[1] X. Chen, Z.-C. Gu, and X.-G. Wen, “Local unitary transformation, long-range
quantum entanglement, wave function renormalization, and topological order,”
Phys. Rev. B, vol. 82, no. 15, p. 155 138, 2010.
[2] X.-G. Wen, “Topological order: From long-range entangled quantum matter to a
unified origin of light and electrons,” ISRN Condensed Matter Physics, vol. 2013,
p. 198 710, 2013.
[3] Z.-C. Gu and X.-G. Wen, “Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order,” Phys. Rev. B, vol. 80, no. 15,
pp. 155 131–, Oct. 2009.
[4] F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, “Symmetry protection of
topological phases in one-dimensional quantum spin systems,” Phys. Rev. B, vol.
85, no. 7, pp. 075 125–, Feb. 2012.
[5] E. Lieb, T. Schultz, and D. Mattis, “Two soluble models of an antiferromagnetic
chain,” Ann. Phys., vol. 16, no. 3, pp. 407–466, 1961.
[6] I. Affleck and E. H. Lieb, “A proof of part of Haldane’s conjecture on spin chains,”
Lett. Math. Phys., vol. 12, no. 1, pp. 57–69, 1986.
[7] M. Oshikawa, M. Yamanaka, and I. Affleck, “Magnetization plateaus in spin chains:
‘‘Haldane gap” for half-integer spins,” Phys. Rev. Lett., vol. 78, pp. 1984–1987, 10
1997.
[8] M. Oshikawa, “Commensurability, excitation gap, and topology in quantum manyparticle systems on a periodic lattice,” Phys. Rev. Lett., vol. 84, no. 7, pp. 1535–
1538, Feb. 2000.
[9] M. B. Hastings, “Lieb-Schultz-Mattis in higher dimensions,” Phys. Rev. B, vol. 69,
no. 10, pp. 104 431–, Mar. 2004.
[10] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, “Topological field theory of time-reversal
invariant insulators,” Phys. Rev. B, vol. 78, no. 19, p. 195 424, 2008.
[11] X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,”
Rev. Mod. Phys., vol. 83, no. 4, pp. 1057–1110, Oct. 2011.
69
[12] C. Callan and J. Harvey, “Anomalies and fermion zero modes on strings and domain
walls,” Nucl. Phys. B, vol. 250, no. 1, pp. 427 –436, 1985.
[13] S. C. Furuya and M. Oshikawa, “Symmetry protection of critical phases and a global
anomaly in 1 + 1 dimensions,” Phys. Rev. Lett., vol. 118, no. 2, pp. 021 601–, Jan.
2017.
[14] G. Y. Cho, C.-T. Hsieh, and S. Ryu, “Anomaly manifestation of Lieb-Schultz-Mattis
theorem and topological phases,” Phys. Rev. B, vol. 96, no. 19, pp. 195 105–, Nov.
2017.
[15] M. A. Metlitski and R. Thorngren, “Intrinsic and emergent anomalies at deconfined
critical points,” Phys. Rev. B, vol. 98, no. 8, p. 085 140, 2018.
[16] G. ’t Hooft, C. Itzykson, A. Jaffe, H. Lehmann, P. K. Mitter, I. M. Singer, and
R. Stora, “Recent Developments in Gauge Theories. Proceedings, Nato Advanced
Study Institute, Cargese, France, August 26 - September 8, 1979,” NATO Sci. Ser. B,
vol. 59, pp.1–438, 1980.
[17] Y. Yao, C.-T. Hsieh, and M. Oshikawa, “Anomaly matching and symmetry-protected
critical phases in SU(N ) spin systems in 1 + 1 dimensions,” Phys. Rev. Lett., vol.
123, no. 18, p. 180 201, 2019.
[18] Y. Yao and M. Oshikawa, “Generalized boundary condition applied to Lieb-SchultzMattis-type ingappabilities and many-body Chern numbers,” Phys. Rev. X, vol. 10,
no. 3, pp. 031 008–, Jul. 2020.
[19] C. Wu, J.-P. Hu, and S.-C. Zhang, “Exact SO(5) symmetry in the spin-3/2 fermionic
system,” Phys. Rev. Lett., vol. 91, no. 18, pp. 186 402–, Oct. 2003.
