Combinatorial theory of the semiclassical evaluation of transport moments. I. Equivalence with the random matrix approach
To study electronic transport through chaotic quantum dots, there are two main theoretical approaches. One involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging. The other treats the transport in the semiclassical approximation and stu...
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description | To study electronic transport through chaotic quantum dots, there are two main theoretical approaches. One involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging. The other treats the transport in the semiclassical approximation and studies correlations among sets of classical trajectories. There are established evaluation procedures within the semiclassical evaluation that, for several linear and nonlinear transport moments to which they were applied, have always resulted in the agreement with random matrix predictions. We prove that this agreement is universal: any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory. The equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs (maps) with certain properties and exhibiting systematic cancellations among their contributions. Remaining trajectory sets can be identified with primitive (palindromic) factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices. The equivalence is proved for systems with and without time reversal symmetry. |
doi_str_mv | 10.1063/1.4826442 |
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We prove that this agreement is universal: any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory. The equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs (maps) with certain properties and exhibiting systematic cancellations among their contributions. Remaining trajectory sets can be identified with primitive (palindromic) factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices. The equivalence is proved for systems with and without time reversal symmetry.</description><identifier>ISSN: 0022-2488</identifier><identifier>EISSN: 1089-7658</identifier><identifier>DOI: 10.1063/1.4826442</identifier><identifier>CODEN: JMAPAQ</identifier><language>eng</language><publisher>New York: American Institute of Physics</publisher><subject>Approximation ; CHAOS THEORY ; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS ; Combinatorial analysis ; Correlation analysis ; DIAGRAMS ; Electron transport ; Equivalence ; EVALUATION ; GRAPH THEORY ; MATRICES ; Matrix ; Matrix theory ; NONLINEAR PROBLEMS ; Physics ; QUANTUM DOTS ; Quantum theory ; RANDOMNESS ; SCATTERING ; SEMICLASSICAL APPROXIMATION ; SYMMETRY ; Thermal expansion ; TRAJECTORIES ; Trajectory analysis</subject><ispartof>Journal of mathematical physics, 2013-11, Vol.54 (11), p.1</ispartof><rights>AIP Publishing LLC</rights><rights>Copyright American Institute of Physics Nov 2013</rights><rights>2013 AIP Publishing LLC.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c451t-ffe39de18e9b19ee6f267bc95bbd86ba98922461e9917f124900de152590a2df3</citedby><cites>FETCH-LOGICAL-c451t-ffe39de18e9b19ee6f267bc95bbd86ba98922461e9917f124900de152590a2df3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://pubs.aip.org/jmp/article-lookup/doi/10.1063/1.4826442$$EHTML$$P50$$Gscitation$$H</linktohtml><link.rule.ids>230,314,776,780,790,881,1553,4498,27901,27902,76127,76133</link.rule.ids><backlink>$$Uhttps://www.osti.gov/biblio/22251491$$D View this record in Osti.gov$$Hfree_for_read</backlink></links><search><creatorcontrib>Berkolaiko, G.</creatorcontrib><creatorcontrib>Kuipers, J.</creatorcontrib><title>Combinatorial theory of the semiclassical evaluation of transport moments. I. Equivalence with the random matrix approach</title><title>Journal of mathematical physics</title><description>To study electronic transport through chaotic quantum dots, there are two main theoretical approaches. One involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging. The other treats the transport in the semiclassical approximation and studies correlations among sets of classical trajectories. There are established evaluation procedures within the semiclassical evaluation that, for several linear and nonlinear transport moments to which they were applied, have always resulted in the agreement with random matrix predictions. We prove that this agreement is universal: any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory. The equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs (maps) with certain properties and exhibiting systematic cancellations among their contributions. Remaining trajectory sets can be identified with primitive (palindromic) factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices. The equivalence is proved for systems with and without time reversal symmetry.