Microscopic diagnosis of universal geometric responses in fractional quantum Hall liquids

Hu, Liangdong; Liu, Zhao; Sheng, D. N.; Haldane, F. D.  M.; Zhu, W.

doi:10.1103/physrevb.103.085103

Microscopic diagnosis of universal geometric responses in fractional quantum Hall liquids

Journal Article · Mon Jan 31 23:00:00 EST 2022 · Physical Review. B

DOI:https://doi.org/10.1103/physrevb.103.085103· OSTI ID:1850792

Hu, Liangdong ^[1]; ^[2]; Sheng, D. N. ^[3]; Haldane, F. D. M. ^[4]; Zhu, W. ^[5]

Westlake Univ., Hangzhou (China); OSTI
Zhejiang Univ., Hangzhou (China)
California State University, Northridge (CSUN), CA (United States)
Princeton Univ., NJ (United States)
Westlake Univ., Hangzhou (China)

Topological quantum liquids contain internal degrees of freedom that are coupled to geometric response. Yet, an explicit and microscopic identification of the geometric response remains difficult. Here, taking notable fractional quantum Hall (FQH) states as typical examples, we systematically investigate a promising protocol—the Dehn-twist deformation on the torus geometry—to probe the geometric response of correlated topological states and establish the relation between such response and universal properties of pertinent states. Based on analytical derivations and numerical simulations, we find that the geometry-induced Berry phase encodes features for a broad class of FQH states at the Laughlin, hierarchy, Halperin and non-Abelian Moore-Read fillings. Our findings conclusively demonstrate that the adiabatic Dehn-twist deformation can faithfully capture rich geometric and topological information, including the Hall viscosity and topological spin of the pertinent FQH state and the chiral central charge of the underlying edge conformal field theory. Our approach provides a powerful way to reveal topological orders of generic FQH states and address previously open questions.

View Accepted Manuscript (DOE)

Research Organization:: Princeton University, NJ (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: SC0002140

OSTI ID:: 1850792

Journal Information:: Physical Review. B, Journal Name: Physical Review. B Journal Issue: 8 Vol. 103; ISSN 2469-9950; ISSN PRBMDO

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (80)