[20] C. Honerkamp and W. Hofstetter, “Ultracold Fermions and the SU(N ) Hubbard
model,” Phys. Rev. Lett., vol. 92, no. 17, pp. 170 403–, Apr. 2004.
[21] M. A. Cazalilla, A. Ho, and M Ueda, “Ultracold gases of ytterbium: Ferromagnetism
and Mott states in an SU(6) Fermi system,” New J. Phys., vol. 11, no. 10, 2009.
[22] A. V. Gorshkov, M Hermele, V Gurarie, C Xu, P. S. Julienne, J Ye, P. Zoller, E.
Demler, M. D. Lukin, and A. Rey, “Two-orbital SU (N ) magnetism with ultracold
alkaline-earth atoms,” Nature physics, vol. 6, no. 4, pp. 289–295, 2010.
[23] S. Taie, R. Yamazaki, S. Sugawa, and Y. Takahashi, “An SU(6) Mott insulator of an
atomic Fermi gas realized by large-spin Pomeranchuk cooling,” Nature Physics, vol.
8, no. 11, pp. 825–830, 2012.
70
[24] G. Pagano, M. Mancini, G. Cappellini, P. Lombardi, F. Sch¨afer, H. Hu, X.-J. Liu,
J. Catani, C. Sias, and M. Inguscio, “A one-dimensional liquid of fermions with
tunable spin,” Nature Physics, vol. 10, no. 3, pp. 198–201, 2014.
[25] F. Scazza, C. Hofrichter, M. H¨ofer, P. De Groot, I. Bloch, and S. F¨olling, “Observation of two-orbital spin-exchange interactions with ultracold SU(N )-symmetric
fermions,” Nature Physics, vol. 10, no. 10, pp. 779–784, 2014.
[26] X Zhang, M Bishof, S. Bromley, C. Kraus, M. Safronova, P Zoller, A. M. Rey,
and J Ye, “Spectroscopic observation of SU(N )-symmetric interactions in Sr orbital
magnetism,” Science, vol. 345, no. 6203, pp. 1467–1473, 2014.
[27] F. C. Alcaraz and M. J. Martins, “Conformal anomaly for the exactly integrable
SU(N ) magnets,” J. Phys. A: Math. Theor., vol. 22, no. 18, pp. L865–L870, 1989.
[28] M. F¨uhringer, S. Rachel, R. Thomale, M. Greiter, and P. Schmitteckert, “DMRG
studies of critical SU(N ) spin chains,” Ann. Phys. (Berlin), vol. 17, no. 12, pp. 922–
936, 2008.
[29] Y. Matsumoto, S. Nakatsuji, K. Kuga, Y. Karaki, N. Horie, Y. Shimura, T. Sakakibara, A. H. Nevidomskyy, and P. Coleman, “Quantum criticality without tuning in
the mixed valence compound beta-YbAlB4 ,” Science, vol. 331, no. 6015, pp. 316–
319, 2011.
[30] G. Uimin, “One-dimensional problem for S=1 with modified antiferromagnetic
Hamiltonian,” JETP Lett., vol. 12, p. 225, 1970.
[31] C. Lai, “Lattice gas with nearestneighbor interaction in one dimension with arbitrary
statistics,” J. Math. Phys., vol. 15, no. 10, pp. 1675–1676, 1974.
[32] B. Sutherland, “Model for a multicomponent quantum system,” Phys. Rev. B, vol.
12, no. 9, pp. 3795–3805, Nov. 1975.
[33] P. Chen, Z.-L. Xue, I. P. McCulloch, M.-C. Chung, C.-C. Huang, and S. K. Yip,
“Quantum critical spin-2 chain with emergent SU(3) symmetry,” Phys. Rev. Lett.,
vol. 114, no. 14, pp. 145 301–, Apr. 2015.
[34] I. Affleck, “Spin gap and symmetry breaking in CuO2 layers and other antiferromagnets,” Phys. Rev. B, vol. 37, pp. 5186–5192, 10 1988.
[35] B. Nachtergaele and R. Sims, “A multi-dimensional Lieb-Schultz-Mattis theorem,”
Communications in Mathematical Physics, vol. 276, no. 2, pp. 437–472, 2007.
