Landau Levels in QCD in an External Magnetic Field

Transcription

1 Landau Levels in QCD in an External Magnetic Field Matteo Giordano Eötvös Loránd University (ELTE) Budapest ELTE particle physics seminar Budapest, 21/09/2016 Based on work in progress with F. Bruckmann, G. Endrődi, S. D. Katz, T. G. Kovács, F. Pittler, and J. Wellnhofer Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

2 Landau Levels in QCD in an External Magnetic Field Matteo Giordano Eötvös Loránd University (ELTE) Budapest ELTE particle physics seminar Budapest, 21/09/2016 Based on work in progress with F. Bruckmann, G. Endrődi, S. D. Katz, T. G. Kovács, F. Pittler, and J. Wellnhofer feat. P. McCartney, J. Page, and S. Wonder Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

3 QCD in an External Magnetic Field External magnetic field B on strongly interacting matter [review: Andersen, Naylor, Tranberg (2016)] heavy ion collisions: colliding ions generate magnetic field which affects the QGP Au+Au, b = 10 fm, s = 200 GeV ( ) 2 eb e2 2Z Au v GeV 2 4π 1 v 2 b [review: Huang (2016)] neutron stars: EOS of hadronic matter/qgp in magnetic field needed to determine the mass/radius ratio for magnetars (strong B field, low rotation frequency) evolution of the early universe: strength of B can affect phase transitions in the early universe, baryogenesis Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

4 Phase Diagram from the Lattice Zero chemical potential (no sign problem), T c (B = 0, µ = 0) = 155 MeV T 130 MeV, T 190 MeV magnetic catalysis: ψψ increases with B 130 MeV < T < 190 MeV inverse magnetic catalysis: ψψ decreases with B Phase diagram: from light-quark condensate (red) or strange quark susceptibility (blue) T c decreases with B [Bali, Bruckmann, Endrődi et al. (2012a), Bali, Bruckmann, Endrődi et al. (2012b)] Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

5 Valence and Sea Effects In a magnetic field B-dependent Dirac operator /D(A, B) ψψ = 1 DA det( /D(A, B) + m)e Sg [A] 1 Tr Z(B) /D(A, B) + m Sg [A] Z(B) = DA det( /D(A, B) + m)e Valence effect: the density of low Dirac modes increases for nonzero B (even if B is not in the determinant) catalysis Sea effect: the determinant suppresses configurations with higher density of low modes inverse catalysis Sea effect wins over valence near T c, where it is most effective inverse magnetic catalysis [Bruckmann, Endrődi, Kovács (2013)] Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

6 Low-Energy Models Analytic calculation in low-energy models for QCD: χpt, (P)NJL, bag model... PNJL OK at T = 0 for eb < 0.3 GeV 2 χpt OK for T < 100 MeV, eb < 0.1 GeV 2 General prediction: magnetic catalysis at all temperatures, T c (B) increases [Bali, Bruckmann, Endrődi et al. (2012b)] Magnetic catalysis mostly attributed to the behaviour of the *scary orchestral sound* Lowest Landau Level (LLL)... what is that? Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

7 Low-Energy Models Analytic calculation in low-energy models for QCD: χpt, (P)NJL, bag model... PNJL OK at T = 0 for eb < 0.3 GeV 2 χpt OK for T < 100 MeV, eb < 0.1 GeV 2 General prediction: magnetic catalysis at all temperatures, T c (B) increases [Bali, Bruckmann, Endrődi et al. (2012b)] Magnetic catalysis mostly attributed to the behaviour of the *scary orchestral sound* Lowest Landau Level (LLL)... what is that? Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

8 Landau Levels: Quantum Mechanics Spinless particle, minimal coupling, uniform magnetic field in the z direction: A = (0, Bx, 0) Since [p y, H] = [p z, H] = 0 H = 1 [ ] px 2 + (p y qbx) 2 + pz 2 2m H = 1 [ ] px 2 + (k y qbx) 2 + kz 2 = 2m ( E= ω c n + 1 ) + k2 z 2 2m, harmonic oscillator {}}{ p 2 x 2m + m 2 ω2 c(x x 0 ) 2 + k2 z 2m ω c = qb m, x 0 = k y qb Degeneracy: finite system with PBC, k y = 2πk L y x 0 = 2πk qbl y < L x within the system 0 k < N B =, k Z and center of force qblx Ly 2π = Φ B 2π Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

9 Landau Levels and Particle Spectrum Energy levels of an otherwise free particle in a uniform magnetic field B (minimal coupling + magnetic moment, Minkowski, relativistic) En,s 2 z (p z ) = pz 2 + qb(2n 2s z + 1) + m 2 Turn on strong interactions: LLs work for weakly coupled particles m 2 π ± (B) = E 2 n,0 (0) = m2 π ± (0) + eb What about quarks? (electrically charged & strongly coupled) [Bali, Bruckmann, Endrődi et al. (2012a)] Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

