Topological susceptibility at finite temperature

Slides:



Advertisements
Similar presentations
Lecture 1: basics of lattice QCD Peter Petreczky Lattice regularization and gauge symmetry : Wilson gauge action, fermion doubling Different fermion formulations.
Advertisements

A method of finding the critical point in finite density QCD
Selected Topics in AdS/CFT lecture 1
Lattice Quantum Chromodynamic for Mathematicians Richard C. Brower Yale University May Day 2007 Tutorial in “ Derivatives, Finite Differences and.
The QCD equation of state for two flavor QCD at non-zero chemical potential Shinji Ejiri (University of Tokyo) Collaborators: C. Allton, S. Hands (Swansea),
Axial symmetry at finite temperature Guido Cossu High Energy Accelerator Research Organization – KEK Lattice Field Theory on multi-PFLOPS computers German-Japanese.
Su Houng Lee Theme: 1.Will U A (1) symmetry breaking effects remain at high T/  2.Relation between Quark condensate and the ’ mass Ref: SHL, T. Hatsuda,
Su Houng Lee 1. Mesons with one heavy quark 2. Baryons with one heavy quark 3. Quarkonium Arguments based on two point function  can be generalized to.
2D and time dependent DMRG
A Chiral Random Matrix Model for 2+1 Flavor QCD at Finite Temperature and Density Takashi Sano (University of Tokyo, Komaba), with H. Fujii, and M. Ohtani.
Test of the Stefan-Boltzmann behavior for T>0 at tree-level of perturbation theory on the lattice DESY Summer Student 2010 Carmen Ka Ki Li Imperial College.
1 Chiral Symmetry Breaking and Restoration in QCD Da Huang Institute of Theoretical Physics, Chinese Academy of
O(N) linear and nonlinear sigma-model at nonzeroT within the auxiliary field method CJT study of the O(N) linear and nonlinear sigma-model at nonzeroT.
Breakdown of the adiabatic approximation in low-dimensional gapless systems Anatoli Polkovnikov, Boston University Vladimir Gritsev Harvard University.
Excited Hadrons: Lattice results Christian B. Lang Inst. F. Physik – FB Theoretische Physik Universität Graz Oberwölz, September 2006 B ern G raz R egensburg.
The XXV International Symposium on Lattice Field Theory 29 July - 5 August 2007, Regensburg, Deutschland K. Miura, N. Kawamoto and A. Ohnishi Hokkaido.
Finite Density with Canonical Ensemble and the Sign Problem Finite Density Algorithm with Canonical Ensemble Approach Finite Density Algorithm with Canonical.
A direct relation between confinement and chiral symmetry breaking in temporally odd-number lattice QCD Lattice 2013 July 29, 2013, Mainz Takahiro Doi.
Exact Results for perturbative partition functions of theories with SU(2|4) symmetry Shinji Shimasaki (Kyoto University) JHEP1302, 148 (2013) (arXiv: [hep-th])
Mass modification of heavy-light mesons in spin-isospin correlated matter Masayasu Harada (Nagoya Univ.) at Mini workshop on “Structure and production.
Exploring Real-time Functions on the Lattice with Inverse Propagator and Self-Energy Masakiyo Kitazawa (Osaka U.) 22/Sep./2011 Lunch BNL.
Yusuke Hama (Univ. Tokyo) Collaborators Tetsuo Hatsuda (Univ. Tokyo)
Lianyi He and Pengfei Zhuang Physics Department, Tsinghua U.
Lattice Fermion with Chiral Chemical Potential NTFL workshop, Feb. 17, 2012 Arata Yamamoto (University of Tokyo) AY, Phys. Rev. Lett. 107, (2011)
Finite N Index and Angular Momentum Bound from Gravity “KEK Theory Workshop 2007” Yu Nakayama, 13 th. Mar (University of Tokyo) Based on hep-th/
