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Emergence of anyons and ground-state propertiesof the anyon gas
Douglas LundholmKTH Stockholm
based on work in collaborations withMichele Correggi, Romain Duboscq, Simon Larson,
Viktor Qvarfordt, Nicolas Rougerie, Robert Seiringer,Jan Philip Solovej
Oslo, January 2018
Emergence of anyons Lundholm Slide 1/37
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Outline of Talk
1 Introduce 2D anyons — ideal or extended
2 Emergence of anyons in physics
3 The ideal anyon gas
4 Discussion
Emergence of anyons Lundholm Slide 2/37
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Identical particles and statistics in 2D
Particle exchange in 2D: Ψ: (R2)N → C
Ψ(x1, . . . ,xj , . . . ,xk, . . . ,xN ) ∈ C
eiαπ ∈ U(1) any phase
α = 0: bosons
α = 1: fermions
xj xk
anyons: ‘fractional’-statistics quasiparticles in confined systems— expected to arise e.g. in fractional quantum Hall systems
∼1970 Souriau; Streater & Wilde . . . Leinaas & Myrheim ’77; Goldin, Menikoff & Sharp ’81; Wilczek ’82 . . .
Reviews by Frohlich ’90, Wilczek ’90, Lerda ’92, Myrheim ’99, Khare ’05, Ouvry ’07, Stern ’08, Hansson et al ’17...
Past rigorous QM studies by Baker, Canright & Mulay ’93, Dell’Antonio, Figari & Teta ’97
Emergence of anyons Lundholm Slide 3/37
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Identical particles and statistics in 2D
Particle exchange in 2D: Ψ: (R2)N → C
|Ψ(x1, . . . ,xj , . . . ,xk, . . . ,xN )|2 = |Ψ(x1, . . . ,xk, . . . ,xj , . . . ,xN )|2
eiαπ ∈ U(1) any phase
α = 0: bosons
α = 1: fermions
xj xk
anyons: ‘fractional’-statistics quasiparticles in confined systems— expected to arise e.g. in fractional quantum Hall systems
∼1970 Souriau; Streater & Wilde . . . Leinaas & Myrheim ’77; Goldin, Menikoff & Sharp ’81; Wilczek ’82 . . .
Reviews by Frohlich ’90, Wilczek ’90, Lerda ’92, Myrheim ’99, Khare ’05, Ouvry ’07, Stern ’08, Hansson et al ’17...
Past rigorous QM studies by Baker, Canright & Mulay ’93, Dell’Antonio, Figari & Teta ’97
Emergence of anyons Lundholm Slide 3/37
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Identical particles and statistics in 2D
Particle exchange in 2D: Ψ: (R2)N → C
Ψ(x1, . . . ,xj , . . . ,xk, . . . ,xN ) = eiθ Ψ(x1, . . . ,xk, . . . ,xj , . . . ,xN )
eiαπ ∈ U(1) any phase
α = 0: bosons
α = 1: fermions
xj xk
anyons: ‘fractional’-statistics quasiparticles in confined systems— expected to arise e.g. in fractional quantum Hall systems
∼1970 Souriau; Streater & Wilde . . . Leinaas & Myrheim ’77; Goldin, Menikoff & Sharp ’81; Wilczek ’82 . . .
Reviews by Frohlich ’90, Wilczek ’90, Lerda ’92, Myrheim ’99, Khare ’05, Ouvry ’07, Stern ’08, Hansson et al ’17...
Past rigorous QM studies by Baker, Canright & Mulay ’93, Dell’Antonio, Figari & Teta ’97
Emergence of anyons Lundholm Slide 3/37
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Identical particles and statistics in 2D
Particle exchange in 2D: Ψ: (R2)N → C
Ψ(x1, . . . ,xj , . . . ,xk, . . . ,xN ) = ±Ψ(x1, . . . ,xk, . . . ,xj , . . . ,xN )
eiαπ ∈ U(1) any phase
α = 0: bosons
α = 1: fermions
xj xk
anyons: ‘fractional’-statistics quasiparticles in confined systems— expected to arise e.g. in fractional quantum Hall systems
∼1970 Souriau; Streater & Wilde . . . Leinaas & Myrheim ’77; Goldin, Menikoff & Sharp ’81; Wilczek ’82 . . .