Conformal Field Theory Di Francesco, Philippe; Mathieu, Pierre; Sénéchal, David Graduate Texts in Contemporary Physics https://doi.org/10.1007/978-1-4612-2256-9	book	January 1997
Hall viscosity from gauge/gravity duality Saremi, Omid; Son, Dam Thanh Journal of High Energy Physics, Vol. 2012, Issue 4 https://doi.org/10.1007/JHEP04(2012)091	journal	April 2012
Holographic spontaneous parity breaking and emergent hall viscosity and angular momentum Son, Dam Thanh; Wu, Chaolun Journal of High Energy Physics, Vol. 2014, Issue 7 https://doi.org/10.1007/JHEP07(2014)076	journal	July 2014
Nonabelions in the fractional quantum hall effect Moore, Gregory; Read, Nicholas Nuclear Physics B, Vol. 360, Issue 2-3 https://doi.org/10.1016/0550-3213(91)90407-O	journal	August 1991
Fault-tolerant quantum computation by anyons Kitaev, A. Yu. Annals of Physics, Vol. 303, Issue 1 https://doi.org/10.1016/S0003-4916(02)00018-0	journal	January 2003
Anyons in an exactly solved model and beyond Kitaev, Alexei Annals of Physics, Vol. 321, Issue 1 https://doi.org/10.1016/j.aop.2005.10.005	journal	January 2006
Geometry of quantum Hall states: Gravitational anomaly and transport coefficients Can, Tankut; Laskin, Michael; Wiegmann, Paul B. Annals of Physics, Vol. 362 https://doi.org/10.1016/j.aop.2015.02.013	journal	November 2015
Fractional quantum Hall physics in topological flat bands Parameswaran, Siddharth A.; Roy, Rahul; Sondhi, Shivaji L. Comptes Rendus Physique, Vol. 14, Issue 9-10 https://doi.org/10.1016/j.crhy.2013.04.003	journal	November 2013
Observation of neutral modes in the fractional quantum Hall regime Bid, Aveek; Ofek, N.; Inoue, H. Nature, Vol. 466, Issue 7306 https://doi.org/10.1038/nature09277	journal	July 2010
Observed quantization of anyonic heat flow Banerjee, Mitali; Heiblum, Moty; Rosenblatt, Amir Nature, Vol. 545, Issue 7652 https://doi.org/10.1038/nature22052	journal	April 2017
Proliferation of neutral modes in fractional quantum Hall states Inoue, Hiroyuki; Grivnin, Anna; Ronen, Yuval Nature Communications, Vol. 5, Issue 1 https://doi.org/10.1038/ncomms5067	journal	June 2014
Observation of a transition from a topologically ordered to a spontaneously broken symmetry phase Samkharadze, N.; Schreiber, K. A.; Gardner, G. C. Nature Physics, Vol. 12, Issue 2 https://doi.org/10.1038/nphys3523	journal	October 2015
Synthesizing a ν=2/3 fractional quantum Hall effect edge state from counter-propagating ν=1 and ν=1/3 states Cohen, Yonatan; Ronen, Yuval; Yang, Wenmin Nature Communications, Vol. 10, Issue 1 https://doi.org/10.1038/s41467-019-09920-5	journal	April 2019
Observation of half-integer thermal Hall conductance Banerjee, Mitali; Heiblum, Moty; Umansky, Vladimir Nature, Vol. 559, Issue 7713 https://doi.org/10.1038/s41586-018-0184-1	journal	June 2018
Fractional (Chern and topological) insulators Neupert, Titus; Chamon, Claudio; Iadecola, Thomas Physica Scripta, Vol. T164 https://doi.org/10.1088/0031-8949/2015/T164/014005	journal	August 2015
Probing the geometry of the Laughlin state Johri, Sonika; Papić, Z.; Schmitteckert, P. New Journal of Physics, Vol. 18, Issue 2 https://doi.org/10.1088/1367-2630/18/2/025011	journal	February 2016
Degeneracy of non-Abelian quantum Hall states on the torus: domain walls and conformal field theory Ardonne, Eddy; Bergholtz, Emil J.; Kailasvuori, Janik Journal of Statistical Mechanics: Theory and Experiment, Vol. 2008, Issue 04 https://doi.org/10.1088/1742-5468/2008/04/P04016	journal	April 2008
Quantum Hall states and conformal field theory on a singular surface Can, T.; Wiegmann, P. Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 49 https://doi.org/10.1088/1751-8121/aa91dc	journal	November 2017