</description><subject>Approximation</subject><subject>CHAOS THEORY</subject><subject>CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS</subject><subject>Combinatorial analysis</subject><subject>Correlation analysis</subject><subject>DIAGRAMS</subject><subject>Electron transport</subject><subject>Equivalence</subject><subject>EVALUATION</subject><subject>GRAPH THEORY</subject><subject>MATRICES</subject><subject>Matrix</subject><subject>Matrix theory</subject><subject>NONLINEAR PROBLEMS</subject><subject>Physics</subject><subject>QUANTUM DOTS</subject><subject>Quantum theory</subject><subject>RANDOMNESS</subject><subject>SCATTERING</subject><subject>SEMICLASSICAL APPROXIMATION</subject><subject>SYMMETRY</subject><subject>Thermal expansion</subject><subject>TRAJECTORIES</subject><subject>Trajectory analysis</subject><issn>0022-2488</issn><issn>1089-7658</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2013</creationdate><recordtype>article</recordtype><recordid>eNp90cFqGzEQBmBRGqib9tA3EPTSFtbVyNJaOhaTNoFALslZaOVZrLArbSRtWr99ZTskh9CeNDAfPxp-Qj4BWwJrV99hKRRvheBvyAKY0s26leotWTDGecOFUu_I-5zvGQNQQizIfhPHzgdbYvJ2oGWHMe1p7A8TzTh6N9icvas7fLTDbIuP4bhPNuQppkLHOGIoeUmvlvTiYfaVYXBIf_uyO8ZUuY0jHW1J_g-105SidbsP5Ky3Q8aPT-85uft5cbu5bK5vfl1tflw3TkgoTd_jSm8RFOoONGLb83bdOS27bqvazmqlORctoNaw7oELzVjlkkvNLN_2q3Py-ZQbc_EmO1_Q7VwMAV0xnHMJQkNVX06qfu5hxlzM6LPDYbAB45xNRWIlmGjFS-AzvY9zCvUGw4FrybTU-n8KhNRCKskPWV9PyqWYc8LeTMmPNu0NMHMo1IB5KrTabyd7uOHYwzN-jOkFmul49T_x6-S_oe2uPg</recordid><startdate>20131101</startdate><enddate>20131101</enddate><creator>Berkolaiko, G.</creator><creator>Kuipers, J.</creator><general>American Institute of Physics</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7U5</scope><scope>8FD</scope><scope>H8D</scope><scope>JQ2</scope><scope>L7M</scope><scope>OTOTI</scope></search><sort><creationdate>20131101</creationdate><title>Combinatorial theory of the semiclassical evaluation of transport moments. I. Equivalence with the random matrix approach</title><author>Berkolaiko, G. ; Kuipers, J.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c451t-ffe39de18e9b19ee6f267bc95bbd86ba98922461e9917f124900de152590a2df3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2013</creationdate><topic>Approximation</topic><topic>CHAOS THEORY</topic><topic>CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS</topic><topic>Combinatorial analysis</topic><topic>Correlation analysis</topic><topic>DIAGRAMS</topic><topic>Electron transport</topic><topic>Equivalence</topic><topic>EVALUATION</topic><topic>GRAPH THEORY</topic><topic>MATRICES</topic><topic>Matrix</topic><topic>Matrix theory</topic><topic>NONLINEAR PROBLEMS</topic><topic>Physics</topic><topic>QUANTUM DOTS</topic><topic>Quantum theory</topic><topic>RANDOMNESS</topic><topic>SCATTERING</topic><topic>SEMICLASSICAL APPROXIMATION</topic><topic>SYMMETRY</topic><topic>Thermal expansion</topic><topic>TRAJECTORIES</topic><topic>Trajectory analysis</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Berkolaiko, G.</creatorcontrib><creatorcontrib>Kuipers, J.</creatorcontrib><collection>CrossRef</collection><collection>Solid State and Superconductivity Abstracts</collection><collection>Technology Research Database</collection><collection>Aerospace Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>OSTI.GOV</collection><jtitle>Journal of mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Berkolaiko, G.</au><au>Kuipers, J.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Combinatorial theory of the semiclassical evaluation of transport moments. I. Equivalence with the random matrix approach</atitle><jtitle>Journal of mathematical physics</jtitle><date>2013-11-01</date><risdate>2013</risdate><volume>54</volume><issue>11</issue><spage>1</spage><pages>1-</pages><issn>0022-2488</issn><eissn>1089-7658</eissn><coden>JMAPAQ</coden><abstract>To study electronic transport through chaotic quantum dots, there are two main theoretical approaches. One involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging. The other treats the transport in the semiclassical approximation and studies correlations among sets of classical trajectories. There are established evaluation procedures within the semiclassical evaluation that, for several linear and nonlinear transport moments to which they were applied, have always resulted in the agreement with random matrix predictions. We prove that this agreement is universal: any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory. The equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs (maps) with certain properties and exhibiting systematic cancellations among their contributions. Remaining trajectory sets can be identified with primitive (palindromic) factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices. The equivalence is proved for systems with and without time reversal symmetry.</abstract><cop>New York</cop><pub>American Institute of Physics</pub><doi>10.1063/1.4826442</doi><tpages>26</tpages><oa>free_for_read</oa></addata></record> |
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subjects | Approximation CHAOS THEORY CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS Combinatorial analysis Correlation analysis DIAGRAMS Electron transport Equivalence EVALUATION GRAPH THEORY MATRICES Matrix Matrix theory NONLINEAR PROBLEMS Physics QUANTUM DOTS Quantum theory RANDOMNESS SCATTERING SEMICLASSICAL APPROXIMATION SYMMETRY Thermal expansion TRAJECTORIES Trajectory analysis |
title | Combinatorial theory of the semiclassical evaluation of transport moments. I. Equivalence with the random matrix approach |
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