Rotor Spectra, Berry Phases, and Monopole Fields: From Antiferromagnets to QCD Uwe-Jens Wiese Bern University LATTICE08, Williamsburg, July 14, 008 S. Chandrasekharan (Duke University) F.-J. Jiang, F. Kämpfer, M. Nyfeler (Bern University) M. Pepe (INFN, Milano University)
Outline From Graphene to Ni x CoO Rotor Spectrum at Half-Filling Rotor Spectrum in the Single-Hole Sector Rotor Spectrum in the QCD Vacuum Sector Rotor Spectrum in the Single-Nucleon Sector Conclusions
Outline From Graphene to Ni x CoO Rotor Spectrum at Half-Filling Rotor Spectrum in the Single-Hole Sector Rotor Spectrum in the QCD Vacuum Sector Rotor Spectrum in the Single-Nucleon Sector Conclusions
The Hubbard model on the honeycomb lattice H = t xy (c xc y + c y c x ) + U x (c xc x 1), c x = (c x, c x ) Local charge and spin operators Q x = c xc x 1, Sx = c x σ c x, [Sx a, Sy b ] = iδ xy ε abc Sx c U(1) Q and SU() s symmetries Q = Q x, S = Sx, [H, Q] = [H, S] = 0 x x
Unbroken SU() s symmetric phase (graphene) at U < U c 3.5 1.5 1 0.5 0-4 -3 - -1 0 1 3 4-3 - -1 0 1 3 Brillouin zone Dispersion relation Effective Dirac Lagrangian for free graphene L = ψ f s γ µ µ ψs f f =α,β s=+,
The t-j model for the antiferromagnetic SU() s broken symmetry phase (Ni x CoO ) at U U c { H = P t (c xc y + c y c x ) + J Sx S y }P. xy xy reduces to the Heisenberg model at half-filling H = J xy Sx S y Effective Goldstone boson field in SU()/U(1) = S e(x) = (e 1 (x), e (x), e 3 (x)), e(x) = 1 Low-energy effective action for magnons S[ e] = d x dt ρ s ( i e i e + 1 ) c t e t e
Fit to predictions in the ε-regime of magnon chiral perturbation theory with βc L, l = (βc/l) 1/3 χ s = M s L β 3 χ u = ρ s 3c { 1 + c ρ s Ll β 1(l) + { 1 + 1 c 3 ρ s Ll β 1 (l) + 1 3 ( ) c [ β1 (l) + 3β (l) ]} ρ s Ll ( ) c [ β (l) 1 ] } ρ s Ll 3 β 1 (l) 6ψ(l) 50 10 4 4680 spins 3344 spins 508 spins 1560 spins 836 spins 40 30 4680 spins 3344 spins 508 spins 1560 spins 836 spins χ S Ja 4 10 3 < W t > 0 1 10 3 10 0 30 40 50 60 70 β J 0 0 30 40 50 60 70 β J M s = 0.689(4), ρ s = 0.10()J, c = 1.97(16)Ja
Outline From Graphene to Ni x CoO Rotor Spectrum at Half-Filling Rotor Spectrum in the Single-Hole Sector Rotor Spectrum in the QCD Vacuum Sector Rotor Spectrum in the Single-Nucleon Sector Conclusions
Rotor spectrum in the δ-regime of magnon chiral perturbation theory with βc L S(S + 1) Z = (S + 1) exp( βe S ), E S = Θ S=0 Moment of inertia Θ = ρ sl c [ 1 + 3.90065 c 4π ρ s L + O ( )] 1 L P. Hasenfratz and F. Niedermayer, Z. Phys. B9 (1993) 91 L β L