Quantized Hall conductivity in two dimensions Laughlin, R. B. Physical Review B, Vol. 23, Issue 10 https://doi.org/10.1103/PhysRevB.23.5632	journal	May 1981
Quantized Hall conductance as a topological invariant Niu, Qian; Thouless, D. J.; Wu, Yong-Shi Physical Review B, Vol. 31, Issue 6 https://doi.org/10.1103/PhysRevB.31.3372	journal	March 1985
Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces Wen, X. G.; Niu, Q. Physical Review B, Vol. 41, Issue 13 https://doi.org/10.1103/PhysRevB.41.9377	journal	May 1990
Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states and p x + i p y paired superfluids Read, N. Physical Review B, Vol. 79, Issue 4 https://doi.org/10.1103/PhysRevB.79.045308	journal	January 2009
Particle-hole symmetry breaking and the ν = 5 2 fractional quantum Hall effect Wang, Hao; Sheng, D. N.; Haldane, F. D. M. Physical Review B, Vol. 80, Issue 24 https://doi.org/10.1103/PhysRevB.80.241311	journal	December 2009
Hall viscosity, orbital spin, and geometry: Paired superfluids and quantum Hall systems Read, N.; Rezayi, E. H. Physical Review B, Vol. 84, Issue 8 https://doi.org/10.1103/PhysRevB.84.085316	journal	August 2011
Edge-mode combinations in the entanglement spectra of non-Abelian fractional quantum Hall states on the torus Liu, Zhao; Bergholtz, Emil J.; Fan, Heng Physical Review B, Vol. 85, Issue 4 https://doi.org/10.1103/PhysRevB.85.045119	journal	January 2012
Emergent many-body translational symmetries of Abelian and non-Abelian fractionally filled topological insulators Bernevig, B. Andrei; Regnault, N. Physical Review B, Vol. 85, Issue 7 https://doi.org/10.1103/PhysRevB.85.075128	journal	February 2012
Band mass anisotropy and the intrinsic metric of fractional quantum Hall systems Yang, Bo; Papić, Z.; Rezayi, E. H. Physical Review B, Vol. 85, Issue 16 https://doi.org/10.1103/PhysRevB.85.165318	journal	April 2012
Quasiparticle statistics and braiding from ground-state entanglement Zhang, Yi; Grover, Tarun; Turner, Ari Physical Review B, Vol. 85, Issue 23 https://doi.org/10.1103/PhysRevB.85.235151	journal	June 2012
Minimal entangled states and modular matrix for fractional quantum Hall effect in topological flat bands Zhu, W.; Sheng, D. N.; Haldane, F. D. M. Physical Review B, Vol. 88, Issue 3 https://doi.org/10.1103/PhysRevB.88.035122	journal	July 2013
Evidence for a ν = 5 / 2 fractional quantum Hall nematic state in parallel magnetic fields Liu, Yang; Hasdemir, S.; Shayegan, M. Physical Review B, Vol. 88, Issue 3 https://doi.org/10.1103/PhysRevB.88.035307	journal	July 2013
Momentum polarization: An entanglement measure of topological spin and chiral central charge Tu, Hong-Hao; Zhang, Yi; Qi, Xiao-Liang Physical Review B, Vol. 88, Issue 19 https://doi.org/10.1103/PhysRevB.88.195412	journal	November 2013
Hall viscosity of hierarchical quantum Hall states Fremling, M.; Hansson, T. H.; Suorsa, J. Physical Review B, Vol. 89, Issue 12 https://doi.org/10.1103/PhysRevB.89.125303	journal	March 2014
Electromagnetic and gravitational responses of two-dimensional noninteracting electrons in a background magnetic field Abanov, Alexander G.; Gromov, Andrey Physical Review B, Vol. 90, Issue 1 https://doi.org/10.1103/PhysRevB.90.014435	journal	July 2014
Guiding-center Hall viscosity and intrinsic dipole moment along edges of incompressible fractional quantum Hall fluids Park, YeJe; Haldane, F. D. M. Physical Review B, Vol. 90, Issue 4 https://doi.org/10.1103/PhysRevB.90.045123	journal	July 2014
Geometry of fractional quantum Hall fluids Cho, Gil Young; You, Yizhi; Fradkin, Eduardo Physical Review B, Vol. 90, Issue 11 https://doi.org/10.1103/PhysRevB.90.115139	journal	September 2014
