Interplay between fractional quantum Hall liquid and crystal phases at low filling

Zuo, Zheng-Wei; Balram, Ajit C.; Pu, Songyang; Zhao, Jianyun; Jolicoeur, Thierry; Wójs, A.; Jain, J. K.

doi:10.1103/physrevb.102.075307

Title: Interplay between fractional quantum Hall liquid and crystal phases at low filling

Journal Article · Wed Aug 19 00:00:00 EDT 2020 · Physical Review. B

DOI:https://doi.org/10.1103/physrevb.102.075307· OSTI ID:1773670

Zuo, Zheng-Wei ^[1];

^[2];

^[3]; Zhao, Jianyun ^[3]; Jolicoeur, Thierry ^[4];

^[5];

^[3]

Pennsylvania State Univ., University Park, PA (United States); Henan Univ. of Science and Technology, Luoyang (China). Henan Key Lab. of Photoelectric Energy Storage Materials and Applications
Inst. of Mathematical Sciences, HBNI, CIT Campus, Chennai (India)
Pennsylvania State Univ., University Park, PA (United States)
Univ. Paris-Saclay, Gif-sur-Yvette (France). Inst. de Physique Théorique
Wrocław Univ. of Science and Technology, Wrocław (Poland)

The nature of the state at low Landau-level filling factors has been a long-standing puzzle in the field of the fractional quantum Hall effect (FQHE). While theoretical calculations suggest that a crystal is favored at filling factors $$\nu$$≲1/6, experiments show, at somewhat elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect at $$\nu$$ 1/7, 2/11, 2/13, 3/19, 1/9, 2/15, and 2/17, which belong to the standard sequences $$\nu$$ =n/(6n ± 1) and $$\nu$$=n/(8n ± 1). To address this paradox, we investigate the nature of some of the low-$$\nu$$ states, specifically $$\nu$$= 1/7, 2/13, and 1/9, by variational Monte Carlo, density matrix renormalization group, and exact diagonalization methods. We conclude that in the thermodynamic limit, these are likely to be incompressible fractional quantum Hall liquids, albeit with strong short-range crystalline correlations. Finally, this suggests a natural explanation for the experimentally observed behavior and a rich phase diagram that admits, in the low-disorder limit, a multitude of crystal-FQHE liquid transitions as the filling factor is reduced.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Pennsylvania State Univ., University Park, PA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES); National Natural Science Foundation of China (NSFC)

Contributing Organization:: Penn State; Henan University of Science and Technology, China; Institute for Mathematical Sciences, India; CNRS France; Wroclaw University, Poland

Grant/Contract Number:: SC0005042; 11604081; 11447008; 2014/14/A/ST3/00654

OSTI ID:: 1773670

Journal Information:: Physical Review. B, Vol. 102, Issue 7; ISSN 2469-9950

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (79)

