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On ℤ2-indices for ground states of fermionic chains

Chris Bourne Hermann Schulz Baldes 東北大学 DOI:10.1142/S0129055X20500282

2020.03.16

概要

For parity-conserving fermionic chains, we review how to associate Z2-indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the Z2-valued spectral flow provides a topological obstruction for two systems to have the same Z2-index. A rudimentary definition of a Z2-phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.

参考文献

[1] H. Araki, On quasifree states of CAR and Bogoliubov automorphisms, Publ. Res. Inst. Math.

Sci. 6, 385–442 (1970/71).

57

[2] H. Araki, D. E. Evans, On a C ∗ -algebra approach to phase transition in the two-dimensional

Ising model. Comm. Math. Phys., 91, no. 4, 489–503 (1983).

[3] H. Araki, T. Matsui, Ground states of the XY -model, Comm. Math. Phys., 101, 213–245 (1985).

[4] M. F. Atiyah, V. Patodi, I. M. Singer, Spectral asymmetry and Riemannian geometry, III. Proc.

Camb. Phil. Soc., 79, 71–99 (1976).

[5] M. F. Atiyah, I. M. Singer. Index theory for skew-adjoint Fredholm operators. Inst. Hautes Etudes

Sci. Publ. Math., 37, 5–26 (1969).

[6] J. Avron, R. Seiler, B. Simon, The index of a pair of projections, J. Funct. Anal., 120, 220–237

(1994).

[7] S. Bachmann, A. Bols, W. De Roeck, M. Fraas, Quantization of conductance in gapped interacting

systems, Ann. Henri Poincar´e, 19, no. 3, 695–708 (2018).

[8] S. Bachmann, A. Bols, W. De Roeck, M. Fraas, A many-body index for quantum charge transport,

Comm. Math. Phys., online first (2019). https://doi.org/10.1007/s00220-019-03537-x

[9] S. Bachmann, W. De Roeck, M. Fraas, The adiabatic theorem and linear response theory for

extended quantum systems, Comm. Math. Phys., 361, no. 3, 997–1027 (2018).

[10] S. Bachmann, S. Michalakis, B. Nachtergaele, R. Sims, Automorphic equivalence of gapped phases

of quantum lattice systems, Comm. Math. Phys., 309, no. 3, 835–871 (2012).

[11] S. Bachmann, B. Nachtergaele, On gapped phases with a continuous symmetry and boundary

operators, J. Stat. Phys. 154, 91–112 (2014).

[12] S. Bachmann, Y. Ogata, C 1 -classification of gapped parent Hamiltonians of quantum spin chains,

Comm. Math. Phys., 338, no. 3, 1011–1042 (2015).

[13] R. J. Baxter, Exactly solved models in statistical mechanics. (Academic Press, Inc. [Harcourt

Brace Jovanovich, Publishers], London, 1982).

[14] S. Beckus, J. Bellissard, Continuity of a the spectrum of a field of self-adjoint operators, Ann.

Henri Poincar´e, 17, 3425–3442 (2016).

[15] J. P. Blaizot, G. Ripka, Quantum Theory of Finite Fermi Systems, (MIT Press, Boston, 1985).

[16] C. Bourne, A. L. Carey, M. Lesch, A. Rennie, The KO-valued spectral flow for skew-adjoint

Fredholm operators, arXiv:1907.04981 (2019).

[17] O. Bratelli, D. R. Robinson, Operators Algebras and Quantum Statistical Mechanics 1, 2nd

edition, (Springer, Berlin, 1997).

[18] O. Bratelli, D. R. Robinson, Operators Algebras and Quantum Statistical Mechanics 2, 2nd

edition, (Springer, Berlin, 1997).

[19] J.-B. Bru, W. de Siqueira Pedra, Lieb-Robinson Bounds for Multi-Commutators and Applications

to Response Theory, (Springer Briefs in Mathematical Physics 13, 2017).

58

[20] N. Bultinck, D. J Williamson, J. Haegeman, F. Verstraete, Fermionic matrix product states and

one-dimensional topological phases, Phys. Rev. B, 95, 075108 (2017).

