VINCENT SACKSTEDER IV

1
Three Alternative Sigma  Models of Disorder and
Their Relative Merits

• VINCENT SACKSTEDER IV
• ASIA PACIFIC CENTER
• FOR THEORETICAL PHYSICS
• LIKE THE ICTP,
• BUT FOR THE ASIA-PACIFIC REGION.
• LOCATED IN SOUTH KOREA.
• PH.D. OBTAINED AT ROME LA SAPIENZA

2
Format of This Talk

• Original Hamiltonians and Ensembles
• Three Models, each Exactly Equivalent to the
  original Ensemble
• Work out SUSY Model, approximations etc.
• Work out the Alternatives. (Comparison with
  SUSY.)
• Summary
• 26 slides a lot to cover.

3
Starting Point Loosely Coupled Grains

• Non-interacting electrons studying statistics of
  an ensemble of matrices containing a random
  component.
• I keep things flexible degrees of freedom live
  on some graph, and the graph topology is
  described by a matrix T.
• Convenient to choose a graph which allows the
  order parameter to be spatially uniform.
• Try to avoid continuum limit, which is not well
  controlled.
• N orbitals per site, N large but not infinite.
• Will require that interactions between sites are
  small this is called the diffusive regime.
  Loosely coupled.

4
Starting Point Loosely Coupled Grains

• Two alternatives
• Anderson-Wegner Hamiltonian H T V.
• K is a constant nearest neighbor Laplacian which
  acts equally on each orbital it commutes with
  the n index specifying the orbital.
• V is a local potential consisting of random
  matrices on each site.
• Band Hamiltonian H
• Both the on-site potential and the couplings
  between sites are random.
• ltHgt 0. ltH Hgt T.
• T has a diagonal piece corresponding to the
  potential, and an off-diagonal piece t
  controlling the kinetics.
• T 1 - t,
• Diffusive regime t ltlt 1 in the momentum basis.
• In zero dimensions the models I will be
  discussing are exactly equivalent!

5
Three Models

• Efetovs SUSY model starts from the TV
  Hamiltonian and converts exactly to a path
  integral with graded matrices.
• Fyodorovs model starts from the same TV
  Hamiltonian and converts to a path integral with
  standard (non-graded) matrices.
• Conversion is exact in zero dimensions, except
  possibly a non-perturbative loophole in
  manipulation of a determinant. Reproduces
  correct zero dimensional result so the loophole
  may be irrelevant.
• As yet have not found the exact path integral in
  D dimensions however an approximate result is
  available.
• Disertoris band model Disertori starts from the
  random band Hamiltonian and converts exactly to a
  path integral with non-graded matrices.
• Follows Fyodorovs program closely.
• Same loophole in the determinant manipulation.

6
Efetov Model

• Efetov converts exactly from vectors to graded 2I
  x 2I matrices containing both bosons and
  Grassmann variables.
• Structure of Q is
• The I x I diagonal blocks are bosonic, while the
  I x I off diagonal blocks are Grassmann
  variables.
• The Grassmann variables may be interpreted as an
  interaction between the fermionic matrix and the
  bosonic matrix.
• A lot of this talk will be about this
  interaction.
• The original SUSY between fermionic and bosonic
  sources is amplified here now we see not just
  symmetry in the observables but also a single
  graded group manifold with continuous
  transformations involving all blocks of Q.
• This is the usual meaning of SUSY, but I want to
  point out that the new SUSY may be broken, while
  the observable SUSY should not be broken if one
  is doing the math right.

7
Efetov Model Before Saddle Point Exact

• Two Parts
• Lagrangian
• Efetov-Wegner Boundary Terms
• Not spatial boundaries instead boundaries of the
  supergroups parameterization.
• Strictly outside the path integral formalism.
• Neglected by the saddle point approximation, but
  in zero dimensions they prove to be important,
  leading order. This is a signal that 1/N is not
  necessarily a foolproof control parameter.
• I have never seen a treatment of these terms
  except in zero dimensions, or after
  dimensionality has been integrated away to create
  a zero dimensional effective model.
• Ivanov and Skvortsovs alternate parameterization
  also features a domain of integration. After
  Grassman variables are integrated away there are
  again boundary terms. Not treated in D
  dimensions?

