landau levels in qcd in an external magnetic field · landau levels in qcd in an external magnetic...

Landau Levels in QCD in an External Magnetic Field

Matteo Giordano

Eotvos Lorand University (ELTE)Budapest

ELTE particle physics seminarBudapest, 21/09/2016

Based on work in progress with

F. Bruckmann, G. Endrodi, S. D. Katz, T. G. Kovacs, F. Pittler, and J. Wellnhofer

feat. P. McCartney, J. Page, and S. Wonder

Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/2016 1 / 26

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 whichaffects the QGP

Au+Au, b = 10 fm,√s = 200 GeV

eB ∼ e2

4π

2ZAuv√1− v2

(2

b

)2

≈ 0.1 GeV2

[review: Huang (2016)]

neutron stars: EOS of hadronic matter/QGP in magnetic field neededto determine the mass/radius ratio for magnetars (strong B field, lowrotation frequency)evolution of the early universe: strength of B can affect phasetransitions in the early universe, baryogenesis


Phase Diagram from the Lattice

Zero chemical potential (no sign problem), Tc(B = 0, µ = 0) = 155 MeV

T ≤ 130 MeV, T & 190 MeVmagnetic catalysis:〈ψψ〉 increases with B

130 MeV < T < 190 MeVinverse magnetic catalysis:〈ψψ〉 decreases with B

Phase diagram:from light-quark condensate (red)

or strange quark susceptibility (blue)

Tc decreases with B

[Bali, Bruckmann, Endrodi et al. (2012a),

Bali, Bruckmann, Endrodi et al. (2012b)]


Valence and Sea Effects

In a magnetic field B-dependent Dirac operator /D(A,B)

〈ψψ〉 =1

Z (B)

∫DA det( /D(A,B) + m)e−Sg [A] Tr

1

/D(A,B) + m

Z (B) =

∫DA det( /D(A,B) + m)e−Sg [A]

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 densityof low modes ⇒ inverse catalysis

Sea effect wins over valence near Tc , where it is most effective⇒ inverse magnetic catalysis

[Bruckmann, Endrodi, Kovacs (2013)]


Low-Energy Models

Analytic calculation in low-energy models for QCD: χPT, (P)NJL, bagmodel. . .

[Bali, Bruckmann, Endrodi et al. (2012b)]

PNJL ∼ OK at T = 0 foreB < 0.3 GeV2

χPT ∼ OK for T < 100 MeV,eB < 0.1 GeV2

General prediction: magneticcatalysis at all temperatures,Tc(B) increases

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, 21/09/2016 5 / 26

Landau Levels: Quantum Mechanics

Spinless particle, minimal coupling, uniform magnetic field in the zdirection: ~A = (0,Bx , 0)

H =1

2m

[p2x + (py − qBx)2 + p2

z

]

Since [py ,H] = [pz ,H] = 0

H =1

2m

[p2x + (ky − qBx)2 + k2

z

]=

harmonic oscillator︷︸︸︷p2x

2m+

m

2ω2c (x − x0)2 +

k2z

2m

E= ωc

(n +

1

2

)+

k2z

2m, ωc =

|qB|m

, x0 =kyqB

Degeneracy: finite system with PBC, ky = 2πkLy

, k ∈ Z and center of force

x0 = 2πkqBLy

< Lx within the system ⇒ 0 ≤ k < NB =qBLxLy

2π = ΦB2π


Landau Levels and Particle Spectrum

Energy levels of an otherwise free particle in a uniform magnetic field B(minimal coupling + magnetic moment, Minkowski, relativistic)

E 2n,sz (pz) = p2

z + qB(2n − 2sz + 1) + m2

Turn on strong interactions: LLs work for weakly coupled particles

[Bali, Bruckmann, Endrodi et al. (2012a)]

m2π±(B) = E 2

n,0(0) = m2π±(0) + eB

What about quarks?(electrically charged

& strongly coupled)


Landau Levels for Free Fermions: 2d Continuum

Dirac Equation (Euclidean), 2d: ( /D + m)ψ = (iλ+ m)ψ

/D = γµDµ, Dµ = ∂µ + iqaµ , aµ = (0,Bx)

Eigenvalues of /D2:

−λ2 = |qB| [2n + 1 + 2szsgn(qB)]︸︷︷︸k=0,1,...

n = 0, 1, . . . , sz = ±12

Degeneracy of λ0, ±λk for k 6= 0 = NB

NB =LxLyqB

2π=

ΦB

2π(quantised in a finite volume with PBC)

λ2n,sz is NB degenerate;

for (n, sz ) = (0,−1), λ20,−1 = 0 → NB zero modes;

for (n, sz ) 6= (0,−1), from 2NB identical modes λ2n,1, λ

2n+1,−1 → ±|λn.sz |, each NB degenerate


Landau Levels for Free Fermions: 4d continuum

Dirac Equation (Euclidean), 4d: ( /D + m)ψ = (iλ+ m)ψ

/D = γµDµ, Dµ = ∂µ + iqaµ , aµ = (0,Bx , 0, 0)

Separation of variables, ψ(x , y , z , t) = φ(x , y)e ipzze ipt t

−λ2 = |qB| [2n + 1 + 2szsgn(qB)]︸︷︷︸k=0,1,...

