One-dimensional limits of the fractional quantum Hall effect Emil J. Bergholtz1 and Anders Karlhede2

1 Max Planck Institute for the Physics of Complex Systems, Dresden, Germany 2 Department of physics, Stockholm University, Stockholm, Sweden

Partly based on collaborations with: Thors Hans Hansson, Emma Wikberg, Maria Hermanns (Stockholm U), Janik Kailasvuori (MPI-PKS), Eddy Ardonne (Nordita), Juha Suorsa and Susanne Viefers (Oslo U)

Introduction

References

Model: Mapping to 1D

Hee =∑

n

∑

k>m

Vkmc†n+mc†n+kcn+m+kcn

ν = 1

ν = 1/3

ν ≤ 1

L1, L2

2

‘Experimental’ situation

Complicated 1D interaction

L1 ∼ L2

6

largeSimple 1D interaction

Solvable, TT limit!

small

↔Smooth development!

1) Tao-Thouless limit

L1 ∼ L2

Vm+p,m → 0 as L1 → 0 if m #= 0

V21

V10

V20

ν = 1/3

ν = 2/5

ν = p/q

6

L1 ∼ L2

Vm+p,m → 0 as L1 → 0 if m #= 0

V21

V10

V20

= 1 /3

= 2 /5

= 3 /7

1 0 0 1 0 0 1 0 0 1 0 0 1 0 0

1 0 0 1 0 1 0 0 1 0 1 0 0 1 0

1 0 0 1 0 1 0 1 0 0 1 01 0

= p / q

6

L1 ∼ L2

Vm+p,m → 0 as L1 → 0 if m #= 0

V21

V10

V20

ν = 1/3

ν = 2/5

ν = p/q

6

L1 ∼ L2

Vm+p,m → 0 as L1 → 0 if m #= 0

V21

V10

V20

ν = 1/3

ν = 2/5

ν = 3/7

1 0 0 1 0 0 1 0 0 1 0 0 1 0 0

ν = p/q

6

(Tao-Thouless (TT) states [7])

Related approaches0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1 0

0.05

0. 1

0.15

0. 2

0.25

0. 3

1/ 3 2/ 3

2/ 5

2/ 7

3/ 5

3/ 7 4/ 7

4/ 9

4/11

4/13

5/ 9

5/11

5/13

5/17

6/11

6/17

7/11

7/17

8/13

!=p/ q

1/ q

1/q

p/q

0.25 0.3 0.35 0.4 0.45 0.50

0.05

0.1

0.15

0.2

0.25

0.3

1/3

2/5

2/7 3/7

3/11

4/9

4/11

4/13 5/135/17 6/17

6/19

7/177/19

8/19

!=p/q

1/q

Experiment, varying B

.....at lower disorder

stability/disorder

Filling fraction

x

x

x

xx

x

x

x xx

x

xx

x xxxxx

x observed fractions [15]

