Three Alternative Sigma Models of Disorder and Their Relative Merits

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. • <H> = 0. <H H> = 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 << 1 in the momentum basis. • In zero dimensions the models I will be discussing are exactly equivalent!

5. Three Models • Efetov’s SUSY model: starts from the T+V Hamiltonian and converts exactly to a path integral with graded matrices. • Fyodorov’s model: starts from the same T+V 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. • Disertori’s band model: Disertori starts from the random band Hamiltonian and converts exactly to a path integral with non-graded matrices. • Follows Fyodorov’s 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 supergroup’s 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 Skvortsov’s 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 VerbaarschotWeidenmuller 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! • Eigenvaluecorrelator 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: • Q’s 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 Efetov’ssupermatrix. • 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 Model:Interaction 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 Model:Zero 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 Model:Zero 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 determinant’s 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 Model:Zero 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 Model:Zero Dimensions • Strong coupling between eigenvalues sharing the same well. • In same well can be removed and then N also! • The determinant’s 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 Model:Zero 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 Model:D 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 << 1). • Saddle point approximation applied to . • Information about the band structure is not truncated!

20. Working out the Band Model:D 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 Q’s), preventing any diffusive approximation. • The diffusive approximation is controlled by ratio of the kinetic energy k to the level splitting=mass. 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 Model:D 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 Q’s 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 Wegner’s 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 Disertori’s 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.

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27. Working out the Fyodorov/Band Model:Zero 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 Green’s 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 Green’s functions we are averaging; for two-point correlator, I equals two. • A bosonic vector S with I N V complex entries. • A Grassmann vector ψ 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 don’t 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 vector’s i index and also the bosonic vector’s 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.