A 4D generalization of the QHE and the emergent relativity principle

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A 4D generalization of the QHE and the emergent
relativity principle

• Tsinghua University
• April, 2004

SC Zhang, Stanford University so5.stanford.edu
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Motivation for the talk

• QFT and condensed matter physics
• QFT has been an extremely useful description for
  the low energy physics in condensed matter
  systems.
• Can condensed matter systems provide insight into
  UV sectors of QFT?
• Non-relativistic quantum many body system at high
  energygtrelativistic QFT at low energy
• He films in 2Dgt21 dim QED
• D-wave superconductivitygtDirac fermions
• 2D quantum Hall effectgt21 Chern-Simons theory
• These examples have profound implications.

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Motivation for the talk (continued)

• The emergent relativity principle
• Einsteins principle of relativity does not need
  to be postulated a priori, but can be derived
  from quantum mechanics of discrete degrees of
  freedom.
• Relativity emerges in the low energy sector,
  due to interaction of many degrees of freedom.
• What do we gain from this approach?
• More natural unification of quantum mechanics and
  relativity.
• Ultraviolet theory is well-defined, no problems
  with infinities.
• If the graviton can emerge in the low energy
  sector, this could give a theory of quantum
  gravity.
• Analog experiments can guide us.

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Plan of the talk

• Review some examples of emergent relativity in
  condensed matter physics
• 2D quantum Hall effect
• 4D quantum Hall effect
• Single particle Hamiltonian
• Quantum liquid
• Emergent 31 dim relativistic physics at the edge
• Exact wave function for a quantum membrane
• Outlook

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Some references

• To see a world in a grain of sand
• Hep-th/0210162, Festschrift to John Wheelers
  90th birthday, published by Cambridge University
  Press
• A 4D generalization of the quantum Hall effect
• Science 294, 823 (2001)

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He film and 21 dim QED

• Elementary excitations of the He liquid sound
  mode and the vortex.

• sound mode?photon, vortex?electric charge.

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He film and 21 dim QED

• At long wave length and low energy, the dynamics
  of a superfluid helium film is described by the
  21 dim QED.

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D-wave superconductivity in D21

• A 2D system of interacting electrons can give
  rise to d-wave superconductivity
• The fermi surface has an energy gap everywhere
  except at four nodal points.

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D-wave superconductivity and Dirac fermion in 21
dim

• At the nodal points, the linearized equations of
  motion is exactly the Dirac equation in 21 dim.

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2D QHE and Chern-Simons theory

• The electron gas in 2D can be transformed into a
  gas of bosons, with an odd number of flux quanta
  attached to them. The bosons then form a
  superfluid state.

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Fundamental concepts from cond-mat theory

• The BCS theory of superconductivitygtconcept of
  broken symmetrygtstandard model
• Scaling near critical pointgtWilsons theory of
  renormalization groupgtasymptotic freedom in QCD
• Quantum Hall effectgt?

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Quantum Hall effect in 2D

• 2D space plus a abelian gauge field
• Landau levels

• Laughlins wave function

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Emergent properties of the QHE

• Fractionally charged quasi-particles with Q1/m.
• Long distance physics completely described by a
  topological quantum field theory, the
  Chern-Simons-Landau-Ginzburg theory.
• Relativistic chiral edge states.
• Non-commutative geometry.
• Quantized Hall conductance

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Non-commutative geometry

• A generalization of the Heisenberg uncertainty
  principle to coordinates.
• Guiding center motion in the lowest-Landau level

• The magnetic length provides a UV cut-off without
  breaking any translational symmetry.

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Can one generalize QHE to higher D?

• In 2D quantum Hall effect

• In higher D, there is no simple antisymmetric 2
  index tensor. However, a generalization for SU(2)
  spin current is possible in D4

• Current is transverse to applied E field, and
  therefore nondissipative. It flows along the
  direction where the spin is pointing. gt perfect
  spin-orbit coupling!

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Quantum Hall effect

• Haldane sphere

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Higher D generalization

• Second Hopf map from S7 to S4

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What is the question?

• Nonabelian holonomy of a SO(5) spinor

• This is the SU(2) gauge potential of a Yang
  monopole, or the conformal mapping of the
  Yang-Mills instanton in D4, constructed by
  Belavin, Polyakov, etc.

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Algebra and geometry

• When is the spatial infinity of a d-dimensional
  Euclidean space, or the equator of a
  d-dimensional sphere isomorphic to a group?
• For d1, the spatial infinity of R1 of is
  isomorphic to Z2 .
• For d2, the spatial infinity of R2 is S1 , which
  is isomorphic to U(1).
• For d4, the spatial infinity of R4 is S3 , which
  is isomorphic to SU(2).
• There is nothing more!

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A SO(5) symmetric Hamiltonian

• All eigenstates fall into the (p,q) irreps of
  SO(5). For a given I, we have p2Iq. I is
  similar to the B field strength, and q is similar
  to Landau level index.