+p2z + p2

t n = 0, 1, . . . , sz = ±12

Degeneracy of λ0, ±λk for k 6= 0 = 2NB (factor of 2 from particle/antiparticle)

NB =LxLyqB

2π=

ΦB

2π(quantised in a finite volume with PBC)

z-momentum: pz = 2πLzkz

t-momentum: at finite temperature T , ABC in the t direction for fermions

pt =π

Lt(2kt + 1) = πT (2kt + 1) (Matsubara frequencies)


Magnetic Catalysis

Simple explanation:

1 after switching on strong interactions spectrum still approximatelyorganised in Landau levels

2 LLL is λ = 0 with degeneracy ∝ B ⇒ B increases the density ρ(0) oflow Dirac modes

3 Chiral condensate 〈ψψ〉 ∝ ρ(0) (Banks-Casher relation), increases dueto increased degeneracy ⇒ magnetic catalysis (valence effect)For nonzero quark masses 〈ψψ〉 ≈ ρ(0)

4 Would lead to Tc(B) > Tc(0), but not clear if this works in thecritical region (sea effects are important there)

. . . is this true? Check on the lattice and curse the day you were born


Lattice Gauge Theory (at Breakneck Speed)

Lattice: replace continuum spacetime with a discrete box

Z =

∫DU

∫Dψ

∫Dψ e−Sg [U]−ψ( /D+m)ψ =

∫DU det( /D + m)e−Sg [U]

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 → Ds

(Ds)xy =1

2

∑

µ

ηµ(x) [Uµ(x)δx+µ y − U−µ(x)δx−µ y ] , ηµ = (−1)∑ν<µ xν

Disadvantages: doublers reduced but not removed, requires rooting trick

Advantages:

Ds anti-Hermitean, purely imaginary spectrum, symmetric wrt origin

numerically cheap

the single-component fermion field can be treated as 2d or 4d if we soplease (and we do)


Landau Levels for Free Fermions: 2d Lattice

Finite lattice with periodic boundary conditions

Magnetic field along z : Uy (x) = e ia2qBx ∈ U(1), Ux = 1

(with exceptions at the lattice boundary)

B-field quantised: NB =LxLyqB

2π=

ΦB

2π∈ Z⇒ qB =

2π

LxLyNB

Rational ratios of charges ⇒ qi = niq, quantised qB

2d free spectrumLL structure spoiled by finitespacing artefacts

Gaps ∼ remnant of the LLstructure

Fractal structure: Hofstadterbutterfly [Hofstadter (1976)]

Colour code: eigenvalues grouped

according to LL degeneracy


Landau Levels for Interacting Fermions: 2d Lattice

Add SU(3) interaction: Uµ(x) = uµ(x)Wµ(x)

uy (x) = e ia2qBx ∈ U(1), ux = 1 (up and down quark q = 2

3,− 1

3)

Wµ(x) ∈ SU(3) integrated over with Haar measure

2d configuration = 2d slice of a 4d configuration

2d interacting spectrumHofstadter butterfly washedaway, gaps disappear. . .

. . . but “lowest Landaulevel” (LLL) survives, widegap: why?

Colour code: eigenvalues groupedaccording to LL degeneracy

(6NB : staggered doubling × 3 colours)


Gap in the Continuum Limit

The gap even survives the continuum limit

10

100

0 5 10 15 20

λ/m

flux quantum (prop. to B)

a=0.082 fm0.062 fm0.050 fm0.041 fm

Continuum limit:

Fixed B and LxLy = (aNs)2 ⇒ flux NB is RG invariant

Spectrum renormalises the quark mass ⇒ λmud

is RG invariant


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


Spin

σxy : z-component of spin (it is not just σz for staggered fermions)

Matrix elements of σxy for 2d eigenstates φi ,j

(σxy )ij = 〈φi |σxy |φj〉

Almost diagonal, appreciably different from zero only in the LLL

In 2d chirality = 2sz : near zero modes with definite chirality?I have heard of that already but I was too busy not listening


Topological Origin of 2d LLL

LLL has topological origin, “robust” under small deformations

Atiyah-Singer index theorem: Qtop = 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

Qtop =q

4π

∫d2x εµνFµν =

q

2πLxLyB = NB

Switching on SU(3) fields changes nothing: same definition of Qtopπ1(SU(3)) trivial

LLL (= zero modes) survives; other LLs are mixed by the SU(3) interaction

Staggered = 2 species, 3 colours: 6NB almost-zero modes (splitting dueto finite-spacing artefacts), almost-definite spin


Can One See Landau Levels from the Spectrum in 4d?

