Constructing a sigma model from semiclassics In collaboration with: - - PowerPoint PPT Presentation

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Constructing a sigma model from semiclassics In collaboration with: - - PowerPoint PPT Presentation

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Constructing a sigma model from semiclassics In collaboration with: Alexander Altland, Petr Braun, Fritz Haake, Stefan Heusler Sebastian Mller Approaches to spectral statistics Semiclassics Approaches to spectral statistics Semiclassics

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Constructing a sigma model from semiclassics

Sebastian Müller

In collaboration with:

Alexander Altland, Petr Braun, Fritz Haake, Stefan Heusler

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Approaches to spectral statistics

Semiclassics

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Approaches to spectral statistics

Semiclassics Sigma model

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Approaches to spectral statistics

Semiclassics Sigma model field-theoretical method for averaging over random matrices, disorder

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Approaches to spectral statistics

Semiclassics Sigma model field-theoretical method for averaging over random matrices, disorder

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Approaches to spectral statistics

Semiclassics Sigma model field-theoretical method for averaging over random matrices, disorder doing combinatorics

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Approaches to spectral statistics

Semiclassics Sigma model field-theoretical method for averaging over random matrices, disorder doing combinatorics Universal results, in agreement with RMT

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Sigma model in RMT

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Sigma model in RMT

Generating function

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Sigma model in RMT

Generating function

Z=〈 ECE−D  EAE−B〉

E = detE−H

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Sigma model in RMT

Generating function write as Gauss integral

Z=〈 ECE−D  EAE−B〉

E = detE−H

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Sigma model in RMT

Generating function write as Gauss integral What about ?

Z=〈 ECE−D  EAE−B〉

E = detE−H

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Sigma model in RMT

Generating function write as Gauss integral What about ?

  • anticommuting (fermionic) variables

Z=〈 ECE−D  EAE−B〉

E = detE−H

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Sigma model in RMT

Generating function write as Gauss integral What about ?

  • replica trick
  • anticommuting (fermionic) variables

Z=〈 ECE−D  EAE−B〉

E = detE−H

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Sigma model in RMT

Generating function write as Gauss integral What about ?

  • replica trick
  • anticommuting (fermionic) variables

Z=〈 ECE−D  EAE−B〉

E = detE−H

E = limr 0 E

−r−1

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Sigma model in RMT

random matrix average

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Sigma model in RMT

random matrix average Gauss integrals

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Sigma model in RMT

random matrix average Gauss integrals traded for integral over matrices

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Sigma model in RMT

random matrix average Gauss integrals traded for integral over matrices

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Sigma model in RMT

random matrix average Gauss integrals traded for integral over matrices

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Sigma model in RMT

random matrix average Gauss integrals traded for integral over matrices

energy differences

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Relevance for semiclassics

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Relevance for semiclassics

non-oscillatory / oscillatory terms:

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Relevance for semiclassics

 Z falls into integrals over two submanifolds

 non-oscillatory

non-oscillatory / oscillatory terms:

 non-oscillatory

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Relevance for semiclassics

 Z falls into integrals over two submanifolds

 non-oscillatory

1

non-oscillatory / oscillatory terms:

Z = Z + Z

2

 non-oscillatory

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Relevance for semiclassics

 Z falls into integrals over two submanifolds

 non-oscillatory

1

non-oscillatory / oscillatory terms:

Z = Z + Z

2

 non-oscillatory

  • non-oscillatory
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Relevance for semiclassics

 Z falls into integrals over two submanifolds

 non-oscillatory • oscillatory

1

non-oscillatory / oscillatory terms:

Z = Z + Z

2

 non-oscillatory

  • non-oscillatory
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Relevance for semiclassics

 Z falls into integrals over two submanifolds

 non-oscillatory • oscillatory

1

non-oscillatory / oscillatory terms:

Z = Z + Z

2

 non-oscillatory

(Berry & Keating, 1990; Keating & S.M., 2007)

  • same relation as with Riemann-Siegel !
  • non-oscillatory
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Relevance for semiclassics

 Z falls into integrals over two submanifolds

 non-oscillatory • oscillatory

1

non-oscillatory / oscillatory terms:

Z = Z + Z

2

 non-oscillatory

(Berry & Keating, 1990; Keating & S.M., 2007)

Z2A,B,C ,D = Z 1A,B,−D,−C

  • same relation as with Riemann-Siegel !
  • non-oscillatory
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Relevance for semiclassics

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Relevance for semiclassics

Analog of diagonal approximation:

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Relevance for semiclassics

Analog of diagonal approximation: rational parametrization

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Relevance for semiclassics

Analog of diagonal approximation: rational parametrization

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Relevance for semiclassics

Analog of diagonal approximation: rational parametrization Keep only Gaussian terms!