Microscopic Verification of Topological Electron-Vortex Binding in the Lowest Landau-Level Crystal State Chang, Chia-Chen; Jeon, Gun Sang; Jain, Jainendra K. Physical Review Letters, Vol. 94, Issue 1 https://doi.org/10.1103/PhysRevLett.94.016809	journal	January 2005
Ground State of Two-Dimensional Electrons in Strong Magnetic Fields and $\frac{1}{3}$ Quantized Hall Effect Yoshioka, D.; Halperin, B. I.; Lee, P. A. Physical Review Letters, Vol. 50, Issue 16 https://doi.org/10.1103/PhysRevLett.50.1219	journal	April 1983
Competing Fractional Quantum Hall and Electron Solid Phases in Graphene Chen, Shaowen; Ribeiro-Palau, Rebeca; Yang, Kang Physical Review Letters, Vol. 122, Issue 2 https://doi.org/10.1103/PhysRevLett.122.026802	journal	January 2019
Fractional quantum Hall states at $\frac{1}{3}$ and $\frac{5}{2}$ filling: Density-matrix renormalization group calculations Zhao, Jize; Sheng, D. N.; Haldane, F. D. M. Physical Review B, Vol. 83, Issue 19 https://doi.org/10.1103/PhysRevB.83.195135	journal	May 2011
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
Competing Crystal Phases in the Lowest Landau Level Archer, Alexander C.; Park, Kwon; Jain, Jainendra K. Physical Review Letters, Vol. 111, Issue 14 https://doi.org/10.1103/PhysRevLett.111.146804	journal	October 2013
Two-Dimensional Electron System in High Magnetic Fields: Wigner Crystal versus Composite-Fermion Liquid Mandal, Sudhansu S.; Peterson, Michael R.; Jain, Jainendra K. Physical Review Letters, Vol. 90, Issue 10 https://doi.org/10.1103/PhysRevLett.90.106403	journal	March 2003
Evidence for two-dimentional quantum Wigner crystal Goldman, V. J.; Santos, M.; Shayegan, M. Physical Review Letters, Vol. 65, Issue 17 https://doi.org/10.1103/PhysRevLett.65.2189	journal	October 1990
Fractional quantum Hall effect in graphene: Quantitative comparison between theory and experiment Balram, Ajit C.; Tőke, Csaba; Wójs, A. Physical Review B, Vol. 92, Issue 7 https://doi.org/10.1103/PhysRevB.92.075410	journal	August 2015
Search for exact local Hamiltonians for general fractional quantum Hall states Sreejith, G. J.; Fremling, M.; Jeon, Gun Sang Physical Review B, Vol. 98, Issue 23 https://doi.org/10.1103/PhysRevB.98.235139	journal	December 2018
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
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
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
Defect motifs for spherical topologies Wales, David J.; McKay, Hayley; Altschuler, Eric L. Physical Review B, Vol. 79, Issue 22 https://doi.org/10.1103/PhysRevB.79.224115	journal	June 2009
State counting for excited bands of the fractional quantum Hall effect: Exclusion rules for bound excitons Balram, Ajit C.; Wójs, Arkadiusz; Jain, Jainendra K. Physical Review B, Vol. 88, Issue 20 https://doi.org/10.1103/PhysRevB.88.205312	journal	November 2013
Odd- and even-denominator fractional quantum Hall states in monolayer WSe2 Shi, Qianhui; Shih, En-Min; Gustafsson, Martin V. Nature Nanotechnology, Vol. 15, Issue 7 https://doi.org/10.1038/s41565-020-0685-6	journal	July 2020
Electrical conductivity and Wigner crystallization Paalanen, M. A.; Willett, R. L.; Ruel, R. R. Physical Review B, Vol. 45, Issue 23 https://doi.org/10.1103/PhysRevB.45.13784	journal	June 1992
Density matrix formulation for quantum renormalization groups White, Steven R. Physical Review Letters, Vol. 69, Issue 19 https://doi.org/10.1103/PhysRevLett.69.2863	journal	November 1992
Hamiltonian theory of the composite-fermion Wigner crystal Narevich, R.; Murthy, Ganpathy; Fertig, H. A. Physical Review B, Vol. 64, Issue 24 https://doi.org/10.1103/PhysRevB.64.245326	journal	December 2001
Microscopic wave functions for the fractional quantized Hall states at $ν = \frac{2}{5} and \frac{2}{7}$ Morf, R.; d'Ambrumenil, N.; Halperin, B. I. Physical Review B, Vol. 34, Issue 4 https://doi.org/10.1103/PhysRevB.34.3037	journal	August 1986
Theoretical search for the nested quantum Hall effect of composite fermions Mandal, Sudhansu S.; Jain, Jainendra K. Physical Review B, Vol. 66, Issue 15 https://doi.org/10.1103/PhysRevB.66.155302	journal	October 2002
Entanglement Entropy in Fermionic Laughlin States Haque, Masudul; Zozulya, Oleksandr; Schoutens, Kareljan Physical Review Letters, Vol. 98, Issue 6 https://doi.org/10.1103/PhysRevLett.98.060401	journal	February 2007
