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# Disorder-Driven Transition in the $\nu =5/2$ Fractional Quantum Hall Effect

## Abstract

The fractional quantum Hall (FQH) effect at the filling number ν = 5/2 is a primary candidate for non-Abelian topological order, while the fate of such a state in the presence of random disorder has not been resolved. We address this open question by implementing an unbiased diagnosis based on numerical exact diagonalization. Here, we calculate the disorder averaged Hall conductance and the associated statistical distribution of the topological invariant Chern number, which unambiguously characterize the disorder-driven collapse of the FQH state. As the disorder strength increases towards a critical value, a continuous phase transition is detected based on the disorder configuration averaged wave function fidelity and the entanglement entropy. In the strong disorder regime, we identify a composite Fermi liquid phase with fluctuating Chern numbers, in striking contrast to the well-known ν = 1/3 case where an Anderson insulator appears. Interestingly, the lowest Landau level projected a local density profile, the wave function overlap, and the entanglement entropy as a function of disorder strength simultaneously signal an intermediate phase, which may be relevant to the recent proposal of a particle-hole Pfaffian state or Pfaffian–anti-Pfaffian puddle state.

- Authors:

- Westlake Univ., Hangzhou (China); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- California State Univ., Northridge, CA (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1571593

- Alternate Identifier(s):
- OSTI ID: 1546824

- Report Number(s):
- LA-UR-18-28368

Journal ID: ISSN 0031-9007; PRLTAO

- Grant/Contract Number:
- 89233218CNA000001

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review Letters

- Additional Journal Information:
- Journal Volume: 123; Journal Issue: 5; Journal ID: ISSN 0031-9007

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 36 MATERIALS SCIENCE; Material Science

### Citation Formats

```
Zhu, Wei, and Sheng, D. N. Disorder-Driven Transition in the ν=5/2 Fractional Quantum Hall Effect. United States: N. p., 2019.
Web. doi:10.1103/PhysRevLett.123.056804.
```

```
Zhu, Wei, & Sheng, D. N. Disorder-Driven Transition in the ν=5/2 Fractional Quantum Hall Effect. United States. doi:10.1103/PhysRevLett.123.056804.
```

```
Zhu, Wei, and Sheng, D. N. Thu .
"Disorder-Driven Transition in the ν=5/2 Fractional Quantum Hall Effect". United States. doi:10.1103/PhysRevLett.123.056804.
```

```
@article{osti_1571593,
```

title = {Disorder-Driven Transition in the ν=5/2 Fractional Quantum Hall Effect},

author = {Zhu, Wei and Sheng, D. N.},

abstractNote = {The fractional quantum Hall (FQH) effect at the filling number ν = 5/2 is a primary candidate for non-Abelian topological order, while the fate of such a state in the presence of random disorder has not been resolved. We address this open question by implementing an unbiased diagnosis based on numerical exact diagonalization. Here, we calculate the disorder averaged Hall conductance and the associated statistical distribution of the topological invariant Chern number, which unambiguously characterize the disorder-driven collapse of the FQH state. As the disorder strength increases towards a critical value, a continuous phase transition is detected based on the disorder configuration averaged wave function fidelity and the entanglement entropy. In the strong disorder regime, we identify a composite Fermi liquid phase with fluctuating Chern numbers, in striking contrast to the well-known ν = 1/3 case where an Anderson insulator appears. Interestingly, the lowest Landau level projected a local density profile, the wave function overlap, and the entanglement entropy as a function of disorder strength simultaneously signal an intermediate phase, which may be relevant to the recent proposal of a particle-hole Pfaffian state or Pfaffian–anti-Pfaffian puddle state.},

doi = {10.1103/PhysRevLett.123.056804},

journal = {Physical Review Letters},

number = 5,

volume = 123,

place = {United States},

year = {2019},

month = {8}

}

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

##
Topological Orders in Rigid States

journal, February 1990

- Wen, X. G.
- International Journal of Modern Physics B, Vol. 04, Issue 02

##
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

##
Nonabelions in the fractional quantum hall effect

journal, August 1991

- Moore, Gregory; Read, Nicholas
- Nuclear Physics B, Vol. 360, Issue 2-3

##
Paired Hall state at half filling

journal, June 1991

- Greiter, Martin; Wen, Xiao-Gang; Wilczek, Frank
- Physical Review Letters, Vol. 66, Issue 24