xx

2) Quantum Hall Circle

1. D.C. Tsui, H.L. Störmer and A.C. Gossard, Phys. Rev. Lett. 48, 1599 (1982). 2. R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 3. J.K. Jain, Phys. Rev. Lett. 63, 199 (1989).4. E.H. Rezayi, and F.D.M. Haldane, Phys. Rev. B 50, 17199 (1994).5. E.J. Bergholtz and Anders Karlhede, Phys. Rev. Lett. 94, 026802 (2005). 6. E.J. Bergholtz and A. Karlhede, J. Stat. Mech. (2006) L040017. R. Tao, and D.J. Thouless, Phys. Rev. B 28, 1142 (1983).8. E.J. Bergholtz, M. Hermanns, T.H.Hansson and A. Karlhede, Phys. Rev. Lett. 99, 256803 (2007).9. E.J. Bergholtz and A. Karlhede, Phys. Rev. B 77, 155308 (2008).10. W.P. Su, and J.R. Schrieffer, Phys. Rev. Lett. 46, 738 (1981).11. F.D.M. Haldane, Phys. Rev. Lett. 51, 605 (1983).12. B.I. Halperin, Phys. Rev. Lett. 52, 1583, 2390(E) (1984).13. S.A. Kivelson, D.-H. Lee, and S.-C. Zhang, Phys. Rev. B 46, 2223 (1992).14. C.A. Lütken, and G.G. Ross, Phys. Rev. B 45, 11837 (1992)15. W. Pan et al., Phys. Rev. B 77, 075307 (2008). 16. P.W. Anderson, Phys. Rev. B 28, 2264 (1983).17. S. Jansen, E.H. Lieb, and R. Seiler, Commun. Math. Phys. 285, 503-535 (2009).18. E. J. Bergholtz, M. Hermanns, T.H.Hansson, A. Karlhede and S.F Viefers, Phys. Rev. B 77,165325 (2008).19. E. Wikberg, E.J. Bergholtz, and A. Karlhede, arXiv:0903.4093, J. Stat. Mech. In press (2009).20. E.J. Bergholtz, J.Kailasvuori, E.Wikberg, T.H.Hansson and A. Karlhede, Phys. Rev. B 74, 081308(R) (2006).21. A. Seidel, and D.-H. Lee, Phys. Rev. Lett. 97, 056804 (2006).22. N. Read, Phys. Rev. B 73, 245334 (2006).23. E. Ardonne, E.J. Bergholtz, J. Kailasvuori, and E. Wikberg, J. Stat. Mech. (2008) P04016.24. E. Ardonne, Phys. Rev. Lett. 102, 180401 (2009).25. G. Moore, and N. Read, Nucl. Phys. B 360, 362 (1991).26. M. Greiter, X.G. Wen, and F. Wilczek, Phys. Rev. Lett. 66, 3205 (1991). 27. Rezayi and Read Phys. Rev. B 59, 8084 (1999).28. B.I. Halperin, P.A. Lee, and N. Read, Phys. Rev. B 47, 7312 (1993).29. E.H. Rezayi, and N. Read, Phys. Rev. Lett. 72, 900 (1994).30. B.A. Bernevig, and F.D.M. Haldane, Phys. Rev. Lett. 100, 246802 (2008); Phys. Rev. B 77, 184502 (2008) etc.31. M. Greiter, Bull. Am. Phys. Soc. 38, 137 (1993).32. N. Regnault, M.O, Goerbig and Th. Jolicoeur, hys. Rev. Lett. 101, 066803 (2008).33. X.-G. Wen, and Z. Wang, Phys. Rev. B 77, 235108 (2008).34. M. Barkeshli and X.-G. Wen, Phys. Rev. B 79, 195132 (2009). 35. A. Seidel and K. Yang, Phys. Rev. Lett. 101, 036804 (2008).36. E.J. Bergholtz and A. Karlhede, J. Stat. Mech. (2009) P04015.37. M.I. Dyakonov, in Recent Trends in Theory of Physical Phenomena in High Magnetic Fields, (Kluwer, 2003).38. A. Karlhede, unpublished.

Gapless phase at half-filling

9.0 8.2 8.4 10.6

10101010....

0.993 0.996 0.995 0.998

L

5.3

(5,2) (6,4) (8,3) (0,0)

8.0 9.4

10101010....

0.998 0.999

L

5.3

(4,2) (5,0)

1

1

7.9

10101010....

0.999

L

5.3

(4,2)

1

Solvable! Solvable!Luttinger liquid

(Overlap)

High overlap with Rezayi-Read wf for all L1 ≥ 5.3

‘Fermi sea’

(Note the contrast to the 1/3 state 100100100... )

↑≡ 10 , ↓≡ 01

H ≈∑

i

(s+i s−i+1 + H.c.)

• Standard theory [2,3] is amazingly successful....

• ...but, mainly based on inspired guess work.

Non-Abelian states

The TT patterns encodes the topological data (fusion rules, quantum numbers, degeneracies etc) of non-abelian QH states in a simple and efficient way [23].

101010101010101010101010110011001100110011001100

10101010100110011001100

100110011001100

or 1010101010

“Bratteli diagram” (here for the k=4, M=0 bosonic Read-Rezayi state)

6

[04]

[13]

[22]

[31]

[40]

!!!"

!!!"

###$!

!!"

!!!"

###$!

!!"

###$

###$

!!!"

!!!"

###$

. . .

. . .

###$!

!!"###$!

!!"###$

. . .

. . .

n = 0 1 2 3 4 5

FIG. 5: Bratelli diagram used to illustrate that for k even quasiparticles can only be inserted in pairs.

un- [40][31][40] = (40)(31)30(40) Nφ ≡ (k − 1)M + 2twisted [04][13][04] = (04)03(13)(04) ≡ −M (mod kM + 2)

[31][22][31] = (31)(22)21(31) Ne ≡ k − 1 ≡ −1 (mod k)...

twisted [31][22][13] = (31)(22)(13)1 Nφ ≡(

k2 − 1

)M + 1

[13][22][31] = (13)1(22)1(31)3 (mod kM + 2)... Ne ≡ k

2 − 1 (mod k) .

