Topological orders of monopoles and hedgehogs: From electronic and magnetic spin-orbit coupling to quarks
Abstract
Topological states of matter are, generally, quantum liquids of conserved topological defects. We establish this by constructing and analyzing topological field theories which introduce gauge fields to describe the dynamics of singularities in the original field configurations. In this work, homotopy groups are utilized to identify topologically protected singularities, and the conservation of their protected number is captured by a topological action term that unambiguously obtains from the given set of symmetries. Stable phases of these theories include quantum liquids with emergent massless Abelian and non-Abelian gauge fields, as well as topological orders with long-range quantum entanglement, fractional excitations, boundary modes, and unconventional responses to external perturbations. This paper focuses on the derivation of topological field theories and basic phenomenological characterization of topological orders associated with homotopy groups πn(Sn), n ≥ 1 . These homotopies govern monopole and hedgehog topological defects in d = n + 1 dimensions, and enable the generalization of both weakly interacting and fractional quantum Hall liquids of vortices to d > 2 . Hedgehogs have not been in the spotlight so far, but they are particularly important defects of magnetic moments because they can be stimulated in realistic systems with spin-orbit coupling, such as chiralmore »
- Authors:
-
[1]
- George Mason Univ., Fairfax, VA (United States); Johns Hopkins Univ., Baltimore, MD (United States)
- Publication Date:
- Research Org.:
- Energy Frontier Research Centers (EFRC) (United States). Institute for Quantum Matter (IQM); Johns Hopkins Univ., Baltimore, MD (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES). Materials Sciences & Engineering Division; National Science Foundation (NSF); USDOE
- OSTI Identifier:
- 1767746
- Alternate Identifier(s):
- OSTI ID: 1607797
- Grant/Contract Number:
- SC0019331; PHY-1607611
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review B
- Additional Journal Information:
- Journal Volume: 101; Journal Issue: 11; Journal ID: ISSN 2469-9950
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; optics; thermal conductivity; defects; charge transport; superconductivity; magnetism and spin physics; quantum information science; materials and chemistry by design; synthesis (novel materials)
Citation Formats
Nikolić, Predrag. Topological orders of monopoles and hedgehogs: From electronic and magnetic spin-orbit coupling to quarks. United States: N. p., 2020.
Web. doi:10.1103/PhysRevB.101.115144.
Nikolić, Predrag. Topological orders of monopoles and hedgehogs: From electronic and magnetic spin-orbit coupling to quarks. United States. https://doi.org/10.1103/PhysRevB.101.115144
Nikolić, Predrag. Mon .
"Topological orders of monopoles and hedgehogs: From electronic and magnetic spin-orbit coupling to quarks". United States. https://doi.org/10.1103/PhysRevB.101.115144. https://www.osti.gov/servlets/purl/1767746.
@article{osti_1767746,
title = {Topological orders of monopoles and hedgehogs: From electronic and magnetic spin-orbit coupling to quarks},
author = {Nikolić, Predrag},
abstractNote = {Topological states of matter are, generally, quantum liquids of conserved topological defects. We establish this by constructing and analyzing topological field theories which introduce gauge fields to describe the dynamics of singularities in the original field configurations. In this work, homotopy groups are utilized to identify topologically protected singularities, and the conservation of their protected number is captured by a topological action term that unambiguously obtains from the given set of symmetries. Stable phases of these theories include quantum liquids with emergent massless Abelian and non-Abelian gauge fields, as well as topological orders with long-range quantum entanglement, fractional excitations, boundary modes, and unconventional responses to external perturbations. This paper focuses on the derivation of topological field theories and basic phenomenological characterization of topological orders associated with homotopy groups πn(Sn), n ≥ 1 . These homotopies govern monopole and hedgehog topological defects in d = n + 1 dimensions, and enable the generalization of both weakly interacting and fractional quantum Hall liquids of vortices to d > 2 . Hedgehogs have not been in the spotlight so far, but they are particularly important defects of magnetic moments because they can be stimulated in realistic systems with spin-orbit coupling, such as chiral magnets and d = 3 topological materials. We predict topological orders in systems with U(1) × Spin(d) symmetry in which fractional electric charge attaches to hedgehogs. Monopoles, the analogous defects of charge or generic U(1) currents, may bind to hedgehogs via Zeeman effect, or effectively emerge in purely magnetic systems. The latter can lead to spin liquids with different topological orders than that of the resonant valence bond spin liquid. Charge fractionalization of quarks in atomic nuclei is also seen as possibly arising from the charge-hedgehog attachment.},
doi = {10.1103/PhysRevB.101.115144},
journal = {Physical Review B},
number = 11,
volume = 101,
place = {United States},
year = {Mon Mar 30 00:00:00 EDT 2020},
month = {Mon Mar 30 00:00:00 EDT 2020}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 4 works
Citation information provided by
Web of Science
Web of Science
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Evidence for magnetic Weyl fermions in a correlated metal
journal, September 2017
- Kuroda, K.; Tomita, T.; Suzuki, M. -T.
