Topological zero modes at finite chemical potential

Transcription

1 Topological zero modes at finite chemical potential Falk Bruckmann (Univ. Regensburg) Sign 2012, Regensburg, Sept with Rudi Rödl, Tin Sulejmanpasic (paper in prep.) Falk Bruckmann Topological zero modes at finite chemical potential 0 / 9

2 Topology in a nut-shell saddle points in Yang-Mills path integral Z = DA µ e S[Aµ] S = d 4 x trf 2 µν minimal for (anti)selfdual F µν = ± F µν ( E = ± B) classical solutions, called instantons Falk Bruckmann Topological zero modes at finite chemical potential 1 / 9

3 Topology in a nut-shell saddle points in Yang-Mills path integral Z = DA µ e S[Aµ] S = d 4 x trfµν 2 minimal for (anti)selfdual F µν = ± F µν ( E = ± B) classical solutions, called instantons topological charge Q = d 4 x trf µν Fµν Z instantons: Q = 1 not deformable into vacuum nonperturbative Falk Bruckmann Topological zero modes at finite chemical potential 1 / 9

4 Topology in a nut-shell saddle points in Yang-Mills path integral Z = DA µ e S[Aµ] S = d 4 x trfµν 2 minimal for (anti)selfdual F µν = ± F µν ( E = ± B) classical solutions, called instantons topological charge Q = d 4 x trf µν Fµν Z instantons: Q = 1 not deformable into vacuum nonperturbative 4d lumps of action density, parameters: location, size Falk Bruckmann Topological zero modes at finite chemical potential 1 / 9

5 Topology in a nut-shell saddle points in Yang-Mills path integral Z = DA µ e S[Aµ] S = d 4 x trfµν 2 minimal for (anti)selfdual F µν = ± F µν ( E = ± B) classical solutions, called instantons topological charge Q = d 4 x trf µν Fµν Z instantons: Q = 1 not deformable into vacuum nonperturbative 4d lumps of action density, parameters: location, size fermions: index theorem for Dirac operator /D[A] Q = 1 left-handed zero mode Q = 1 right-handed zero mode Falk Bruckmann Topological zero modes at finite chemical potential 1 / 9

6 Semiclassical approach to QCD superposition of N instantons and N anti-instantons: N, N =O(V ) interactions: classical, quantum fluctuations, moduli space metric and fermion determinant det /D (in background A µ ): dominant in the basis of exact (anti)instanton zero modes: ψ0,i /D ψ 0,j Sij... overlap matrix chirality and antihermiticity of /D ( 0 X S = X 0 )... anti-hermitian N.B. nontrivial weight for X Gaussian weight: chiral RMT (with µ, too) chiral symmetry breaking lowest eigenvalues of /D resp. S = band of quasi zero modes nonzero density near λ = 0 Banks-Casher = chiral condensate Falk Bruckmann Topological zero modes at finite chemical potential 2 / 9

7 Chemical potential implementation: /D(µ) = γ ν ( ν ia ν ) + µγ 0 not of definite hermiticity det complex sign problem compensation by opposite µ: semiclassics: /D (µ) = /D( µ) (1) also overlap matrix S(µ) not of definite hermiticity sign problem S(µ) governs fermionic interaction between instantons and anti-instantons at µ > 0 first of all: need zero modes ψ 0 (µ) in top. backgrounds Falk Bruckmann Topological zero modes at finite chemical potential 3 / 9

8 Technicalities and our approach for a general operator /D(µ): e.g. Kanazawa, Wettig, Yamamoto 2011 bi-orthonormalization between right eigenmodes and left eigenmodes (1) = right eigenmodes of /D( µ) d 4 x ψ m(x; µ)ψ n (x; µ) = δ mn findings: densities/profiles ψ m(x; µ)ψ m (x; µ)... 2 generically complex! (0) SU(2) pseudoreal profiles are real, still not necessarily positive (1) analytic continuation from imag. µ (/D still antiherm.) to real µ: conventional orthonormalization bi-orthonormalization (2) ψ 0 at imag. µ explicitly known for finite T instantons = calorons dummy ADHM-Nahm formalism; Garcia-Perez et al Falk Bruckmann Topological zero modes at finite chemical potential 4 / 9

9 Zero modes for calorons general caloron (nontrivial holonomy = asymptotic Polyakov loop): N c constituents dyons = magn. monopoles Kraan, van Baal; Lee, Lu 1998 SU(2) caloron with Q = 1: two lumps in action density sizes=o( T ), distance T (parameter) Falk Bruckmann Topological zero modes at finite chemical potential 5 / 9

