QCD in external magnetic fields

Gergely Endrodi

Goethe University of Frankfurt

81. Jahrestagung der DPGMunster, 28. March 2017

in collaboration with

Gunnar Bali, Falk Bruckmann, Zoltan Fodor, Matteo Giordano,Sandor Katz, Tamas Kovacs, Stefan Krieg, Ferenc Pittler,Andreas Schafer, Kalman Szabo, Jacob Wellnhofer

Outline

I preface: magnetic fields and latticesI lattice quantum chromodynamics

I role of Landau levelsI order parameterI magnetic catalysis

I nonzero temperatureI phase diagramI applications

I summary

1 / 21

Magnetic fields and lattices

Experiments

I conductance of graphene lattices in strong magnetic fields[Ponomarenko et al Nature 497 ’13, Dean et al Nature 497 ’13][Hunt et al Science 340 ’13]

[Ponomarenko et al ’13] [Dean et al ’13]

2 / 21

Landau versus Bloch

• free charged particleI exposed to magnetic field in continuum space: Landau orbits

I on a (crystal) lattice: Bloch waves

• what happens if the two are combined?

3 / 21

Landau versus Bloch

• free charged particleI exposed to magnetic field in continuum space: Landau orbitsI on a (crystal) lattice: Bloch waves

• what happens if the two are combined?

3 / 21

Landau versus Bloch

• free charged particleI exposed to magnetic field in continuum space: Landau orbitsI on a (crystal) lattice: Bloch waves

• what happens if the two are combined?

3 / 21

Hofstadter’s butterfly [Hofstadter ’76]

• relativistic electrons in a magnetic field in 2D [Endrodi ’14]

Bmax ∝ 1/a24 / 21

Hofstadter’s butterfly at low magnetic fields

I Landau levels are visible at low B [Endrodi ’14]

λ2 = 2n · B5 / 21

Hofstadter’s butterfly at low magnetic fields

I Landau levels are visible at low B [Endrodi ’14]

λ2 = 2n · B5 / 21

Hofstadter’s butterfly at low magnetic fields

I Landau levels are visible at low B [Endrodi ’14]

λ2 = 2n · B5 / 21

Hofstadter’s butterfly at low magnetic fields

I Landau levels are visible at low B [Endrodi ’14]

λ2 = 2n · B5 / 21

Translation to lattice QCD

From quantum mechanics to QCD

I non-interacting electrons strongly interacting quarkssmears out butterfly

I quantum mechanics quantum field theory with pathintegral

Z =∫DAµDψDψ e−S

S =∫

d4x[1

4TrFµνFµν + ψ( /D + m)ψ], /D = γµ(∂µ+igsAµ+iqAµ)

I crystal lattice as a regulator

pmax ≈ 1/a

continuum limit: a→ 0

I energies (measureable) Dirac eigenvalues(not measureable)

6 / 21

The butterfly in lattice QCD

I continuum limit a→ 0⇒ lowest Landau-level can be separated in full QCD(related to topology) [Bruckmann, Endrodi et al ’16]

7 / 21

The butterfly in lattice QCD

I continuum limit a→ 0⇒ lowest Landau-level can be separated in full QCD(related to topology) [Bruckmann, Endrodi et al ’16] 7 / 21

The phases of QCD

Chiral condensate

I physical observable in QCD:condensate of quarks with mass m

ψψ =∑λ

mλ2 + m2

I for m = 0: order parameter for chiral symmetry breaking

SUL(Nf )× SUR(Nf ) ψψ−−→ SULR(Nf )

8 / 21

Chiral condensate

I physical observable in QCD:condensate of quarks with mass m

ψψ =∑λ

mλ2 + m2

I for m = 0: order parameter for chiral symmetry breaking

SUL(Nf )× SUR(Nf ) ψψ−−→ SULR(Nf )

