen 95, Sec. 1.1);
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.
(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.
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.
under construction
chiral?spin liquid (see Wikipedia)
$\mathbb{Z}/2$?spin liquid (see Wikipedia)
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 [doi:10.1080/00018739500101566, arXiv:cond-mat/9506066]
The additional requirement that the Berry connection be non-abelian (and/or the presence of anyons):
Suggestion that topological order goes along with long-range entanglement:
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$]$
Jason Alicea, Matthew Fisher, Marcel Franz, Yong-Baek Kim, Strongly Interacting Topological Phases, report on Banff workshop 15w5051 (2015) [pdf, pdf]
Xiao-Gang Wen, Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89 41004 (2017) [arXiv:1610.03911, doi:10.1103/RevModPhys.89.041004]
Textbook accounts:
Bei Zeng, Xie Chen, Duan-Lu Zhou, Xiao-Gang Wen:
Topological Order and Long-Range Entanglement, 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]
Masaki Oshikawa, Yong Baek Kim, Kirill Shtengel, Chetan Nayak, Sumanta Tewari, Topological degeneracy of non-Abelian states for dummies, Annals of Physics 322 6 (2007) 1477-1498 [doi:10.1016/j.aop.2006.08.001, arXiv:cond-mat/0607743]
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]
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)
V. Yu. Irkhin, Yu. N. Skryabin, Topological states in strongly correlated systems, Journal of Superconductivity and Novel Magnetism, 35 8 (2022) 2141-2151 [arXiv:2209.04336, doi:10.1007/s10948-022-06251-3]
See also:
Xiao-Gang Wen: Vacuum Degeneracy of Chiral Spin State in Compactified Spaces, Phys. Rev. B 40 7387 (1989) [doi:10.1103/PhysRevB.40.7387]
Xiao-Gang Wen: Topological Orders in Rigid States, Int. J. Mod. Phys. B 4 239 (1990) [doi:10.1142/S0217979290000139, pdf]
Xiao-Gang Wen, Qian Niu: Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces, Phys. Rev. B 41 9377 (1990) [doi:10.1103/PhysRevB.41.9377]
E. Keski-Vakkuri, Xiao-Gang Wen: Ground state structure of hierarchical QH states on torus and modular transformation, Int. J. Mod. Phys. B 7 4227 (1993) [doi:10.1142/S0217979293003644, arXiv:hep-th/9303155]
Definite examples of topologically ordered materials remain somewhat elusive, but a few candidates have been discussed.
[Manning-Coe & Bradlyn 2023:] outside of the fractional quantum Hall effect, the connection between the microscopic Hamiltonian for interacting electrons and the topological order of the ground state remains elusive.
Argument that superconductors are actually topologically ordered phases:
On topological semimetals with degenerate ground states:
On topological semimetals which gap to topologically ordered phases:
Chong Wang, Lei Gioia, Anton A. Burkov, Fractional Quantum Hall Effect in Weyl Semimetals, Phys. Rev. Lett. 124 096603 (2020) [doi:10.1103/PhysRevLett.124.096603]
Manisha Thakurathi, Anton A. Burkov, Theory of the fractional quantum Hall effect in Weyl semimetals, Phys. Rev. B 101 235168 (2020) [doi:10.1103/PhysRevB.101.235168, arXiv:2005.13545]
“while we focused on fully gapped states […] it also makes sense to consider gapless strongly correlated states with the same property. Such states may be accessed easily within the formalism, presented above
Dan Sehayek, Manisha Thakurathi, Anton A. Burkov, Charge density waves in Weyl semimetals, Phys. Rev. B 102 115159 (2020) [doi:10.1103/PhysRevB.102.115159]
Jinmin Yi, Xuzhe Ying, Lei Gioia, Anton A. Burkov, Topological order in interacting semimetals, Phys. Rev. B 107 115147 (2023) [arXi:2301.03628, doi:10.1103/PhysRevB.107.115147]
Hongshuang Liu, Jiashuo Liang, Taiyu Sun, Liying Wang: Recent progress in topological semimetal and its realization in Heusler compounds, Materials Today Physics 41 (2024) 101343 [doi:10.1016/j.mtphys.2024.101343]
On detection of topological order by observing modular transformations on the ground state:
On thermodynamic stability of topological order:
Zohar Nussinov, Gerardo Ortiz: Autocorrelations and Thermal Fragility of Anyonic Loops in Topologically Quantum Ordered Systems, Phys. Rev. B 77 x (2008) Phys. Rev. B 77, 064302 [doi:10.1103/PhysRevB.77.064302, arXiv:0709.2717, arXiv:0709.2717]
Zohar Nussinov, Gerardo Ortiz: Symmetry and Topological Order, Proceedings of the National Academy of Sciences 106 40 (2009) 16944-16949 [doi:10.1073/pnas.0803726105, arXiv:cond-mat/0605316]