8
Working out the Efetov model

• Zero dimensions
• Arithmetic already extremely complicated see
  Verbaarschot Weidenmuller and Zirnbauer for the
  most thorough treatment.
• Apply saddle point approximation using N as a
  large parameter fixes the eigenvalues of Q,
  which turn out not to have Grassmann components.
• Eigenvalues given by
• No eigenvalue repulsion!
• Any interactions between the fermionic and
  bosonic sectors is hidden in the remaining
  angular integrations!
• Eigenvalue correlator contains a smooth part and
  an oscillatory part
• Boundary terms contribute only to the smooth
  part.
• Boundary terms must be added in by hand outside
  of the saddle point treatment.
• We will see later that in equivalent formulations
  the oscillatory part is a signal of eigenvalue
  splitting caused by an eigenvalue repulsion.

9
Working out the Efetov Model in D Dimensions

• Boundary terms neglected except after reducing
  the model to zero dimensions.
• Uncontrolled approximation.
• Saddle Point Equations
• Cconstrain eigenvalues after constraint we have
  a SUSY sigma model.
• Saddle point equations are typically evaluated
  based on a diffusive assumption that the kinetic
  energy is small compared to the scattering
  self-energy. Mathematically uncontrolled.
• All band structure information is lost.
• Switch to a continuum model, uncontrolled.
• 1/N control parameter is overshadowed by
  kinetics.
• Also one should integrate out the fluctuations in
  the eigenvalues to obtain their contributions to
  the Lagrangian. Has this been done?
• Possible to do better, in a controlled fashion,
  preserve band info, etc?
• Again no sign of eigenvalue interactions.

10
Working out the Efetov Model in D Dimensions

• Sigma Model
• Qs eigenvalues are now constrained.
• Immediately one does effective field theory,
  obtaining THE SUSY SIGMA MODEL
• Assume SUSY is not broken, though Mirlin and
  Fyodorov discussed it being broken in the
  delocalized phase.
• Standard EFT assumptions diffusive assumption,
  no topological terms, smallest number of spatial
  derivatives dominates, etc.
• All band structure information is lost.
• Switch to a continuum model with rotational
  symmetry, uncontrolled.
• 1/N control parameter overshadowed by kinetics.
• Possible to do better, in a controlled fashion,
  preserve band info, etc?
• Q is generally treated perturbatively except in
  special geometries.

11
Fyodorov and Band Models

• Convert to two I x I bosonic matrices and
  .
• Similar to the bosonic blocks of Efetovs
  supermatrix.
• Idea and technique due to Fyodorov.
• Conversion is exact in zero dimensions, and the
  two models are identical in zero dimensions.
• Band model is exact in any dimension.
• Fyodorov (Wegner model) has not been done exactly
  in other dimensions.
• The constant kinetic operator allows the n index
  at one site to couple to the n index at another
  site, complicating the process of integrating out
  the original vector degrees of freedom.
• Actually I think maybe this can be done but one
  has to introduce additional non-local matrices
  which are coupled to .

12
BandModel

• No Boundary Terms!!
• Bosonic observable
• Fermionic observable
• Write down for later.

13
Working out the Fyodorov/Band ModelInteraction
Term

• The determinant is the only interaction between
  and , which are otherwise independent.
• The determinant is superficially weighted by 1/N
  and a naive analysis will neglect it.
• The determinant can be rewritten in terms of
  Grassmann variables.
• Obtain (formally) a picture similar to SUSY.
• However in this case the Grassmann lagrangian is
  quadratic, while in the Efetov model (even before
  saddle point) the Grassmann matrices figure in a
  logarithm/determinant and therefore are hard to
  integrate.

14
Working out the Fyodorov/Band ModelZero
Dimensions

• Do angular integrations first, and do them
  exactly. A lot easier than in SUSY case, and
  scalable to any I x I.
• Eigenvalue sector of the theory has a very
  similar structure even if angular integrations
  are saved for last.
• Consolidate eigenvalues of and into a
  vector x.
• Limits of integration on the eigenvalues
  remain.

15
Working out the Fyodorov/Band ModelZero
Dimensions

• Three different ways of doing the saddle point.
• First way Treat the determinant as a prefactor.
• Eigenvalues are non-interacting, same as in
  Efetov model
• At lowest order the Van der Monde determinant is
  zero!
• Must calculate perturbative corrections to the
  saddle point. Thousands of terms, hundreds of
  which are non-zero. Almost all cancel due to the
  determinants antisymmetry.
• First contributions weighted by .
  Correct final result.
• The dominant saddle points are ones with two
  eigenvalues and two eigenvalues.
• Impossible to generalize this approach to D
  dimensions because the determinant is a
  polynomial of order .
• Graph the potential and eigenvalues.
• Second way treat determinant perturbatively.
  Fails.