Spinful composite fermions in a negative effective field Davenport, Simon C.; Simon, Steven H. Physical Review B, Vol. 85, Issue 24 https://doi.org/10.1103/PhysRevB.85.245303	journal	June 2012
Crystallization in the Fractional Quantum Hall Regime Induced by Landau-Level Mixing Zhao, Jianyun; Zhang, Yuhe; Jain, J. K. Physical Review Letters, Vol. 121, Issue 11 https://doi.org/10.1103/PhysRevLett.121.116802	journal	September 2018
Working principles of doping-well structures for high-mobility two-dimensional electron systems Chung, Yoon Jang; Rosales, K. A. Villegas; Baldwin, K. W. Physical Review Materials, Vol. 4, Issue 4 https://doi.org/10.1103/PhysRevMaterials.4.044003	journal	April 2020
A note contrasting two microscopic theories of the fractional quantum Hall effect Jain, Jainendra K. Indian Journal of Physics, Vol. 88, Issue 9 https://doi.org/10.1007/s12648-014-0491-9	journal	May 2014
Ground-State Phase Diagram of 2D Electrons in a High Landau Level: A Density-Matrix Renormalization Group Study Shibata, Naokazu; Yoshioka, Daijiro Physical Review Letters, Vol. 86, Issue 25 https://doi.org/10.1103/PhysRevLett.86.5755	journal	June 2001
Microwave resonances in low-filling insulating phases of two-dimensional electron and hole systems Engel, L. W.; Li, C. -C; Shahar, D. Physica E: Low-dimensional Systems and Nanostructures, Vol. 1, Issue 1-4 https://doi.org/10.1016/S1386-9477(97)00025-8	journal	January 1997
Wigner crystal versus Hall insulator Manoharan, H. C.; Shayegan, M. Physical Review B, Vol. 50, Issue 23 https://doi.org/10.1103/PhysRevB.50.17662	journal	December 1994
Conduction threshold and pinning frequency of magnetically induced Wigner solid Williams, F. I. B.; Wright, P. A.; Clark, R. G. Physical Review Letters, Vol. 66, Issue 25 https://doi.org/10.1103/PhysRevLett.66.3285	journal	June 1991
Pinning mode resonances of new phases of 2D electron systems in high magnetic fields Sambandamurthy, G.; Wang, Zhihai; Lewis, R. M. Solid State Communications, Vol. 140, Issue 2 https://doi.org/10.1016/j.ssc.2006.05.053	journal	October 2006
Density Matrix Renormalization Group Study of Incompressible Fractional Quantum Hall States Feiguin, A. E.; Rezayi, E.; Nayak, C. Physical Review Letters, Vol. 100, Issue 16 https://doi.org/10.1103/PhysRevLett.100.166803	journal	April 2008
Fractional quantum Hall effect at $ν = 2 + 4 / 9$ Balram, Ajit C.; Wójs, A. Physical Review Research, Vol. 2, Issue 3 https://doi.org/10.1103/PhysRevResearch.2.032035	journal	August 2020
Transition from a fractional quantum Hall liquid to an electron solid at Landau level filling $ν = \frac{1}{3}$ in tilted magnetic fields Pan, W.; Csáthy, G. A.; Tsui, D. C. Physical Review B, Vol. 71, Issue 3 https://doi.org/10.1103/PhysRevB.71.035302	journal	January 2005
Quantitative study of large composite-fermion systems Jain, J. K.; Kamilla, R. K. Physical Review B, Vol. 55, Issue 8 https://doi.org/10.1103/PhysRevB.55.R4895	journal	February 1997
The density-matrix renormalization group in the age of matrix product states Schollwöck, Ulrich Annals of Physics, Vol. 326, Issue 1 https://doi.org/10.1016/j.aop.2010.09.012	journal	January 2011
Experimental studies of the $v = \frac{1}{5}$ hierarchy in the fractional quantum Hall effect Mallett, J. R.; Clark, R. G.; Nicholas, R. J. Physical Review B, Vol. 38, Issue 3 https://doi.org/10.1103/PhysRevB.38.2200	journal	July 1988
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
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
Neutral collective modes in spin-polarized fractional quantum Hall states at filling factors $\frac{1}{3}$ , $\frac{2}{5}$ , $\frac{3}{7}$ , and $\frac{4}{9}$ Majumder, Dwipesh; Mandal, Sudhansu S. Physical Review B, Vol. 90, Issue 15 https://doi.org/10.1103/PhysRevB.90.155310	journal	October 2014
Repulsive interactions in quantum Hall systems as a pairing problem Ortiz, G.; Nussinov, Z.; Dukelsky, J. Physical Review B, Vol. 88, Issue 16 https://doi.org/10.1103/PhysRevB.88.165303	journal	October 2013
Liquid-solid transition and the fractional quantum-Hall effect Lam, Pui K.; Girvin, S. M. Physical Review B, Vol. 30, Issue 1 https://doi.org/10.1103/PhysRevB.30.473	journal	July 1984