Reviews by Frohlich ’90, Wilczek ’90, Lerda ’92, Myrheim ’99, Khare ’05, Ouvry ’07, Stern ’08, Hansson et al ’17...
Past rigorous QM studies by Baker, Canright & Mulay ’93, Dell’Antonio, Figari & Teta ’97
Emergence of anyons Lundholm Slide 3/37
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Identical particles and statistics in 2D
Particle exchange in 2D: Ψ: (R2)N → C
Ψ(x1, . . . ,xj , . . . ,xk, . . . ,xN ) = eiαπΨ(x1, . . . ,xk, . . . ,xj , . . . ,xN )
eiαπ ∈ U(1) any phase
α = 0: bosons
α = 1: fermionsxj xk
anyons: ‘fractional’-statistics quasiparticles in confined systems— expected to arise e.g. in fractional quantum Hall systems
∼1970 Souriau; Streater & Wilde . . . Leinaas & Myrheim ’77; Goldin, Menikoff & Sharp ’81; Wilczek ’82 . . .
Reviews by Frohlich ’90, Wilczek ’90, Lerda ’92, Myrheim ’99, Khare ’05, Ouvry ’07, Stern ’08, Hansson et al ’17...
Past rigorous QM studies by Baker, Canright & Mulay ’93, Dell’Antonio, Figari & Teta ’97
Emergence of anyons Lundholm Slide 3/37
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Identical particles and statistics in 2D
Particle exchange in 2D: Ψ: (R2)N → C
Ψ(x1, . . . ,xj , . . . ,xk, . . . ,xN ) = eiαπΨ(x1, . . . ,xk, . . . ,xj , . . . ,xN )
eiαπ ∈ U(1) any phase
α = 0: bosons
α = 1: fermionsxj xk
anyons: ‘fractional’-statistics quasiparticles in confined systems— expected to arise e.g. in fractional quantum Hall systems
∼1970 Souriau; Streater & Wilde . . . Leinaas & Myrheim ’77; Goldin, Menikoff & Sharp ’81; Wilczek ’82 . . .
Reviews by Frohlich ’90, Wilczek ’90, Lerda ’92, Myrheim ’99, Khare ’05, Ouvry ’07, Stern ’08, Hansson et al ’17...
Past rigorous QM studies by Baker, Canright & Mulay ’93, Dell’Antonio, Figari & Teta ’97
Emergence of anyons Lundholm Slide 3/37
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Modelling anyons concretely — anyon gauge
ei2pαπ ei(2p+1)απ
p p
Think: free kinetic energy T0 = ~22m
∑Nj=1(−i∇j)2 acting on multi-valued
Ψα := Uα Ψ0, U :=∏j<k
eiφjk =∏j<k
zj − zk|zj − zk|
.
Emergence of anyons Lundholm Slide 4/37
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Modelling anyons concretely — anyon gauge
ei2pαπ ei(2p+1)απ
p p
Think: free kinetic energy T0 = ~22m
∑Nj=1(−i∇j)2 acting on multi-valued
Ψα := Uα Ψ0, U :=∏j<k
eiφjk =∏j<k
zj − zk|zj − zk|
.
