Constructing a sigma model from semiclassics
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1 Constructing a sigma model from semiclassics In collaboration with: Alexander Altland, Petr Braun, Fritz Haake, Stefan Heusler Sebastian Müller
2 Approaches to spectral statistics Semiclassics
3 Approaches to spectral statistics Semiclassics Sigma model
4 Approaches to spectral statistics Semiclassics Sigma model field-theoretical method for averaging over random matrices, disorder
5 Approaches to spectral statistics Semiclassics Sigma model field-theoretical method for averaging over random matrices, disorder
6 Approaches to spectral statistics Semiclassics Sigma model field-theoretical method for averaging over random matrices, disorder doing combinatorics
7 Approaches to spectral statistics Semiclassics Sigma model field-theoretical method for averaging over random matrices, disorder doing combinatorics Universal results, in agreement with RMT
8 Sigma model in RMT
9 Sigma model in RMT Generating function
10 Sigma model in RMT Generating function Z= E C E D E A E B E = det E H
11 Sigma model in RMT Generating function Z= E C E D E A E B write E = det E H as Gauss integral
12 Sigma model in RMT Generating function Z= E C E D E A E B E = det E H write as Gauss integral What about?
13 Sigma model in RMT Generating function Z= E C E D E A E B E = det E H write as Gauss integral What about? anticommuting (fermionic) variables
14 Sigma model in RMT Generating function Z= E C E D E A E B E = det E H write as Gauss integral What about? anticommuting (fermionic) variables replica trick
15 Sigma model in RMT Generating function Z= E C E D E A E B E = det E H write as Gauss integral What about? anticommuting (fermionic) variables replica trick E = lim r 0 E r 1
16 Sigma model in RMT random matrix average
17 Sigma model in RMT random matrix average Gauss integrals
18 Sigma model in RMT random matrix average Gauss integrals traded for integral over matrices
19 Sigma model in RMT random matrix average Gauss integrals traded for integral over matrices
20 Sigma model in RMT random matrix average Gauss integrals traded for integral over matrices
21 Sigma model in RMT random matrix average Gauss integrals traded for integral over matrices energy differences
22 Relevance for semiclassics
23 Relevance for semiclassics non-oscillatory / oscillatory terms:
24 Relevance for semiclassics non-oscillatory / oscillatory terms: Z falls into integrals over two submanifolds non-oscillatory
25 Relevance for semiclassics non-oscillatory / oscillatory terms: Z falls into integrals over two submanifolds Z = Z + Z 1 2 non-oscillatory
26 Relevance for semiclassics non-oscillatory / oscillatory terms: Z falls into integrals over two submanifolds Z = Z + Z 1 2 non-oscillatory non-oscillatory
27 Relevance for semiclassics non-oscillatory / oscillatory terms: Z falls into integrals over two submanifolds Z = Z + Z 1 2 non-oscillatory oscillatory non-oscillatory
28 Relevance for semiclassics non-oscillatory / oscillatory terms: Z falls into integrals over two submanifolds Z = Z + Z 1 2 non-oscillatory oscillatory non-oscillatory same relation as with Riemann-Siegel! (Berry & Keating, 1990; Keating & S.M., 2007)
29 Relevance for semiclassics non-oscillatory / oscillatory terms: Z falls into integrals over two submanifolds Z = Z + Z 1 2 non-oscillatory oscillatory non-oscillatory same relation as with Riemann-Siegel! Z 2 A, B, C, D = Z 1 A, B, D, C (Berry & Keating, 1990; Keating & S.M., 2007)
30 Relevance for semiclassics
31 Relevance for semiclassics Analog of diagonal approximation:
32 Relevance for semiclassics Analog of diagonal approximation: rational parametrization
33 Relevance for semiclassics Analog of diagonal approximation: rational parametrization
34 Relevance for semiclassics Analog of diagonal approximation: rational parametrization Keep only Gaussian terms!
35 Relevance for semiclassics Keep all terms perturbation theory
36 Relevance for semiclassics Keep all terms perturbation theory
37 Relevance for semiclassics Keep all terms perturbation theory
38 Relevance for semiclassics Keep all terms perturbation theory encounters = vertices
39 Relevance for semiclassics Keep all terms perturbation theory encounters = vertices links = propagator lines
40 Constructing a sigma model from semiclassics
41 Constructing a sigma model from semiclassics Replica trick also works in semiclassics!
42 Constructing a sigma model from semiclassics Replica trick also works in semiclassics! r `original` pseudo-orbits
43 Constructing a sigma model from semiclassics Replica trick also works in semiclassics! r `original` pseudo-orbits r `partner` pseudo-orbits
44 Constructing a sigma model from semiclassics Replica trick also works in semiclassics! r `original` pseudo-orbits r `partner` pseudo-orbits differing in encounters
45 Drawing orbits
46 Drawing orbits 1 Draw encounters
47 Drawing orbits 1 Draw encounters
48 Drawing orbits 2 Choose pseudo-orbits (colors)
49 Drawing orbits 3 Connect!
50 Drawing orbits 1 Draw encounters
51 Drawing orbits 1 Draw encounters write for each entrance port, for each exit port
52 Drawing orbits 1 Draw encounters write for each entrance port, for each exit port
53 Drawing orbits 2 Choose pseudo-orbit (colors)
54 Drawing orbits 2 Choose pseudo-orbit (colors) choose indices according to pseudo-orbits
55 Drawing orbits 2 Choose pseudo-orbit (colors) choose indices according to pseudo-orbits
56 Drawing orbits 3 Connect!
57 Drawing orbits Connect! 3 numbers of entrances & corresp. exits must coincide
58 Drawing orbits Connect! 3 numbers of entrances & corresp. exits must coincide possible connections = factorial
59 Drawing orbits Connect! 3 numbers of entrances & corresp. exits must coincide possible connections = factorial Gaussian integral with powers
60 Drawing orbits Connect! 3 numbers of entrances & corresp. exits must coincide possible connections = factorial Gaussian integral with powers
61 Drawing orbits Connect! 3 numbers of entrances & corresp. exits must coincide possible connections = factorial Gaussian integral with powers also put in energy differences
62 Result
63 Result Summation gives
64 Result Summation gives
65 Result Summation gives Agreement with sigma model, RMT
66 Result Summation gives Agreement with sigma model, RMT Difference to ballistic sigma model:
67 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)
68 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
69 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
70 Outlook
71 Outlook possible extension: localization e.g. in long wires
72 Outlook possible extension: localization e.g. in long wires semiclassical contributions changed (diffusion)
73 Outlook possible extension: localization e.g. in long wires semiclassical contributions changed (diffusion) expect one-dimension sigma model
74 Conclusions
75 Conclusions Semiclassics
76 Conclusions Semiclassics Sigma model
77 Conclusions Semiclassics Sigma model Universality
78 Conclusions Semiclassics Sigma model Universality Same relation between non-osc. & osc. contributions
79 Conclusions Semiclassics Sigma model Universality Same relation between non-osc. & osc. contributions encounters = perturbation series
80 Conclusions Semiclassics Sigma model Universality Same relation between non-osc. & osc. contributions encounters = perturbation series count link connections using matrix integral
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