10 Landau Levels for Free Fermions: 2d Continuum Dirac Equation (Euclidean), 2d: ( /D + m)ψ = (iλ + m)ψ /D = γ µ D µ, D µ = µ + iqa µ, a µ = (0, Bx) Eigenvalues of /D 2 : λ 2 = qb [2n s z sgn(qb)] }{{} k=0,1,... n = 0, 1,..., s z = ± 1 2 Degeneracy of λ 0, ±λ k for k 0 = N B N B = L xl y qb 2π = Φ B 2π (quantised in a finite volume with PBC) λ 2 n,s z is N B degenerate; for (n, s z ) = (0, 1), λ 2 0, 1 = 0 N B zero modes; for (n, s z ) (0, 1), from 2N B identical modes λ 2 n,1, λ2 n+1, 1 ± λn.sz, each N B degenerate Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

11 Landau Levels for Free Fermions: 4d continuum Dirac Equation (Euclidean), 4d: ( /D + m)ψ = (iλ + m)ψ /D = γ µ D µ, D µ = µ + iqa µ, a µ = (0, Bx, 0, 0) Separation of variables, ψ(x, y, z, t) = φ(x, y)e ipz z e iptt λ 2 = qb [2n s z sgn(qb)] +pz 2 + pt 2 }{{} n = 0, 1,..., s z = ± 1 2 k=0,1,... Degeneracy of λ 0, ±λ k for k 0 = 2N B N B = L xl y qb = Φ B (quantised in a finite volume with PBC) 2π 2π (factor of 2 from particle/antiparticle) z-momentum: p z = 2π L z k z t-momentum: at finite temperature T, ABC in the t direction for fermions p t = π L t (2k t + 1) = πt (2k t + 1) (Matsubara frequencies) Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

12 Magnetic Catalysis Simple explanation: 1 after switching on strong interactions spectrum still approximately organised in Landau levels 2 LLL is λ = 0 with degeneracy B B increases the density ρ(0) of low Dirac modes 3 Chiral condensate ψψ ρ(0) (Banks-Casher relation), increases due to increased degeneracy magnetic catalysis (valence effect) For nonzero quark masses ψψ ρ(0) 4 Would lead to T c (B) > T c (0), but not clear if this works in the critical region (sea effects are important there)... is this true? Check on the lattice and curse the day you were born Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

13 Lattice Gauge Theory (at Breakneck Speed) Lattice: replace continuum spacetime with a discrete box Z = DU Dψ D ψ e Sg [U] ψ( /D+m)ψ Sg [U] = DU det( /D + m)e No problems with the gauge action, manifest gauge invariance preserved Fermions are problematic: doublers, chiral symmetry, personality disorders Staggered discretisation: single-component fermion field, /D D s (D s ) xy = 1 η µ (x) [U µ (x)δ x+ˆµ y U µ (x)δ x ˆµ y ], η µ = ( 1) ν<µ xν 2 µ Disadvantages: doublers reduced but not removed, requires rooting trick Advantages: D s anti-hermitean, purely imaginary spectrum, symmetric wrt origin numerically cheap the single-component fermion field can be treated as 2d or 4d if we so please (and we do) Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

14 Landau Levels for Free Fermions: 2d Lattice Finite lattice with periodic boundary conditions Magnetic field along z: U y (x) = e ia2 qbx U(1), U x = 1 B-field quantised: N B = L xl y qb 2π 2d free spectrum (with exceptions at the lattice boundary) = Φ B 2π Z qb = 2π L x L y N B Rational ratios of charges q i = n i q, quantised qb LL structure spoiled by finite spacing artefacts Gaps remnant of the LL structure Fractal structure: Hofstadter butterfly [Hofstadter (1976)] Colour code: eigenvalues grouped according to LL degeneracy Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

15 Landau Levels for Interacting Fermions: 2d Lattice Add SU(3) interaction: U µ (x) = u µ (x)w µ (x) u y (x) = e ia2qbx U(1), u x = 1 (up and down quark q = 2 3, 1 3 ) W µ (x) SU(3) integrated over with Haar measure 2d configuration = 2d slice of a 4d configuration 2d interacting spectrum Hofstadter butterfly washed away, gaps disappear but lowest Landau level (LLL) survives, wide gap: why? Colour code: eigenvalues grouped according to LL degeneracy (6N B : staggered doubling 3 colours) Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