Infinite density matrix renormalization group for multicomponent quantum Hall systems Zaletel, Michael P.; Mong, Roger S. K.; Pollmann, Frank Physical Review B, Vol. 91, Issue 4 https://doi.org/10.1103/PhysRevB.91.045115	journal	January 2015
Low-energy effective theory in the bulk for transport in a topological phase Bradlyn, Barry; Read, N. Physical Review B, Vol. 91, Issue 12 https://doi.org/10.1103/PhysRevB.91.125303	journal	March 2015
Topological characterization of the non-Abelian Moore-Read state using density-matrix renormalization group Zhu, W.; Gong, S. S.; Haldane, F. D. M. Physical Review B, Vol. 92, Issue 16 https://doi.org/10.1103/PhysRevB.92.165106	journal	October 2015
Phase diagram of ν = 1 2 + 1 2 bilayer bosons with interlayer couplings Liu, Zhao; Vaezi, Abolhassan; Repellin, Cécile Physical Review B, Vol. 93, Issue 8 https://doi.org/10.1103/PhysRevB.93.085115	journal	February 2016
Torsional anomalies, Hall viscosity, and bulk-boundary correspondence in topological states Hughes, Taylor L.; Leigh, Robert G.; Parrikar, Onkar Physical Review D, Vol. 88, Issue 2 https://doi.org/10.1103/PhysRevD.88.025040	journal	July 2013
Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States Li, Hui; Haldane, F. D. M. Physical Review Letters, Vol. 101, Issue 1 https://doi.org/10.1103/PhysRevLett.101.010504	journal	July 2008
Finite-Layer Thickness Stabilizes the Pfaffian State for the 5/2 Fractional Quantum Hall Effect: Wave Function Overlap and Topological Degeneracy Peterson, Michael. R.; Jolicoeur, Th.; Das Sarma, S. Physical Review Letters, Vol. 101, Issue 1 https://doi.org/10.1103/PhysRevLett.101.016807	journal	July 2008
Fractional Quantum Hall State at ν = 5 2 and the Moore-Read Pfaffian Storni, M.; Morf, R. H.; Das Sarma, S. Physical Review Letters, Vol. 104, Issue 7 https://doi.org/10.1103/PhysRevLett.104.076803	journal	February 2010
Geometrical Description of the Fractional Quantum Hall Effect Haldane, F. D. M. Physical Review Letters, Vol. 107, Issue 11 https://doi.org/10.1103/PhysRevLett.107.116801	journal	September 2011
Hall Viscosity and Electromagnetic Response Hoyos, Carlos; Son, Dam Thanh Physical Review Letters, Vol. 108, Issue 6 https://doi.org/10.1103/PhysRevLett.108.066805	journal	February 2012
Characterizing Topological Order by Studying the Ground States on an Infinite Cylinder Cincio, L.; Vidal, G. Physical Review Letters, Vol. 110, Issue 6 https://doi.org/10.1103/PhysRevLett.110.067208	journal	February 2013
Topological Characterization of Fractional Quantum Hall Ground States from Microscopic Hamiltonians Zaletel, Michael P.; Mong, Roger S. K.; Pollmann, Frank Physical Review Letters, Vol. 110, Issue 23 https://doi.org/10.1103/PhysRevLett.110.236801	journal	June 2013
Identifying Non-Abelian Topological Order through Minimal Entangled States Zhu, W.; Gong, S. S.; Haldane, F. D. M. Physical Review Letters, Vol. 112, Issue 9 https://doi.org/10.1103/PhysRevLett.112.096803	journal	March 2014
Density-Curvature Response and Gravitational Anomaly Gromov, Andrey; Abanov, Alexander G. Physical Review Letters, Vol. 113, Issue 26 https://doi.org/10.1103/PhysRevLett.113.266802	journal	December 2014
Thermal Hall Effect and Geometry with Torsion Gromov, Andrey; Abanov, Alexander G. Physical Review Letters, Vol. 114, Issue 1 https://doi.org/10.1103/PhysRevLett.114.016802	journal	January 2015
Framing Anomaly in the Effective Theory of the Fractional Quantum Hall Effect Gromov, Andrey; Cho, Gil Young; You, Yizhi Physical Review Letters, Vol. 114, Issue 1 https://doi.org/10.1103/PhysRevLett.114.016805	journal	January 2015