(12)

Each unit cell will in itself necessarily satisfy (??) whereas for the sandwiched sites in boldface numbers taken together,Nφ and Ne might differ from (??), as is the case in (12). Inspection of these few typical cases makes it straightforwardto generalize to arbitrary even k. Then, the map (??) allows us to generalize to arbitrary M . The general result isgiven to the right in (12). These results are consitent with the consistency results (??) and (??) with n = 2.

Some vacuum compositions of multiple pairs can be combined from the single pair vacuum compositions. Whatwe will see is that taking two untwisted or two twisted vacuum compositions give an untwisted composition whereascombining one untwisted with one twisted gives a twisted vacuum composition. The modular results for Nφ and Ne

will sum in an analogous way. Examples of this and also of a vacuum composition that can not be derived from thesingle pair compositions but nonetheless still conform to the same untwisted or twisted modular results are given in

eq:keven4holes

[31][22][31][40][31] = (31)(22)21(31)30(40)(31) Nφ ≡ −2M

(mod kM + 2)[40][31][22][31][40] = (40)(31)(22)21(31)30(40) Ne ≡ −2 (mod k)

...[31][22][31][22][13] = (31)(22)21(31)(22)(13)1 Nφ ≡

(k2 − 2

)M + 1

(modkM + 2)[40][31][22][13][04] = (40)(31)(22)(13)(04)0 Ne ≡ k

2 − 2 (mod k) ....

(13)

In general on will have that for p pairs of quasiholes

untwisted Nφ ≡ −pM (mod kM + 2)Ne ≡ −p (mod k)

twisted Nφ ≡ −(p− 1)M + (k2 − 1)M + 1 (mod kM + 2)

Ne ≡ −(p− 1) + k2 − 1 (mod k)

(14)

• Identify (two dual) solvable limits of the many-body problem.

• Analyze possible universality classes.

• Provide explicit connections to 1D phenomena. • Show/argue that the solutions are adiabatically connected to the experimental regime.

• Provide a framework for proposing new topological phases.

System:The quantum Hall (QH) system - cold electrons in two dimensions in a perpendicular magnetic field - is a striking example of a system where unexpected phenomena emerge at low energies [1].

• Only experimentally realized system exhibiting topological order!• Extremely rich system: precision and universality, fractional charge and statistics, topological q-bits (?), ...

Challenge:

Our approach:

The most central problem in the understanding of the fractional QH system is that of how (strongly) interacting electrons organize in a spin-polarized Landau level

...aims to...

• Connection to integrable 1D models.

• Candidate ground states a la Laughlin for all filling fractions.

• Exact particle-hole conjugates - also for the bulk! [38]

Further applications:

e∗ = −e/3

(domain wall separating degenerate groundstates)

100100100100100100100100100100100100100100100100100100100101001001001001010010010010010010100100100100100100

-2e-3e

100100100100100100100100100 groundstate10010010100100100100100100 -e/3

Adding a finite number of quasiparticles (or holes) gives sequences of new ground states, where the excitations condense and stay as far apart as possible. This can be done at any filling and provides a fractal structure (complete devil’s staircase) of states and a microscopic derivation [8] of the Haldane-Halperin hierarchy [11,12]. Moreover, one can calculate the gap at any filling - it depends only on the denominator and motivates the phase diagram [13,14], and the stability of states in agreement with experiment [15], as shown below.

Add many quasielectrons 10 to 1/3, lowest energy states:

100101001010010100101001010010100101...

(102)510, ν = 6/17

(102)310, ν = 4/11

10210, ν = 2/5

2

100100100101001001001001010010010010010...(102)510, ν = 6/17

(102)310, ν = 4/11

10210, ν = 2/5

2

100100100100100100100100100100100100...

ν = 1/(2m + 1) = 1/3, 1/5, . . .

ν = 1/(2m + 1)

ν = 2/5, 4/11, 10/21, . . .

ν =1

2m + 1

ν =2

5,

4

11,10

21. . .

e! = e/(2m + 1)

ν = 1/2, 1/4, 3/8, . . .

ν = p/q

ν = p/q, q

ν = p/(2mp + 1)

1..0.....0..1↔ 0..1.....1..0

1......1

ν = 1/3

e! = ±e/q

100 = 102, ν = 1/3

1

1001001001001001010010010010010010010...(102)510, ν = 6/17

(102)310, ν = 4/11

10210, ν = 2/5

2

Fractional charge

Abelian hierarchy

But: this holds only for odd q! (and the lowest LL)

–the crucial question(! = 1)

Nb: The TT limit can also be taken in other geometires - eg by squeezing a sphere!

