Phys. Rev. Lett. 101, (2008) Quantum Transport in Chaotic Cavities and Painlevé Transcendents
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1 30 Phys. Rev. Lett. 101, (2008) Quantum Transort in Chaotic Cavities and Painlevé Transcendents Eugene Kanzieer 1 and Vladimir Al. Osiov 2 1 H.I.T. Holon Institute of Technology, Israel 2 Universität Duisburg-Essen, Germany IV Brunel Worksho on Random Matrix Theory, December 20, 2008
2 29 Outline Brief Intro: Painlevé roerty and aearance of Painlevé transcendents in hysics 2D Ising model 1D imenetrable Bose gas Growth models New! Painlevé transcendents in quantum transort roblems: Cumulants of Landauer conductance Landauer conductance and its cumulants: Known results Integrable theory of quantum transort in chaotic cavities Conclusions / Oen roblems
3 28 Preface: Painlevé functions in hysics Painlevé transcendents and their aearance in hysics Rational function of its arguments E. Picard (1889) All movable singularities are restricted to oles (no movable branch oints) (a) linear 2nd order DEs (b) Weierstrass DE (c) Riccati DE [ Painlevé equations P 6 I P VI ] nonlinear secial functions P. Painlevé (1900,1902) B. Gambier (1905) R. Fuchs (1910)
4 27 Preface: Painlevé functions in hysics Painlevé transcendents and their aearance in hysics σp III σp V fascinating roerties [ Painlevé equations P 6 I P VI ] nonlinear secial functions P. Painlevé (1900,1902) B. Gambier (1905) R. Fuchs (1910) P. Clarkson, Painleve equations nonlinear secial functions, J. Com. Al. Math. 153, 127 (2003)
5 26 Preface: Painlevé functions in hysics Painlevé transcendents and their aearance in hysics 2D Ising model T. Wu, B. McCoy, C. Tracy, and E. Barouch (1976) σp III
6 26 Preface: Painlevé functions in hysics Painlevé transcendents and their aearance in hysics 2D Ising model T. Wu, B. McCoy, C. Tracy, and E. Barouch (1976) σp III
7 25 Preface: Painlevé functions in hysics Painlevé transcendents and their aearance in hysics M. Girardeau (1960) A. Lenard (1964) Imenetrable Bose gas M. Jimbo, T. Miwa, Y. Môri, and M. Sato (1980) T. Esslinger ETH Zürich σp V
8 24 Preface: Painlevé functions in hysics Painlevé transcendents and their aearance in hysics Growth models in (1+1)D (oriented digital boiling) J. Gravner, C. Tracy, and H. Widom (2001) Universal regime of shae fluctuations σp II
9 24 Preface: Painlevé functions in hysics Painlevé transcendents and their aearance in hysics Growth models in (1+1)D (oriented digital boiling) J. Gravner, C. Tracy, and H. Widom (2001) Universal regime of shae fluctuations σp II
10 23 Outline Brief Intro: Painlevé roerty and aearance of Painlevé transcendents in hysics 2D Ising model 1D imenetrable Bose gas Growth models New! Painlevé transcendents in quantum transort roblems: Cumulants of Landauer conductance
11 22 New: Painlevé in quantum transort!! Painlevé transcendents in quantum transort roblems cavity leads couling Left reservoir Right reservoir S. Oberholzer, Universität Bazel Landauer conductance
12 21 New: Painlevé in quantum transort!! Painlevé transcendents in quantum transort roblems cavity leads couling Bohigas Conjecture + Quantum regime Ehrenfest time Electron dwell time Left reservoir Right reservoir Landauer conductance
13 20 New: Painlevé in quantum transort!! Painlevé transcendents in quantum transort roblems Main Result σp V Left reservoir Right reservoir Landauer conductance
14 19 Outline Brief Intro: Painlevé roerty and aearance of Painlevé transcendents in hysics 2D Ising model 1D imenetrable Bose gas Growth models New! Painlevé transcendents in quantum transort roblems: Cumulants of Landauer conductance Landauer conductance and its cumulants: Known results
15 18 Landauer Conductance Landauer conductance and its cumulants: Known results cavity leads couling Scattering matrix aroach Left reservoir Right reservoir Landauer conductance
16 18 Landauer Conductance Landauer conductance and its cumulants: Known results cavity leads couling Scattering matrix aroach Left reservoir Right reservoir Landauer conductance
17 18 Landauer Conductance Landauer conductance and its cumulants: Known results cavity leads couling Scattering matrix aroach Left reservoir Right reservoir Landauer conductance
