New phases of quantum matter and topology
Nobel Prize in Physics for 2016
DOI:
https://doi.org/10.15407/visn2016.12.063Keywords:
topological phase transitions, topological phases, topological invariants, Hall conductivity, spin chainsAbstract
In 2016 the Nobel Committee in Physics awarded Prof. D.J. Thouless (1/2 prize), Prof. J.M. Kosterlitz (1/4 prize) and Prof. F.D.M. Haldane (1/4 prize) for “theoretical discovery of topological phase transitions and topological phases of matter”. We discuss new ideas and the results of papers where (i) topological phase transitions (Berezinskii–Kosterlitz–Thouless phase transitions) in two-dimensional condensed matter were theoretically predicted, (ii) a deep connection between quantization of the Hall conductivity in 2D systems with violated T-invariance and topological quantities (Chern invariant) was revealed (Thouless–Kohmoto–Nightingale–den Nijs), (iii) new quantum phase (Haldane phase) of spin chains with integer spin was predicted. Main attention was given to qualitative explanation of the predicted new phenomena. We follow the interconnections between the cited works of Nobel laureates and low-dimensional models of relativistic quantum field theory where the crucial role of topological invariants in the special phases of quantum matter was first noted.
References
Peierls R. Bemerkungen über Umwandlungstemperaturen (Remarks on transition temperatures). Helv. Phys. Acta. 1934. 7(Suppl. 2): 81.
Landau L.D. By the theory of phase transitions. JETP. 1937. 7:627.
Mermin N.D., Wagner H. Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. 1966. 17(26): 1133.https://doi.org/10.1103/PhysRevLett.17.1133
Hohenberg P.C. Existence of long-range order in one and two dimensions. Phys. Rev. 1967. 158(2): 383.https://doi.org/10.1103/PhysRev.158.383
Bogolyubov N.N. Quasi-averages in problems of statistical mechanics. (Dubna: JINR, 1961).
Stanley H.E., Kaplan T.A. Possibility of a phase transition for the two-dimensional Heisenberg model. Phys. Rev. Lett. 1966. 17(17): 913.https://doi.org/10.1103/PhysRevLett.17.913
Stanley H.E. Dependence of critical properties on dimensionality of spins. Phys. Rev. Lett. 1968. 20(12): 589.https://doi.org/10.1103/PhysRevLett.20.589
Moore M.A. Additional evidence for a phase transition in the plane-rotator and classical Heisenberg models for two-dimensional lattices. Phys. Rev. Lett. 1969. 23(15): 861.https://doi.org/10.1103/PhysRevLett.23.861
Berezinskii V.L. Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group. I. Classical systems. JETP. 1971. 32(3): 493.
Berezinskii V.L. Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. Quantum systems. JETP. 1972. 34(3): 610.
Kosterlitz J.M., Thouless D.J. Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory). J. Phys. C. 1972. 5(11): 124.https://doi.org/10.1088/0022-3719/5/11/002
Kosterlitz J.M., Thouless D.J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C. 1973. 6(7): 1181.https://doi.org/10.1088/0022-3719/6/7/010
Halperin B.I., Nelson D.R. Theory of two-dimensional melting. Phys. Rev. Lett. 1978. 41(2): 519.https://doi.org/10.1103/PhysRevLett.41.519
Young A.P. Melting and the vector Coulomb gas in two dimensions. Phys. Rev. B. 1979. 19(4):1855.https://doi.org/10.1103/PhysRevB.19.1855
Beasley M.R., Mooij J.E., Orlando T.P. Possibility of vortex-antivortex pair dissociation in two-dimensional superconductors. Phys Rev. Lett. 1979. 42(17): 1165.https://doi.org/10.1103/PhysRevLett.42.1165
Doniach S., Huberman B.A. Topological excitations in two-dimensional superconductors. Phys. Rev. Lett. 1979. 42(17): 1169.https://doi.org/10.1103/PhysRevLett.42.1169
von Klitzing K., Dorda G., Pepper M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 1980. 45(6): 494.https://doi.org/10.1103/PhysRevLett.45.494
von Klitzing K. The quantized Hall effect. In: Nobel Lectures in Physics 1981–1990. (World Scientific Publishing Company, 1993).
