Probing the geometry of the Laughlin state
Abstract
It has recently been pointed out that phases of matter with intrinsic topological order, like the fractional quantum Hall states, have an extra dynamical degree of freedom that corresponds to quantum geometry. Here we perform extensive numerical studies of the geometric degree of freedom for the simplest example of fractional quantumHall states—the filling v = 1/3 Laughlin state.We perturb the system by a smooth, spatially dependent metric deformation and measure the response of the Hall fluid, finding it to be proportional to the Gaussian curvature of the metric. Further, we generalize the concept of coherent states to formulate the bulk off-diagonal long range order for the Laughlin state, and compute the deformations of the metric in the vicinity of the edge of the system. We introduce a ‘pair amplitude’ operator and show that it can be used to numerically determine the intrinsic metric of the Laughlin state. Furthermore, these various probes are applied to several experimentally relevant settings that can expose the quantum geometry of the Laughlin state, in particular to systems with mass anisotropy and in the presence of an electric field gradient.
- Authors:
-
- Princeton Univ., Princeton, NJ (United States)
- Univ. of Leeds, Leeds (United Kingdom)
- Karlsruhe Institute of Technology, Eggenstien-Leopoldshafen (Gemany); Heinrich-Heine-Univ., Dusseldorf (Germany)
- Publication Date:
- Research Org.:
- Princeton Univ., NJ (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES)
- OSTI Identifier:
- 1255131
- Grant/Contract Number:
- SC0002140
- Resource Type:
- Accepted Manuscript
- Journal Name:
- New Journal of Physics
- Additional Journal Information:
- Journal Volume: 18; Journal Issue: 2; Journal ID: ISSN 1367-2630
- Publisher:
- IOP Publishing
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Citation Formats
Johri, Sonika, Papic, Z., Schmitteckert, P., Bhatt, R. N., and Haldane, F. D. M. Probing the geometry of the Laughlin state. United States: N. p., 2016.
Web. doi:10.1088/1367-2630/18/2/025011.
Johri, Sonika, Papic, Z., Schmitteckert, P., Bhatt, R. N., & Haldane, F. D. M. Probing the geometry of the Laughlin state. United States. https://doi.org/10.1088/1367-2630/18/2/025011
Johri, Sonika, Papic, Z., Schmitteckert, P., Bhatt, R. N., and Haldane, F. D. M. Fri .
"Probing the geometry of the Laughlin state". United States. https://doi.org/10.1088/1367-2630/18/2/025011. https://www.osti.gov/servlets/purl/1255131.
@article{osti_1255131,
title = {Probing the geometry of the Laughlin state},
author = {Johri, Sonika and Papic, Z. and Schmitteckert, P. and Bhatt, R. N. and Haldane, F. D. M.},
abstractNote = {It has recently been pointed out that phases of matter with intrinsic topological order, like the fractional quantum Hall states, have an extra dynamical degree of freedom that corresponds to quantum geometry. Here we perform extensive numerical studies of the geometric degree of freedom for the simplest example of fractional quantumHall states—the filling v = 1/3 Laughlin state.We perturb the system by a smooth, spatially dependent metric deformation and measure the response of the Hall fluid, finding it to be proportional to the Gaussian curvature of the metric. Further, we generalize the concept of coherent states to formulate the bulk off-diagonal long range order for the Laughlin state, and compute the deformations of the metric in the vicinity of the edge of the system. We introduce a ‘pair amplitude’ operator and show that it can be used to numerically determine the intrinsic metric of the Laughlin state. Furthermore, these various probes are applied to several experimentally relevant settings that can expose the quantum geometry of the Laughlin state, in particular to systems with mass anisotropy and in the presence of an electric field gradient.},
doi = {10.1088/1367-2630/18/2/025011},
journal = {New Journal of Physics},
number = 2,
volume = 18,
place = {United States},
year = {Fri Feb 05 00:00:00 EST 2016},
month = {Fri Feb 05 00:00:00 EST 2016}
}
Web of Science
Works referenced in this record:
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
Nobel Lecture: The fractional quantum Hall effect
journal, July 1999
- Stormer, Horst L.