18 17 Landauer Conductance Landauer conductance and its cumulants: Known results BPC Scattering matrix aroach Left reservoir Right reservoir Landauer conductance
19 16 Landauer Conductance Landauer conductance and its cumulants: Known results Scattering matrix aroach Semiclassical arguments: Blümel & Smilansky (1990) Microscoic justification: Brouwer (1995) Early (exact) calculation of moments/cumulants: Baranger & Mello (1994) 1 st & 2 nd cumulants
20 15 Landauer Conductance Landauer conductance and its cumulants: Known results exonential growth!! Symmetric functions Selberg integral All moments Novaes (2008) 3 rd & 4 th cumulants Savin, Sommers & Wieczorek (2007) Semiclassical arguments: Blümel & Smilansky (1990) Microscoic justification: Brouwer (1995) Early (exact) calculation of moments/cumulants: Baranger & Mello (1994) 1 st & 2 nd cumulants
21 14 Outline Brief Intro: Painlevé roerty and aearance of Painlevé transcendents in hysics 2D Ising model 1D imenetrable Bose gas Growth models New! Painlevé transcendents in quantum transort roblems: Cumulants of Landauer conductance Landauer conductance and its cumulants: Known results Integrable theory of quantum transort in chaotic cavities
22 13 Landauer Conductance Integrable theory of quantum transort (Landauer conductance) BPC Scattering matrix aroach Left reservoir Right reservoir Landauer conductance
23 12 Landauer Conductance Integrable theory of quantum transort (Landauer conductance) Cumulant generating function Truncate! (Zyczkowski & Sommers, 2000) Itzykson-Zuber formula, but: high degeneracy of C-matrices BPC Left reservoir Right reservoir Landauer conductance
24 11 Landauer Conductance Integrable theory of quantum transort (Landauer conductance) Cumulant generating function Truncate! (Zyczkowski & Sommers, 2000) Itzykson-Zuber formula, but: high degeneracy of C-matrices
25 10 Landauer Conductance Integrable theory of quantum transort (Landauer conductance) Cumulant generating function Ga formation robability (LUE) Tracy & Widom (1994)
26 09 Landauer Conductance Integrable theory of quantum transort (Landauer conductance) Cumulant generating function! Main Result
27 08 New: Painlevé in quantum transort!! Integrable theory of quantum transort (Landauer conductance) cavity leads couling Bohigas Conjecture + Quantum regime Ehrenfest time Electron dwell time Left reservoir Right reservoir Landauer conductance
28 07 New: Painlevé in quantum transort!! Painlevé transcendents in quantum transort roblems Main Result σp V Left reservoir Right reservoir Landauer conductance
29 06 New: Painlevé in quantum transort!! Consequences / further results Conductance cumulants obey a nonlinear recurrence equation Novaes (2008)
30 05 New: Painlevé in quantum transort!! Consequences / further results Entire conductance distribution function follows from the Toda Lattice Conductance robability density function
31 04 New: Painlevé in quantum transort!! Consequences / further results Asymtotic analysis of conductance cumulants Asymtotic analysis of conductance distribution (deviations from the Gaussian law) Statistics of the noise ower as a function of bias voltage and the temerature Joint statistics of Landauer conductance and the noise ower V Osiov RMT
32 03 Outline Brief Intro: Painlevé roerty and aearance of Painlevé transcendents in hysics 2D Ising model 1D imenetrable Bose gas Growth models New! Painlevé transcendents in quantum transort roblems: Cumulants of Landauer conductance Landauer conductance and its cumulants: Known results Integrable theory of quantum transort in chaotic cavities Conclusions / Oen roblems
33 02 New: Painlevé in quantum transort!! Conclusions / Oen roblems Main Result σp V Left reservoir Right reservoir Landauer conductance
34 01 New: Painlevé in quantum transort!! Conclusions / Oen roblems Non-ideal contacts (Poisson kernel) Lossy quantum transort (electrons escaing through the third lead) Full counting statistics Other symmetry classes (β=1 and β=4) V Osiov RMT
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