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Landau levels on S4

• Exact solution of the ground state wave function

• Ground degeneracy

• Ground state energy

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Connections among math concepts

• Summary

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Many body wave function

• Thermodynamic limit

• The problem is effectively six dimensional, given
  by orbital space S4 times the isospin space S2.
  CP3 is locally isomorphic to S4 x S2
• Filling fraction is given by

S2
S4
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Integer QHE

• Slater determinant for u1

• Density correlation function

• An incompressible quantum liquid!

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Fractional QHE

• Powers of Slater determinant

• This is a homogeneous polynomial of the SO(5)
  spinor with degree pmp, therefore, the filling
  factor is u1/ m3.
• This quantum liquid has elementary excitations
  with fractional charge q1/ m3.
• Fractionalization is possible in higher D!!!

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Single particle Lagrangian

• In 2D QHE, the single particle Lagrangian is
  given by

• The 4D generalization is given by

• Locally, the problem is isomorphic to 3
  independent planes of 2D QHE, but the problem is
  SO(5) symmetric!

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The topological field theory description

• What is the topological field theory description
  of the 4D QHE liquid? What is the generalization
  of the Chern-Simons term in 21 dim?

• It is a SU(2) non-abelian Chern-Simons theory in
  41 dimensions.

• This theory can also be obtained from a 61 dim
  abelian Chern-Simons theory, via dimensional
  reduction.

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Noncommutative geometry

• Noncommutative coordinates in lll

• Magnetic translations commute up to a U(1)
  factor.
• Generalization

• Magnetic translations commute up to a SU(2)
  factor.
• Unification of space, time, spin and the quantum!

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Fractional statistics

• In the 21 dimensional QHE, quasi-particles have
  fractional statistics, generated by the
  Chern-Simons term.
• In the 41 dimensional QHE, there are membrane (2
  brane) like excitations with fractional
  statistics.

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Quantized space, no time

• Clocks can only be built from energy level
  differences. In the LLL, there is no energy level
  differences, and there is no time evolution!
• Space is quantized.
• The theory contains only topological degrees of
  freedom.
• Dynamics comes only from the edge!

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Bulk and edge excitations

• So far there is no clock and no dynamics in the
  bulk.
• In an incompressible liquid, it costs high energy
  to change the volume of the liquid, but it costs
  low energy to change the shape of the liquid.
  Therefore, low energy excitations are confined at
  the edge.

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Edge state of 2D QHE

• In the lowest Landau level, a particle and a hole
  pair moves along a linear track.
• y l02 qx
• In the lll, the only energy is the potential
  energy. Since V(y)V y, one obtains a
  relativistic dispersion
• Ec qx

y
x
V
y
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Particle-hole pair excitations
Extremal dipole states maximal dipole distance
in x5, point-like particle in 3D
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A theory of higher spin particles?
Ripple waves On the boundary

• HelicityT1- T2

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Maxwell and Einstein equations

• Particle-hole operators at the edge

• For S1, this reduces to the Maxwells equation
  with
• For S2, we obtain linearized Einstein equation.

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Embarrassment of riches!

• Higher spin particles Sgt3
• String theory also contains higher spin states,
  but they are massive!
• Non-extremal dipole states
• They are nonlocal extended states
• We need a spin gap for higher spin states!
• Recent work shows that particles may cluster to
  form 2-branes, reducing the entropy at the edge.

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Motivation to study quantum membrane

• Slater determinant wave function and the Jastrow
  wave functions.
• In the 2D case, they are related by the van der
  Monde identity for the u1 case.
• The LLL wave functions have more symmetry than
  SO(5), in fact they have SU(4) symmetry of the
  CP3 manifold.
• SU(4) can be reduced to SP(4)SO(5) through the
  symplectic metric.
• Particles may cluster to form 2-branes, reducing
  the entropy at the edge.

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Construction of the membrane wave function

• The symplectic metric

• The generalization of the Laughlin-Haldane wave
  function

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Graphical representation

• When m is odd, this is a fermionic wave function.
• It is a SO(5) singlet.
• When the product is expanded, y(i) occurs
  pm(N-1) times. Therefore, N scales like p, which
  scales like R2. It is a membrane!

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Plasma analogy

• Quantum particles on CP3.

• Attractive on S4, repulsive on S2. A membrane on
  CP3!

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Magic dimensions
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Conclusions

• A 4D generalization of the QHE.
• Well defined cut-off at the magnetic length,
  without violating the rotational symmetries.
• Relativity emerges at the boundary, including
  massless vector and tensor fields.
• What is special about D1,2,4?
• Spatial infinity Z2, U(1) and SU(2).
• Future directions
• Fermionic excitations.
• Fully interacting edge theory.
• Experiments related to spintronics.

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QFT and cond-mat physics

• Relativistic quantum field theorygt quantum
  mechanics
• Quantum many body physicsgtrelativistic quantum
  field theory.
• Logical structure of physics may be more complex
  than a reductionistic hierarchical structure.