Zohar Nussinov, Gerardo Ortiz: A symmetry principle for Topological Quantum Order, Annals of Physics 324 5 (2009) 977-1057 [doi:10.1016/j.aop.2008.11.002, arXiv:cond-mat/0702377]
Zohar Nussinov, Gerardo Ortiz: Sufficient symmetry conditions for Topological Quantum Order, PNAS 106 40 (2009) 16944-16949 [doi:10.1073/pnas.080372610]
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
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 in “reciprocal momentum space” via twisted equivariant K-theory:
Discussion of quantum measurement of topologically ordered states:
On entanglement entropy in arithmetic Chern-Simons theory:
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):
and in the example of Chern-Simons theory:
Discussion in the dimer model:
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:
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
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]
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:
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:
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:
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:
Arguments realizing such anyonic topological order in the worldvolume-field theory on M5-branes:
Via KK-compactification on closed 3-manifolds (Seifert manifolds) analogous to the 3d-3d correspondence (which instead uses hyperbolic 3-manifolds):
Gil Young Cho, Dongmin Gang, Hee-Cheol Kim: M-theoretic Genesis of Topological Phases, J. High Energ. Phys. 2020 115 (2020) [arXiv:2007.01532, doi:10.1007/JHEP11(2020)115]
Shawn X. Cui, Paul Gustafson, Yang Qiu, Qing Zhang, From Torus Bundles to Particle-Hole Equivariantization, Lett Math Phys 112 15 (2022) [doi:10.1007/s11005-022-01508-3, arXiv:2106.01959]
Shawn X. Cui, Yang Qiu, Zhenghan Wang, From Three Dimensional Manifolds to Modular Tensor Categories, Commun. Math. Phys. 397 (2023) 1191–1235 [doi:10.1007/s00220-022-04517-4, arXiv:2101.01674]
Federico Bonetti, Sakura Schäfer-Nameki, Jingxiang Wu, $MTC[M_3,G]$: 3d Topological Order Labeled by Seifert Manifolds [arXiv:2403.03973]
Relation to ZX-calculus:
On detection of topological order by observing modular transformations on the ground state:
See also:
The theory of anyonic topologically ordered quantum materials is often discussed assuming periodic boundary conditions, making the space of positions of a given anyon a torus. (While this is a dubious assumption for position-space anyons in actual experiment, the intended ground state-degeracy crucially depends on this assumption.)
In fact, the notion of topological order was introduced already assuming torus-shaped materials:
Xiao-Gang Wen: Vacuum Degeneracy of Chiral Spin State in Compactified Spaces, Phys. Rev. B 40 7387 (1989) [doi:10.1103/PhysRevB.40.7387]
Xiao-Gang Wen: Topological Orders in Rigid States, Int. J. Mod. Phys. B 4 239 (1990) [doi:10.1142/S0217979290000139, pdf]
Xiao-Gang Wen, Qian Niu: Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces, Phys. Rev. B 41 9377 (1990) [doi:10.1103/PhysRevB.41.9377]
E. Keski-Vakkuri, Xiao-Gang Wen: Ground state structure of hierarchical QH states on torus and modular transformation, Int. J. Mod. Phys. B 7 4227 (1993) [doi:10.1142/S0217979293003644, arXiv:hep-th/9303155]
Further discussion along these lines:
Zhu-Xi Luo, Yu-Ting Hu, Yong-Shi Wu: On Quantum Entanglement in Topological Phases on a Torus, Phys. Rev. B 94 075126 (2016) [doi:10.1103/PhysRevB.94.075126, arXiv:1603.01777]
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]
Explicit discussion of anyons on tori:
Xiao-Gang Wen, E. Dagotto, Eduardo Fradkin: Anyons on a torus, Phys. Rev. B 42 (1990) 6110 [doi:10.1103/PhysRevB.42.6110]
Roberto Iengo, Kurt Lechner: Quantum mechanics of anyons on a torus, Nuclear Physics B 346 2–3 (1990) 551-575 [doi:10.1016/0550-3213(90)90292-L, spire:28316]
Roberto Iengo, Kurt Lechner: Exact results for anyons on a torus, Nuclear Physics B 364 3 (1991) 551-583 [doi:10.1016/0550-3213(91)90277-5]
Yasuhiro Hatsugai, Mahito Kohmoto, Yong-Shi Wu: Anyons on a torus: Braid group, Aharonov-Bohm period, and numerical study, Phys. Rev. B 43 (1991) 10761 [doi:10.1103/PhysRevB.43.10761]
Yutaka Hosotani, Choon-Lin Ho: Anyons on a Torus, AIP Conf. Proc. 272 (1992) 1466–1469 [doi:10.1063/1.43444, arXiv:hep-th/9210112]
Choon-Lin Ho, Yutaka Hosotani: Anyon equation on a torus, International Journal of Modern Physics A 07 23 (1992) 5797-5831 [doi:10.1142/S0217751X92002647]
Ikuo Ichinose, Toshiyuki Ohbayashi: Exactly soluble model of multispecies anyons and the braid group on a torus, Nucl.Phys. B 419 (1994) 529-552 [doi:10.1016/0550-3213(94)90343-3]