- Nature Materials, Vol. 16, Issue 11
Continuum of quantum fluctuations in a three-dimensional S = 1 Heisenberg magnet
journal, October 2018
- Plumb, K. W.; Changlani, Hitesh J.; Scheie, A.
- Nature Physics, Vol. 15, Issue 1
Topological BF field theory description of topological insulators
journal, June 2011
- Cho, Gil Young; Moore, Joel E.
- Annals of Physics, Vol. 326, Issue 6
Artificial light and quantum order in systems of screened dipoles
journal, September 2003
- Wen, Xiao-Gang
- Physical Review B, Vol. 68, Issue 11
Correlated topological insulators and the fractional magnetoelectric effect
journal, May 2011
- Swingle, B.; Barkeshli, M.; McGreevy, J.
- Physical Review B, Vol. 83, Issue 19
Experimental Realization of a Three-Dimensional Topological Insulator, Bi2Te3
journal, June 2009
- Chen, Y. L.; Analytis, J. G.; Chu, J.-H.
- Science, Vol. 325, Issue 5937, p. 178-181
Ground States of Quantum Antiferromagnets in Two Dimensions
journal, May 2002
- Sachdev, Subir; Park, Kwon
- Annals of Physics, Vol. 298, Issue 1
A time-reversal invariant topological phase at the surface of a 3D topological insulator
journal, September 2013
- Bonderson, Parsa; Nayak, Chetan; Qi, Xiao-Liang
- Journal of Statistical Mechanics: Theory and Experiment, Vol. 2013, Issue 09
Classification of Bosonic Topological Orders: The Case When Pointlike Excitations Are All Bosons
journal, June 2018
- Lan, Tian; Kong, Liang; Wen, Xiao-Gang
- Physical Review X, Vol. 8, Issue 2
Non-Fermi-Liquid and Topological States with Strong Spin-Orbit Coupling
journal, November 2013
- Moon, Eun-Gook; Xu, Cenke; Kim, Yong Baek
- Physical Review Letters, Vol. 111, Issue 20
Rods of Neutron Scattering Intensity in : Compelling Evidence for Significant Anisotropic Exchange in a Magnetic Pyrochlore Oxide
journal, May 2011
- Thompson, Jordan D.; McClarty, Paul A.; Rønnow, Henrik M.
- Physical Review Letters, Vol. 106, Issue 18
Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors
journal, August 2006
- Qi, Xiao-Liang; Wu, Yong-Shi; Zhang, Shou-Cheng
- Physical Review B, Vol. 74, Issue 8
Quantization and periodicity of the axion action in topological insulators
journal, December 2010
- Vazifeh, M. M.; Franz, M.
- Physical Review B, Vol. 82, Issue 23
A PCAC puzzle: π0→γγ in the σ-model
journal, March 1969
- Bell, J. S.; Jackiw, R.
