Holographic Fermions with Lattices

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Holographic Fermions with Lattices

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04/25/2013, ?????????????
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G. Horowitz, J. Santos and D. Tong Optical
Conductivity with Holographic Lattices. JHEP 1207
(2012) 168 ,ArXiv1204.0519. Further Evidence for
Lattice-Induced Scaling.JHEP 1211 (2012) 102,
ArXiv1209.1098. G. Horowitz and J.
Santos General Relativity and the
Cuprates arXiv1302.6586

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• Holographic Fermionic Liquid with Lattices
• arXiv1304.2128 

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Outlines

• Preliminary Applications of AdS/CFT to CMT
• Introduction Why lattices?
• How to find a lattice background?
• Holographic Fermions with lattices
• Prospects

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Applications of AdS/CFT to CMT

• Theoretical foundation
• A p2 dimensional theory of quantum
  gravity may be described by a p1 dimensional
  quantum field theory without gravity.

Large N gauge theories in D-dim
(Semi-)Classical gravity in D1-dim
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Applications of AdS/CFT to CMT

• Bulk/boundary correspondence
• More specifically

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Bulk??????
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• Eg.1Holographic superconductors

The action of matter in the bulk
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• Holographic superconducting phase

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• Eg.2Holographic (Non-)Fermi-like Liquid

The retarded Green function
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Introduction Why lattices?

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Introduction Why lattices?

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Introduction Why lattices?
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Introduction Why lattices?

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?delta???????????????????
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How to find a lattice background?

• Two methods

1?Scalar lattice Simulating lattices with
periodic scalar field
with potential
2?Ionic lattice directly introducing a
periodic chemical potential
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How to find a lattice background?

• 4D Framework

Equations of motion
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How to find a lattice background?

• 4D Framework

Scalar field with periodic behavior
Lattice constant
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How to find a lattice background?

• 4D Setup

Ansatz of variables
No change!
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RN black holes
Temperature
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How to find a lattice background?

• Crucial technical issues in AdS/CMT with
  lattices

1?Numerically solve the background equations with
appropriate boundary and gauge conditions
2?Numerically solve the perturbation equations
over the background.
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How to find a lattice background?

• DeTurck method

1?Einstein-DeTurck equation
Here a reference metric is the RN black hole
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How to find a lattice background?

• DeTurck method

2?To guarantee the numerical result is a
solution to Einstein equation

1. The convergence of the solutions

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How to find a lattice background?

• Boundary conditions

1?Conformal symmetry at infinity (z0)
Remark Such an assignment must be consistent
with the asymptotic behavior of
the EOM!
2?Regular conditions on horizon (z1)
Remark To me it is not clear yet if such a
regular condition will definitely
lead to a unique solution!
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How to find a lattice background?

• Numerical methods in solving equations

1?(pseudo)spectral method
Change the partial differential equations
into nonlinear algebraic equations by
pseudospectral collocation approximation
X direction Fourier series
Z direction Chebyshev polynomials
2?Newton-Raphson method
Change nonlinear algebraic equations into
linear algebraic equations and then solve then
with simple command Linearsolve in Mathematica
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How to find a lattice background?

• The numerical results examples

1?Scalar lattice
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How to find a lattice background?

• The numerical results

2?charge density
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How to find a lattice background?

• The numerical results examples

2?Ionic lattice
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Holographic fermions with lattices

• Contents

1?Consider a Fermionic field over a lattice,
solving the Dirac equations numerically.
2?Locating the position of the Fermi surface via
the standard holographic dictionary.
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Holographic fermions with lattices

• The setup

Background
Remark a) it is a linear, no need of Newton
method. b) it is first-order,
only fixing the boundary condition on one side.
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Holographic fermions with lattices

• Writing down the Dirac equations explicitly

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Holographic fermions with lattices

• The spectral method

Boundary condition at the horizon (z1)
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Holographic fermions with lattices

• Read off the retarded Green function

The asymptotic behavior of EOM at infinity
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Holographic fermions with lattices

• The numerical results

1?Parameters for the background
2?A parameter for perturbations
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Holographic fermions with lattices

• The numerical results

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Holographic fermions with lattices

• The shape of the Fermi surface is ellipse!

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Holographic fermions with lattices

• Some other properties

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Holographic fermions with lattices

• The numerical results on band gap

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Holographic fermions with lattices

• The numerical results on band gap

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Summary

• New results on holographic fermions when lattice
  is introduced

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Prospects

• On holographic fermions with lattices
• On applications of lattices to other topics

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1. Weyl????????????????
2. ?????AdS3/CFT2(?????????????)
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4. ???????/?????

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