Tsui D.C., Stormer H.L., Gossard A.C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 1982. 48(22): 1559.https://doi.org/10.1103/PhysRevLett.48.1559
Laughlin R.B. Quantized Hall conductivity in two dimensions. Phys. Rev. B. 1981. 23(10): 5632.https://doi.org/10.1103/PhysRevB.23.5632
Thouless D.J., Kohmoto M., Nightingale M.P., den Nijs M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 1982. 49(6): 405.https://doi.org/10.1103/PhysRevLett.49.405
Haldane F.D.M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the "Parity Anomaly". Phys. Rev. Lett. 1988. 61(18): 2015.https://doi.org/10.1103/PhysRevLett.61.2015
Chang C.-Z., Zhang J., Feng X., Shen J., Zhang Z., Guo M., Li K., Ou Y., Wei P., Wang L.-L., Ji Z.-Q., Feng Y., Ji S., Chen X., Jia J., Dai X., Fang Z., Zhang S.-C., He K., Wang Y., Lu L., Ma X.-C., Xue Q.-K. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science. 2013. 340(6129): 167.https://doi.org/10.1126/science.1234414
Chang C.-Z., Zhao W., Kim D.Y., Zhang H., Assaf B.A., Heiman D., Zhang S.-C., Liu C., Chan M.H.W., Moodera J.S. High-precision realization of robust quantum anomalous Hall state in a hard ferromagnetic topological insulator. Nat. Mater. 2015. 14: 473.https://doi.org/10.1038/nmat4204
Jotzu G., Messer M., Desbuquois R., Lebrat M., Uehlinger T., Greif D., Esslinger T. Experimental realization of the topological Haldane model with ultracold fermions. Nature. 2014. 515: 237.https://doi.org/10.1038/nature13915
Haldane F.D.M. Continuum dynamics of the 1-D Heisenberg antiferromagnet: identification with the O(3) nonlinear sigma model. Phys. Lett. A. 1983. 93(9): 464.https://doi.org/10.1016/0375-9601(83)90631-X
Haldane F.D.M. Nonlinear field theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis Neel state. Phys. Rev. Lett. 1983. 50(15): 1153.https://doi.org/10.1103/PhysRevLett.50.1153
Bethe H. Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Physik. 1931. 71: 205.https://doi.org/10.1007/BF01341708
Belavin A.A., Polyakov A.M. Metastable states of two-dimensional isotropic ferromagnet. JETP Letters. 1975. 22(10): 245.
Pohlmeyer K. Integrable Hamiltonian systems and interactions through quadratic constraints. Commun. Math. Phys. 1976. 46(3): 207.https://doi.org/10.1007/BF01609119
Polyakov A.M. Hidden symmetry of the two-dimensional chiral fields. Phys. Lett. B. 1977. 72(2): 224.https://doi.org/10.1016/0370-2693(77)90707-9
Coleman S., Weinberg E. Radiative corrections as the origin of spontaneous symmetry breaking. Phys. Rev. D. 1973. 7: 1888.https://doi.org/10.1103/PhysRevD.7.1888
Krive I.V., Rozhavskii A.S. Fractional charge in quantum field theory and solid-state physics. Sov. Phys. Usp. 1987. 30: 370.https://doi.org/10.1070/PU1987v030n05ABEH002884
Polyakov A.M. Compact gauge fields and the infrared catastrophe. Phys. Lett. B. 1975. 59(1): 82.https://doi.org/10.1016/0370-2693(75)90162-8
Belavin A.A., Polyakov A.M., Tyupkin Yu.S., Schwartz A.S. Pseudoparticle solutions of the Yang-Mills equations. Phys. Lett. B. 1975. 59(1): 85.https://doi.org/10.1016/0370-2693(75)90163-X
Shankar R., Read N. The nonlinear sigma model is massless. Nucl. Phys. B. 1990. 336(3): 457.https://doi.org/10.1016/0550-3213(90)90437-I
Buyers W.J.L., Morra R.M., Armstrong R.L., Hogan M.J., Gerlach P., Hirakawa K. Experimental evidence for the Haldane gap in a spin-1 nearly isotropic antiferromagnetic chain. Phys. Rev. Lett. 1986. 56(4): 371.https://doi.org/10.1103/PhysRevLett.56.371