No: in 4d the p2z + p2

t contribution makes it impossible to identify LL fromthe spectrum even in the free continuum case (people softly crying in the background)

But: factorisation of the eigenmodes

0

0.5

1

1.5

2

2.5

3

0 100 200 300 400 500 600

eige

nval

ue

mode number

012

ψ(j)k,pz ,pt

(x , y , z , t) = φ(j)k (x , y)e ipzze ipt t

− /D22dφ

(j)k (x , y) = |qB|kφ(j)

k (x , y)

j = 1, . . . ,NB , k = 2[n + szsgn(qB)] + 1

To extract LL from 4d eigenmode ψ: diagonalise − /D22d and project on one

of its eigenvectors localised on a z , t slice, φ(j)k,z0t0

(x , y , z , t)

|qB|k = 〈ψ| − /D22d |φ(j)

k,z0t0〉〈φ(j)

k,z0t0|ψ〉

φ(j)k,z0t0

(x , y , z , t) = φ(j)k (x , y)δz z0δt t0


Overlap with LLL

On the lattice Landau level structure washed away already in 2dStill, LLL survives in 2d: how to check its physical effects?

Idea: determine the 2d eigenmodes on each slice, identify the LLL on eachslice, 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|φj ,tz〉〈φj ,tz |ψ〉

φj,zt : jth 2d mode>0 of Ds restricted to zt slice

(all 2d modes, all slices form complete 4d basis)

Jump at j = 3NB , also in thecontinuum: LLL is different

True for low 4d modes, disappears higher up in

the spectrum, but LLL still different


Two-Dimensional Basis


LLL Projector

To extract LLL: project on 2d LLL on all slices

Π0(B) =∑

tz slices

∑

1≤j≤3Nb

|φj ,tz(B)〉〈φj ,tz(B)|+ |φj ,tz(B)〉〈φj ,tz(B)|

φ: positive 2d modes, φ: negative 2d modes

Π0 allows to define LLL-restricted observables:

OΓ = ψΓψ → OLLLΓ = ψΠ0ΓΠ0ψ

〈OLLLΓ 〉ψ,ψ,A = 〈Tr ΓΠ0M

−1Π0〉A M = Ds + m

Only valence effect considered, no sea effects which require modificationsof the fermion determinant in the partition function(very interesting, but considerably harder)


LLL Condensate I

Quark condensate: Γ = 1

full condensate: 〈ψψ〉B = 〈TrM−1〉Bprojected condensate: 〈ψψLLL〉B = 〈TrΠ0M

−1Π0〉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) = mud [〈ψψ〉B − 〈ψψ〉0]


LLL Condensate II

Projected condensate: we cannot subtract 〈ψψLLL〉0 = 0

Define the projector

Π0(B) =∑

tz slices

∑

1≤j≤3NB

|φj ,tz(0)〉〈φj ,tz(0)|+ |φj ,tz(0)〉〈φj ,tz(0)|

∆〈ψψLLL〉(B) = 〈ψΠ0(B)ψ〉B − 〈ψΠ0(B)ψ〉0Measures the variation due to the change of the first 3NB 2d modes on each slice caused by B

Should cure additive divergenceAdditive 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

∆〈ψψLLL〉(B)

∆〈ψψ〉(B)=〈ψΠ0(B)ψ〉B − 〈ψΠ0(B)ψ〉0

〈ψψ〉B − 〈ψψ〉0Matteo Giordano (ELTE) Landau Levels in QCD Budapest, 21/09/2016 23 / 26

Two-Dimensional Basis (Reprise)


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 = 13

Below Tc

nt = 6

nt = 8

nt = 10

nt = 120.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2

eB = (πT )2

∆〈ψψ(B

)〉 LLL

∆〈ψψ(B

)〉

eB[GeV2

]

T = 123MeV

(larger nt = finer lattice)

Finite continuum limit(it seems)

Ratio increases with B

Ratio decreases with T

Reaches ∼ 0.5 around1÷ 1.5 GeV2, large field






Around Tc

nt = 6

nt = 8

nt = 10

nt = 120.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0 0.5 1 1.5 2

eB = (πT )2

∆〈ψψ(B

)〉 LLL

∆〈ψψ(B

)〉

eB[GeV2

]

T = 170MeV











Above Tc

nt = 6

nt = 8

nt = 10

nt = 120.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.5 1 1.5 2

eB = (πT )2

∆〈ψψ(B

)〉 LLL

∆〈ψψ(B

)〉

eB[GeV2

]

T = 189MeV











Above Tc

nt = 6

nt = 8

nt = 10

nt = 12-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2

eB = (πT )2∆〈ψψ(B

)〉 LLL

∆〈ψψ(B

)〉

eB[GeV2

]

T = 214MeV







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 ofthe 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?)


References

I J. O. Andersen, W. R. Naylor and A. Tranberg, Rev. Mod. Phys. 88 (2016)025001

I X. G. Huang, Rept. Prog. Phys. 79 (2016) 076302

I G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg,A. Schafer and K. K. Szabo, JHEP 02 (2012) 044

I G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz and A. Schafer,Phys. Rev. D 86 (2012) 071502

I F. Bruckmann, G. Endrodi and T. G. Kovacs, JHEP 04 (2013) 112

I D. R. Hofstadter, Phys. Rev. B 14 (1976) 2239

I 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

I F. Zappa, Waka/Jawaka (1972); The Grand Wazoo (1972)


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