71
[36] S. A. Parameswaran, A. M. Turner, D. P. Arovas, and A. Vishwanath, “Topological
order and absence of band insulators at integer filling in non-symmorphic crystals,”
Nature Physics, vol. 9, pp. 299–303, 2013.
[37] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized global symmetries,” JHEP, vol. 2015, 2015.
[38] H. Watanabe, “Insensitivity of bulk properties to the twisted boundary condition,”
Phys. Rev. B, vol. 98, no. 15, p. 155 137, 2018.
[39] E. Witten, “Fermion path integrals and topological phases,” Rev. Mod. Phys., vol.
88, no. 3, pp. 035 001–, Jul. 2016.
[40] ——, “The “parity” anomaly on an unorientable manifold,” Phys. Rev. B, vol. 94,
no. 19, pp. 195 150–, Nov. 2016.
[41] B. A. Bernevig and T. L. Hughes, Topological insulators and topological superconductors.
Princeton University Press, 2013.
[42] S.-Q. Shen, Topological Insulators: Dirac Equation in Condensed Matters. Springer,
2013.
[43] E. Witten, “Global gravitational anomalies,” Comm. Math. Phys., vol. 100, no. 2,
pp. 197–229, 1985.
[44] K. Fujikawa and H. Suzuki, Path integrals and quantum anomalies. Oxford University Press on Demand, 2004, vol. 122 0198529139.
[45] ——, “Anomalies, local counter terms and bosonization,” Physics reports, vol. 398,
no. 4-6, pp. 221–243, 2004.
[46] H. Georgi, Lie Algebras In Particle Physics. Boca Raton: CRC Press, 2000.
[47] I. Affleck, “Critical behaviour of SU(n) quantum chains and topological non-linear
sigma-models,” Nucl. Phys. B, vol. 305, no. 4, pp. 582–596, 1988.
[48] I. Affleck and F. D. M. Haldane, “Critical theory of quantum spin chains,”
Phys. Rev. B, vol. 36, no. 10, pp. 5291–5300, Oct. 1987.
[49] R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology,”
Commun. Math. Phys., vol. 129, no. 2, pp. 393–429, 1990.
[50] I. Affleck,
ous results
T.
on
Kennedy, E. H. Lieb, and
valence-bond ground states in
72
H. Tasaki, “Rigorantiferromagnets,” in
Condensed Matter Physics and Exactly Soluble Models. Springer, 2004, pp. 249–
252.
[51] I. Zaliznyak, L.-P. Regnault, and D Petitgrand, “Neutron-scattering study of the dynamic spin correlations in CsNiCl3 above N´eel ordering,” Phys. Rev. B, vol. 50, no.
21, p. 15 824, 1994.
[52] S.-H. Lee, C Broholm, W Ratcliff, G Gasparovic, Q Huang, T. Kim, and S.-W.
Cheong, “Emergent excitations in a geometrically frustrated magnet,” Nature, vol.
418, no. 6900, pp. 856 1476–4687, 2002.
[53] I. A. Zaliznyak and S.-H. Lee, “Magnetic neutron scattering,” Tech. Rep., 2004.
´ A. P´erigo, F. Bergner, S. Disch, A. Heinemann, S.
[54] S. M¨uhlbauer, D. Honecker, E.
Erokhin, D. Berkov, C. Leighton, and M. R. Eskildsen, “Magnetic small-angle neutron scattering,” Rev. Mod. Phys., vol. 91, no. 1, p. 015 004, 2019.
[55] H. Alloul, J. Bobroff, M. Gabay, and P. J. Hirschfeld, “Defects in correlated metals
and superconductors,” Rev. Mod. Phys., vol. 81, no. 1, pp. 45–108, Jan. 2009.
[56] E. Witten, “Non-abelian bosonization in two dimensions,” in Bosonization. World
Scientific, 1994, pp. 201–218.
[57] S. G. Naculich and H. J. Schnitzer, “Duality between SU(N )k and SU(k)N WZW
models,” Nucl. Phys. B, vol. 347, pp. 687–742, 1990.
[58] E. Kiritsis and V. Niarchos, “Large-N limits of 2d CFTs, quivers and AdS3 duals,”
JHEP, vol. 04, p. 113, 2011.
[59] A. Kapustin and N. Seiberg, “Coupling a QFT to a TQFT and duality,” JHEP, vol.
2014, no. 4, p. 1, 2014.