Possible persistence of fractional quantum Hall effect down to ultralow fillings Peterson, Michael R.; Jain, Jainendra K. Physical Review B, Vol. 68, Issue 19 https://doi.org/10.1103/PhysRevB.68.195310	journal	November 2003
Transition from an Electron Solid to the Sequence of Fractional Quantum Hall States at Very Low Landau Level Filling Factor Pan, W.; Stormer, H. L.; Tsui, D. C. Physical Review Letters, Vol. 88, Issue 17 https://doi.org/10.1103/PhysRevLett.88.176802	journal	April 2002
Observation of a reentrant insulating phase near the 1/3 fractional quantum Hall liquid in a two-dimensional hole system Santos, M. B.; Suen, Y. W.; Shayegan, M. Physical Review Letters, Vol. 68, Issue 8 https://doi.org/10.1103/PhysRevLett.68.1188	journal	February 1992
Correlation Lengths of the Wigner-Crystal Order in a Two-Dimensional Electron System at High Magnetic Fields Ye, P. D.; Engel, L. W.; Tsui, D. C. Physical Review Letters, Vol. 89, Issue 17 https://doi.org/10.1103/PhysRevLett.89.176802	journal	October 2002
Composite fermions in a negative effective magnetic field: A Monte Carlo study Möller, Gunnar; Simon, Steven H. Physical Review B, Vol. 72, Issue 4 https://doi.org/10.1103/PhysRevB.72.045344	journal	July 2005
Melting of a 2D quantum electron solid in high magnetic field Chen, Yong P.; Sambandamurthy, G.; Wang, Z. H. Nature Physics, Vol. 2, Issue 7 https://doi.org/10.1038/nphys322	journal	June 2006
The half-filled Landau level: The case for Dirac composite fermions Geraedts, S. D.; Zaletel, M. P.; Mong, R. S. K. Science, Vol. 352, Issue 6282 https://doi.org/10.1126/science.aad4302	journal	April 2016
Comparison of the density-matrix renormalization group method applied to fractional quantum Hall systems in different geometries Hu, Zi-Xiang; Papić, Z.; Johri, S. Physics Letters A, Vol. 376, Issue 30-31 https://doi.org/10.1016/j.physleta.2012.05.031	journal	June 2012
Crystallization of the incompressible quantum-fluid state of a two-dimensional electron gas in a strong magnetic field Levesque, D.; Weis, J. J.; MacDonald, A. H. Physical Review B, Vol. 30, Issue 2 https://doi.org/10.1103/PhysRevB.30.1056	journal	July 1984
Sharp tunnelling resonance from the vibrations of an electronic Wigner crystal Jang, Joonho; Hunt, Benjamin M.; Pfeiffer, Loren N. Nature Physics, Vol. 13, Issue 4 https://doi.org/10.1038/nphys3979	journal	December 2016
The density-matrix renormalization group Schollwöck, U. Reviews of Modern Physics, Vol. 77, Issue 1 https://doi.org/10.1103/RevModPhys.77.259	journal	April 2005
Transport of a sliding Wigner crystal in the four flux composite fermion regime Zhang, Chi; Du, Rui-Rui; Manfra, M. J. Physical Review B, Vol. 92, Issue 7 https://doi.org/10.1103/PhysRevB.92.075434	journal	August 2015
Commensurability Oscillations of Composite Fermions Induced by the Periodic Potential of a Wigner Crystal Deng, H.; Liu, Y.; Jo, I. Physical Review Letters, Vol. 117, Issue 9 https://doi.org/10.1103/PhysRevLett.117.096601	journal	August 2016
XXIV. On the structure of the atom: an investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results to the theory of atomic structure Thomson, J. J. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 7, Issue 39 https://doi.org/10.1080/14786440409463107	journal	March 1904
Magnetic-Field-Induced Insulating Phases at Large $r_{s}$ Csáthy, G. A.; Noh, Hwayong; Tsui, D. C. Physical Review Letters, Vol. 94, Issue 22 https://doi.org/10.1103/PhysRevLett.94.226802	journal	June 2005
Effect of Landau-level mixing on quantum-liquid and solid states of two-dimensional hole systems Santos, M. B.; Jo, J.; Suen, Y. W. Physical Review B, Vol. 46, Issue 20 https://doi.org/10.1103/PhysRevB.46.13639	journal	November 1992
Collective excitations of composite fermions across multiple Λ levels Majumder, Dwipesh; Mandal, Sudhansu S.; Jain, Jainendra K. Nature Physics, Vol. 5, Issue 6 https://doi.org/10.1038/nphys1275	journal	May 2009
Excitonic instability and termination of fractional quantum Hall effect Kamilla, Rajiv K.; Jain, J. K. Physical Review B, Vol. 55, Issue 20 https://doi.org/10.1103/PhysRevB.55.R13417	journal	May 1997
Composite Fermions Jain, Jainendra K. https://doi.org/10.1017/CBO9780511607561	book	January 2007
On the Interaction of Electrons in Metals Wigner, E. Physical Review, Vol. 46, Issue 11 https://doi.org/10.1103/PhysRev.46.1002	journal	December 1934