Idea: vary L1! [4-6]

TT limit:

States:

Hamiltonian:

ψk ∼ eik 2πL1

xe−(y−k 2πL1

)2/2

1y

x

y

x

k= 1 2 3 ....

L1/π

5

I

Bz

x

Vy

! ! ! ! ! ! ! ! ! ! ! ! ! ! !

+ + + + + + + + + + + + + + + + + + +

FIG. 1: Sketch of the Hall experiment. The 2DEG is exposed to a strong perpendicular magnetic field Bz. A current Ix is

passed through the sample along the x-direction, and the resulting transverse voltage Vy measured for varying values of the

magnetic field.

FIG. 2: Sketch of the (integer) quantum Hall effect. The Hall resistance as function of the magnetic field is quantized, i.e.

exhibits plateaux. (The result predicted by the classical Hall effect corresponds to a straight line through the centres of the

plateaux.) The longitudinal resistivity is zero except at transitions between plateaux. Courtesy of D.R. Leadley, Warwick

University 1997.

(bulk) excitations. The integer effect was discovered in 1980 by von Klitzing et al[15]; two years later, using even

cleaner samples, Tsui and collaborators reported the discovery of the fractional effect[16] at ν = 1/3. Since then, with

the fabrication of ever-higher mobility samples, a large number of fractions have been observed[17]. The quantization

of the Hall resistance turned out to be extremely exact (to at least ten parts in a billion), which has led to the

introduction of a new standard of resistance, with the so-called von Klitzing constant RK = h/e2, roughly equal to

25812.8 ohms, as the fundamental unit.

Physically, the number ν in (1) corresponds to the Landau level filling fraction at the center of the corresponding

plateau[18]. In other words, the IQHE occurs when an integer number of Landau levels is filled, while the FQHE

A possible state at

ν = 1

ν = 1/3

ν ≤ 1

L1, L2

2

ν = 1

ν = 1/3

ν ≤ 1

L1, L2

1 0 0 1 1 0 0 0 1 0 0 0 0 1 0

2

ν = 1/(2m + 1) = 1/3, 1/5, . . .

ν = 1/(2m + 1)

ν = 2/5, 4/11, 10/21, . . .

ν =1

2m + 1

ν =2

5,

4

11,10

21. . .

e! = e/(2m + 1)

ν = 1/2, 1/4, 3/8, . . .

ν = p/q

ν = p/q, q

ν = p/(2mp + 1)

1..0.....0..1↔ 0..1.....1..0

1......1

ν = 1/3

e! = ±e/q

100 = 102, ν = 1/3

1

(all ee-terms that preserve position of CM)

(eg Coulomb V(r)=e /r)2

No kinetic energy!

k+m k-m

Vk,m

A Landau level (LL) is 1D! With periodic b.c. using Landau gauge this is explicit:

Nb: Exact mapping of a single Landau level!

• Hopping amplitudes become exponentially suppressed.

• Only ‘electrostatic’ terms remain! [5,6].

• The occupation number basis diagonalize the interaction!

• Ground states obtained by separating the electrons as much as possible:

For small enough L1, the overlap between the single particle states vanish and...

Condensates and phase diagram

ν = 1/(2m + 1) = 1/3, 1/5, . . .

ν = 1/(2m + 1)

ν = 2/5, 4/11, 10/21, . . .

ν =1

2m + 1

ν =2

5,

4

11,10

21. . .

e! = e/(2m + 1)

ν = 1/2, 1/4, 3/8, . . .

ν = p/q

ν = p/q, q

ν = p/(2mp + 1)

1..0.....0..1↔ 0..1.....1..0

1......1

ν = 1/3

1

Quasielectrons with charge -e/3 (at ) are obtained by inserting 10:

Charge is detremined by the Su-Schriefer argument [10]:

Adiabatic continuity

The abelian hierarchy emerge as the solution of a generic repulsive two-body interaction in the TT limit [6,8,9].

ν = 1/(2m + 1) = 1/3, 1/5, . . .

ν = 1/(2m + 1)

ν = 2/5, 4/11, 10/21, . . .

ν =1

2m + 1

ν =2

5,

4

11,10

21. . .

e! = e/(2m + 1)

ν = 1/2, 1/4, 3/8, . . .

ν = p/q

ν = p/q, q

ν = p/(2mp + 1)

1..0.....0..1↔ 0..1.....1..0

1......1

ν = 1/3

1

ν = 1/(2m + 1) = 1/3, 1/5, . . .

ν = 1/(2m + 1)

ν = 2/5, 4/11, 10/21, . . .

ν =1

2m + 1

ν =2

5,

4

11,10

21. . .

e! = e/(2m + 1)

ν = 1/2, 1/4, 3/8, . . .