Emergence of anyons Lundholm Slide 4/37
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Modelling anyons concretely — magnetic gauge
Bosons (Ψ∈L2sym) in R2 with Aharonov-Bohm magnetic interactions:
Tα :=~2
2m
N∑j=1
D2j , Dj = −i∇j+αAj(xj), Aj(x) =
∑k 6=j
(x− xk)⊥
|x− xk|2
These are ideal anyons. One can also model R-extended anyons:
ARj (x) :=
∑k 6=j
(x− xk)⊥
|x− xk|2R, |x|R := max{|x|, R}
⇒ curlαARj = 2πα
∑k 6=j
1BR(xk)
πR2
R→0−−−→ 2πα∑k 6=j
δxk
We would like to understand the N -anyon ground state Ψ0 and energy
E0(N) := inf spec HN , HN = Tα + V =
N∑j=1
(~2
2mD2j + V (xj)
)
Emergence of anyons Lundholm Slide 5/37
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Modelling anyons concretely — magnetic gauge
Bosons (Ψ∈L2sym) in R2 with Aharonov-Bohm magnetic interactions:
Tα :=~2
2m
N∑j=1
D2j , Dj = −i∇j+αAj(xj), Aj(x) =
∑k 6=j
(x− xk)⊥
|x− xk|2
These are ideal anyons. One can also model R-extended anyons:
ARj (x) :=
∑k 6=j
(x− xk)⊥
|x− xk|2R, |x|R := max{|x|, R}
⇒ curlαARj = 2πα
∑k 6=j
1BR(xk)
πR2
R→0−−−→ 2πα∑k 6=j
δxk
We would like to understand the N -anyon ground state Ψ0 and energy
E0(N) := inf spec HN , HN = Tα + V =
N∑j=1
(~2
2mD2j + V (xj)
)
Emergence of anyons Lundholm Slide 5/37
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Modelling anyons concretely — magnetic gauge
Bosons (Ψ∈L2sym) in R2 with Aharonov-Bohm magnetic interactions:
Tα :=~2
2m
N∑j=1
D2j , Dj = −i∇j+αAj(xj), Aj(x) =
∑k 6=j
(x− xk)⊥
|x− xk|2
These are ideal anyons. One can also model R-extended anyons:
ARj (x) :=
∑k 6=j
(x− xk)⊥
|x− xk|2R, |x|R := max{|x|, R}
⇒ curlαARj = 2πα
∑k 6=j
1BR(xk)
πR2
R→0−−−→ 2πα∑k 6=j
δxk
We would like to understand the N -anyon ground state Ψ0 and energy
E0(N) := inf spec HN , HN = Tα + V =
N∑j=1
(~2
2mD2j + V (xj)
)Emergence of anyons Lundholm Slide 5/37
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How to create an anyon in the lab?
• Need several particles!
• Need 2D!
Emergence of anyons Lundholm Slide 6/37
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How to create an anyon in the lab?
• Need several particles!
• Need 2D!
Emergence of anyons Lundholm Slide 6/37
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How to create anyons in the lab?
DL, Rougerie, Phys. Rev. Lett., 2016 — avoids usual Berry phase argument of Arovas, Schrieffer, Wilczek, 1984cf. e.g. Forte, 1991
zj
,wk
∈ C
←→ strong pot.
B
′
• Two species of particles in a plane (bosons or fermions)
• Strong perpendicular magnetic field B ⇒ LLL
• Strong repulsion between particles ⇒ Laughlin state
⇒ Effective Hamiltonian with a reduced magnetic field andα = α0 + 1/n
Emergence of anyons Lundholm Slide 7/37
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How to create anyons in the lab?
DL, Rougerie, Phys. Rev. Lett., 2016 — avoids usual Berry phase argument of Arovas, Schrieffer, Wilczek, 1984cf. e.g. Forte, 1991
zj ,wk∈ C
←→ strong pot.
B
′
• Two species of particles in a plane (bosons or fermions)
• Strong perpendicular magnetic field B ⇒ LLL
• Strong repulsion between particles ⇒ Laughlin state
⇒ Effective Hamiltonian with a reduced magnetic field andα = α0 + 1/n
Emergence of anyons Lundholm Slide 7/37
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How to create anyons in the lab?
DL, Rougerie, Phys. Rev. Lett., 2016 — avoids usual Berry phase argument of Arovas, Schrieffer, Wilczek, 1984cf. e.g. Forte, 1991
zj ,wk∈ C
←→ strong pot.
B
′
• Two species of particles in a plane (bosons or fermions)
• Strong perpendicular magnetic field B ⇒ LLL
• Strong repulsion between particles ⇒ Laughlin state
⇒ Effective Hamiltonian with a reduced magnetic field andα = α0 + 1/n
Emergence of anyons Lundholm Slide 7/37
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How to create anyons in the lab?