- Reviews of Modern Physics, Vol. 71, Issue 4
Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations
journal, May 1983
- Laughlin, R. B.
- Physical Review Letters, Vol. 50, Issue 18
Fractional Statistics and the Quantum Hall Effect
journal, August 1984
- Arovas, Daniel; Schrieffer, J. R.; Wilczek, Frank
- Physical Review Letters, Vol. 53, Issue 7
Geometrical Description of the Fractional Quantum Hall Effect
journal, September 2011
- Haldane, F. D. M.
- Physical Review Letters, Vol. 107, Issue 11
Effective-Field-Theory Model for the Fractional Quantum Hall Effect
journal, January 1989
- Zhang, S. C.; Hansson, T. H.; Kivelson, S.
- Physical Review Letters, Vol. 62, Issue 1
Composite-fermion approach for the fractional quantum Hall effect
journal, July 1989
- Jain, J. K.
- Physical Review Letters, Vol. 63, Issue 2
Shift and spin vector: New topological quantum numbers for the Hall fluids
journal, August 1992
- Wen, X. G.; Zee, A.
- Physical Review Letters, Vol. 69, 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 quantized Hall effect
journal, July 1986
- von Klitzing, Klaus
- Reviews of Modern Physics, Vol. 58, Issue 3
Viscosity of Quantum Hall Fluids
journal, July 1995
- Avron, J. E.; Seiler, R.; Zograf, P. G.
- Physical Review Letters, Vol. 75, Issue 4
Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states and paired superfluids
journal, January 2009
- Read, N.
- Physical Review B, Vol. 79, Issue 4
Hall viscosity, orbital spin, and geometry: Paired superfluids and quantum Hall systems
journal, August 2011
- Read, N.; Rezayi, E. H.
- Physical Review B, Vol. 84, Issue 8
Guiding-center Hall viscosity and intrinsic dipole moment along edges of incompressible fractional quantum Hall fluids
journal, July 2014
- Park, YeJe; Haldane, F. D. M.
- Physical Review B, Vol. 90, Issue 4
Thermal Hall Effect and Geometry with Torsion
journal, January 2015
- Gromov, Andrey; Abanov, Alexander G.
- Physical Review Letters, Vol. 114, Issue 1
Geometric Adiabatic Transport in Quantum Hall States
journal, August 2015
- Klevtsov, S.; Wiegmann, P.
- Physical Review Letters, Vol. 115, Issue 8
Framing Anomaly in the Effective Theory of the Fractional Quantum Hall Effect
journal, January 2015
- Gromov, Andrey; Cho, Gil Young; You, Yizhi
- Physical Review Letters, Vol. 114, Issue 1
Geometry of fractional quantum Hall fluids
journal, September 2014
- Cho, Gil Young; You, Yizhi; Fradkin, Eduardo
- Physical Review B, Vol. 90, Issue 11
Geometry of quantum Hall states: Gravitational anomaly and transport coefficients
journal, November 2015
- Can, Tankut; Laskin, Michael; Wiegmann, Paul B.
- Annals of Physics, Vol. 362
Band mass anisotropy and the intrinsic metric of fractional quantum Hall systems
journal, April 2012
- Yang, Bo; Papić, Z.; Rezayi, E. H.
- Physical Review B, Vol. 85, Issue 16
Fractional quantum Hall states in two-dimensional electron systems with anisotropic interactions
journal, July 2012
- Wang, Hao; Narayanan, Rajesh; Wan, Xin
- Physical Review B, Vol. 86, Issue 3
Incompressible states of dirac fermions in graphene with anisotropic interactions
journal, January 2014
- Apalkov, Vadim M.; Chakraborty, Tapash
- Solid State Communications, Vol. 177
Aspects of anisotropic fractional quantum Hall effect in phosphorene
journal, October 2015
- Ghazaryan, Areg; Chakraborty, Tapash
- Physical Review B, Vol. 92, Issue 16
Fractional quantum Hall effect in a tilted magnetic field
journal, June 2013
- Papić, Z.