##
Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level

journal, March 1999

- Read, N.; Rezayi, E.
- Physical Review B, Vol. 59, Issue 12

##
Fault-tolerant quantum computation by anyons

journal, January 2003

- Kitaev, A. Yu.
- Annals of Physics, Vol. 303, Issue 1

##
Non-Abelian anyons and topological quantum computation

journal, September 2008

- Nayak, Chetan; Simon, Steven H.; Stern, Ady
- Reviews of Modern Physics, Vol. 80, Issue 3

##
Observation of an even-denominator quantum number in the fractional quantum Hall effect

journal, October 1987

- Willett, R.; Eisenstein, J. P.; Störmer, H. L.
- Physical Review Letters, Vol. 59, Issue 15

##
Exact Quantization of the Even-Denominator Fractional Quantum Hall State at $\mathit{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5\mathrm{}/2$ Landau Level Filling Factor

journal, October 1999

- Pan, W.; Xia, J. -S.; Shvarts, V.
- Physical Review Letters, Vol. 83, Issue 17

##
Activation gaps of fractional quantum Hall effect in the second Landau level

journal, February 2008

- Choi, H. C.; Kang, W.; Das Sarma, S.
- Physical Review B, Vol. 77, Issue 8

##
Experimental studies of the fractional quantum Hall effect in the first excited Landau level

journal, February 2008

- Pan, W.; Xia, J. S.; Stormer, H. L.
- Physical Review B, Vol. 77, Issue 7

##
Quasi-Particle Properties from Tunneling in the Formula Fractional Quantum Hall State

journal, May 2008

- Radu, I. P.; Miller, J. B.; Marcus, C. M.
- Science, Vol. 320, Issue 5878

##
Observation of a quarter of an electron charge at the ν = 5/2 quantum Hall state

journal, April 2008

- Dolev, M.; Heiblum, M.; Umansky, V.
- Nature, Vol. 452, Issue 7189

##
Magnetic-Field-Tuned Aharonov-Bohm Oscillations and Evidence for Non-Abelian Anyons at $\nu =5/2$

journal, October 2013

- Willett, R. L.; Nayak, C.; Shtengel, K.
- Physical Review Letters, Vol. 111, Issue 18

##
Measurement of filling factor 5/2 quasiparticle interference with observation of charge e/4 and e/2 period oscillations

journal, May 2009

- Willett, R. L.; Pfeiffer, L. N.; West, K. W.
- Proceedings of the National Academy of Sciences, Vol. 106, Issue 22

##
Observation of neutral modes in the fractional quantum Hall regime

journal, July 2010

- Bid, Aveek; Ofek, N.; Inoue, H.
- Nature, Vol. 466, Issue 7306

##
Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect

journal, April 2000

- Read, N.; Green, Dmitry
- Physical Review B, Vol. 61, Issue 15

##
Transition from Quantum Hall to Compressible States in the Second Landau Level: New Light on the $\nu \phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5\mathrm{}/2$ Enigma

journal, February 1998

- Morf, R. H.
- Physical Review Letters, Vol. 80, Issue 7

##
Incompressible Paired Hall State, Stripe Order, and the Composite Fermion Liquid Phase in Half-Filled Landau Levels

journal, May 2000

- Rezayi, E. H.; Haldane, F. D. M.
- Physical Review Letters, Vol. 84, Issue 20

##
Spontaneous Particle-Hole Symmetry Breaking in the $\nu =5/2$ Fractional Quantum Hall Effect

journal, October 2008

- Peterson, Michael R.; Park, Kwon; Das Sarma, S.
- Physical Review Letters, Vol. 101, Issue 15