16 Gap in the Continuum Limit The gap even survives the continuum limit 100 a=0.082 fm fm fm fm λ/m flux quantum (prop. to B) Continuum limit: Fixed B and L x L y = (an s ) 2 flux N B is RG invariant Spectrum renormalises the quark mass λ m ud is RG invariant Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

17 Localisation Near the Gap Another way to see the gap: localisation IPR = sites φ 4 (IPR) 1 size of the mode Modes localised near the gap, extended elsewhere Gap fluctuates from configuration to configuration Mixing is difficult at the edges of the gap Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

18 Spin σ xy : z-component of spin (it is not just σ z for staggered fermions) Matrix elements of σ xy for 2d eigenstates φ i,j (σ xy ) ij = φ i σ xy φ j Almost diagonal, appreciably different from zero only in the LLL In 2d chirality = 2s z : near zero modes with definite chirality? I have heard of that already but I was too busy not listening Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

19 Topological Origin of 2d LLL LLL has topological origin, robust under small deformations Atiyah-Singer index theorem: Q top = n n + In 2d also vanishing theorem : n n + = 0, i.e., either n or n + nonzero [Kiskis (1977); Nielsen and Schroer (1977); Ansourian (1977)] In 2d topological charge = flux of Abelian field Q top = q d 2 x ɛ µν F µν = q 4π 2π L xl y B = N B Switching on SU(3) fields changes nothing: same definition of Q top π 1 (SU(3)) trivial LLL (= zero modes) survives; other LLs are mixed by the SU(3) interaction Staggered = 2 species, 3 colours: 6N B almost-zero modes (splitting due to finite-spacing artefacts), almost-definite spin Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

20 Can One See Landau Levels from the Spectrum in 4d? No: in 4d the p 2 z + p 2 t contribution makes it impossible to identify LL from the spectrum even in the free continuum case (people softly crying in the background) But: factorisation of the eigenmodes eigenvalue mode number ψ (j) k,p z,p t (x, y, z, t) = φ (j) k (x, y)eipz z e iptt /D 2 2dφ (j) k (x, y) = qb kφ(j) k (x, y) j = 1,..., N B, k = 2[n + s z sgn(qb)] + 1 To extract LL from 4d eigenmode ψ: diagonalise /D 2 2d and project on one of its eigenvectors localised on a z, t slice, φ (j) k,z 0 t 0 (x, y, z, t) qb k = ψ /D 2 2d φ (j) k,z 0 t 0 φ (j) k,z 0 t 0 ψ φ (j) k,z 0 t 0 (x, y, z, t) = φ (j) k (x, y)δ z z 0 δ t t0 Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

21 Overlap with LLL On the lattice Landau level structure washed away already in 2d Still, LLL survives in 2d: how to check its physical effects? Idea: determine the 2d eigenmodes on each slice, identify the LLL on each slice, and project the 4d mode on the union of the LLLs Does this make sense? Yes! (people overwhelmed by enthusiasm, losing consciousness ) O(j) = tz slices ψ φ j,tz φ j,tz ψ φ j,zt : jth 2d mode>0 of D s restricted to zt slice (all 2d modes, all slices form complete 4d basis) Jump at j = 3N B, also in the continuum: LLL is different True for low 4d modes, disappears higher up in the spectrum, but LLL still different Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

22 Overlap with LLL On the lattice Landau level structure washed away already in 2d Still, LLL survives in 2d: how to check its physical effects? Idea: determine the 2d eigenmodes on each slice, identify the LLL on each slice, and project the 4d mode on the union of the LLLs Does this make sense? Yes! (people overwhelmed by enthusiasm, losing consciousness ) O(j) = tz slices ψ φ j,tz φ j,tz ψ φ j,zt : jth 2d mode>0 of D s restricted to zt slice (all 2d modes, all slices form complete 4d basis) Jump at j = 3N B, also in the continuum: LLL is different True for low 4d modes, disappears higher up in the spectrum, but LLL still different Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

23 Two-Dimensional Basis Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

24 LLL Projector To extract LLL: project on 2d LLL on all slices Π 0 (B) = φ j,tz (B) φ j,tz (B) + φ j,tz (B) φ j,tz (B) tz slices 1 j 3N b φ: positive 2d modes, φ: negative 2d modes Π 0 allows to define LLL-restricted observables: O Γ = ψγψ O LLL Γ = ψπ 0 ΓΠ 0 ψ O LLL Γ ψ, ψ,a = Tr ΓΠ 0M 1 Π 0 A M = D s + m Only valence effect considered, no sea effects which require modifications of the fermion determinant in the partition function (very interesting, but considerably harder) Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