Competition between composite-fermion-crystal and liquid orders at $ν = 1 ∕ 5$ Chang, Chia-Chen; Töke, Csaba; Jeon, Gun Sang Physical Review B, Vol. 73, Issue 15 https://doi.org/10.1103/PhysRevB.73.155323	journal	April 2006
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
Microwave resonance and weak pinning in two-dimensional hole systems at high magnetic fields Li, C. -C.; Yoon, J.; Engel, L. W. Physical Review B, Vol. 61, Issue 16 https://doi.org/10.1103/PhysRevB.61.10905	journal	April 2000
Composite Fermions in the Hilbert Space of the Lowest Electronic Landau Level Jain, J. K.; Kamilla, R. K. International Journal of Modern Physics B, Vol. 11, Issue 22 https://doi.org/10.1142/S0217979297001301	journal	September 1997
Quantum liquid versus electron solid around ν=1/5 Landau-level filling Jiang, H. W.; Willett, R. L.; Stormer, H. L. Physical Review Letters, Vol. 65, Issue 5 https://doi.org/10.1103/PhysRevLett.65.633	journal	July 1990
Structure and dynamics of spherical crystals characterized for the Thomson problem Wales, David J.; Ulker, Sidika Physical Review B, Vol. 74, Issue 21 https://doi.org/10.1103/PhysRevB.74.212101	journal	December 2006
Static and dynamic properties of a two-dimensional Wigner crystal in a strong magnetic field Maki, Kazumi; Zotos, Xenophon Physical Review B, Vol. 28, Issue 8 https://doi.org/10.1103/PhysRevB.28.4349	journal	October 1983
Laughlin-Jastrow-correlated Wigner crystal in a strong magnetic field Yi, Hangmo; Fertig, H. A. Physical Review B, Vol. 58, Issue 7 https://doi.org/10.1103/PhysRevB.58.4019	journal	August 1998
Band structure of the fractional quantum Hall effect Dev, Gautam; Jain, J. K. Physical Review Letters, Vol. 69, Issue 19 https://doi.org/10.1103/PhysRevLett.69.2843	journal	November 1992
Evidence for the Fractional Quantum Hall State at $ν = \frac{1}{7}$ Goldman, V. J.; Shayegan, M.; Tsui, D. C. Physical Review Letters, Vol. 61, Issue 7 https://doi.org/10.1103/PhysRevLett.61.881	journal	August 1988
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
Wigner crystals in the lowest Landau level at low-filling factors Yang, Kun; Haldane, F. D. M.; Rezayi, E. H. Physical Review B, Vol. 64, Issue 8 https://doi.org/10.1103/PhysRevB.64.081301	journal	August 2001
Fractional Quantum Hall Effect and Wigner Crystal of Interacting Composite Fermions Liu, Yang; Kamburov, D.; Hasdemir, S. Physical Review Letters, Vol. 113, Issue 24 https://doi.org/10.1103/PhysRevLett.113.246803	journal	December 2014
Composite-fermion approach for the fractional quantum Hall effect Jain, J. K. Physical Review Letters, Vol. 63, Issue 2 https://doi.org/10.1103/PhysRevLett.63.199	journal	July 1989
Experimental studies of the fractional quantum Hall effect in the first excited Landau level Pan, W.; Xia, J. S.; Stormer, H. L. Physical Review B, Vol. 77, Issue 7 https://doi.org/10.1103/PhysRevB.77.075307	journal	February 2008
Evidence for Two Different Solid Phases of Two-Dimensional Electrons in High Magnetic Fields Chen, Yong P.; Lewis, R. M.; Engel, L. W. Physical Review Letters, Vol. 93, Issue 20 https://doi.org/10.1103/PhysRevLett.93.206805	journal	November 2004
rf conductivity of a two-dimensional electron system at small Landau-level filling factors Paalanen, M. A.; Willett, R. L.; Littlewood, P. B. Physical Review B, Vol. 45, Issue 19 https://doi.org/10.1103/physrevb.45.11342	journal	May 1992

Similar Records

The two-dimensional Wigner solid transition at finite magnetic field

Miscellaneous · Fri Jan 01 00:00:00 EST 1993 · OSTI ID:1773670

Price, R D

Z_{n}

superconductivity of composite bosons and the

7 / 3

fractional quantum Hall effect

Journal Article · Fri Mar 20 00:00:00 EDT 2020 · Physical Review Research · OSTI ID:1773670

Balram, Ajit C.; Jain, J. K.; Barkeshli, Maissam

New results of ultra-low temperature experiments in the second Landau level of a two-dimensional electron system.

Conference · Tue Jul 01 00:00:00 EDT 2003 · OSTI ID:1773670

Pan, Wei; Vicente, C; Stormer, H L; +7 more

Related Subjects

75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
Wigner crystal
fractional quantum Hall effect

Title: Interplay between fractional quantum Hall liquid and crystal phases at low filling

Citation Formats

References (79)

Similar Records

Related Subjects