Symmetries in the Gross-Neveu Phase Diagram

Gerald Dunne

University of Connecticut

crystalline phases of GN models: gap equation and integrable hierarchies

energy-reflection symmetry of periodic QES systems

G.Başar &GD, arxiv:0803.1501, PRL 100, 200404 (2008) arxiv:0806.2659, PRD 78, 065022 (2008) G.Başar, GD & M.Thies, arxiv: 0903.1868, PRD in pressF.Correa, GD & M.Plyushchay, arxiv: 0904.2768

GD & M.Shifman, hep-th/0204224, Ann. Phys. 299, 143 (2002)

Gross-Neveu Models

LGN = ψ̄ i ∂/ψ +g2

2(ψ̄ψ

)2

LNJL = ψ̄ i ∂/ ψ +g2

2

[(ψ̄ψ

)2 +(ψ̄iγ5ψ

)2]

GN2

χGN2

NJL2

phase diagram?(T, µ)

Gross/Neveu, 1974

mB =2π

m DHN, 1975

ψ → γ5 ψ

ψ → eiα γ5ψ

• renormalizable; large Nf limit• asymptotically free• chiral symmetry breaking• dynamical mass generation• self-bound baryonic states

Wolff, 1985

uniform condensate

Phase diagram of Gross-Neveu model

lattice analysis of GN2

Karsch et al 1986

1√2≈ .71

2π≈ .64

Wolff, 1985

uniform condensate

Phase diagram of Gross-Neveu model

Thies & Urlichs, 2005

periodic, crystalline,

phase

lattice GN2 model

de Forcrand/Wenger 2006

0 0.4 0.8 1.2µ/!

0

0

0.2

0.4

0.6

T/!0

2π≈ .64

trans-polyacetylene = GN2 Su, Schreiffer, Heeger, 1979

dimerization = discrete chiral symmetry of GN model

polaron crystal Brazovskii, 1980; Horovitz, 1981

Condensed matter analogues

inhomogeneous superconductors and ferromagnetism

1 dim. Peierls-Fröhlich electron-phonon modelMertsching/Fischbeck, 1981; Belokolos et al, 1981

magnetic field = µ

Machida/Nakanishi, 1984

inhomogeneous gap equation : GN2

Thies/Urlichs (2005): finite-gap potentials: V± = Σ2 ± Σ′

Σ(x) = m νsn(m x; ν) cn(m x; ν)

dn(m x; ν)kink crystal:

Σ(x)g2N

=δ

δΣ(x)ln det [∂/ + Σ(x)]

V± = Σ2 ± Σ′DHN(1975): inverse scattering

reflectionless potentials:

Σ(x) = m tanh(m x)single kink:

⇒

ν = 0.999

ν = 0.01

kink crystal Σ(x) = m νsn(m x; ν) cn(m x; ν)

dn(m x; ν)

complex gap equation : NJL2

Shei (1976): inv. scattering: reflectionless Dirac system

∆(x) = mcosh

(m sin( θ

2 ) x− i θ2

)

cosh(m sin( θ

2 ) x) twisted kink

GD & Basar (2008): finite-gap Dirac system

∆(x) = Aσ

(A x + iK′ − i θ

2

)

σ (A x + iK′) σ(i θ2

) eiQx twisted kink crystal

∆ =Σ − iΠ

∆(x)g2N

=δ

δ∆∗(x)ln det

[∂/ + (Σ(x)− iγ5 Π(x))

]

2ϕ

twisted kink crystal: general solution of NJL2 gap equation

real kink crystal

spiral crystal

single-particle spectra

∆(x) = Aσ

(A x + iK′ − i θ

2

)

σ (A x + iK′) σ(i θ2

) eiQx

solving the (complex) gap equation

=(−i d

dx ∆(x)∆∗(x) i d

dx

)

Bogoliubov/de Gennes hamiltonian

resolvent : Gorkov Green’s function R(x;E) ≡ 〈x| 1H − E

|x〉

ρ(E) =1π

Im∫

dx trR(x;E + iε)spectral function

∆(x)g2N

=δ

δ∆∗(x)ln det

[∂/ + (Σ(x)− iγ5 Π(x))

]

H = −iγ5 d

dx+ γ0Σ(x) + iγ1Π(x)

∂

∂xR(x;E)σ3 = i

[(E −∆(x)

∆∗(x) −E

), R(x;E)σ3

]Eilenberger eqn:

solving the (complex) gap equation

=(−i d

dx ∆(x)∆∗(x) i d

dx

)