- Il Nuovo Cimento A, Vol. 60, Issue 1
Models of three-dimensional fractional topological insulators
journal, December 2012
- Maciejko, Joseph; Qi, Xiao-Liang; Karch, Andreas
- Physical Review B, Vol. 86, Issue 23
Topological properties and dynamics of magnetic skyrmions
journal, December 2013
- Nagaosa, Naoto; Tokura, Yoshinori
- Nature Nanotechnology, Vol. 8, Issue 12
gauge theory of electron fractionalization in strongly correlated systems
journal, September 2000
- Senthil, T.; Fisher, Matthew P. A.
- Physical Review B, Vol. 62, Issue 12
Anyons on a torus
journal, October 1990
- Wen, X. G.; Dagotto, E.; Fradkin, E.
- Physical Review B, Vol. 42, Issue 10
Quantum Ice: A Quantum Monte Carlo Study
journal, February 2012
- Shannon, Nic; Sikora, Olga; Pollmann, Frank
- Physical Review Letters, Vol. 108, Issue 6
Physics of Three-Dimensional Bosonic Topological Insulators: Surface-Deconfined Criticality and Quantized Magnetoelectric Effect
journal, February 2013
- Vishwanath, Ashvin; Senthil, T.
- Physical Review X, Vol. 3, Issue 1
Anomalous Hall effect
journal, May 2010
- Nagaosa, Naoto; Sinova, Jairo; Onoda, Shigeki
- Reviews of Modern Physics, Vol. 82, Issue 2
Nonabelions in the fractional quantum hall effect
journal, August 1991
- Moore, Gregory; Read, Nicholas
- Nuclear Physics B, Vol. 360, Issue 2-3
Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
journal, October 2010
- Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang
- Physical Review B, Vol. 82, Issue 15
Absence of U(1) spin liquids in two dimensions
journal, November 2003
- Herbut, Igor F.; Seradjeh, Babak H.; Sachdev, Subir
- Physical Review B, Vol. 68, Issue 19
Two-Dimensional Magnetotransport in the Extreme Quantum Limit
journal, May 1982
- Tsui, D. C.; Stormer, H. L.; Gossard, A. C.
- Physical Review Letters, Vol. 48, Issue 22
Generalized Modular Transformations in Topologically Ordered Phases and Triple Linking Invariant of Loop Braiding
journal, September 2014
- Jiang, Shenghan; Mesaros, Andrej; Ran, Ying
- Physical Review X, Vol. 4, Issue 3
Three-Triplet Model with Double Symmetry
journal, August 1965
- Han, M. Y.; Nambu, Y.
- Physical Review, Vol. 139, Issue 4B
Comment on “Spin Transport Properties of the Quantum One-Dimensional Non-Linear Sigma Model”
journal, August 2000
- Sachdev, Subir; Damle, Kedar
- Journal of the Physical Society of Japan, Vol. 69, Issue 8
Unusual Liquid State of Hard-Core Bosons on the Pyrochlore Lattice
journal, January 2008
- Banerjee, Argha; Isakov, Sergei V.; Damle, Kedar
- Physical Review Letters, Vol. 100, Issue 4
Disordered, spin liquid, and valence-bond ordered phases of kagome lattice quantum Ising models with transverse field and dynamics
journal, August 2005
- Nikolić, P.
- Physical Review B, Vol. 72, Issue 6
Classification of two-dimensional fermionic and bosonic topological orders
journal, March 2015
- Gu, Zheng-Cheng; Wang, Zhenghan; Wen, Xiao-Gang
- Physical Review B, Vol. 91, Issue 12
Topological invariants for gauge theories and symmetry-protected topological phases
journal, April 2015
- Wang, Chenjie; Levin, Michael
- Physical Review B, Vol. 91, Issue 16
Non-Abelian string and particle braiding in topological order: Modular representation and -dimensional twisted gauge theory
journal, January 2015
- Wang, Juven C.; Wen, Xiao-Gang
- Physical Review B, Vol. 91, Issue 3
Quantum Field Theory of Particles with Both Electric and Magnetic Charges
journal, December 1968
- Zwanziger, Daniel
- Physical Review, Vol. 176, Issue 5
Axial-Vector Vertex in Spinor Electrodynamics
journal, January 1969
- Adler, Stephen L.