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Relevance for semiclassics

Keep all terms perturbation theory

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Relevance for semiclassics

Keep all terms perturbation theory

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Relevance for semiclassics

Keep all terms perturbation theory

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Relevance for semiclassics

Keep all terms perturbation theory encounters =

vertices

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Relevance for semiclassics

Keep all terms perturbation theory

links =

propagator lines

encounters =

vertices

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Constructing a sigma model from semiclassics

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Constructing a sigma model from semiclassics

Replica trick also works in semiclassics!

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Constructing a sigma model from semiclassics

Replica trick also works in semiclassics! r `original` pseudo-orbits

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Constructing a sigma model from semiclassics

Replica trick also works in semiclassics! r `original` pseudo-orbits r `partner` pseudo-orbits

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Constructing a sigma model from semiclassics

Replica trick also works in semiclassics! r `original` pseudo-orbits r `partner` pseudo-orbits differing in encounters

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Drawing orbits

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Drawing orbits

Draw encounters

1

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Drawing orbits

Draw encounters

1

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Drawing orbits

Choose pseudo-orbits (colors)

2

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Drawing orbits

Connect!

3

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Drawing orbits

Draw encounters

1

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Drawing orbits

Draw encounters

1

write for each entrance port,

for each exit port

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Drawing orbits

Draw encounters

1

write for each entrance port,

for each exit port

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Drawing orbits

Choose pseudo-orbit (colors)

2

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Drawing orbits

Choose pseudo-orbit (colors)

2

choose indices according to pseudo-orbits

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Drawing orbits

Choose pseudo-orbit (colors)

2

choose indices according to pseudo-orbits

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Drawing orbits

Connect!

3

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Drawing orbits

Connect!

3

  • numbers of entrances & corresp.

exits must coincide

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Drawing orbits

Connect!

3

  • numbers of entrances & corresp.

exits must coincide

  • possible connections = factorial
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Drawing orbits

Connect!

3

  • numbers of entrances & corresp.

exits must coincide

  • possible connections = factorial

Gaussian integral with powers

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Drawing orbits

Connect!

3

  • numbers of entrances & corresp.

exits must coincide

  • possible connections = factorial

Gaussian integral with powers

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Drawing orbits

Connect!

3

  • numbers of entrances & corresp.

exits must coincide

  • possible connections = factorial

also put in energy differences Gaussian integral with powers

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Result

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Result

Summation gives

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Result

Summation gives

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Result

Summation gives Agreement with sigma model, RMT

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Result

Summation gives Agreement with sigma model, RMT

Difference to ballistic sigma model:

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Result

Summation gives Agreement with sigma model, RMT

Difference to ballistic sigma model:

Muzykantskii & Khmel'nitskii, JETP Lett. (1995) Andreev, Agam, Simons & Altshuler, PRL (1996) Jan Müller, Micklitz, Altland (2007)

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Result

Summation gives Agreement with sigma model, RMT

Difference to ballistic sigma model:

Muzykantskii & Khmel'nitskii, JETP Lett. (1995) Andreev, Agam, Simons & Altshuler, PRL (1996) Jan Müller, Micklitz, Altland (2007)

  • perturbative
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Result

Summation gives Agreement with sigma model, RMT

Difference to ballistic sigma model:

Muzykantskii & Khmel'nitskii, JETP Lett. (1995) Andreev, Agam, Simons & Altshuler, PRL (1996) Jan Müller, Micklitz, Altland (2007)

  • perturbative
  • no problems due to regularisation
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Outlook

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Outlook

possible extension: localization e.g. in long wires

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Outlook

possible extension: localization e.g. in long wires

  • semiclassical contributions changed (diffusion)
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Outlook

possible extension: localization e.g. in long wires

  • semiclassical contributions changed (diffusion)
  • expect one-dimension sigma model
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Conclusions

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Conclusions

Semiclassics

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Conclusions

Semiclassics Sigma model

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Conclusions

Semiclassics Sigma model Universality

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Conclusions

Semiclassics Sigma model Universality

  • Same relation between non-osc. & osc. contributions
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Conclusions

Semiclassics Sigma model Universality

  • Same relation between non-osc. & osc. contributions
  • encounters = perturbation series
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Conclusions

Semiclassics Sigma model Universality

  • Same relation between non-osc. & osc. contributions
  • encounters = perturbation series
  • count link connections using matrix integral