Su Houng Lee Theme: 1.Will U A (1) symmetry breaking effects remain at high T 2.Relation between Quark condensate and the ’ mass Ref: SHL, T. Hatsuda,
Su Houng Lee Theme: Relation between Quark condensate and the ’ mass Ref: SHL, T. Hatsuda, PRD 54, R1871 (1996) Y. Kwon, SHL, K. Morita, G. Wolf, PRD86,
MEM analysis of the QCD sum rule and its Application to the Nucleon spectrum Tokyo Institute of Technology Keisuke Ohtani Collaborators : Philipp Gubler,
Eigo Shintani (KEK) (JLQCD Collaboration) KEKPH0712, Dec. 12, 2007.
Komaba seminarT.Umeda (Tsukuba)1 A study of charmonium dissociation temperatures using a finite volume technique in the lattice QCD T. Umeda and H. Ohno.
Instanton vacuum at finite density Hyun-Chul Kim Department of Physics Inha University S.i.N. and H.-Ch.Kim, Phys. Rev. D 77, (2008) S.i.N., H.Y.Ryu,
Vlasov Equation for Chiral Phase Transition
1 Approaching the chiral limit in lattice QCD Hidenori Fukaya (RIKEN Wako) for JLQCD collaboration Ph.D. thesis [hep-lat/ ], JLQCD collaboration,Phys.Rev.D74:094505(2006)[hep-
Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998) Criterion.
CPOD2011 , Wuhan, China 1 Isospin Matter Pengfei Zhuang Tsinghua University, Beijing ● Phase Diagram at finite μ I ● BCS-BEC Crossover in pion superfluid.
1 Localization and Critical Diffusion of Quantum Dipoles in two Dimensions U(r)-random ( =0) I.L. Aleiener, B.L. Altshuler and K.B. Efetov Quantum Particle.
Introduction instanton molecules and topological susceptibility Random matrix model Chiral condensate and Dirac spectrum A modified model and Topological.
1 NJL model at finite temperature and chemical potential in dimensional regularization T. Fujihara, T. Inagaki, D. Kimura : Hiroshima Univ.. Alexander.
Syo Kamata Rikkyo University In collaboration with Hidekazu Tanaka.
Matter-antimatter coexistence method for finite density QCD
Spectral functions in functional renormalization group approach
Study of the structure of the QCD vacuum
Collective Excitations in QCD Plasma
Nc=2 lattice gauge theories with
Non-Hermitian quantum mechanics and localization
Thermodynamics of QCD in lattice simulation with improved Wilson quark action at finite temperature and density WHOT-QCD Collaboration Yu Maezawa (Univ.
Spontaneous chiral symmetry breaking on the lattice
Speaker: Takahiro Doi (Kyoto University)
NGB and their parameters
Adjoint sector of MQM at finite N
mesons as probes to explore the chiral symmetry in nuclear matter
Chiral phase transition in magnetic field
Study of Aoki phase in Nc=2 gauge theories
Chapter III Dirac Field Lecture 1 Books Recommended:
Strangeness and charm in hadrons and dense matter, YITP, May 15, 2017
Color Superconductivity in dense quark matter
Lattice QCD, Random Matrix Theory and chiral condensates
Takashi Umeda (Hiroshima Univ.) for WHOT-QCD Collaboration
quark angular momentum in lattice QCD
Exact vector channel sum rules at finite temperature
Lecture 23. Systems with a Variable Number of Particles
Chengfu Mu, Peking University
Pavel Buividovich (Regensburg University)
SOC Fermi Gas in 1D Optical Lattice —Exotic pairing states and Topological properties 中科院物理研究所 胡海平 Collaborators : Chen Cheng, Yucheng Wang, Hong-Gang.
Towards Understanding the In-medium φ Meson with Finite Momentum
Nuclear Forces - Lecture 5 -
Jun Nishimura (KEK, SOKENDAI) JLQCD Collaboration:
A possible approach to the CEP location
EoS in 2+1 flavor QCD with improved Wilson fermion
FOR RANDOMLY PERTURBED Martin Dvořák, Pavel Cejnar
Presentation transcript:

Topological susceptibility at finite temperature in a random matrix model Munehisa Ohtani (Univ. Regensburg) with C. Lehner, T. Wettig (Univ. Regensburg) T. Hatsuda (Univ. of Tokyo) Introduction Chiral condensate in RM model T dependence of Dirac spectrum A modified model and Topological susceptibility Summary 28 Nov. @ U-Tokyo, Komaba

Introduction  Chiral symmetry breaking and instanton molecules 32 p2  E.-M.Ilgenfritz & E.V.Shuryak PLB325(1994)263  Chiral symmetry breaking and instanton molecules _ Banks-Casher rel: y y  = -pr (0) _ y y : chiral restoration # of I-I  : Formation of instanton molecules ? where r (l) = 1/V S d(l- ln) = -1/p Im Tr( l-D+ie )-1 Index Theorem: -1  tr FF = N+ - N- ~ 32 p2  0 mode of +(-) chirality associated with an isolated (anti-) instanton T = 0.91Tc T = 1.09Tc T = 1.01Tc  V.Weinberg’s talk @ Lattice 07 quasi 0 modes begin to have a non-zero eigenvalue r (0) becomes sparse

Instanton molecules &Topological susceptibility isolated (anti-)instantons at low T (anti-)instanton molecule at high T q(x)2 q(x)2 topological charge density q(x) d4x q(x)2 decreases as T  The formation of instanton molecules suggests decreasing topological susceptibility as T  d4x q(x)2 = 1/V d4yd4x(q(x)2 + q(y)2 )/2  1/V d4yd4x q(x)q(y) = Q2/V

Applications of Random Matrix  Energy levels of highly excited states in nucleus  E.P. Wigner, Ann. Math.53(1951)36; M. Mehta, Random Matrices (1991) universality and symmetry  Conductance fluctuation in mesoscopic systems  D.R. Hofstadter, Phys. Rev. B14 (1976) 2239 # fluc. of e- levels for a mesoscopic system (with L s.t. coherence length > L > mean free path of e-)  Spectral density of chUE  Andreev reflection e- e- e- hole metal superconductor  R. Opperman, Physica A167 (1990) 301 Classification of ensembles  Symmetries as Time rev. spin rotation etc  2D quantum gravity, zeros of Riemann z function,…  P. DiFrancesco et.al., J. Phys. Rep. 254 (1995) 1; A.M. Odlyzko, Math. of Comp. 48 (1987) 273.

Random matrix model / / / /  A.D.Jackson & J.J.M.Verbaarschot, PRD53(1996)  Random matrix model at T  0 Chiral symmetry: {DE , g5} = 0 Hermiticity: DE†= DE / ZQCD =  P det(iDE + mf ) YM / / / f with iDRM = 0 iW iW† 0 W  CN - × N + ZRM = S e-Q2/2Nt  DW e-N/2S2trW†W P det(iDRM + mf ) Q f | + g0 pT The lowest Matsubara freq. quasi 0 mode basis, i.e. topological charge: Q = N+ - N- Chiral restoration and Topological susceptibility

Hubbard Stratonovitch transformation  T.Wettig, A.Schäfer, H.A.Weidenmüller, PLB367(1996) 1) ZRM rewritten with fermions y 2) integrate out random matrix W Action with 4-fermi int. 3) introduce auxiliary random matrix S  CNf × Nf 4) integrate out y ZRM = S e-Q2/2Nt  DS e- N /2S2trS†S det S + m ipT (N - |Q|)/2det(S + m)|Q| Q ipT S† + m | In case of Nf = 1, integration by S can be carried out exactly.

Thermodynamic limit of chiral condensate _ y y  = m lnZRM /VNf = det(iDRM + m) tr (iDRM + m)-1/det(iDRM + m)  _ _ y y / y y 0 Fixed m = 0.1/S slow convergence  analytic calculation for N   N   N = 24 N = 23 N = 25, 26, 27, 28 T / Tc

Saddle point equations dim. of matrix N  N+ + N- ( V) plays a role of “1/ h ” The saddle point eqs. for S, Q/N become exact in the thermodynamic limit. ZRM = S e-Q2/2Nt  DS e- N /2S2trS†S det S + m ipT (N - |Q|)/2det(S + m)|Q| Q ipT S† + m Saddle pt. eq: (S - |Q|/Nm)((S + m)2-Q2/N2m2 + p2T2) = (1-|Q|/N)(S + m-|Q|/Nm)

The 2nd order transition in the chiral limit Chiral condensate _  y y  = m lnZRM /VNf =1 N tr S0 + m ipT -1 whereS0 : saddle pt. value VNf ipT S0† + m (Q = 0 at the saddle pt.) _ _  y y/  y y 0 T / Tc m S The 2nd order transition in the chiral limit