##
Particle-hole symmetry breaking and the $\nu ={\scriptstyle \frac{5}{2}}$ fractional quantum Hall effect

journal, December 2009

- Wang, Hao; Sheng, D. N.; Haldane, F. D. M.
- Physical Review B, Vol. 80, Issue 24

##
Fractional Quantum Hall State at $\nu =\frac{5}{2}$ and the Moore-Read Pfaffian

journal, February 2010

- Storni, M.; Morf, R. H.; Das Sarma, S.
- Physical Review Letters, Vol. 104, Issue 7

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

journal, December 2016

- Zhu, W.; Liu, Zhao; Haldane, F. D. M.
- Physical Review B, Vol. 94, Issue 24

##
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

##
Fractional quantum Hall states at $\frac{1}{3}$ and $\frac{5}{2}$ 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

##
Particle-Hole Symmetry and the Pfaffian State

journal, December 2007

- Levin, Michael; Halperin, Bertrand I.; Rosenow, Bernd
- Physical Review Letters, Vol. 99, Issue 23

##
Particle-Hole Symmetry and the $\nu =\frac{5}{2}$ Quantum Hall State

journal, December 2007

- Lee, Sung-Sik; Ryu, Shinsei; Nayak, Chetan
- Physical Review Letters, Vol. 99, Issue 23

##
Breaking of Particle-Hole Symmetry by Landau Level Mixing in the $\nu =5/2$ Quantized Hall State

journal, March 2011

- Rezayi, Edward H.; Simon, Steven H.
- Physical Review Letters, Vol. 106, Issue 11

##
Infinite density matrix renormalization group for multicomponent quantum Hall systems

journal, January 2015

- Zaletel, Michael P.; Mong, Roger S. K.; Pollmann, Frank
- Physical Review B, Vol. 91, Issue 4

##
Phase Diagram of the $\nu =5/2$ Fractional Quantum Hall Effect: Effects of Landau-Level Mixing and Nonzero Width

journal, April 2015

- Pakrouski, Kiryl; Peterson, Michael R.; Jolicoeur, Thierry
- Physical Review X, Vol. 5, Issue 2

##
Landau Level Mixing and the Ground State of the $\nu =5/2$ Quantum Hall Effect

journal, July 2017

- Rezayi, Edward H.
- Physical Review Letters, Vol. 119, Issue 2

##
Observation of half-integer thermal Hall conductance

journal, June 2018

- Banerjee, Mitali; Heiblum, Moty; Umansky, Vladimir
- Nature, Vol. 559, Issue 7713

##
Stabilization of the Particle-Hole Pfaffian Order by Landau-Level Mixing and Impurities That Break Particle-Hole Symmetry

journal, August 2016

- Zucker, P. T.; Feldman, D. E.
- Physical Review Letters, Vol. 117, Issue 9

##
Symmetry enforced non-Abelian topological order at the surface of a topological insulator

journal, April 2014

- Chen, Xie; Fidkowski, Lukasz; Vishwanath, Ashvin
- Physical Review B, Vol. 89, Issue 16

##
Is the Composite Fermion a Dirac Particle?

journal, September 2015

- Son, Dam Thanh
- Physical Review X, Vol. 5, Issue 3

##
Paired states in half-filled Landau levels

journal, June 2017

- Milovanović, M. V.
- Physical Review B, Vol. 95, Issue 23

##
Numerical exploration of trial wave functions for the particle-hole-symmetric Pfaffian

journal, August 2018

- Mishmash, Ryan V.; Mross, David F.; Alicea, Jason
- Physical Review B, Vol. 98, Issue 8

##
Parton construction of a wave function in the anti-Pfaffian phase

journal, July 2018

- Balram, Ajit C.; Barkeshli, Maissam; Rudner, Mark S.
- Physical Review B, Vol. 98, Issue 3

##
Theory of Disorder-Induced Half-Integer Thermal Hall Conductance

journal, July 2018

- Mross, David F.; Oreg, Yuval; Stern, Ady
- Physical Review Letters, Vol. 121, Issue 2

