Universal Transport Properties in the Quantum Hall...
Transcript of Universal Transport Properties in the Quantum Hall...
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Universal Transport PropertiesUniversal Transport Propertiesin the in the
Quantum Hall EffectQuantum Hall Effect
● Chern-Simons effective action: bulk and edge
● Transport due to chiral and gravitational anomalies
● Wen-Zee term, ‘orbital spin’ and Hall viscosity
● Orbital spin at the edge and its universality
Andrea Cappelli(INFN & Florence Univ.)
with L. Maffi (Florence Univ.)
OutlineOutline
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Edge excitationsEdge excitations
edge ~ Fermi surface: linearize energy
relativistic field theory in (1+1) dimensions with chiral excitations (X.G.Wen, '89)
Weyl fermion (non interacting)
Interacting fermion chiral boson (Luttinger liquid)
Incompressible fluid
t
½
Fermi surface
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Chern-Simons action & Hall CurrentChern-Simons action & Hall Current
● Hall current is topological
● Sources of field are anyons
● Needs boundary action chiral boson CFT
● Bulk topological theory is tantamount to conformal field theory on boundary
Introduce Wen's hydrodynamic matter field and current
Density
Hall current
Laughlin state
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Boundary CFT and chiral anomalyBoundary CFT and chiral anomaly● edge states are chiral fermions/bosons
● chiral anomaly: boundary charge is not conserved
● bulk (B) and boundary (b) compensate: anomaly inflow
● adiabatic flux insertion (Laughlin)
spectral flow
● edge chiral anomaly: exact quantization of the Hall current
universal transport coefficient
bulk-boundary correspondence
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Thermal current and gravitational anomalyThermal current and gravitational anomaly● gravitational anomaly in the chiral edge theory
● Casimir energy and ‘Casimir matter current’
● specific heat and thermal current
● unbalanced edges
● It has been measured
● bulk-boundary correspondence: gravitational Chern-Simons action
(Read, Green ’00; A.C., Huerta, Zemba '02)(Heiblum et al. ‘14-’19; Wang et al. ‘18)
metric background in (2+1)-d
(Stone ‘12; Gromov et al. ‘15)
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Wen-Zee-FrWen-Zee-Frööhlich actionhlich action
● consider spatial metric only and corresponding spin connection
● further transport coefficient
● electron ‘orbital spin’, e.g. on n-th Landau level
(Avron et al., Read et al.)
Wen-Zee shift
Hall viscosity
strain
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Hall viscosityHall viscosity
● Constant stirring creates an orthogonal static force, non dissipative
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Is Hall viscosity universal?Is Hall viscosity universal?
● If YES:
- further universal ‘geometric’ transport coefficient orbital spin
- suggests that ‘composite fermion’ excitations are extended objects
renewed interest, following Haldane, Read, D.T. Son, Wiegmann,…
● BUT: is not related to an anomaly!
● how to understand?
A: study quantities related to in the edge CFT and check universality
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Bulk-boundary correspondence for Bulk-boundary correspondence for
● add boundary terms to the Wen-Zee action
extrinsic curvature needed for Euler characteristic
● predicts ground-state values for charge and spin in the edge CFT
● BUT:
- g-s values are not universal, can be tuned to zero (~ local terms in the 2d action)
- which boundary conditions?
- bulk non-universality: this action suggests that islands with different values can be made in the bulk without closing the gap
(Gromov, Jensen, Abanov ‘16)
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Orbital spin at the edgeOrbital spin at the edge
● Explicitly study of Landau levels for integer filling
● limit to the edge: momentum and radius
● wavefunctions of level are gaussian-localized at with spread
● shift outward by is absent for straight boundary (extrinsic curvature) orbital spin i.e. up to const.
● wavefunctions pushed up in confining potential acquire higher energy for higher Landau levels
(AC, Maffi `18)
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Fermion CFT at the edgeFermion CFT at the edge
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● n-th level branch is displaced by
● Can be accounted for by chemical potential shifts
● how to fill the Fermi sea?
- top: least energy (smooth boundary)
- bottom: same momentum (sharp boundary)
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● analyze the cases:
i) single branch
no physical effect of orbital spin
ii) left: independent (non-interacting) branches, as e.g. integer Hall effect, or connected to a reservoir ( )
no effect of
iii) right: overlapping (interacting) branches, e.g. sharp boundary & hierarchical Hall states, for isolated Hall droplets
differences are observable and universal
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● Excited state w.r.t. standard CFT ground state
● agreement with WZ action (up to const.)
● orbital spins identified up to const.
● Results extend to fractional (hierarchical) filling in the bosonic CFT
● How to measure?
● By Coulomb blockade (tunneling in a isolated droplet at zero bias)
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Squeeze area by
quadrupole deformation
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ConclusionsConclusions● charge and heat currents are associated to anomalies of the edge theory
universal transport coefficients
● Hall viscosity, proportional to average orbital spin is not universal
● differences of orbital spin are universal and can be observed with edge physics in isolated Hall droplets
● possible experiment: Coulomb blockade in the isolated droplet with quadrupole deformation
Not discussed
● extension of analysis to fractional fillings by using area-preserving diffeomorphisms (W-infinity algebra)
● other physical effects of orbital spin for QHE on a conical surface (metric singularities are sources of ) (Laskin, Chiu, Can, Wiegmann ‘16-’18)
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Quantum incompressible fluidsQuantum incompressible fluids ● Area-preserving diffeomorphisms of incompressible fluids
A = constant
● Fluctuations of the fluid are described by diffs, generated by Poisson brackets
classical
quantum Moyal brackets
algebra (GMP sin-algebra)
● Fully understood in the edge CFT It reproduces/predicts Jain hierarchy
minimal models (A.C., Trugenberger, Zemba '96)
A A
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● Bulk fluctuations in lowest Landau level are non-local:
● can be expressed in terms of fields of increasing spin, traceless & symmetric
● Recover Wen hydrodynamic field plus correction
● Spin-one and spin-two fields parameterize matter fluctuations
(Iso, Karabali, Sakita)
+ gauge symmetry