Universal Wave-Function Overlap and Universal Topological Data from Generic Gapped Ground States Moradi, Heidar; Wen, Xiao-Gang Physical Review Letters, Vol. 115, Issue 3 https://doi.org/10.1103/PhysRevLett.115.036802	journal	July 2015
Geometric Adiabatic Transport in Quantum Hall States Klevtsov, S.; Wiegmann, P. Physical Review Letters, Vol. 115, Issue 8 https://doi.org/10.1103/PhysRevLett.115.086801	journal	August 2015
Stabilization of the Particle-Hole Pfaffian Order by Landau-Level Mixing and Impurities That Break Particle-Hole Symmetry Zucker, P. T.; Feldman, D. E. Physical Review Letters, Vol. 117, Issue 9 https://doi.org/10.1103/PhysRevLett.117.096802	journal	August 2016
Emergent Conformal Symmetry and Geometric Transport Properties of Quantum Hall States on Singular Surfaces Can, T.; Chiu, Y. H.; Laskin, M. Physical Review Letters, Vol. 117, Issue 26 https://doi.org/10.1103/PhysRevLett.117.266803	journal	December 2016
Hydrodynamic Electron Flow and Hall Viscosity Scaffidi, Thomas; Nandi, Nabhanila; Schmidt, Burkhard Physical Review Letters, Vol. 118, Issue 22 https://doi.org/10.1103/PhysRevLett.118.226601	journal	June 2017
Investigating Anisotropic Quantum Hall States with Bimetric Geometry Gromov, Andrey; Geraedts, Scott D.; Bradlyn, Barry Physical Review Letters, Vol. 119, Issue 14 https://doi.org/10.1103/PhysRevLett.119.146602	journal	October 2017
Unified Description of the Classical Hall Viscosity Holder, Tobias; Queiroz, Raquel; Stern, Ady Physical Review Letters, Vol. 123, Issue 10 https://doi.org/10.1103/PhysRevLett.123.106801	journal	September 2019
Two-Dimensional Magnetotransport in the Extreme Quantum Limit Tsui, D. C.; Stormer, H. L.; Gossard, A. C. Physical Review Letters, Vol. 48, Issue 22 https://doi.org/10.1103/PhysRevLett.48.1559	journal	May 1982
Quantized Hall Conductance in a Two-Dimensional Periodic Potential Thouless, D. J.; Kohmoto, M.; Nightingale, M. P. Physical Review Letters, Vol. 49, Issue 6 https://doi.org/10.1103/PhysRevLett.49.405	journal	August 1982
Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations Laughlin, R. B. Physical Review Letters, Vol. 50, Issue 18 https://doi.org/10.1103/PhysRevLett.50.1395	journal	May 1983
Fractional Quantization of the Hall Effect: A Hierarchy of Incompressible Quantum Fluid States Haldane, F. D. M. Physical Review Letters, Vol. 51, Issue 7 https://doi.org/10.1103/PhysRevLett.51.605	journal	August 1983
Fractional Statistics and the Quantum Hall Effect Arovas, Daniel; Schrieffer, J. R.; Wilczek, Frank Physical Review Letters, Vol. 53, Issue 7 https://doi.org/10.1103/PhysRevLett.53.722	journal	August 1984
Many-Particle Translational Symmetries of Two-Dimensional Electrons at Rational Landau-Level Filling Haldane, F. D. M. Physical Review Letters, Vol. 55, Issue 20 https://doi.org/10.1103/PhysRevLett.55.2095	journal	November 1985
Edge channels for the fractional quantum Hall effect Beenakker, C. W. J. Physical Review Letters, Vol. 64, Issue 2 https://doi.org/10.1103/PhysRevLett.64.216	journal	January 1990
Paired Hall state at half filling Greiter, Martin; Wen, Xiao-Gang; Wilczek, Frank Physical Review Letters, Vol. 66, Issue 24 https://doi.org/10.1103/PhysRevLett.66.3205	journal	June 1991
Transport evidence for phase separation into spatial regions of different fractional quantum Hall fluids near the boundary of a two-dimensional electron gas Chang, A. M.; Cunningham, J. E. Physical Review Letters, Vol. 69, Issue 14 https://doi.org/10.1103/PhysRevLett.69.2114	journal	October 1992
Shift and spin vector: New topological quantum numbers for the Hall fluids Wen, X. G.; Zee, A. Physical Review Letters, Vol. 69, Issue 6 https://doi.org/10.1103/PhysRevLett.69.953	journal	August 1992