- Physical Review, Vol. 177, Issue 5
Topological Insulators in Three Dimensions
journal, March 2007
- Fu, Liang; Kane, C. L.; Mele, E. J.
- Physical Review Letters, Vol. 98, Issue 10, Article No. 106803
String-net condensation: A physical mechanism for topological phases
journal, January 2005
- Levin, Michael A.; Wen, Xiao-Gang
- Physical Review B, Vol. 71, Issue 4
Exactly soluble models for fractional topological insulators in two and three dimensions
journal, December 2011
- Levin, Michael; Burnell, F. J.; Koch-Janusz, Maciej
- Physical Review B, Vol. 84, Issue 23
Holographic fractional topological insulators
journal, October 2010
- Hoyos, Carlos; Jensen, Kristan; Karch, Andreas
- Physical Review D, Vol. 82, Issue 8
Resonating valence bonds: A new kind of insulator?
journal, February 1973
- Anderson, P. W.
- Materials Research Bulletin, Vol. 8, Issue 2
Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets
journal, May 2014
- Gingras, M. J. P.; McClarty, P. A.
- Reports on Progress in Physics, Vol. 77, Issue 5
Can quarks have integer charge?
journal, July 1981
- Iijima, B.; Jaffe, R. L.
- Physical Review D, Vol. 24, Issue 1
Topological invariants of time-reversal-invariant band structures
journal, March 2007
- Moore, J. E.; Balents, L.
- Physical Review B, Vol. 75, Issue 12, Article No. 121306(R)
Correspondence between two-dimensional bosons and a bulk superconductor in a magnetic field
journal, February 1989
- Fisher, Matthew P. A.; Lee, D. H.
- Physical Review B, Vol. 39, Issue 4
Effective hydrodynamic field theory and condensation picture of topological insulators
journal, April 2016
- Chan, AtMa P. O.; Kvorning, Thomas; Ryu, Shinsei
- Physical Review B, Vol. 93, Issue 15
Exotic Order in Simple Models of Bosonic Systems
journal, December 2002
- Motrunich, O. I.; Senthil, T.
- Physical Review Letters, Vol. 89, Issue 27
Interaction Driven Subgap Spin Exciton in the Kondo Insulator
journal, January 2015
- Fuhrman, W. T.; Leiner, J.; Nikolić, P.
- Physical Review Letters, Vol. 114, Issue 3
Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates
journal, May 2011
- Wan, Xiangang; Turner, Ari M.; Vishwanath, Ashvin
- Physical Review B, Vol. 83, Issue 20
U(1)×SU(2)-gauge invariance of non-relativistic quantum mechanics, and generalized Hall effects
journal, September 1992
- Fröhlich, J.; Studer, U. M.
- Communications in Mathematical Physics, Vol. 148, Issue 3
Non-Abelian statistics versus the Witten anomaly
journal, September 2011
- McGreevy, John; Swingle, Brian
- Physical Review D, Vol. 84, Issue 6
New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance
journal, August 1980
- Klitzing, K. v.; Dorda, G.; Pepper, M.
- Physical Review Letters, Vol. 45, Issue 6
The topological theory of defects in ordered media
journal, July 1979
- Mermin, N. D.
- Reviews of Modern Physics, Vol. 51, Issue 3
Stability of spin liquids in two dimensions
journal, December 2004
- Hermele, Michael; Senthil, T.; Fisher, Matthew P. A.