- Physical Review B, Vol. 87, Issue 24
Field theory of the quantum Hall nematic transition
journal, September 2013
- Maciejko, J.; Hsu, B.; Kivelson, S. A.
- Physical Review B, Vol. 88, Issue 12
Theory of Nematic Fractional Quantum Hall States
journal, December 2014
- You, Yizhi; Cho, Gil Young; Fradkin, Eduardo
- Physical Review X, Vol. 4, Issue 4
Geometry of compressible and incompressible quantum Hall states: Application to anisotropic composite-fermion liquids
journal, December 2013
- Yang, Kun
- Physical Review B, Vol. 88, Issue 24
Splitting of the Fermi Contour of Quasi-2D Electrons in Parallel Magnetic Fields
journal, June 2015
- Mueed, M. A.; Kamburov, D.; Shayegan, M.
- Physical Review Letters, Vol. 114, Issue 23
Composite Fermions with Tunable Fermi Contour Anisotropy
journal, May 2013
- Kamburov, D.; Liu, Yang; Shayegan, M.
- Physical Review Letters, Vol. 110, Issue 20
Density matrix formulation for quantum renormalization groups
journal, November 1992
- White, Steven R.
- Physical Review Letters, Vol. 69, Issue 19
Finitely correlated states on quantum spin chains
journal, March 1992
- Fannes, M.; Nachtergaele, B.; Werner, R. F.
- Communications in Mathematical Physics, Vol. 144, Issue 3
Thermodynamic Limit of Density Matrix Renormalization
journal, November 1995
- Östlund, Stellan; Rommer, Stefan
- Physical Review Letters, Vol. 75, Issue 19
Class of ansatz wave functions for one-dimensional spin systems and their relation to the density matrix renormalization group
journal, January 1997
- Rommer, Stefan; Östlund, Stellan
- Physical Review B, Vol. 55, Issue 4
Ground-State Phase Diagram of 2D Electrons in a High Landau Level: A Density-Matrix Renormalization Group Study
journal, June 2001
- Shibata, Naokazu; Yoshioka, Daijiro
- Physical Review Letters, Vol. 86, Issue 25
Density Matrix Renormalization Group Study of Incompressible Fractional Quantum Hall States
journal, April 2008
- Feiguin, A. E.; Rezayi, E.; Nayak, C.
- Physical Review Letters, Vol. 100, Issue 16
Density matrix renormalization group for bosonic quantum Hall effect
journal, March 2010
- Kovrizhin, D. L.
- Physical Review B, Vol. 81, Issue 12
Fractional quantum Hall states at and filling: Density-matrix renormalization group calculations
journal, May 2011
- Zhao, Jize; Sheng, D. N.; Haldane, F. D. M.
- Physical Review B, Vol. 83, Issue 19
Comparison of the density-matrix renormalization group method applied to fractional quantum Hall systems in different geometries
journal, June 2012
- Hu, Zi-Xiang; Papić, Z.; Johri, S.
- Physics Letters A, Vol. 376, Issue 30-31
Exact matrix product states for quantum Hall wave functions
journal, December 2012
- Zaletel, Michael P.; Mong, Roger S. K.
- Physical Review B, Vol. 86, Issue 24
Matrix product states for trial quantum Hall states
journal, April 2013
- Estienne, B.; Papić, Z.; Regnault, N.
- Physical Review B, Vol. 87, Issue 16
Topological Characterization of Fractional Quantum Hall Ground States from Microscopic Hamiltonians
journal, June 2013
- Zaletel, Michael P.; Mong, Roger S. K.; Pollmann, Frank
- Physical Review Letters, Vol. 110, Issue 23
Fractional Quantization of the Hall Effect: A Hierarchy of Incompressible Quantum Fluid States
journal, August 1983
- Haldane, F. D. M.