Probability distribution of magnetization M 3 = S 3 p(m 3 ) = 1 Z exp( βe S ) S M 3 Perfect agreement without additional adjustable parameters 1 10 1 Honeycomb Lattice, 836 Spins, βj = 60 1 10 0 S z = 0: 0.5444 S z = 1: 0.1909 S z = : 0.0338 S z = 3: 0.0083 S z = 0: 0.5445(9) S z = 1: 0.191(5) S z = : 0.0336(3) S z = 3: 0.0079(6) p(m 3 ) 1 10-1 1 10-1 10-3 -4-0 4 M 3
Outline From Graphene to Ni x CoO Rotor Spectrum at Half-Filling Rotor Spectrum in the Single-Hole Sector Rotor Spectrum in the QCD Vacuum Sector Rotor Spectrum in the Single-Nucleon Sector Conclusions
Hole dispersion in the t-j model 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.0-4 -3 - -1 0 1 3 4 0-1 - -3 1 3 Effective Lagrangian for Holes L = f =α,β s=+, [ Mψ f s ψ f s + ψ f s D t ψ f s + 1 M D iψ f s D i ψ f s Covariant derivative coupling to composite magnon gauge field D µ ψ f ±(x) = [ µ ± iv 3 µ(x) ] ψ f ±(x) ]
Effective Lagrangian for quantum mechanical rotor L = Θ t e t e+ f =α,β Ψ f [ E( p) i t + v 3 t σ 3 ] Ψ f, Ψ(t) = Polar coordinates for the staggered magnetization e = (sin θ cos ϕ, sin θ sin ϕ, cos θ) v 3 t = sin θ tϕ Canonically conjugate momenta Θ t θ = p θ, Θ t ϕ = 1 sin θ (p ϕ + ia ϕ ) Abelian monopole Berry gauge field ( ψ f + (t) ψ f (t) ) A θ = 0, A ϕ = i sin θ σ 3, F θϕ = θ A ϕ ϕ A θ = i sin θ σ 3
Rotor Hamiltonian in the single-hole sector H = 1 { 1 Θ sin θ θ[ sin θ θ ] + 1 } sin θ ( ϕ A ϕ ) + E( p) = 1 ( J 1 ) + E( p) Θ 4 Angular momentum operators ( J ± = exp(±iϕ) ± θ + i cot θ ϕ 1 tan θ ) σ 3, J 3 = i ϕ σ 3 Energy spectrum E j = 1 Θ [ j(j + 1) 1 4 Wave functions are monopole harmonics ] + E( p), j { 1, 3, 5,...} Y ± 1 (θ, ϕ) = 1 sin θ,± 1 π exp(±iϕ), Y ± 1 (θ, ϕ) = 1 cos θ, 1 π
Outline From Graphene to Ni x CoO Rotor Spectrum at Half-Filling Rotor Spectrum in the Single-Hole Sector Rotor Spectrum in the QCD Vacuum Sector Rotor Spectrum in the Single-Nucleon Sector Conclusions
Rotor Lagrangian in massless N f = QCD L = Θ ] [ 4 Tr t U t U, Θ = FπL 3 Rotor spectrum Rotor quantum numbers E l = j L(j L + 1) + j R (j R + 1) Θ = l(l + ) Θ j L = j R, l = j L + j R {0, 1,,...} Degeneracy g = (j L + 1)(j R + 1) = (l + 1) H. Leutwyler, Phys. Lett. B189 (1987) 197
Outline From Graphene to Ni x CoO Rotor Spectrum at Half-Filling Rotor Spectrum in the Single-Hole Sector Rotor Spectrum in the QCD Vacuum Sector Rotor Spectrum in the Single-Nucleon Sector Conclusions
Effective Lagrangian for quantum mechanical rotor L = Θ ] [ [ 4 Tr t U t U + Ψ E( p) i t iv t i g ] A M ( σ p)a t Ψ Gauge and vector fields composed of pion fields U = u v t = 1 ( ) u t u + u t u, a t = 1 ( ) u t u u t u i Spherical coordinates for the pion field U = cos α + i sin α e α τ, e α = (sin θ cos ϕ, sin θ sin ϕ, cos θ), e θ = (cos θ cos ϕ, cos θ sin ϕ, sin θ), e ϕ = ( sin ϕ, cos ϕ, 0) Concrete form of gauge and vector fields v t = i sin α ( tθ e ϕ sin θ t ϕ e θ ) τ, ( t α a t = e α + sin α tθ e θ + sin α sin θ ) tϕ e ϕ τ