- Physical Review B, Vol. 70, Issue 21
Correlated Quantum Phenomena in the Strong Spin-Orbit Regime
journal, March 2014
- Witczak-Krempa, William; Chen, Gang; Kim, Yong Baek
- Annual Review of Condensed Matter Physics, Vol. 5, Issue 1
Mott physics and band topology in materials with strong spin–orbit interaction
journal, March 2010
- Pesin, Dmytro; Balents, Leon
- Nature Physics, Vol. 6, Issue 5
Quantum spin liquids: a review
journal, November 2016
- Savary, Lucile; Balents, Leon
- Reports on Progress in Physics, Vol. 80, Issue 1
Quantum Spin Hall Insulator State in HgTe Quantum Wells
journal, November 2007
- Konig, M.; Wiedmann, S.; Brune, C.
- Science, Vol. 318, Issue 5851, p. 766-770
Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
journal, December 2006
- Bernevig, B. A.; Hughes, T. L.; Zhang, S.-C.
- Science, Vol. 314, Issue 5806, p. 1757-1761
Experimental signatures of three-dimensional fractional topological insulators
journal, December 2012
- Swingle, Brian
- Physical Review B, Vol. 86, Issue 24
Gapped symmetry preserving surface state for the electron topological insulator
journal, September 2013
- Wang, Chong; Potter, Andrew C.; Senthil, T.
- Physical Review B, Vol. 88, Issue 11
Dual Dirac Liquid on the Surface of the Electron Topological Insulator
journal, November 2015
- Wang, Chong; Senthil, T.
- Physical Review X, Vol. 5, Issue 4
Anomaly manifestation of Lieb-Schultz-Mattis theorem and topological phases
journal, November 2017
- Cho, Gil Young; Hsieh, Chang-Tse; Ryu, Shinsei
- Physical Review B, Vol. 96, Issue 19
Majorana Fermions and Non-Abelian Statistics in Three Dimensions
journal, January 2010
- Teo, Jeffrey C. Y.; Kane, C. L.
- Physical Review Letters, Vol. 104, Issue 4
Topological Entanglement Entropy
journal, March 2006
- Kitaev, Alexei; Preskill, John
- Physical Review Letters, Vol. 96, Issue 11
Topological transitions among skyrmion- and hedgehog-lattice states in cubic chiral magnets
journal, March 2019
- Fujishiro, Y.; Kanazawa, N.; Nakajima, T.
- Nature Communications, Vol. 10, Issue 1
Two-dimensional heavy fermions on the strongly correlated boundaries of Kondo topological insulators
journal, December 2014
- Nikolić, Predrag
- Physical Review B, Vol. 90, Issue 23
Topological field theory of time-reversal invariant insulators
journal, November 2008
- Qi, Xiao-Liang; Hughes, Taylor L.; Zhang, Shou-Cheng
- Physical Review B, Vol. 78, Issue 19, Article No. 195424
Inducing a Magnetic Monopole with Topological Surface States
journal, February 2009
- Qi, X. -L.; Li, R.; Zang, J.
- Science, Vol. 323, Issue 5918
Quantum Phases of Vortices in Rotating Bose-Einstein Condensates
journal, August 2001
- Cooper, N. R.; Wilkin, N. K.; Gunn, J. M. F.
- Physical Review Letters, Vol. 87, Issue 12
Braiding Statistics of Loop Excitations in Three Dimensions
journal, August 2014
- Wang, Chenjie; Levin, Michael
- Physical Review Letters, Vol. 113, Issue 8
Gauge fields in real and momentum spaces in magnets: monopoles and skyrmions
journal, December 2012
- Nagaosa, N.; Yu, X. Z.; Tokura, Y.
- Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 370, Issue 1981
Large anomalous Nernst effect at room temperature in a chiral antiferromagnet
journal, July 2017
- Ikhlas, Muhammad; Tomita, Takahiro; Koretsune, Takashi
- Nature Physics, Vol. 13, Issue 11
Weyl and Dirac semimetals in three-dimensional solids
journal, January 2018
- Armitage, N. P.; Mele, E. J.; Vishwanath, Ashvin
- Reviews of Modern Physics, Vol. 90, Issue 1
Fractional Topological Insulators in Three Dimensions
journal, December 2010
- Maciejko, Joseph; Qi, Xiao-Liang; Karch, Andreas
- Physical Review Letters, Vol. 105, Issue 24
First-Order Phase Transitions in Superconductors and Smectic- Liquid Crystals
journal, February 1974
- Halperin, B. I.; Lubensky, T. C.; Ma, Shang-keng
- Physical Review Letters, Vol. 32, Issue 6
Quantized Hall Conductance in a Two-Dimensional Periodic Potential
journal, August 1982
- Thouless, D. J.; Kohmoto, M.; Nightingale, M. P.