DL, Rougerie, Phys. Rev. Lett., 2016 — avoids usual Berry phase argument of Arovas, Schrieffer, Wilczek, 1984cf. e.g. Forte, 1991
zj ,wk∈ C
←→ strong pot.
B
′
• Two species of particles in a plane (bosons or fermions)
• Strong perpendicular magnetic field B ⇒ LLL
• Strong repulsion between particles ⇒ Laughlin state
⇒ Effective Hamiltonian with a reduced magnetic field andα = α0 + 1/n
Emergence of anyons Lundholm Slide 7/37
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How to create anyons in the lab?
DL, Rougerie, Phys. Rev. Lett., 2016 — avoids usual Berry phase argument of Arovas, Schrieffer, Wilczek, 1984cf. e.g. Forte, 1991
zj
,wk
∈ C
←→ strong pot.
B′
• Two species of particles in a plane (bosons or fermions)
• Strong perpendicular magnetic field B ⇒ LLL
• Strong repulsion between particles ⇒ Laughlin state
⇒ Effective Hamiltonian with a reduced magnetic field andα = α0 + 1/n
Emergence of anyons Lundholm Slide 7/37
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How to create anyons in the lab?
DL, Rougerie, Phys. Rev. Lett., 2016 — avoids usual Berry phase argument of Arovas, Schrieffer, Wilczek, 1984cf. e.g. Forte, 1991
zj
,wk
∈ C
←→ strong pot.
B′
• Two species of particles in a plane (bosons or fermions)
• Strong perpendicular magnetic field B ⇒ LLL
• Strong repulsion between particles ⇒ Laughlin state
⇒ Effective Hamiltonian with a reduced magnetic field andα = α0 + 1/n
Emergence of anyons Lundholm Slide 7/37
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How to create anyons in the lab?
DL, Rougerie, Phys. Rev. Lett., 2016 — avoids usual Berry phase argument of Arovas, Schrieffer, Wilczek, 1984
Take two different species of quantum particles in a strongmagnetic field B > 0: ‘tracer’ particles at xj=1...M ∈ R2
in a large sea of ‘bath’ particles at yk=1...N ∈ R2, N �M .
HM+N = L2sym(R2M )⊗ L2
sym(R2N )
HM+N = HM ⊗ 1 + 1⊗HN +
M∑j=1
N∑k=1
W12(xj − yk),
HM =
M∑j=1
1
2m
(pxj + eA(xj)
)2+
∑1≤i<j≤M
W11(xi − xj),
HN =
N∑k=1
1
2(pyk + A(yk))
2 +∑
1≤i<j≤NW22(yi − yj)
Emergence of anyons Lundholm Slide 8/37
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How to create anyons in the lab?
DL, Rougerie, Phys. Rev. Lett., 2016 — avoids usual Berry phase argument of Arovas, Schrieffer, Wilczek, 1984
Take two different species of quantum particles in a strongmagnetic field B > 0: ‘tracer’ particles at xj=1...M ∈ R2
in a large sea of ‘bath’ particles at yk=1...N ∈ R2, N �M .
HM+N = L2sym(R2M )⊗ L2
sym(R2N )
HM+N = HM ⊗ 1 + 1⊗HN +
M∑j=1
N∑k=1
W12(xj − yk),
HM =
M∑j=1
1
2m
(−i∇xj +
eB
2x⊥j
)2
+∑
1≤i<j≤MW11(xi − xj),
HN =
N∑k=1
1
2
(−i∇yk +
B
2y⊥k
)2
+∑
1≤i<j≤NW22(yi − yj)
Emergence of anyons Lundholm Slide 8/37
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How to create anyons in the lab?
Ansatz: Ψ(X,Y) = Φ(X)cqh(X)Ψqh(X,Y),
with a Laughlin wave function coupled to quasi-holes at xj ≡ zj :
Ψqh(X,Y) =
M∏j=1
N∏k=1
(zj−wk)q∏
1≤i<j≤N(wi−wj)n e−B
∑Nj=1 |wj |2/4.