[21] N. Bultinck, D. J. Williamson, J. Haegeman, F. Verstraete, Fermionic projected entangled-pair

states and topological phases, J. Phys. A, 51, 025202 (2017).

[22] J. S. Calder´

on-Garc´ıa, A. F. Reyes-Lega, Majorana fermions and orthogonal complex structures,

Modern Phys. Lett. A, 33, no. 14, 1840001, 12 pages (2018).

[23] A. L. Carey, D. M. O’Brien, Automorphisms of the infinite-dimensional Clifford algebra and the

Atiyah-Singer mod 2 index, Topology, 22, 437–448 (1983).

[24] A. L. Carey, J. Phillips, H. Schulz-Baldes, Spectral flow for skew-adjoint Fredholm operators, J.

Spec. Theory, 9, no. 1, 137–170 (2019).

[25] A. L. Carey, H. Schulz-Baldes, Spectral flow of monopole insertion in topological insulators,

Comm. Math. Phys., 370, no. 3, 895–923 (2019).

[26] M. Cha, P. Naaijkens, B. Nachtergaele, On the stability of charges in infinite quantum spin

systems, Comm. Math. Phys., 373, 219–264 (2020).

[27] G. De Nittis, H. Schulz-Baldes, Spectral flows associated to flux tubes, Ann. Henri Poincar´e, 17,

1–35 (2016).

[28] G. De Nittis, H. Schulz-Baldes, The non-commutative topology of two-dimensional dirty superconductors, J. Geometry and Physics, 124, 100–123 (2018).

[29] J. Dixmier, C ∗ -Algebras, (North Holland Publishing Company, Amsterdam, 1977).

[30] N. Doll, H. Schulz-Baldes, N. Waterstraat, Parity as Z2 -valued spectral flow, Bull. London Math.

Soc., 51, 836–852 (2019).

[31] S. Doplicher, R. Longo, Standard and split inclusions of von Neumann algebras. Invent. Math.,

75, 493–536 (1984).

[32] D. E. Evans, Y. Kawahigashi, Quantum Symmetries and Operator Algebras, (Oxford University

Press, Oxford, 1998).

[33] L. Fidkowski, A. Kitaev, Topological phases of fermions in one dimension, Phys. Rev. B, 83,

075103 (2011).

[34] D. S. Freed, M. J. Hopkins, Reflection positivity and invertible topological phases, Geom. Topol.,

to appear, arXiv:1604.06527.

[35] A. Giuliani, V. Mastropietro, M. Porta, Universality of the Hall conductivity in interacting electron systems, Comm. Math. Phys. 349, no. 3, 1107–1161 (2017): .

[36] J. Großmann and H. Schulz-Baldes, Index pairings in presence of symmetries with applications

to topological insulators, Comm. Math. Phys., 343, no. 2, 477–513 (2016).

[37] A. Kapustin, A. Turzillo, M. You, Spin topological field theory and fermionic matrix product

states, Phys. Rev. B, 98, 125101 (2018).

59

[38] M. B. Hastings, An area law for one-dimensional quantum systems, J. Stat. Mech. Theory Exp.,

no. 8, P08024, 14 pages (2007).

[39] M. B. Hastings, T. Koma, Spectral gap and exponential decay of correlations, Comm. Math.

Phys., 265, 781–804 (2006).

[40] M. B. Hastings, S. Michalakis, Quantization of Hall conductance for interacting electrons on a

torus, Comm. Math. Phys., 334, 433–471 (2015).

[41] T. Kato, Perturbation Theory for Linear Operators, 2nd edition, (Springer-Verlag, Berlin, 1976).

[42] H. Katsura, D. Schuricht, M. Takahashi. Exact ground states and topological order in interacting

Kitaev/Majorana chains. Phys. Rev. B, 92, 115137 (2015).

[43] K. Kawabata, R. Kobayashi, N. Wu, H. Katsura. Exact zero modes in twisted Kitaev chains.

Phys. Rev. B, 95, 195140 (2017).

[44] A. Kitaev, Unpaired Majorana fermions in quantum wires, Physics-Uspekhi, 44, 131 (2001).

[45] A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Physics, 321, no. 1, 2–111

(2006).