[60] A.
Kapustin,
“D-branes
in
ArXiv preprint hep-th/9909089, 1999.
topologically
nontrivial
B-field,”
[61] N. Seiberg, “Thoughts about quantum field theory,” Strings 2019, 2019.
[62] A. Altland and B. D. Simons, Condensed matter field theory. Cambridge University
Press, 2010, ISBN: 113948513X.
[63] M. Greiter and S. Rachel, “Valence bond solids for SU(N ) spin chains: Exact
models, spinon confinement, and the Haldane gap,” Phys. Rev. B, vol. 75, no. 18,
pp. 184 441–, May 2007.
73
[64] M. Greiter, S. Rachel, and D. Schuricht, “Exact results for SU(3) spin chains: Trimer
states, valence bond solids, and their parent Hamiltonians,” Phys. Rev. B, vol. 75, no.
6, p. 060 401, 2007.
[65] P. Francesco, P. Mathieu, and D. S´en´echal, Conformal field theory. Springer Science
& Business Media, 2012.
[66] I. Affleck, “Exact critical exponents for quantum spin chains, non-linear sigmamodels at theta=pi and the quantum Hall effect,” Nuclear Physics B, vol. 265, no.
3, pp. 409–447, 1986.
[67] J. M. Kosterlitz, “The critical properties of the two-dimensional XY model,”
Journal of Physics C: Solid State Physics, vol. 7, no. 6, pp. 1046–1060, 1974.
[68] I Affleck, D Gepner, H. J. Schulz, and T Ziman, “Critical behaviour of
spin-s Heisenberg antiferromagnetic chains: Analytic and numerical results,”
Journal of Physics A: Mathematical and General, vol. 22, no. 5, pp. 511–529, 1989.
[69] J. L. Cardy, “Logarithmic corrections to finite-size scaling in strips,”
Journal of Physics A: Mathematical and General, vol. 19, no. 17, pp. L1093–
L1098, 1986.
[70] K.
Majumdar
and
M.
Mukherjee,
“Logarithmic
corrections
to
finite-size
spectrum
of
SU(N )
symmetric
quantum
chains,”
Journal of Physics A: Mathematical and General, vol. 35, no. 38, pp. L543–
L549, 2002.
[71] M. Srednicki, Quantum Field Theory, 1st. Cambridge University Press, 2007.
[72] R. Kobayashi, K. Shiozaki, Y. Kikuchi, and S. Ryu, “Lieb-Schultz-Mattis type theorem with higher-form symmetry and the quantum dimer models,” Phys. Rev. B, vol.
99, no. 1, pp. 014 402–, Jan. 2019.
[73] C. Schweigert, “On moduli spaces of flat connections with non-simply connected
structure group,” Nucl. Phys. B, vol. 492, no. 3, pp. 743–755, 1997.
[74] A. Borel, R. Friedman, J. W. Morgan, and J.
Almost commuting elements in compact Lie groups.
American
Soc., 2002.
W. Morgan,
Mathematical
[75] V. G. Kac and A. V. Smilga, “Vacuum structure in supersymmetric
Yang–Mills
theories
with
any
gauge
group,”
in
The Many Faces of the Superworld: Yuri Golfand Memorial Volume.
World
Scientific, 2000, pp. 185–234.
74
[76] X. Chen and A. Vishwanath, “Towards gauging time-reversal symmetry: A tensor
network approach,” Phys. Rev. X, vol. 5, no. 4, p. 041 034, 2015.
[77] H. Watanabe, H. C. Po, A. Vishwanath, and M. Zaletel, “Filling constraints
for spin-orbit coupled insulators in symmorphic and nonsymmorphic crystals,”
Proc. Natl. Acad. Sci. USA, vol. 112, no. 47, pp. 14 551–14 556, 2015.
[78] G. Misguich, C. Lhuillier, M. Mambrini, and P. Sindzingre, “Degeneracy of the
ground-state of antiferromagnetic spin-1/2 Hamiltonians,” Eur. Phys. J. B, vol. 26,
p. 167, 2002.
[79] S. C. Furuya and Y. Horinouchi, “Translation constraints on quantum phases with
twisted boundary conditions,” Phys. Rev. B, vol. 100, no. 17, pp. 174 435–, Nov.
2019.
75
...
論文の公開元へ