Claim: for N �M :
〈Ψ, HM+NΨ〉 ≈⟨
Φ, HeffM Φ
⟩+BN/2,
where
HeffM =
M∑j=1
1
2m
(pxj +
B
2(e− 1
n)x⊥j + αAR
j (xj)
)2
+∑
1≤i<j≤MW11(xi−xj)
is an effective Hamiltonian describing M anyons with α = 1/n,R =
√2/B.
Emergence of anyons Lundholm Slide 9/37
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How to create anyons in the lab?
Ansatz: Ψ(X,Y) = Φ(X)cqh(X)Ψqh(X,Y),
with a Laughlin wave function coupled to quasi-holes at xj ≡ zj :
Ψqh(X,Y) =
M∏j=1
N∏k=1
(zj−wk)q∏
1≤i<j≤N(wi−wj)n e−B
∑Nj=1 |wj |2/4.
Claim: for N �M :
〈Ψ, HM+NΨ〉 ≈⟨
Φ, HeffM Φ
⟩+BN/2,
where
HeffM =
M∑j=1
1
2m
(pxj +
B
2(e− 1
n)x⊥j + αAR
j (xj)
)2
+∑
1≤i<j≤MW11(xi−xj)
is an effective Hamiltonian describing M anyons with α = 1/n,R =
√2/B.
Emergence of anyons Lundholm Slide 9/37
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How to create anyons in the lab?
DL, Rougerie, Phys. Rev. Lett., 2016 — avoids usual Berry phase argument of Arovas, Schrieffer, Wilczek, 1984
zj ,wk∈ CB
′
• Two species of particles in a plane (bosons or fermions)
• Strong perpendicular magnetic field B ⇒ LLL
• Strong repulsion between particles ⇒ Laughlin state
Ψ(z,w) = Φ(z)c(z)∏j,k
(zj − wk)q∏i<k
(wi − wk)n e−B|w|2/4
⇒ Effective Hamiltonian for Φ with a reduced magnetic field andα = α0 + 1/n
Emergence of anyons Lundholm Slide 10/37
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How to create anyons in the lab?
DL, Rougerie, Phys. Rev. Lett., 2016 — avoids usual Berry phase argument of Arovas, Schrieffer, Wilczek, 1984
zj
,wk
∈ CB′
• Two species of particles in a plane (bosons or fermions)
• Strong perpendicular magnetic field B ⇒ LLL
• Strong repulsion between particles ⇒ Laughlin state
Ψ(z,w) = Φ(z)c(z)∏j,k
(zj − wk)q∏i<k
(wi − wk)n e−B|w|2/4
⇒ Effective Hamiltonian for Φ with a reduced magnetic field andα = α0 + 1/n
Emergence of anyons Lundholm Slide 10/37
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How to create anyons in the lab?
DL, Rougerie, Phys. Rev. Lett., 2016 — avoids usual Berry phase argument of Arovas, Schrieffer, Wilczek, 1984
zj
,wk
∈ C
B′
• Two species of particles in a plane (bosons or fermions)
• Strong perpendicular magnetic field B ⇒ LLL
• Strong repulsion between particles ⇒ Laughlin state
Ψ(z,w) = Φ(z)c(z)∏j,k
(zj − wk)q∏i<k
(wi − wk)n e−B|w|2/4
⇒ Effective Hamiltonian for Φ with a reduced magnetic field andα = α0 + 1/n
Emergence of anyons Lundholm Slide 10/37
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Understanding the zero-temperature ideal anyon gas
cold bosons cold fermions
anyons?
2-body: Leinaas, Myrheim, 1977; Wilczek, 1982; Arovas, Schrieffer, Wilczek, Zee, 19853- and 4-body numerics: Sporre, Verbaarschot, Zahed, 1991-92; Murthy, Law, Brack, Bhaduri, 1991Approximations: average-field theory, lowest Landau level, diluteHundreds of papers...
Emergence of anyons Lundholm Slide 11/37
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Compare with the ideal Bose gas in 2D
Know: Ψ0 = ⊗Nϕ0, ϕ0 lowest state of H1 = −∆R2 + V (x)
The free Bose gas in a box Q = [0, L]2:
H1 = (−∆Q)Dirichlet, ϕ0(x, y) = sin(πx/L) sin(πy/L),
E0(N,L) =〈Ψ0, HNΨ0〉‖Ψ0‖2
= Nλ0 = N2π2
L2= 2π2%
⇒ Energy per area:
E0(N,L)
L2=
2π2%
L2→ 0,
as N →∞ and L→∞ with fixed density % = N/L2.
