1. Duality and Quantum Hall Systems
Duality in Your
Everyday Life
with Rim Dib, Brian Dolan and C.A. Lutken

2. Outline Motivation and Preliminaries What Needs Explaining
The virtue of robust low-energy explanations
What is the Quantum Hall Effect?
What Needs Explaining
Transitions among Quantum Hall Plateaux
A phenomenological explanation
A symmetry interpretation: the group G0(2)
From whence the symmetry?
Statistics shifts and particle/vortex duality
Spinoffs
New Predictions: Bosons and Gq(2) Invariance
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Quantum Hall Duality

3. Why Stay Awake? Field theory: Condensed matter physics:
Many of the most interesting theories have recently been found to enjoy duality symmetries.
eg: Electromagnetism with charges and monopoles has symmetry under their interchange together with e replaced by 2π/e.
Quantum Hall Systems are the first laboratory systems with this kind of symmetry.
May learn about new kinds of duality.
Condensed matter physics:
Duality helps identify robust low-energy features
eg why are some QH features so accurate?
Suggests existence of new kinds of Quantum Hall Systems
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4. What is the Quantum Hall Effect?
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5. 2-d Electron Systems
Girvin, cond-mat/
Electrons/holes can be trapped on a two-dimensional surface using an electrostatic potential well.
Occurs in real semiconducting devices.
Knobs: T, B, n
Carrier densities may be adjusted by varying a gate voltage
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6. The Conductivity Tensor
Crossed electric and magnetic fields, or anisotropy, give currents not parallel to applied E fields
Described, for weak fields, by a conductivity tensor, skl,or a resistivity tensor, rkl = (s-1)kl
skl = slk : Ohmic conductivity
skl = - slk : Hall conductivity
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7. Isotropic 2d Systems Special Case: Isotropic Systems:
Choose: E = E ex, B = B ez
Alternatively, in the upper-half complex plane:
s := sxy + i sxx
Resistivity: r := rxy + i rxx = -1/s
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8. Classical Hall Effect
Lorentz force on moving charges induces a transverse current.
Implies rxy proportional to B:
Where r0 = m/(n e2 t0); n = n/(eB).
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9. Free-particle Quantum Description
Free electrons in magnetic fields form evenly-spaced simple-harmonic Landau levels (LL), labelled by n, k
Energy: wc = eB/m, Size: 1/l2 = eB, Centre: yc = l2 k
Each LL is extremely degenerate in absence of interactions.
Number of available states proportional to sample area.
Fraction of LL filled with N electrons is n = n/eB.
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10. Experiments: Stormer: Physica B177 (92) 401
Experiment: at low temperatures rxy is not proportional to B.
Plateaux with sxy = ke2/h, with k an odd-denominator fraction
Quantized with extraordinary accuracy!
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11. A Quantum Hall Puzzle
Why should sxx vanish and sxy be so accurately quantized in units of e2/h?
A microscopic picture:
Interactions and disorder broaden bands and localize electrons.
Conductivity independent of filling requires most electrons in band to be localized, so as not to conduct.
Narrow energy range in each band containing extended states causes jumps between plateaux.
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12. Quantization: The Corbino Geometry
Girvin, cond-mat/
Laughlin’s gauge-invariance argument:
Insertion of magnetic flux induces radial charge transfer
One flux quantum gives a ‘large’ gauge transformation.
Induces level shift with transfer of a charge quantum
Laughlin, PRB
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13. Other Experiments: Flow Between Plateaux
1. Semicircle Law as B is varied. 2. Transitions between which plateaux? 3. Universality of Critical Points 4. Superuniversality of Critical Exponents 5. rxx  1/rxx Duality
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14. Varying External Parameters
Varying B causes transitions from one plateau to the next.
Varying T removes the area between the plateaux
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15. Temperature Dependence
Yang et.al., cond-mat/
For those B where sxy lies on a plateau, rxx goes to zero at low temperatures.
sxy = -n e2/h ; n = -p/q
Hall Insulator: for very large B, rxx instead goes to infinity at low temperatures.
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16. Critical Points At plateaux, rxx falls as T falls.
Yang et.al., cond-mat/
At plateaux, rxx falls as T falls.
For large B, rxx eventually rises as T falls, (and so is an insulator).
The critical field, Bc, for the plateau-insulator transition, occurs when rxx is constant as T falls.
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17. Critical Resistivity is Universal
Yang et.al., cond-mat/
Positions of fixed points are independent of sample.
e.g. For n =1/(2n+1) to Hall Insulator transitions:
rcxx = h/e2 = 25.3 kΩ
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18. Critical Scaling Near critical points predict universal form:
x = localization length of electrons
Conductivity approaches its critical value as a universal power law as a function of:
temperature if x varies as T-p
system size, if x varies as L
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19. Superuniversality
Wei et. al., PRL
Slope of Hall resistivity between plateaux diverges like 1/Tb
Same power of T regardless of which plateaux are involved!
Width of transition region vanishes like Tb
Same power of T (up to sign) as for previous divergence
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20. B Dependence of Flow For n=1 to Insulator transition:
Shahar et.al., cond-mat/
For n=1 to Insulator transition:
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21. Deviations from Scaling?
Shahar et.al., cond-mat/
Some disordered samples do not seem to have scaling form.
Temperature scaling should imply rxx is a function only of the combination:
Dn/Tk
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22. Flow in the Conductivity Plane
As T goes to zero (IR), sxx flows to zero.
For plateaux sxy ≠ 0 so flow is to fractions on the real axis away from origin.
For Hall insulator sxy also flows to zero, corresponding to flow to the origin.