Rotor Hamiltonian in the single-nucleon sector H = E( p) 1 Θ { 1 sin α ( α A α )[ sin α( α A α )] 1 + sin α sin θ ( 1 θ A θ )[ sin θ( θ A θ )] + sin α sin θ ( ϕ A ϕ ) Non-Abelian monopole Berry gauge field (Λ = g A p /M) A α = i Λ ( σ e p) e α τ, A θ = i (sin α e ϕ + Λ ) ( σ e p) sin α e θ τ, A ϕ = i ( sin α sin θ e θ + Λ ) ( σ e p) sin α sin θ e ϕ τ F αθ = α A θ θ A α + [A α, A θ ] = i 1 Λ F θϕ = θ A ϕ ϕ A θ + [A θ, A ϕ ] = i 1 Λ F ϕα = ϕ A α α A ϕ + [A ϕ, A α ] = i 1 Λ sin α e ϕ τ, sin α sin θ e α τ, sin α sin θ e θ τ }
Rotor Hamiltonian with Λ = g A p /M H = 1 ( J + K Θ 3 ) + 1 4 Θ C = i( σ e p ) ( e α α + 1 sin θ e θ θ + (ΛC + 34 Λ ), 1 sin α sin θ e ϕ ϕ tan α e α commutes with chiral rotations JL = 1 ( ) J K, JR = 1 ( ) J + K, C = J + K + 3 4 Energy spectrum E j = 1 Θ [ j (j + ) + Λ 1 ] + E( p), j = j ± Λ ) τ Rotor quantum numbers and degeneracies j L = j R ± 1 { 1, j = j L+j R, 3, 5 } (,..., g = j + 1 ) ( j + 3 )
Rotor Spectrum as a function of Λ = g A p /M E 40 35 30 5 0 j = 0.5, +Λ j = 0.5, -Λ j = 1.5, +Λ j = 1.5, -Λ j =.5, +Λ j =.5, -Λ j = 3.5, +Λ j = 3.5, -Λ 15 10 5 0-3 - -1 0 1 3 Λ Remarkably, for Λ = ±1 the non-abelian field strength vanishes and E j (±1) = 1 Θ j (j + ) with j = j ± 1. The QCD rotor spectrum then looks like the one of in the vacuum sector, although the system now has baryon number one.
Outline From Graphene to Ni x CoO Rotor Spectrum at Half-Filling Rotor Spectrum in the Single-Hole Sector Rotor Spectrum in the QCD Vacuum Sector Rotor Spectrum in the Single-Nucleon Sector Conclusions
Conclusions There are intriguing analogies between antiferromagnets and QCD. Fermions have characteristic effects on the rotor spectrum. The rotor problem tests the effective theory nonperturbatively. Perturbative matching of Λ to the infinite volume effective theory is necessary before g A could be extracted from the rotor level splitting. Interesting related work A. Ali-Khan et al., Nucl. Phys. B689 (004) 175 W. Detmold and M. Savage, Phys. Lett. B599 (004) 3 P. F. Bedaque, H. W. Griesshammer, and G. Rupak, Phys. Rev. D71 (005) 054015 G. Colangelo, A. Fuhrer, and C. Haefeli, Nucl. Phys. Proc. Suppl. 153 (006) 41 BGR collaboration, P. Hasenfratz et al., PoS (LATTICE 007) 077 JLQCD collaboration, H. Fukaya et al., PoS (LATTICE 007) 077, Phys. Rev. D76 (007) 054503, Phys. Rev. Lett. 98 (007) 17001