Pfaffian quantum Hall state made simple: Multiple vacua and domain walls on a thin torus

We analyze the Moore-Read Pfaffian state on a thin torus. The known sixfold degeneracy is realized by two inequivalent crystalline states with a four- and twofold degeneracy, respectively. The fundamental quasihole and quasiparticle excitations are domain walls between these vacua, and simple counting arguments give a Hilbert space of dimension 2n−1 for 2n−k holes and k particles at fixed positions and assign each a charge ±e∕4. This generalizes the known properties of the hole excitations in the Pfaffian state as deduced using conformal field theory techniques. Numerical calculations using a model Hamiltonian and a small number of particles support the presence of a stable phase with degenerate vacua and quarter-charged domain walls also away from the thin-torus limit. A spin-chain Hamiltonian encodes the degenerate vacua and the various domain walls.

Note:

4 pages, 1 figure. Published, minor changes

References(27)

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[1]

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

et al.

- Phys.Rev.Lett. 59 (1987) 1776-1779
•
DOI:
- 10.1103/PhysRevLett.59.1776

[2]

Nonabelions in the fractional quantum Hall effect

Gregory W. Moore
(
- Yale U.
)
,
N. Read
(
- Yale U.
)

- Nucl.Phys.B 360 (1991) 362-396
•
DOI:
- 10.1016/0550-3213(91)90407-O

[3]

Paired Hall state at half filling

- Phys.Rev.Lett. 66 (1991) 3205-3208
•
DOI:
- 10.1103/PhysRevLett.66.3205

[3]

On paired Hall states

- Nucl.Phys.B 374 (1992) 567-614
•
DOI:
- 10.1016/0550-3213(92)90401-V

[4]

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

E.H. Rezayi
(
- Cal State, L.A.
)
,
F.D.M. Haldane
(
- Princeton U.
)

- Phys.Rev.Lett. 84 (2000) 4685-4688
•
e-Print:
- cond-mat/9906137
•
DOI:
- 10.1103/PhysRevLett.84.4685

[5]

2 n -quasihole states realize 2 n −1 -dimensional spinor braiding statistics in paired quantum Hall states

- Nucl.Phys.B 479 (1996) 529-553
•
e-Print:
- cond-mat/9605145
•
DOI:
- 10.1016/0550-3213(96)00430-0

[6]

Quasiholes and fermionic zero modes of paired fractional quantum Hall states: The mechanism for non-Abelian statistics

N. Read
(
- Yale U.
)
,
E. Rezayi
(
- Cal State, L.A.
)

- Phys.Rev.B 54 (1996) 16864-16887
•
e-Print:
- cond-mat/9609079
•
DOI:
- 10.1103/PhysRevB.54.16864

[7]

Many body systems with nonAbelian statistics

- Nucl.Phys.B 374 (1992) 615-646
•
DOI:
- 10.1016/0550-3213(92)90402-W

[8]

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

N. Read
(
- Yale U.
)
,
Dmitry Green
(
- Yale U.
)

- Phys.Rev.B 61 (2000) 10267
•
e-Print:
- cond-mat/9906453
•
DOI:
- 10.1103/PhysRevB.61.10267

[9]

See e.g

M.H. Freedman
,
A. Kitaev
,
M. Larsen
,
Z. Wang

- Bull.Am.Math.Soc. 40 (2003) 31

[9]

Fault tolerant quantum computation by anyons

A.Yu. Kitaev
(
- Landau Inst.
)

- Annals Phys. 303 (2003) 2-30
•
e-Print:
- quant-ph/9707021
•
DOI:
- 10.1016/S0003-4916(02)00018-0

[10]

E.J. Bergholtz
,
A. Karlhede

- Phys.Rev.Lett. 94 (2005) 26802

[11]

J. Stat Mech. L04001

E.J. Bergholtz
,
A. Karlhede

[12]

Composite fermion approach for the fractional quantum Hall effect

J.K. Jain
(
- Yale U.
)

- Phys.Rev.Lett. 63 (1989) 199-202
•
DOI:
- 10.1103/PhysRevLett.63.199

[13]

E.H. Rezayi
,
F.D.M. Haldane

- Phys.Rev.B 50 (1994) 17199

[14]

A. Seidel
,
H. Fu
,
D.-H. Lee
,
J.M. Leinaas
,
J. Moore

- Phys.Rev.Lett. 95 (2005) 266405

[15]

Conformal Field Theory of Composite Fermions

- Phys.Rev.Lett. 98 (2007) 076801
•
e-Print:
- cond-mat/0603125
•
DOI:
- 10.1103/PhysRevLett.98.076801

[16]

E.H. Rezayi
,
N. Read

- Phys.Rev.Lett. 72 (1994) 900

[17]

Transition from Quantum Hall to Compressible States in the Second Landau Level: New Light on the nu=5/2 Enigma

R.H. Morf
(
- PSI, Villigen
)

- Phys.Rev.Lett. 80 (1998) 1505-1508
•
e-Print:
- cond-mat/9809024
•
DOI:
- 10.1103/PhysRevLett.80.1505

[18]

FRACTIONALLY CHARGED EXCITATIONS IN CHARGE DENSITY WAVE SYSTEMS WITH COMMENSURABILITY. 3.

W.P. Su
(
- UC, Santa Barbara and
- Santa Barbara, KITP
)
,
J.R. Schrieffer
(
- UC, Santa Barbara and
- Santa Barbara, KITP
)

- Phys.Rev.Lett. 46 (1981) 11, 738
•
DOI:
- 10.1103/PhysRevLett.46.738

[19]

talk at theAPS March meeting, unpublished

F.D.M. Haldane

[20]

Abelian and Non-Abelian Hall Liquids and Charge-Density Wave: Quantum Number Fractionalization in One and Two Dimensions

Alexander Seidel
(
- UC, Berkeley and
- LBL, Berkeley
)
,
Dung Hai Lee
(
- UC, Berkeley and
- LBL, Berkeley
)

- Phys.Rev.Lett. 97 (2006) 5, 056804
•
e-Print:
- cond-mat/0604465
•
DOI:
- 10.1103/PhysRevLett.97.056804

[21]

The Pfaffian states on the torus and the crystal states A and B have the same quantum numbers. The simultaneously conserved symmetries are generated by T1 and T 2 2 and their eigenstates are A and B± = B ± T 2 2 is easy to see that B± can be used in the construction of the domain walls instead of B and T 2 2 B

B. It

[22]

Alternatively, notice that moving the primed electron in the configuration 0101011’001100110 a single site gives 01010101’01100110, i.e. the domain wall moves four sites and thus carries charge -e/4

[23]

Having an exact definition of the position of a domain wall is not necessary for these arguments, but there is a rather natural prescription using the B± states introduced in note 19. With this definition the state counting for fixed domain wall positions agree with the one in the text which is however easier to visualize

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