Randomness at the edge: Theory of quantum Hall transport at filling ν=2/3 Kane, C. L.; Fisher, Matthew P. A.; Polchinski, J. Physical Review Letters, Vol. 72, Issue 26 https://doi.org/10.1103/PhysRevLett.72.4129	journal	June 1994
Viscosity of Quantum Hall Fluids Avron, J. E.; Seiler, R.; Zograf, P. G. Physical Review Letters, Vol. 75, Issue 4 https://doi.org/10.1103/PhysRevLett.75.697	journal	July 1995
Transition from Quantum Hall to Compressible States in the Second Landau Level: New Light on the ν = 5 / 2 Enigma Morf, R. H. Physical Review Letters, Vol. 80, Issue 7 https://doi.org/10.1103/PhysRevLett.80.1505	journal	February 1998
Incompressible Paired Hall State, Stripe Order, and the Composite Fermion Liquid Phase in Half-Filled Landau Levels Rezayi, E. H.; Haldane, F. D. M. Physical Review Letters, Vol. 84, Issue 20 https://doi.org/10.1103/PhysRevLett.84.4685	journal	May 2000
Topological Entanglement Entropy Kitaev, Alexei; Preskill, John Physical Review Letters, Vol. 96, Issue 11 https://doi.org/10.1103/PhysRevLett.96.110404	journal	March 2006
Detecting Topological Order in a Ground State Wave Function Levin, Michael; Wen, Xiao-Gang Physical Review Letters, Vol. 96, Issue 11 https://doi.org/10.1103/PhysRevLett.96.110405	journal	March 2006
Particle-Hole Symmetry and the Pfaffian State Levin, Michael; Halperin, Bertrand I.; Rosenow, Bernd Physical Review Letters, Vol. 99, Issue 23 https://doi.org/10.1103/PhysRevLett.99.236806	journal	December 2007
Phase Diagram of the ν = 5 / 2 Fractional Quantum Hall Effect: Effects of Landau-Level Mixing and Nonzero Width Pakrouski, Kiryl; Peterson, Michael R.; Jolicoeur, Thierry Physical Review X, Vol. 5, Issue 2 https://doi.org/10.1103/PhysRevX.5.021004	journal	April 2015
Bimetric Theory of Fractional Quantum Hall States Gromov, Andrey; Son, Dam Thanh Physical Review X, Vol. 7, Issue 4 https://doi.org/10.1103/PhysRevX.7.041032	journal	November 2017
Non-Abelian anyons and topological quantum computation Nayak, Chetan; Simon, Steven H.; Stern, Ady Reviews of Modern Physics, Vol. 80, Issue 3 https://doi.org/10.1103/RevModPhys.80.1083	journal	September 2008
Topological flat band Models and Fractional Chern Insulators Bergholtz, Emil J.; Liu, Zhao International Journal of Modern Physics B, Vol. 27, Issue 24 https://doi.org/10.1142/S021797921330017X	journal	September 2013
The Ground State Structure and Modular Transformations of Fractional Quantum hall States on a Torus Keski-Vakkuri, Esko; Wen, Xiao-Gang International Journal of Modern Physics B, Vol. 07, Issue 25 https://doi.org/10.1142/S0217979293003644	journal	November 1993

Similar Records

Fractional quantum Hall bilayers at half filling: Tunneling-driven non-Abelian phase

Journal Article · Thu Dec 29 23:00:00 EST 2016 · Physical Review. B · OSTI ID:1535814

Effective field theory and projective construction for Z{sub k} parafermion fractional quantum Hall states

Journal Article · Thu Apr 15 00:00:00 EDT 2010 · Physical Review. B, Condensed Matter and Materials Physics · OSTI ID:21366726

Investigating Anisotropic Quantum Hall States with Bimetric Geometry

Journal Article · Thu Oct 05 00:00:00 EDT 2017 · Physical Review Letters · OSTI ID:1541231

Related Subjects

75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
Exact diagonalization
Fractional quantum Hall effect
Geometric & topological phases
Materials Science
Non-Abelian models
Physics

Microscopic diagnosis of universal geometric responses in fractional quantum Hall liquids

Citation Formats

References (80)

Similar Records

Related Subjects