[46] J. Kurmann, H. Thomas, G. M¨

uller, Antiferromagnetic long-range order in the anisotropic quantum spin chain, Physica A, 112, 235 (1982).

[47] T. Matsui, The split property and symmetry breaking of the quantum spin chain, Comm. Math.

Phys., 218, 393–416 (2001).

[48] T. Matsui, Boundedness of entanglement entropy and split property of quantum spin chains, Rev.

Math. Phys., 26, no. 9, 1350017 (2013).

[49] S. Michalakis, J. P. Zwolak, Stability of frustration-free Hamiltonians, Comm. Math. Phys., 322,

no. 2, 277–302 (2013).

[50] D. Monaco, S. Teufel, Adiabatic currents for interacting fermions on a lattice, Rev. Math. Phys.,

31, no. 3, 1950009 (2019).

[51] A. Moon, Automorphic equivalence preserves the split property, J. Funct. Anal., 277, no. 10,

3653–3672 (2019).

[52] A. Moon, B. Nachtergaele, Stability of gapped ground state phases of spins and fermions in one

dimension, J. Math. Phys., 59, 091415 (2018).

[53] B. Nachtergaele, R. Sims, A. Young, Lieb–Robinson bounds, the spectral flow, and stability of

the spectral gap for lattice fermion systems, in Mathematical Problems in Quantum Physics, Vol.

717 of Contemp. Math., (Amer. Math. Soc., Providence, 2018).

[54] B. Nachtergaele, R. Sims, A. Young, Quasi-locality bounds for quantum lattice systems. Part 1.

Lieb–Robinson bounds, quasi-local maps, and spectral flow automorphisms, J. Math. Phys., 60,

061101 (2019).

60

[55] P. T. Nam, M. Napiorkowski, J. P. Solovej, Diagonalization of bosonic quadratic Hamiltonians

by Bogoliubov transformations, J. Funct. Anal., 270, 4340–4368 (2016).

[56] Y. Ogata, A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization III, Comm. Math. Phys., 352, no. 3, 1205–1263 (2017).

[57] Y. Ogata, A Z2 -index of symmetry protected topological phases with time reversal symmetry

for quantum spin chains, Comm. Math. Phys., online first (2019). https://doi.org/10.1007/

s00220-019-03521-5

[58] Y. Ogata, A Z2 -index of symmetry protected topological phases with reflection symmetry for

quantum spin chains, arXiv:1904.01669 (2019).

[59] Y. Ogata, A classification of pure states on quantum spin chains satisfying the split property with

on-site finite group symmetries, arXiv:1908.08621 (2019).

[60] Y. Ogata, H. Tasaki, Lieb-Schultz-Mattis type theorems for quantum spin chains without continuous symmetry, Comm. Math. Phys., 372, 951–962 (2019).

[61] J. Phillips, Self-adjoint Fredholm operators and spectral flow. Canad. Math. Bull., 39, 460–467

(1996).

[62] E. Prodan, H. Schulz-Baldes, Bulk and Boundary Invariants for Complex Topological Insulators:

From K-Theory to Physics, (Springer International, Cham, 2016).

[63] D. Shale, W. F. Stinespring, Spinor representations of infinite orthogonal groups, J. Math. Mech.,

14, 315–322 (1965).

[64] K. Shiozaki, H. Shapourian, S. Ryu, Many-body topological invariants in fermionic symmetryprotected topological phases: Cases of point group symmetries, Phys. Rev. B, 95, 205139 (2017).

[65] M. Takesaki, Theory of Operator Algebras I, (Springer, Berlin, 2002).

[66] A. Turzillo, M. You, Fermionic matrix product states and one-dimensional short-range entangled

phases with antiunitary symmetries, Phys. Rev. B, 99, 035103 (2019).

[67] A. Wassermann, Operator algebras and conformal field theory. III. Fusion of positive energy

representations of LSU(N ) using bounded operators, Invent. Math., 133, no. 3, 467–538 (1998).

[68] D. P. Williams, Crossed Products of C ∗ -algebras, (American Mathematical Society, Rhode Island,

2007).

[69] C. Z. Xiong, Minimalist approach to the classification of symmetry protected topological phases,

J. Phys. A, 51, no. 44, 445001, 71 pages (2018).

61

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