10 Zero modes for calorons general caloron (nontrivial holonomy = asymptotic Polyakov loop): N c constituents dyons = magn. monopoles Kraan, van Baal; Lee, Lu 1998 zero mode ψ 0 supported by twisted dyon profiles: SU(2) caloron with Q = 1: two lumps in action density sizes=o( T ), distance T (parameter) µ = 0 (the other dyon supports ψ 0 periodic in Eucl. time) Falk Bruckmann Topological zero modes at finite chemical potential 5 / 9

11 Zero modes for calorons general caloron (nontrivial holonomy = asymptotic Polyakov loop): N c constituents dyons = magn. monopoles Kraan, van Baal; Lee, Lu 1998 zero mode ψ 0 supported by twisted dyon profiles: SU(2) caloron with Q = 1: two lumps in action density sizes=o( T ), distance T (parameter) µ = 0 (the other dyon supports ψ 0 periodic in Eucl. time) µ = 2.4T analytic expr. for all components stronger peak & negative surrounding Falk Bruckmann Topological zero modes at finite chemical potential 5 / 9

12 Zero modes for dyons from calorons in the large distance limit vev parameter: (A a 0 )2 v zero mode profile: ψ 0 (x; µ)ψ 0(x; µ) 1 vr sinh(vr) trigonometric(µr) µ = 0, µ = 2.4T log plot matches radial ansatz following Jackiw, Rebbi 1976 at the core: (1 + 4µ2 v 2 ) 2... stronger peak asymptotics: exponential 1 sinh(vr) e vr oscillations, zeros at distance: π 2µ Falk Bruckmann Topological zero modes at finite chemical potential 6 / 9

13 Zero modes for instantons from calorons in the zero temperature limit size parameter: ρ zero mode profile: ψ 0 (x; µ)ψ 0(x; µ) 1 + ρ2 r 2 sin 2 (µr) r 2 + t 2 + ρ 2 matches ansaetze Aragao de Carvalho 1981, Cristoforetti 2011 at the core: ( 2 ρ 2 + µ 2 ) 2... stronger peak asymptotics: algebraic t 2, r 4 (for µ = 0 was: t 6, r 6 ) (wrong ψ 0 (x; µ)ψ 0(x; µ) would not be normalizable) spatial oscillations, zeros at distance: π 2µ Falk Bruckmann Topological zero modes at finite chemical potential 7 / 9

14 Lattice results caloron discretised on a lattice (plus cooling) staggered Dirac op.: 4 tastes, no exact zero modes, num. cheap lowest eigenvalues from ARPACK (also lowest singular values) Falk Bruckmann Topological zero modes at finite chemical potential 8 / 9

15 Lattice results caloron discretised on a lattice (plus cooling) staggered Dirac op.: 4 tastes, no exact zero modes, num. cheap lowest eigenvalues from ARPACK (also lowest singular values) Imag (λ/t ) Spectrum D s (µ = 2.4 T ) Free case Caloron 4x O(10 7 ) i Real (λ/t ) 4 modes with imag. parts of O(10 7 )T artefacts negligible higher modes close to those of free case = Matsubara frequencies (summed) profiles agree very well with continuum Falk Bruckmann Topological zero modes at finite chemical potential 8 / x 4x 4x 8x

16 Summary zero modes at finite temperature and imag. µ analytic continuation to real µ zero modes for general calorons large distance or zero temperature zero modes for dyons and instantons systematic and fully analytical, agree with ansaetze real profiles for SU(2) stronger peaks at the cores, oscillation in space confirmed on the lattice with staggered Dirac operator Falk Bruckmann Topological zero modes at finite chemical potential 9 / 9

17 Summary zero modes at finite temperature and imag. µ analytic continuation to real µ zero modes for general calorons large distance or zero temperature zero modes for dyons and instantons systematic and fully analytical, agree with ansaetze real profiles for SU(2) stronger peaks at the cores, oscillation in space confirmed on the lattice with staggered Dirac operator Outlook: compute overlap matrix elements for semiclassical approach damping by oscillations novel interactions of top. objects instantons molecules... Rapp, Schafer, Shuryak, Velkovsky 1999 details on manifestation of sign problem, testing ground for new methods Falk Bruckmann Topological zero modes at finite chemical potential 9 / 9