25 LLL Condensate I Quark condensate: Γ = 1 full condensate: projected condensate: ψψ B = Tr M 1 B ψψ LLL B = Tr Π 0 M 1 Π 0 B Quark condensate affected by additive and multiplicative renormalisation Full condensate: additive divergence is B-independent, subtract ψψ B=0 ψψ (B) = ψψ B ψψ 0 Multiplicative divergence cured multiplying by quark mass r ψψ (B) = m ud [ ψψ B ψψ 0 ] Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

26 LLL Condensate II Projected condensate: we cannot subtract ψψ LLL 0 = 0 Define the projector Π 0 (B) = φ j,tz (0) φ j,tz (0) + φ j,tz (0) φ j,tz (0) tz slices 1 j 3N B ψψ LLL (B) = ψπ 0 (B)ψ B ψ Π 0 (B)ψ 0 Measures the variation due to the change of the first 3N B 2d modes on each slice caused by B Should cure additive divergence Additive div comes from large eigenmodes, where SU(3) interaction can be neglected: in that case one can show explicitly that this quantity is finite in the continuum Multiplicative divergence: if the LLL-overlap has a finite continuum limit (and it seems it does) same as in the full case, cancels in ratios ψψ LLL (B) ψψ (B) = ψπ 0 (B)ψ B ψ Π 0 (B)ψ 0 ψψ B ψψ 0 Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

27 Two-Dimensional Basis (Reprise) Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

28 Preliminary Numerical Results: Quark Condensate How much of the change in the condensate comes from the LLL? No B in the fermion determinant Down quark condensate, q = 1 3 Below T c ψψ(b) LLL ψψ(b) eb = (πt ) 2 T = 123MeV 0.4 nt = 6 nt = nt = 10 nt = eb [ GeV 2] Finite continuum limit (it seems) Ratio increases with B Ratio decreases with T Reaches 0.5 around GeV 2, large field (larger n t = finer lattice) Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

29 Preliminary Numerical Results: Quark Condensate How much of the change in the condensate comes from the LLL? No B in the fermion determinant Down quark condensate, q = 1 3 Around T c ψψ(b) LLL ψψ(b) eb = (πt ) 2 T = 170MeV nt = 6 nt = 8 nt = 10 nt = eb [ GeV 2] Finite continuum limit (it seems) Ratio increases with B Ratio decreases with T Reaches 0.5 around GeV 2, large field (larger n t = finer lattice) Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

30 Preliminary Numerical Results: Quark Condensate How much of the change in the condensate comes from the LLL? No B in the fermion determinant Down quark condensate, q = 1 3 Above T c ψψ(b) LLL ψψ(b) T = 189MeV eb = (πt ) nt = 6 nt = nt = 10 nt = eb [ GeV 2] Finite continuum limit (it seems) Ratio increases with B Ratio decreases with T Reaches 0.5 around GeV 2, large field (larger n t = finer lattice) Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

31 Preliminary Numerical Results: Quark Condensate How much of the change in the condensate comes from the LLL? No B in the fermion determinant Down quark condensate, q = 1 3 Above T c ψψ(b) LLL ψψ(b) T = 214MeV eb = (πt ) 2 0 nt = nt = 8 nt = 10 nt = eb [ GeV 2] Finite continuum limit (it seems) Ratio increases with B Ratio decreases with T Reaches 0.5 around GeV 2, large field (larger n t = finer lattice) Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

32 Conclusions and Outlook What happens to Landau levels in QCD? Landau level structure washed out, both in 2d and 4d, but in 2d lowest Landau level survives (topological reasons) and 4d observables seem to feel the 2d LLL: sizeable fraction of the change in the (valence) condensate comes from it Importance of LLL increases with B and decreases with T Open issues: other observables (e.g. spin) sea effects (inverse catalysis?) Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26

33 References J. O. Andersen, W. R. Naylor and A. Tranberg, Rev. Mod. Phys. 88 (2016) X. G. Huang, Rept. Prog. Phys. 79 (2016) G. S. Bali, F. Bruckmann, G. Endrődi, Z. Fodor, S. D. Katz, S. Krieg, A. Schäfer and K. K. Szabó, JHEP 02 (2012) 044 G. S. Bali, F. Bruckmann, G. Endrődi, Z. Fodor, S. D. Katz and A. Schäfer, Phys. Rev. D 86 (2012) F. Bruckmann, G. Endrődi and T. G. Kovács, JHEP 04 (2013) 112 D. R. Hofstadter, Phys. Rev. B 14 (1976) 2239 J. Kiskis, Phys. Rev. D 15 (1977) 2329; N. K. Nielsen and B. Schroer, Nucl. Phys. B 127 (1977) 493; M. M. Ansourian, Phys. Lett. B 70 (1977) 301 F. Zappa, Waka/Jawaka (1972); The Grand Wazoo (1972) Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/ / 26