- Physical Review Letters, Vol. 51, Issue 7
Finite-Size Studies of the Incompressible State of the Fractionally Quantized Hall Effect and its Excitations
journal, January 1985
- Haldane, F. D. M.; Rezayi, E. H.
- Physical Review Letters, Vol. 54, Issue 3
Edge reconstruction in the fractional quantum Hall regime
journal, September 2003
- Wan, Xin; Rezayi, E. H.; Yang, Kun
- Physical Review B, Vol. 68, Issue 12
Edge reconstruction of fractional quantum Hall liquids with spin degrees of freedom
journal, November 2013
- Zhang, Yuhui; Hu, Zi-Xiang; Yang, Kun
- Physical Review B, Vol. 88, Issue 20
Concept of Off-Diagonal Long-Range Order and the Quantum Phases of Liquid He and of Superconductors
journal, October 1962
- Yang, C. N.
- Reviews of Modern Physics, Vol. 34, Issue 4
Off-diagonal long-range order, oblique confinement, and the fractional quantum Hall effect
journal, March 1987
- Girvin, S. M.; MacDonald, A. H.
- Physical Review Letters, Vol. 58, Issue 12
Off-Diagonal Long-Range Order in Fractional Quantum-Hall-Effect States
journal, October 1988
- Rezayi, E. H.; Haldane, F. D. M.
- Physical Review Letters, Vol. 61, Issue 17
Nature of the long-range order in the quantum Hall effect regime
journal, April 1990
- Chakraborty, Tapash; von der Linden, Wolfgang
- Physical Review B, Vol. 41, Issue 11
Laughlin state on stretched and squeezed cylinders and edge excitations in the quantum Hall effect
journal, December 1994
- Rezayi, E. H.; Haldane, F. D. M.
- Physical Review B, Vol. 50, Issue 23
Edge properties of principal fractional quantum Hall states in the cylinder geometry
journal, September 2012
- Soulé, Paul; Jolicoeur, Thierry
- Physical Review B, Vol. 86, Issue 11
Fractional quantization of Hall conductance
journal, July 1983
- Tao, R.; Thouless, D. J.
- Physical Review B, Vol. 28, Issue 2
Theory of the quantized hall effect
journal, July 1984
- Thouless, D. J.
- Surface Science, Vol. 142, Issue 1-3
Model Fractional Quantum Hall States and Jack Polynomials
journal, June 2008
- Bernevig, B. Andrei; Haldane, F. D. M.
- Physical Review Letters, Vol. 100, Issue 24
Works referencing / citing this record:
Numerical study of anisotropy in a composite Fermi liquid
journal, May 2017
- Ippoliti, Matteo; Geraedts, Scott D.; Bhatt, R. N.
- Physical Review B, Vol. 95, Issue 20
Connection between Fermi contours of zero-field electrons and composite fermions in two-dimensional systems
journal, July 2017
- Ippoliti, Matteo; Geraedts, Scott D.; Bhatt, R. N.
- Physical Review B, Vol. 96, Issue 4
Numerical study of anisotropy in a composite Fermi liquid
text, January 2017
- Ippoliti, Matteo; Geraedts, Scott D.; Bhatt, R. N.
- arXiv
Chiral Gravitons in Fractional Quantum Hall Liquids
text, January 2019
- Liou, Shiuan-Fan; Haldane, F. D. M.; Yang, Kun
- arXiv
Interaction-dependent anisotropy of fractional quantum Hall states
text, January 2019
- Krishna, Akshay; Chen, Fan; Ippoliti, Matteo
- arXiv
Liouville perturbation theory for Laughlin state and Coulomb gas
text, January 2020
- Nemkov, Nikita; Klevtsov, Semyon
- arXiv
On the Laughlin function and its perturbations
journal, July 2019
- Rougerie, Nicolas
- Séminaire Laurent Schwartz — EDP et applications