##
Topological order from disorder and the quantized Hall thermal metal: Possible applications to the $\nu =5/2$ state

journal, July 2018

- Wang, Chong; Vishwanath, Ashvin; Halperin, Bertrand I.
- Physical Review B, Vol. 98, Issue 4

##
Theory of the disordered $\nu =\frac{5}{2}$ quantum thermal Hall state: Emergent symmetry and phase diagram

journal, April 2018

- Lian, Biao; Wang, Juven
- Physical Review B, Vol. 97, Issue 16

##
Striped quantum Hall state in a half-filled Landau level

journal, May 2016

- Wan, Xin; Yang, Kun
- Physical Review B, Vol. 93, Issue 20

##
$\nu =5/2$ Fractional Quantum Hall State in the Presence of Alloy Disorder

journal, March 2014

- Deng, Nianpei; Gardner, G. C.; Mondal, S.
- Physical Review Letters, Vol. 112, Issue 11

##
Disorder-Driven Collapse of the Mobility Gap and Transition to an Insulator in the Fractional Quantum Hall Effect

journal, June 2003

- Sheng, D. N.; Wan, Xin; Rezayi, E. H.
- Physical Review Letters, Vol. 90, Issue 25

##
Mobility gap in fractional quantum Hall liquids: Effects of disorder and layer thickness

journal, August 2005

- Wan, Xin; Sheng, D. N.; Rezayi, E. H.
- Physical Review B, Vol. 72, Issue 7

##
Quantum Entanglement as a Diagnostic of Phase Transitions in Disordered Fractional Quantum Hall Liquids

journal, November 2016

- Liu, Zhao; Bhatt, R. N.
- Physical Review Letters, Vol. 117, Issue 20

##
Quantized Hall Conductance in a Two-Dimensional Periodic Potential

journal, August 1982

- Thouless, D. J.; Kohmoto, M.; Nightingale, M. P.
- Physical Review Letters, Vol. 49, Issue 6

##
Quantized Hall conductance as a topological invariant

journal, March 1985

- Niu, Qian; Thouless, D. J.; Wu, Yong-Shi
- Physical Review B, Vol. 31, Issue 6

##
A Topological Look at the Quantum Hall Effect

journal, August 2003

- Avron, Joseph E.; Osadchy, Daniel; Seiler, Ruedi
- Physics Today, Vol. 56, Issue 8

##
Topological invariant and the quantization of the Hall conductance

journal, April 1985

- Kohmoto, Mahito
- Annals of Physics, Vol. 160, Issue 2

##
Chern number and edge states in the integer quantum Hall effect

journal, November 1993

- Hatsugai, Yasuhiro
- Physical Review Letters, Vol. 71, Issue 22

##
Localization, wave-function topology, and the integer quantized Hall effect

journal, February 1988

- Arovas, Daniel P.; Bhatt, R. N.; Haldane, F. D. M.
- Physical Review Letters, Vol. 60, Issue 7

##
Particle-hole symmetry in the anomalous quantum Hall effect

journal, May 1984

- Girvin, S. M.
- Physical Review B, Vol. 29, Issue 10

##
Universal conductance in the lowest Landau level

journal, January 1993

- Huo, Y.; Hetzel, R. E.; Bhatt, R. N.
- Physical Review Letters, Vol. 70, Issue 4

##
Topological Entanglement Entropy

journal, March 2006

- Kitaev, Alexei; Preskill, John
- Physical Review Letters, Vol. 96, Issue 11

##
Detecting Topological Order in a Ground State Wave Function

journal, March 2006

- Levin, Michael; Wen, Xiao-Gang
- Physical Review Letters, Vol. 96, Issue 11

##
Impact of Disorder on the $5/2$ Fractional Quantum Hall State

journal, May 2011

- Pan, W.; Masuhara, N.; Sullivan, N. S.
- Physical Review Letters, Vol. 106, Issue 20

##
$\nu =5/2$ fractional quantum Hall state in low-mobility electron systems: Different roles of disorder

journal, August 2013

- Gamez, Gerardo; Muraki, Koji
- Physical Review B, Vol. 88, Issue 7

##
Insulating and Fractional Quantum Hall States in the First Excited Landau Level

journal, January 2002

- Eisenstein, J. P.; Cooper, K. B.; Pfeiffer, L. N.
- Physical Review Letters, Vol. 88, Issue 7