ν = p/q

ν = p/q, q

ν = p/(2mp + 1)

1..0.....0..1↔ 0..1.....1..0

1......1

ν = 1/3

e! = ±e/q

1

• At minimal domain walls carry charge • Fractionally charged particle-hole pair excitations have lowest energy, which depends on the size of the charge (ie only on q).

Example:

Claim: TT-states 100, ... develop smoothly into abelian bulk QH-states for all odd q! [6] (No phase transition as L1 increases)

• Same qualitative properties (gap, excitation structure etc.) [16,6,9]

• Proven for Laughlin states and pseudopotential interaction! [4,9,17]

• Numerical studies (exact diagonalization and DMRG).

• New unique bulk wave function (CFT construction) for any state that is obtained by successive condensation of quasielectrons [8,18].

L1 → 0• Gives TT-states in as • Gives Laughlin/Jain wave functions where these exist• Supported by numerics for simplest non-L/J state (at 4/11). • Same structure from a different perspective.

Depending on the interaction and the filling transitions from the TT states may occur. One possibility is a transition to non-abelian states with an enhanced ground state degeneracy. These can be understood as...

• Instabilities at finite L1 (for a two-body interaction) [19].• Ground states of multi-particle interactions [20-24].• L1→0 limits of the bulk wave functions [20].

Moore-Read state [25]

H =12

∫ 2π

0dϕdϕ′ : ρ(ϕ)V (ϕ− ϕ′)ρ(ϕ′) :

Consider QH problem on a cylinder and let L1→∞ while keeping N fixed. The direction along the cylinder can then be integrated out and one finds [36]:

Circle limit:

Phases:

|ΨE〉 =∏

i

ψ†(ϕi)|0〉, E =12{∑

j,k

V (ϕk − ϕj)−∑

j

V (ϕj)}

•Wigner crystals for small filling fractions [36,37].

• Gapped QH states (abelian and non-abelian) at intermediate fillings - these are adiabatically connected to the bulk [36].

This is interacting electrons on a circle without a kinetic term, and is trivially solved:

But, non-trivial physics appears as implementing a filling fraction implies a constraint on the allowed momenta.

Another possible instability is to gapless states - at half-filling this is well understood [5,6,9].

Generalizations

Ground states:

Crucial for non-abelian statistics!

~ 2n/2 degenerate states for n qh ‘s at fixed positions.

Cf. 3-body interaction [26]: Never 3 particles on 4 consecutive sites.

e101010101010101010101010101010101010101010101010101010100110011001100101010101001100110011001010

‘Half’ quasiholes as domain walls:

e/4 e/4e/4e/4

Quasihole degeneracies:

• Six-fold degenerate ground state.

• ’Second generation’ of charge fractionalization [20,21].

• Inequivalent ways of forming domain wall excitations [20,21]

(This model was introduced as a toy model of the FQHE in Ref. 37, and derived as a limit thereof in Ref. 36.)

• The TT-states serve as input in the recent Jack polynomial constructions [30] of (old and new) QH states. Related to this there is a squeezing rule [31,30], where the TT states play the role of highest weight configurations, from which all other configurations are obtained via inward squeezing (this gives a non-trivial selection rule that reduces the Hilbert space size [30-32]).

• In the ”pattern of zeros” approach [33,34], the TT patterns are interpreted in terms of the vanishing properties of the wave functions as (clusters of) particles approach each other. This gives a way of finding new classes of states, and a framework to calculate many properties thereof. (On a technical level, extracting the pattern of zeros is analogous to taking the TT limits of wave functions [6,18,35].)

• The results for the Moore-Read state generalize directly to more complex non-abelian states [22,23], such as the Read-Rezayi states [27].

• ...and suggest a route to finding new non-abelian phases, see eg Ref 24.

• The ground state patterns are naturally organized in Bratteli diagrams (see Fig), which reveal the fusion structure of the states and their excitations.

Towards the bulk

• TT-state 10101010.... ground state for very small L1.• But: metallic, gapless state in the bulk (experiment and theory [28]) !?

Luttinger liquid phase• 10101010.... is annihilated by the shortest range hopping, V21: 1001 0110

• The TT state melts at finite L1 (for sufficiently short range interactions).

• Low energy sector contained in ‘spin space’

• After the melting a gapless Luttinger liquid phase is realized.

• Our microscopic solution has striking similarities (weakly interacting neutral dipoles etc) with the composite fermion [3] description of this state [28].

By comparing to the Rezayi-Read [29] wave function we establish that the Luttinger phase is smoothly connected to the bulk state.