Alexei Kitaev: Fault-tolerant quantum computation by anyons, Annals of Physics 303 1 (2003) 2-30 [doi:10.1016/S0003-4916(02)00018-0, arXiv:quant-ph/9707021]
(introducing the toric code)
Songyang Pu, Jainendra K. Jain: Composite anyons on a torus, Phys. Rev. B 104 (2021) 115135 [doi:10.1103/PhysRevB.104.115135, arXiv:2106.15705]
Steven H. Simon: Anyon Vacuum on a Torus and Quantum Memory, Section 4.3 in: Topological Quantum, Oxford University Press (2023) [ISBN:9780198886723, pdf, webpage]
Shang Liu: Anyon quantum dimensions from an arbitrary ground state wave function, Nature Communications 15 (2024) 5134 [doi:10.1038/s41467-024-47856-7, arXiv:2304.13235]
But anyonic states may alternatively be localized in more abstract spaces. Anyons localized not in position space but in “reciprocal momentum space”, namely on the Brillouin torus of quasi-momenta of electrons in a crystal, are considered in
Also proposals to classify (free or interacting) topological phases of matter by topological quantum field theory mean to consider them on all base space topologies, including tori. This idea may originate around:
Anton Kapustin: Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology [spire:1283873, arXiv:1403.1467]
“Our basic assumption is that a gapped state of matter with short-range interactions can be put on a curved space-time of arbitrary topology […] At short distances a system is usually defined on a regular lattice, with short-range interactions. However, if we allow for disorder, then dislocations in the lattice are possible, and more general triangulations also become possible”
Anton Kapustin: Bosonic Topological Insulators and Paramagnets: a view from cobordisms [spire:1292830, arXiv:1404.6659]
“SPT phases are usually defined on a spatial lattice, while time may or may not be discretized. In the effective action approach we want to allow space-time to have an arbitrary topology, thus we discretize both space and time and regard the system as being defined on a general triangulation $K$ of a $d$-dimensional manifold $X$.”
and is implicit also in the proposal of classification via invertible field theory of:
Daniel S. Freed, Michael J. Hopkins: Section 9.3 of: Reflection positivity and invertible topological phases, Geom. Topol. 25 (2021) 1165-1330 [doi:10.2140/gt.2021.25.1165, arXiv:1604.06527]
Kazuya Yonekura: On the cobordism classification of symmetry protected topological phases, Commun. Math. Phys. 368 (2019) 1121–1173 [doi:10.1007/s00220-019-03439-y, arXiv:1803.10796]
The broad idea that TQFT is the right language to speak about anyonic topological order is now often stated as if self-evident, e.g. in:
Critical commentary on the assumption of non-trivial topology in position space appears in the following (whose authors then suggest that using extended TQFT may ameliorate the problem, p. 2):
Davide Gaiotto, Theo Johnson-Freyd: Condensations in higher categories [spire:1736539, arXiv:1905.09566]
“This relationship between gapped condensed matter systems and TQFTs is perplexing, particularly so if one takes a “global” approach to TQFTs, defining them à la Atiyah 1988 in terms of partition functions attached to non-trivial Euclidean space-time manifolds and spaces of states attached to non-trivial space manifolds. From that perspective, matching a given lattice system to a TQFT would require identifying a lot of extra structure to be added to the definition of the lattice system in order to define it on discretizations of non-trivial space manifolds and to define adiabatic evolutions analogous to non-trivial space-time manifolds.”
“If one takes the information-theoretic perspective that a phase of matter is fully characterized by the local entanglement properties of the ground-state wavefunction of the system, with no reference to a time evolution, then even more work may be needed.”
“These concerns are not just abstract. Given some phase of matter, in the lab or in a computer, it is hard to extract the data which would pin down the corresponding TQFT, or even know if the TQFT exists. For example, we can hardly place a three-dimensional material on a non-trivial space-manifold. We can only try to simulate that by employing judicious collections of defects in flat space.”
Research groups:
Univ. of Maryland, joint quantum institute
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
Last revised on August 24, 2024 at 15:07:08. See the history of this page for a list of all contributions to it.