- Physical Review Letters, Vol. 49, Issue 6
Pyrochlore photons: The spin liquid in a three-dimensional frustrated magnet
journal, February 2004
- Hermele, Michael; Fisher, Matthew P. A.; Balents, Leon
- Physical Review B, Vol. 69, Issue 6
Skyrmion Lattice in a Chiral Magnet
journal, February 2009
- Muhlbauer, S.; Binz, B.; Jonietz, F.
- Science, Vol. 323, Issue 5916
Theory of the kagome lattice Ising antiferromagnet in weak transverse fields
journal, January 2005
- Nikolić, P.; Senthil, T.
- Physical Review B, Vol. 71, Issue 2
Classification of Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions
journal, April 2019
- Lan, Tian; Wen, Xiao-Gang
- Physical Review X, Vol. 9, Issue 2
Quantum Excitations in Quantum Spin Ice
journal, October 2011
- Ross, Kate A.; Savary, Lucile; Gaulin, Bruce D.
- Physical Review X, Vol. 1, Issue 2
Time-reversal symmetry breaking and spontaneous Hall effect without magnetic dipole order
journal, December 2009
- Machida, Yo; Nakatsuji, Satoru; Onoda, Shigeki
- Nature, Vol. 463, Issue 7278
Weyl Semimetal in a Topological Insulator Multilayer
journal, September 2011
- Burkov, A. A.; Balents, Leon
- Physical Review Letters, Vol. 107, Issue 12
Topological Order in a Correlated Three-Dimensional Topological Insulator
journal, January 2014
- Maciejko, Joseph; Chua, Victor; Fiete, Gregory A.
- Physical Review Letters, Vol. 112, Issue 1
Symmetry-respecting topologically ordered surface phase of three-dimensional electron topological insulators
journal, September 2015
- Metlitski, Max A.; Kane, C. L.; Fisher, Matthew P. A.
- Physical Review B, Vol. 92, Issue 12
Connection of Spin and Statistics for Charge-Monopole Composites
journal, May 1976
- Goldhaber, Alfred S.
- Physical Review Letters, Vol. 36, Issue 19
Fractional Topological Insulators of Cooper Pairs Induced by the Proximity Effect
journal, April 2013
- Nikolić, Predrag; Duric, Tanja; Tešanović, Zlatko
- Physical Review Letters, Vol. 110, Issue 17
Entanglement Monotonicity and the Stability of Gauge Theories in Three Spacetime Dimensions
journal, April 2014
- Grover, Tarun
- Physical Review Letters, Vol. 112, Issue 15
Field Corrections to Induced Statistics
journal, January 1989
- Goldhaber, Alfred S.; Mackenzie, R.; Wilczek, Frank
- Modern Physics Letters A, Vol. 04, Issue 01
Sine-Gordon theory of the non-Néel phase of two-dimensional quantum antiferromagnets
journal, August 1989
- Zheng, Wei; Sachdev, Subir
- Physical Review B, Vol. 40, Issue 4
Phase Transition in a Lattice Model of Superconductivity
journal, November 1981
- Dasgupta, C.; Halperin, B. I.
- Physical Review Letters, Vol. 47, Issue 21
Layer Construction of 3D Topological States and String Braiding Statistics
journal, December 2014
- Jian, Chao-Ming; Qi, Xiao-Liang
- Physical Review X, Vol. 4, Issue 4
Origin of Gauge Bosons from Strong Quantum Correlations
journal, December 2001
- Wen, Xiao-Gang
- Physical Review Letters, Vol. 88, Issue 1