ρ(E) =1π

Im∫

dx trR(x;E + iε)spectral function

∆(x)g2N

=δ

δ∆∗(x)ln det

[∂/ + (Σ(x)− iγ5 Π(x))

]

H = −iγ5 d

dx+ γ0Σ(x) + iγ1Π(x)

∆(x) = −N g2 TrD,E

[γ0(1 + γ5) R(x;E)

]

ln det[...] = − 1β

∫dE ρ(E) ln

(1 + e−β(E−µ)

)two views of gap equation:

gap equation NLSE

R(x;E) = N (E)

a(E) + |∆(x)|2 b(E)∆(x)− i∆′(x)

b(E)∆∗(x) + i∆′ ∗(x) a(E) + |∆(x)|2

ansatz, from gap equation

∆′′ − 2|∆|2 ∆ + i (b− 2E) ∆′ − 2 (a− Eb) ∆ = 0

Eilenberger equation

NLSE : exactly soluble; also for exact spectral function

∆(x)g2N

=δ

δ∆∗(x)ln det

[∂/ + (Σ(x)− iγ5 Π(x))

]

small gap

medium gap

large gap

all gap no gap

0

0.2

0.4

0.2 0.4 0.8 1 1.2 1.4 1.6

LOFF kink

kink crystal

crossing the boundaries ...

something completely different ...

Duality and energy-reflection symmetry

Quasi-exactly-soluble QM models M. Shifman, ITEP lectures

portion of the spectrum known algebraically

periodic QES systems : energy-reflection duality

H=polynomial in sl(2,R) generators

energy-reflection symmetry M. Shifman & A. Turbiner, 1998

GD & M. Shifman, 2002

ν = 0.1 ν = 0.9

dual potentialsstrong coupling <--> weak coupling

V (x) = J(J + 1)ν sn2(x; ν)− 12J(J + 1)Lamé potential:

H = J2x + ν J2

y −12J(J + 1)1

QES: J bands; edges determined algebraically

ν

E

1− ν

−E

V (x) = J(J + 1)ν sn2(x; ν)− 12J(J + 1)

∆Etop ∼8J Γ(J + 1/2)

4J√

πΓ(J)(ν)Jpert. th:

Etop ∼J(J + 1)

2

(1− 2

√1− ν√

J(J + 1)+

2− ν

2J(J + 1)+ . . .

)

WKB :

Ebottom ∼ −J(J + 1)

2

(1− 2

√ν√

J(J + 1)+

1 + ν

2J(J + 1)+ . . .

)

pert. th:

∆Ebottom ∼8J Γ(J + 1/2)

4J√

πΓ(J)(1− ν)Jinstanton :

... E[ν]↔ −E[1− ν]perturbative/nonperturbative duality

back to Gross-Neveu ...

ν ↔ 1− ν

dense↔ dilute

V± = Σ2 ± Σ′

= −m2 ν + 2m2 ν

{sn2(m (x + K/2); ν)sn2(m x; ν)

Σ(x) = m νsn(m x; ν) cn(m x; ν)

dn(m x; ν)

Energy-reflection symmetry in the GN2 phase diagram

kink crystal condensate:

0

0.2

0.4

0.2 0.4 0.8 1 1.2 1.4 1.6

LOFF kink

kink crystal

ν = 0

ν = 1

GN phase diagram: dense <--> dilute duality

massive

massless

crystal

GL crystal

Ginzburg-Landau expansion

LGL = c0 + c2|∆|2 + c3Im [∆(∆′)∗] + c4

[|∆|4 + |∆′|2

]

+c5Im[(

∆′′ − 3|∆|2∆)(∆′)∗

]

+c6

[2|∆|6 + 8|∆|2|∆′|2 + 2Re

((∆′)2(∆∗)2

)+ |∆′′|2

]+ . . .

gap equation: all-orders Ginzburg-Landau expansion

[an(x)]NLSE = αn|∆(x)|2 + βn

NLSE entire hierarchy satisfied

=∑

n

cn(T, µ) an(x)

GN2: mKdV hierarchy

NJL2: AKNS hierarchy

Conclusions

• there is a lot of symmetry in the GN phase diagram

• integrable hierachies: GN2 = mKdV ; NJL2 = AKNS

• thermodynamics: crystalline phases

• energy/reflection symmetry = dense/dilute duality

Happy Birthday Misha!

Basar,GD,Thies, 2009:

Schön/Thies, 2000

“chiral spiral”

twisted kink crystal --> chiral spiral

NJL2 phase diagram

∆ = A e2iµ x