Critical behaviour occurs at the bifurcation points
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23. Semicircle Law
Varying B for fixed T causes transitions between plateaux.
In clean systems flow is along a semicircle in the s plane.
In r plane this corresponds to flow with rxx varying from 0 to infinity, and rxy fixed.
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24. Semicircle Evidence I Hilke et al, cond-mat/9810217 11/23/2018
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25. Semicircle Evidence II
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26. rxx to 1/rxx Duality
Shahar et.al., Science , cond-mat/
As rxx varies (for fixed rxy) across the critical point, rxx = 1, it goes into its inverse for opposite filling factors.
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27. An underlying symmetry: phenomenology
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28. Symmetry and Superuniversality
Superuniversality strongly suggests a symmetry:
Resulting symmetry group: G0(2) (plus particle-hole interchange)
Lütken & Ross: postulate symmetry acting on complex s.
Motivated by Cardy’s discovery of SL(2,Z) invariant CFT.
Kivelson, Lee & Zhang use Chern-Simons-Landau-Ginzburg theory to argue for symmetry acting on filling fractions, n
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29. Symmetry Implies Flow Lines
C.B., R. Dib, B. Dolan, cond-mat/
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30. Flow Lines Imply QH Observations
B. Dolan, cond-mat/ ;
C.B., R. Dib, B. Dolan, cond-mat/
Flows along vertical lines and semicircles centred on real axis.
Implies observed properties of transitions:
IR attractive fixed points are fractions p/q, with q odd.
Semicircle Law
Transitions from p1/q1 to p2/q2 are allowed if and only if
(p1 q2 - p2 q1) = ±1.
Universal Critical Points: s* = (1+i)/2
Superuniversality of critical exponents
Contains duality: rxx to 1/rxx
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31. Quasi-microscopic symmetry: Particle/Vortex Duality
C.B., B. Dolan, hep-th/
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32. Microscopic Symmetry Generators
Assume the EM response is governed by the motion of low-energy quasi-particles and/or vortices.
Identify the three symmetry generators as:
ST2S(s) = s/(1-2s): Addition of 2p statistics;
TST2S(s) = (1-s)/(1-2s): Particle-vortex interchange;
P(s) = 1 – s*: Particle – hole interchange.
Goal: calculate how EM response changes under these operations.
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33. Definition of Dual Systems
In 2 space dimensions, weakly-interacting particles and vortices resemble one another.
EM response due to motion.
Characterized by number, mass, charge/vorticity
One symmetry generator is particle-vortex duality
System A:
N particles of mass m
Dual to A:
N vortices of mass m
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34. Low-Energy Dynamics Degrees of Freedom: Systems with particles:
EM Probe: A
Statistics Field: a
Systems with particles:
Particle positions: x
Charge, Mass
Systems with vortices:
Vortex positions: y
Zeroes in order parameter for U(1)EM breaking
Goldstone Mode: f
Phase of order parameter
Mediates long-range Magnus force between vortices (unless eaten)
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35. Relating EM Response for Duals
Integrating out particle/vortex positions gives effective theory of EM response.
Can relate dual EM responses without being able to evaluate path integrals explicitly.
Relies on equivalent form taken by Lp and Lv.
Ditto for q g q + 2p.
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36. Linear Response Functions
For weak probes can restrict to quadratic power of Am
Response function P has convenient representn as a complex variables.
Can relate response functions for dual systems.
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37. EM Response and Duality
Momentum dependence of  distinguishes EM response:
Superconductors
A nonzero
Insulators
A zero
Conductors
Square root branch cut
Conductor  Conductor
Superconductor  Insulator
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38. Duality for Conductors
PV duality acts on conductivity plane.
Ditto for addition of 2p statistics.
Duality not symmetry of Hamiltonian but of RG flow:
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39. Duality Commutes with Flow
Suppose a system with given conductivities corresponds to a system of N quasiparticles, and its dual is N vortices.
Response to changes in B,T computed by asking how N objects of mass m respond, and so is the same for system and its dual.
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40. rxx to 1/rxx Duality
Shahar et.al., Science
Particle-vortex interchange corresponds to the group element which maps the circle from 0 to 1 to itself.
rxx  1/rxx
Particle/vortex interchange for duals implies filling factors are related by
Dn  - Dn
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41. Duality for Bosons
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42. Duality for Boson Charge Carriers
Choose particles to be bosons:
Resulting symmetry group: Gq(2) (plus particle-hole interchange)
Predictions for the QHE can be carried over in whole cloth
Several precursor applications of this symmetry, or its subgroups
Lütken
Wilczek & Shapere
Rey & Zee
Fisher
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43. Gq(2)-Invariant Flow
Duality predicts flow lines, and flow lines predict effects to be expected:
Flow in IR to even integers and fractions p/q, with pq = even
Semicircle Law as for fermions
Rule for which transitions are allowed.
Universal Critical Points:
eg s* = i (metal/insulator transition?)
Superuniversality at the critical points
Duality: sxx to 1/sxx when sxy = 0
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44. Gq(2)-Invariant Flow
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45. The Bottom Line
Particle-Vortex Duality relates the EM response of dual systems
Implies symmetry of flow, not of Hamiltonian
Combined with statistics shift generates large discrete group
Works well for QHE when applied to fermions
Many predictions (semicircles etc) follow robustly from G0(2) symmetry implied for the flow in QH systems.
Implies many new predictions for QH systems when applied to bosons.
Eventually testable with superconducting films, Josephson Junction arrays, etc.
Potentially many more implications…
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46. fin
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