Leutwyler-Smilga model and Random Matrix Using singular value decomposition of S + m  V-1ULV, ZRM is rewritten with the part. func. ZL-S of c eff. theory for 0-momentum Goldstone modes ZL-S(Q,L) = DU e N S2trRe mLU-Q2/2Nt detUQ  H.Leutwyler, A.Smilga, PRD46(1992) ZRM(Q) = NQ  DL ZL-S(Q,L) e-N/2 S2trL2det(L2 + p2T2)N/2 det(L2 + p2T2)|Q|/2 detL|Q| U : nonlinear representation of pions L : determines chiral condensate dL : fluctuations of sigma

meson masses in RMT ms2 S2 mp2 S2 m S m S T / Tc T / Tc Plausible chiral properties

Eigenvalue distribution of Dirac operator _ r (l) = 1/V S d(l- ln) = -1/p Im Tr( l-D+ie )-1 = 1/p Re  y y |m -il _ y y  = m lnZ /VNf = Tr( iD+m )-1 ( pr (l) T / Tc l S r (0) becomes sparse as T  instanton molecule ?

Suppression of topological susceptibility Expansion by Q / N : - ln Z(Q)/Z(0) = b Q2 / N + O (Q3 / N2) a |Q| + b Q2 / N + O (Q3 / N2) × 1 1  0 (as N  ) 2 N sinh a/2   Q2 = 1 1 N 2b in RMM a  0 forT > 0 m S Q / N - ln Z(Q)/Z(0) T = 0 T > 0  T/Tc as N   Unphysical suppression of  at T  0 in RMM

Origin of the unphysical suppression ZRM = S e-Q2/2Nt  DS e- N /2S2trS†S det S + m ipT (N - |Q|)/2det(S + m)|Q| Q ipT S† + m SVD of S+m ZL-S(Q,L) = DU e N S2trRe mLU-Q2/2Nt detUQ  H.Leutwyler, A.Smilga, PRD46(1992) ZRM(Q) = NQ  DL ZL-S(Q,L) e-N/2 S2trL2det(L2 + p2T2)N/2 det(L2 + p2T2)|Q|/2 detL|Q| This factor suppresses c We claim to tune NQ so as to cancel the factor at the saddle point.

Modified Random Matrix model We propose a modified model: ZmRM = S  DL ZL-S(Q,L) e-N/2 S2trL2det(L2 + p2T2)N/2 Q where _  y y  in the conventional model is reproduced. cancelled factor =1 at Q = 0 i.e. saddle pt. eq. does not change (  c at T = 0 in the conventional model is reproduced. ( cancelled factor =1 also at T = 0 i.e. quantities at T = 0 do not change  c at T > 0 is not suppressed in the thermodynamic limit.

topological susceptibility in the modified model c = 1 + Nf 1-1 t m(m+L0)S whereL0 : saddle pt. value m S = 0.1 m S = 0.01 2 T / Tc m S  B.Alles, M.D’Elia, A.Di Giacomo, PLB483(2000) · Decreasing c as T  · Comparable with lattice results

Overlap operator and RMT Ginsparg-Wilson rel: {Dov , g5} = a Dov g5 Dov g5-Hermiticity: Dov†= g5 Dov g5 Dov= 1/a (1+g5 sign(g5iDW))  With eigenfunctions Dovn = lnn , we can show that ln + ln* =  n*(Dov+ g5Dovg5) n = a n* g5Dovg5Dov n= a ln* ln Dov (g5 n ) = g5 Dov†n = ln* (g5 n )  Hermitian operator g5iDW : diagonalized by unitary matrix sign(g5iDW) = U† diag(-1,-1,-1,1) U U = exp(i at cf c) =  Dov =  iDRM ( at T=0 ) as a  0

Summary and outlook Chiral restoration and topological susceptibility c are studied in a random matrix model  formation of instanton molecules connects them via Banks-Casher relation and the index theorem. Conventional random matrix model : 2nd order chiral transition & unphysical suppression of c for T >0 in the thermodynamic limit. We propose a modified model in which y y  & c|T=0 are same as in the original model, c at T >0 is well-defined and decreases as T increases.  consistent with instanton molecule formation, lattice results Outlook: To find out the random matrix with quasi 0 mode basis from which the modified model are derived, Extension to finite chemical potential, Nf dependence … _