8 / 21

Chiral condensate

I physical observable in QCD:condensate of quarks with mass m

ψψ =∑λ

mλ2 + m2

I for m = 0: order parameter for chiral symmetry breaking

SUL(Nf )× SUR(Nf ) ψψ−−→ SULR(Nf )

8 / 21

Phases of QCD

I chiral symmetry breaking and confinement go hand in hand

ψψ ⇔ PI can be probed by the temperature

[Borsanyi et al ’10, Bruckmann, Endrodi et al ’13]

[Aoki, Endrodi et al ’06]

9 / 21

Phases of QCD

I chiral symmetry breaking and confinement go hand in hand

ψψ ⇔ PI can be probed by the temperature

[Borsanyi et al ’10, Bruckmann, Endrodi et al ’13]

[Aoki, Endrodi et al ’06]

9 / 21

Phases of QCD

I chiral symmetry breaking and confinement go hand in hand

ψψ ⇔ PI can be probed by the temperature

[Borsanyi et al ’10, Bruckmann, Endrodi et al ’13] [Aoki, Endrodi et al ’06]

9 / 21

Condensate in magnetic fields

Landau-levels

lowest Landau-level:I λ well below other eigenvaluesI multiplicity is proportional to magnetic flux Φ = B · L2

higher Landau-levels:I λ >

√2B

• if B sufficiently large, only lowest LL matters10 / 21

Chiral condensate

I physical observable in QCD:condensate of quarks with mass m

ψψ =∑λ

mλ2 + m2

I for B � m2, lowest LL approximation

ψψ ∼ BL2 ∑λ∈ lowest LL

mλ2 + m2

I implication: ψψ increases with B‘magnetic catalysis’ [Gusynin et al ’95]

11 / 21

Magnetic catalysis at T = 0

I magnetic catalysis from first principles[Bali, Bruckmann, Endrodi, Fodor, Katz, Schafer ’12]

lowest LL

I magnetic catalysis is a robust concept: captured by NJL,χPT, AdS/CFT and further models [Andersen, Naylor ’14]

12 / 21

Magnetic catalysis at T = 0

I magnetic catalysis from first principles[Bali, Bruckmann, Endrodi, Fodor, Katz, Schafer ’12]

lowest LL

I magnetic catalysis is a robust concept: captured by NJL,χPT, AdS/CFT and further models [Andersen, Naylor ’14]

12 / 21

Magnetic catalysis at T = 0

I magnetic catalysis from first principles[Bali, Bruckmann, Endrodi, Fodor, Katz, Schafer ’12]

lowest LL

I magnetic catalysis is a robust concept: captured by NJL,χPT, AdS/CFT and further models [Andersen, Naylor ’14]

12 / 21

Summary

lowest Landau-level

⇓

magnetic catalysis

13 / 21

Nonzero temperature

Condensate at nonzero temperatures

• going to finite temperatures changes the response qualitativelyinverse magnetic catalysis around Tc[Bruckmann, Endrodi, Kovacs ’13]

14 / 21

Condensate at nonzero temperatures

• going to finite temperatures changes the response qualitativelyinverse magnetic catalysis around Tc[Bruckmann, Endrodi, Kovacs ’13]

14 / 21

Condensate at nonzero temperatures

• going to finite temperatures changes the response qualitativelyinverse magnetic catalysis around Tc[Bruckmann, Endrodi, Kovacs ’13]

14 / 21

Phase diagram

• impact on the QCD phase diagram: Tc(B) decreases[Bali, Bruckmann, Endrodi, Fodor, Katz, Krieg, Schafer, Szabo ’11]

15 / 21

Phase diagram for very strong fields

I go to even stronger B-fieldsI nature of transition changes from crossover to first order

[Endrodi ’15]

16 / 21

Applications

Applications: heavy-ion collisions

I off-central events generate magnetic fields[Kharzeev, McLerran, Warringa ’07]