16
Working out the Fyodorov/Band ModelZero
Dimensions

• Third way of handling the saddle point
• Include the determinant in the saddle point
  equations.
• Now the eigenvalues repel they are 1-D fermions.
• The repulsion causes eigenvalues which in are in
  the same well to split, with typical splitting of
  order .
• Graph the new picture.
• If there are N eigenvalues in a well, then for
  every original saddle point there are now N!
  saddle points.
• Associated minus signs from the determinant
  antisymmetrize the sum over saddle points.
  Therefore only the antisymmetric part of
• 
  contributes.
• The oscillatory part of the two level correlator
  comes from the antisymmetric part of
  , which comes from the level splitting.
• The smooth part comes from the antisymmetric part
  of , which comes from an
  interaction between the energies and the
  determinant.

17
Working out the Fyodorov/Band ModelZero
Dimensions

• Strong coupling between eigenvalues sharing the
  same well.
• In same well can be removed and then N
  also!
• The determinants contribution to the Hessian is
  as strong as the rest of the Lagrangian.
• Superficially 1/N, but two factors of the
  splitting in the denominator.
• No 1/N parameter to control perturbation theory
  impossible to sum the vacuum diagrams.
• However interactions between the two wells are
  controlled by 1/N.
• The strong coupling is responsible for the
  splitting which is necessary for the oscillatory
  part of the correlator. It is real and one must
  be very careful to not omit it.
• Why is this hidden in the SUSY sigma model even
  in D dimensions?

18
Working out the Fyodorov/Band ModelZero
Dimensions

• The interacting saddle point reproduces the
  correct two level correlator at leading order.
• When the energy spacing is small the
  non-perturbative issue can be absorbed into two
  parameters the overall normalization W, and the
  weight of the antisymmetry in .
• Final result

19
Working out the Band ModelD Dimensions

• Remember Exact results so far and no boundary
  terms SUSY must be preserved at this point.
• If spatial fluctations in Q are small
• Related to Thouless energy, coherence length,
  etc.
• Skip sigma model altogether.
• Treat spatial fluctuations perturbatively and
  integrate them out. (Both eigenvalues and
  angular variables.) Integration produces an
  effective potential. Finish with a theory in
  terms of apply the same methods as for
  zero dimensions.
• Clean separation of
• Small fluctuations approximation.
• Diffusive approximation (k ltlt 1).
• Saddle point approximation applied to .
• Information about the band structure is not
  truncated!

20
Working out the Band ModelD Dimensions

• If spatial fluctuations in Q are small
• Find that the determinant changes the diffusion
  constant.
• Superficially the determinant should not make any
  change 1/N.
• Coupling between fluctuations in the eigenvalues
  which share a well is not controlled by 1/N.
• Must use the interacting saddle point because the
  non-interacting saddle point gives a zero mass
  (exact cancellation of the Qs), preventing any
  diffusive approximation.
• The diffusive approximation is controlled by
  ratio of the kinetic energy k to the level
  splittingmass. Therefore must require that
  . So we need to keep N
  finite!
• Does the Efetov sigma model take into account
  this mass issue or hint that is a
  control parameter?

21
Working out the Band ModelD Dimensions

• If spatial fluctuations in Q are not small
• Can not treat angular variables perturbatively.
• Can integrate out the eigenvalue fluctuations,
  producing new terms in the Lagrangian.
• Looking at eigenvalues within a particular well,
  the center of mass is controlled by 1/N and
  integrating it out will produce terms in the
  Lagrangian weighted by 1/N. Therefore one can
  rigorously derive a sigma model where only the
  two centers of mass are constrained.
• Other fluctuations in the eigenvalues are not
  controlled by 1/N. One can still write a sigma
  model where all eigenvalues are frozen but this
  will be an uncontrolled truncation of the correct
  sigma model.

22
Working out the Band Model

• Alternative Formulation Use Grassmann variables
  to rewrite the determinant as
• Defers coupling between and .
• Saddle point approximation is now completely
  controlled by 1/N can integrate out all
  eigenvalue fluctuations producing new terms
  weighted by 1/N.
• New terms will include quartic Grassmann
  interactions.
• Can 1/N terms be dropped, and when? There can be
  issues about take N large before/after taking
  epsilon to zero. Epsilon controls the angular
  integration in this model.