##
Density dependence of the $\nu ={\scriptstyle \frac{5}{2}}$ energy gap: Experiment and theory

journal, January 2010

- Nuebler, J.; Umansky, V.; Morf, R.
- Physical Review B, Vol. 81, Issue 3

##
Nonconventional Odd-Denominator Fractional Quantum Hall States in the Second Landau Level

journal, December 2010

- Kumar, A.; Csáthy, G. A.; Manfra, M. J.
- Physical Review Letters, Vol. 105, Issue 24

##
Theory of the half-filled Landau level

journal, March 1993

- Halperin, B. I.; Lee, Patrick A.; Read, Nicholas
- Physical Review B, Vol. 47, Issue 12

##
The half-filled Landau level: The case for Dirac composite fermions

journal, April 2016

- Geraedts, S. D.; Zaletel, M. P.; Mong, R. S. K.
- Science, Vol. 352, Issue 6282

##
Berry Phase and Model Wave Function in the Half-Filled Landau Level

journal, October 2018

- Geraedts, Scott D.; Wang, Jie; Rezayi, E. H.
- Physical Review Letters, Vol. 121, Issue 14

##
Berry Curvature on the Fermi Surface: Anomalous Hall Effect as a Topological Fermi-Liquid Property

journal, November 2004

- Haldane, F. D. M.
- Physical Review Letters, Vol. 93, Issue 20

##
Topological Delocalization of Two-Dimensional Massless Dirac Fermions

journal, October 2007

- Nomura, Kentaro; Koshino, Mikito; Ryu, Shinsei
- Physical Review Letters, Vol. 99, Issue 14

##
Impurity Scattering in Carbon Nanotubes – Absence of Back Scattering –

journal, May 1998

- Ando, Tsuneya; Nakanishi, Takeshi
- Journal of the Physical Society of Japan, Vol. 67, Issue 5

##
Entanglement in many-body systems

journal, May 2008

- Amico, Luigi; Fazio, Rosario; Osterloh, Andreas
- Reviews of Modern Physics, Vol. 80, Issue 2

##
*Colloquium* : Area laws for the entanglement entropy

journal, February 2010

- Eisert, J.; Cramer, M.; Plenio, M. B.
- Reviews of Modern Physics, Vol. 82, Issue 1

##
Quantum entanglement in condensed matter systems

journal, August 2016

- Laflorencie, Nicolas
- Physics Reports, Vol. 646

##
Entanglement Entropy in Fermionic Laughlin States

journal, February 2007

- Haque, Masudul; Zozulya, Oleksandr; Schoutens, Kareljan
- Physical Review Letters, Vol. 98, Issue 6

##
Entanglement scaling of fractional quantum Hall states through geometric deformations

journal, July 2010

- Läuchli, Andreas M.; Bergholtz, Emil J.; Haque, Masudul
- New Journal of Physics, Vol. 12, Issue 7

##
Evolution of quantum entanglement with disorder in fractional quantum Hall liquids

journal, September 2017

- Liu, Zhao; Bhatt, R. N.
- Physical Review B, Vol. 96, Issue 11

##
Charge-Density-Wave Ordering in Half-Filled High Landau Levels

journal, August 1999

- Rezayi, E. H.; Haldane, F. D. M.; Yang, Kun
- Physical Review Letters, Vol. 83, Issue 6

##
Spontaneous Breakdown of Translational Symmetry in Quantum Hall Systems: Crystalline Order in High Landau Levels

journal, December 2000

- Haldane, F. D. M.; Rezayi, E. H.; Yang, Kun
- Physical Review Letters, Vol. 85, Issue 25

##
Role of disorder in half-filled high Landau levels

journal, October 2002

- Sheng, D. N.; Wang, Ziqiang; Friedman, B.
- Physical Review B, Vol. 66, Issue 16