I strength: B = 1015 T ≈ 1020Bearth ≈ 5m2π

I impact: chiral magnetic effect, anisotropies, elliptic flow,hydrodynamics with B, . . .reviews: [Fukushima ’12] [Kharzeev, Landsteiner, Schmitt, Yee ’14][Kharzeev ’15]

17 / 21

Applications: magnetars

I neutron stars with strong surface magnetic fields[Duncan, Thompson ’92]

I strength on surface: B = 1010 TI strength in core (?): B = 1014 T ≈ 1019Bearth ≈ 0.5m2

π

I impact: convectional processes, mass-radius relation,gravitational collapse/merger [Anderson et al ’08], . . .

18 / 21

Applications: early universe

I large-scale intergalactic magnetic fields 10 µG = 10−9 TI origin in the early universeI generation through a phase transition: electroweak epoch

B ≈ 1019 T [Vachaspati ’91, Enqvist, Olesen ’93]

19 / 21

Applications: low-energy models

I effective theories / low energy models of QCD:incorporate relevant degrees of freedom / mechanismsfor example PNJL model [Gatto, Ruggieri ’11]

eB=19

eB=15

eB=10

eB=0

160 170 180 190 200 2100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

T HMeVL

S�S

0

I models work at low T but not around TcI phase diagram just the opposite of the lattice resultsI some important ingredient is missed [Andersen, Naylor ’14]

[Braun et al ’14, Muller et al ’15, . . . ]20 / 21

Applications: low-energy models

I effective theories / low energy models of QCD:incorporate relevant degrees of freedom / mechanismsfor example PNJL model [Gatto, Ruggieri ’11]

C+ΧSB

D+ΧSR

TT

Χ

P

TB=0 =175 MeV

0 5 10 15 200.8

0.9

1.0

1.1

1.2

eB�mΠ2

TΧ,TPHM

eVL

I models work at low T but not around TcI phase diagram just the opposite of the lattice resultsI some important ingredient is missed [Andersen, Naylor ’14]

[Braun et al ’14, Muller et al ’15, . . . ]20 / 21

Summary

C+ΧSB

D+ΧSR

TT

Χ

P

TB=0 =175 MeV

0 5 10 15 200.8

0.9

1.0

1.1

1.2

eB�mΠ2

TΧ,TPHM

eVL

21 / 21

Backup

Hofstadter’s butterfly [Hofstadter ’76]

I true fractal structure (if the lattice is infinite)I energies accumulate into bands if flux Φ ∈ Q

(2π/a2 and qB are commensurable)I energies isomorphic to the Cantor set if Φ 6∈ Q

(2π/a2 and qB are incommensurable)

1 / 2

Hofstadter’s butterfly [Hofstadter ’76]

I true fractal structure (if the lattice is infinite)I energies accumulate into bands if flux Φ ∈ Q

(2π/a2 and qB are commensurable)I energies isomorphic to the Cantor set if Φ 6∈ Q

(2π/a2 and qB are incommensurable)

1 / 2

Mechanism behind MC and IMC

• two competing mechanisms at finite B[Bruckmann,Endrodi,Kovacs ’13]

I direct (valence) effect B ↔ qfI indirect (sea) effect B ↔ qf ↔ g⟨

ψψ(B)⟩∝∫DAµ e−Sg det( /D(B,A) + m)︸ ︷︷ ︸

sea

Tr[( /D(B,A) + m)−1

]︸ ︷︷ ︸

valence

BB BB

2 / 2

Mechanism behind MC and IMC

• two competing mechanisms at finite B[Bruckmann,Endrodi,Kovacs ’13]

I direct (valence) effect B ↔ qfI indirect (sea) effect B ↔ qf ↔ g⟨

ψψ(B)⟩∝∫DAµ e−Sg det( /D(B,A) + m)︸ ︷︷ ︸

sea

Tr[( /D(B,A) + m)−1

]︸ ︷︷ ︸

valence

2 / 2