Borl trasform a tool for symmtry-braig phoma? M Zirbaur SFB/TR, Gdas Spt, 9 Itroductio: spotaous symmtry braig, ordr paramtr, collctiv fild mthods Borl trasform for itractig frmios ampl: frromagtic ordr Borl trasform for radom matrics SUSY, Voiculscu R-trasform, boso-boso sctor
Itroductio Motivatio by Prspctiv
Symmtry braig Statistical mchaics of fild ϕ : μ ϕ i spit of μ ϕ G y y μ g ϕ dg M with symmtry group G global symmtry I ifiit volum, th thrmal uilibrium stat may spotaously bra th symmtry G Log-rag ordr i corrlatio fuctios : cost Eampl : th frromagtic phas of a spi-magtic systm spotaously bras rotatio symmtry, G d SO 3
Th ttboo ampl of a scalar fild / ] [ ] [ ] [ l ] [ ] [ j F j d S j F j d - j Z j F j Z d d φ δ δ ϕ ϕ φ φ ϕ + + Γ ] [ Lgdr trasform : fuctios poit coctd gratig fuctio for is Fr rgy fuctioal fild: tral Partitio fuctio i th prsc of g, Isig modl fild Scalar
Why us Lgdr trasform? Prturbatio thory : graphical pasio of Γ is simplr oly P-irrducibl graphs cotribut Spotaous symmtry braig sigald by apparac of critical poits δ Γ δφ for φ Γ may rmai aalytic v i th ifiit-volum limit Ladau thory
Collctiv-fild mthods Hubbard-Stratoovich trasformatio V it K + y y K y y, y Λ Bosoizatio Dirac frmios i + dimsios bosoizatio by gaug forms Ladau thory of ordr paramtr phomological! w? Borl trasformatio
Basic otio of Borl trasform ˆ i! / ˆ : U p a d Z p Z a dp p Z Z p a p Z p p π trasform : Ivrs Borl trasform Borl som polyomial cosidr ampl : Th simplst
Gralizatio trasform has ivrs Borl Thm Littlma, MZ: by trasform of Dfi Borl, masur Haar matrics, uitary, Lbsgu masur matrics, positiv Hrmitia matrics, compl ˆ GL, ˆ Hol, Tr Tr Tr M PQ M P P Q d Q f P f Q dp P Q f Q f M f Q d Q d M dp dp M M M M μ μ μ
Borl Trasform: Itractig Frmios
Sttig Quatum systm of itractig frmios grad caoical partitio fuctio Z Tr with β H μ Us cohrt-stat path itgral i trms of frmio filds ψ, ψ rprstatio Lt ψ, ψ b a wll motivatd 'ordr paramtr' g, Coopr pairig with d-wav symmtry which is pctd to acuir a ozro pctatio valu How to procd?
Mai ida Augmt th Hamiltoia by couplig to a tral fild P via th ordr paramtr H H + j Λ P j α α α ψ j, ψ j Sum is ovr lattic Λ of cubs j coars graiig : Borl trasform of partitio fuctio schmatic: Zˆ [ Q] Z [ P] PQ dp Ivrs Borl trasform schmatic: Z [ P] Zˆ [ Q] + PQ dq
Eampl: frromagt { } { } GL O U O GL > J J g g g X g X / Q Q Q M / J P J P P M J T τ τ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ whr by twistd cojugatio : acts o ad Lt spac is Ambit cub : of magtizatio spi paramtr Ordr a
Eampl: frromagt cot d Itroduc couplig : H H + For ach cub j Λ lt ad P j M I J j Λ Q j M Hrm Tr + P j j I U Borl trasform : Zˆ [ Q] : M Λ Z [ U whr Q U Jτ U Pτ U ], U U j Tr for JP j d P j ach cub Ivrs Borl trasform : Z [ P] M Λ Zˆ [ Q] j Tr P j Q j dμ Q j
Borl Trasform for Radom Matrics
Mai ida Spctral corrlatios ar codd i th "partitio fuctio" j Dt p Dt p, j, j H H Partitio fuctio tds to radial suprmatri, Z P : fuctio of { SDt P H } Trasform schmatic: Zˆ Q Ivrs schmatic: Z P Z P, Zˆ Q, itgrat ovr Rimaia symmtric suprspacs DP DQ + STr PQ STr PQ
Why Borl trasform? What' s th advatag? Z p Zˆ { Dt p H } { Tr l ph } { Tr l ph } is bad approimatio wh p ar spctrum { Dt p H } { Tr l ph } p is good approimatio if p pt away from spctrum Idtity pricipl?! p dp dp
Voiculscu R-trasform Cosidr g z lim th avrag trac of { Tr z H } z a g z a rsolvt, z / Voiculscu' s R-trasform is dfid by ivrsio : : + R z Liarity : [Hrmitia R A+ B R if th law of A+ UBU A + R B radom matrics if A, B fr A, B ar fr i th limit is idpdt of U U ] Powr sris : R c fr cumulats c
Eampl: Gaussia Uitary Esmbl dμ R GUE GUE H p Tr H / g GUE z dh, z ± z 4
Th mai objctiv Zˆ GUE Q DP SDt STr PQ Q p GUE { SDt P H } STr Q / Grad si rl uivrsality cojctur rformulatd : Γ Q : lim stays holomorphic for ay ' small' dformatio of Epct : Γ Q STr l Zˆ Q STr l Q c Q / GUE o for sum of Wigr ad uitary smbls Erdös t al, Tao & Vu Madt & MZ
Radom Matrics: Boso-Boso Sctor
Small Cosidr Z p Borl trasform { Dt p H } Zˆ π i with Z p p dp If th spctral masur of H for covrgs waly to a compactly supportd masur, th: Thm Guiot, Maida; 4 : for small ough, lim l Zˆ l + c / Huristic : do saddl aalysis o Zˆ π i b a l p dν p dp
Guiot ad Maida
Larg Uitary smbl dμ H Tr V H dh Zˆ π i c π, j { Dt p H } j V j j π, orthogoal polyomial of d, p V dp of dgr For V aalytic, uiformly cov, lt > g b or < g a Madt, MZ 9 : lim l Zˆ l + c /
Othr symmtry classs: AI Prop Brgr, Eyard, 8; MZ, 9 : Lt H, Q b ral symmtric matrics with Lt SO whr r H r Q + Q H Tr Qg op H g < Th with dμ g, Dt dμ g Haar masur, ad o th uitary symmtric matrics M c + + / Dt P P dν P Q H dν P iv masur / O This rsult allows to stablish Borl trasform ad its ivrs for { Z P Dt / P H } M c, Tr P / U!,
Summary Borl trasform loos itriguig as a mthod to costruct ffctiv fild thoris for symmtry-bro phass of mattr frromagtism, suprcoductivity, For crtai radom matri smbls th larg- limit of th SUSY-Borl trasform of th partitio fuctio is dtrmid by th Voiculscu R-trasform Si rl grad uivrsality cojctur is rformulatd as a cojctur of holomorphicity of Borl trasform Qustio : Ca SUSY-BT b computd i a cotrolld way, say for th Adrso modl?