23
Working Out the Band Model

• Sigma model with constrained Q eigenvalues
• Should be able to drop couplings between the two
  wells. Maybe tricks for a 1-D fermion gas.
  Many-body theory.
• If spatial fluctuations are small, they can be
  integrated away, giving quartic terms. Next one
  should find some way of integrating out the
  fermions (from many body theory), and then treat
  Qs zero modes (including the saddle point
  equations) last of all.

24
Fyodorov Model

• Gives every indication that the same issues of
  eigenvalue repulsion and strong coupling are the
  same in Wegners model as in the band model.
• Still hope for a finding way of doing the
  conversion exactly well worth the effort!
• In the meantime, not discernibly worse than SUSY
  model
• Both require discernment about whether SUSY
  should be broken.
• Both rely heavily on EFT arguments, lose band
  structure, etc.
• Exact form of Fyodorov Lagrangian and observables
  not available except within diffusive assumption
  on the other hand the SUSY model drops the
  boundary terms.

25
Summary

• Fyodorov and Disertoris models are superior
• Allow real control over the diffusive
  approximation.
• Retain band structure information.
• Avoid boundary terms and their truncation.
• Allow clear handling of terms which are
  superficially 1/N but in fact are of leading
  order.
• Highlight repulsion between eigenvalues and lack
  of 1/N control over this.
• Offer new formulations amenable to treatment by
  well-known many body theory techniques.
• Much easier to do the integrations avoid
  complications of graded groups.
• Still hope for exact form for Wegner (Fyodorov)
  model.

26
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27
Working out the Fyodorov/Band ModelZero
Dimensions

• Second way of handling the saddle point
• Do not include the determinant in the saddle
  point equations because it is 1/N.
• Treat the determinant as part of the action and
  expand it perturbatively around the saddle point.
• Non-starter determinant is zero at the naïve
  saddle point. Moreover the Hessian of its
  logarithm is infinite!
• In D-dimensions and a small sample the null
  determinant issue can be circumvented. However
  the Hessian issue will translate to a zero mass
  and cause complete breakdown of any diffusive
  assumption.

28
Derivation of the Models

• Want to compute the electronic density and its
  correlations.
• Closely connected to the on-site Greens
  function.
• Rewrite in terms of determinants
• The symmetry between the fermionic and bosonic
  derivatives can be called supersymmetry
• All three models possess this supersymmetry of
  observables.

29
Derivation of the Models

• Rewrite determinants as Gaussian integrals.
• I is the number of Greens functions we are
  averaging for two-point correlator, I equals
  two.
• A bosonic vector S with I N V complex entries.
• A Grassmann vector ? with INV complex entries.
• Diagonal matrix E containing the energies, and L
  giving the signs of the imaginary parts of E.

30
Derivation of the Models

• Average over disorder generates a quartic
  interaction.

31
Derivation of the Models

• All orbitals are treated the same, so the n index
  occurs only in sums.
• We choose observables which dont know about n.
• When the energy levels are the same, the E matrix
  is proportional to the identity and there are
  exact symmetries under global matrix
  transformations of the Grassmann vectors i
  index and also the bosonic vectors j index.
• Therefore the physical degrees of freedom are
  matrices not vectors.
• Proviso The energy levels must be close enough
  to each other.
• How close is close enough?
• Conversion to matrices is where models begin to
  differ.

32
Abstract

• Wegner's model of weakly coupled metallic grains
  is believed to capture the physics of Anderson
  localization, but has resisted full mathematical
  control in two or more dimensions.  In zero
  dimensions it can be exactly transformed to two
  different matrix models, the first involving
  graded (supersymmetric) matrices.  The second was
  introduced more recently by Fyodorov and avoids
  the use of graded matrices.  The SUSY model was
  generalized long ago to D dimensions and results
  in the famous supersymmetric sigma model. However
  the generalization process lacks mathematical
  rigor, relying heavily on effective field theory
  ideas and an implicit continuum limit. In this
  talk I will briefly review the SUSY
  generalization process and its difficulties and
  then describe what can be learned from
  generalizing Fyodorov's model to D dimensions.  
  Lastly and most importantly I will present my
  work on a third model originated by Spencer and
  Disertori which is identical to Fyodorov's model
  in zero dimensions but which opens new
  mathematical possibilities in D dimensions.