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
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:
⇒
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
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 ...
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
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
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
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