Emergence of anyons Lundholm Slide 12/37
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Compare with the ideal Bose gas in 2D
Know: Ψ0 = ⊗Nϕ0, ϕ0 lowest state of H1 = −∆R2 + V (x)
The free Bose gas in a box Q = [0, L]2:
H1 = (−∆Q)Dirichlet, ϕ0(x, y) = sin(πx/L) sin(πy/L),
E0(N,L) =〈Ψ0, HNΨ0〉‖Ψ0‖2
= Nλ0 = N2π2
L2= 2π2%
⇒ Energy per area:
E0(N,L)
L2=
2π2%
L2→ 0,
as N →∞ and L→∞ with fixed density % = N/L2.
Emergence of anyons Lundholm Slide 12/37
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Compare with the ideal Bose gas in 2D
Know: Ψ0 = ⊗Nϕ0, ϕ0 lowest state of H1 = −∆R2 + V (x)
The free Bose gas in a box Q = [0, L]2:
H1 = (−∆Q)Dirichlet, ϕ0(x, y) = sin(πx/L) sin(πy/L),
E0(N,L) =〈Ψ0, HNΨ0〉‖Ψ0‖2
= Nλ0 = N2π2
L2= 2π2%
⇒ Energy per area:
E0(N,L)
L2=
2π2%
L2→ 0,
as N →∞ and L→∞ with fixed density % = N/L2.
Emergence of anyons Lundholm Slide 12/37
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Compare with the ideal Fermi gas in 2D
Know: Ψ0 =∧N−1k=0 ϕk, ϕk lowest states of H1 = −∆R2 + V (x)
The free Fermi gas in a box Q ⊂ R2: (Weyl asymptotics)
E0(N,L) =N−1∑k=0
λk ∼ 2πN2
L2= 2π%2 L2
⇒ Thomas–Fermi approximation: (Thomas, Fermi, 1927 — precursor to modern DFT)
〈Ψ0, (Tα=1 + V )Ψ0〉 ≈∫R2
(2π%Ψ0(x)2 + V (x)%Ψ0(x)
)dx
The Lieb–Thirring inequality: (Lieb, Thirring, 1975)
〈Ψ, (Tα=1 + V )Ψ〉 ≥∫R2
(CLT %Ψ(x)2 + V (x)%Ψ(x)
)dx
Emergence of anyons Lundholm Slide 13/37
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Compare with the ideal Fermi gas in 2D
Know: Ψ0 =∧N−1k=0 ϕk, ϕk lowest states of H1 = −∆R2 + V (x)
The free Fermi gas in a box Q ⊂ R2: (Weyl asymptotics)
E0(N,L) =
N−1∑k=0
λk ∼ 2πN2
L2= 2π%2 L2
⇒ Thomas–Fermi approximation: (Thomas, Fermi, 1927 — precursor to modern DFT)
〈Ψ0, (Tα=1 + V )Ψ0〉 ≈∫R2
(2π%Ψ0(x)2 + V (x)%Ψ0(x)
)dx
The Lieb–Thirring inequality: (Lieb, Thirring, 1975)
〈Ψ, (Tα=1 + V )Ψ〉 ≥∫R2
(CLT %Ψ(x)2 + V (x)%Ψ(x)
)dx
Emergence of anyons Lundholm Slide 13/37
Page 35
Compare with the ideal Fermi gas in 2D
Know: Ψ0 =∧N−1k=0 ϕk, ϕk lowest states of H1 = −∆R2 + V (x)
The free Fermi gas in a box Q ⊂ R2: (Weyl asymptotics)
E0(N,L) =
N−1∑k=0
λk ∼ 2πN2
L2= 2π%2 L2
⇒ Thomas–Fermi approximation: (Thomas, Fermi, 1927 — precursor to modern DFT)
〈Ψ0, (Tα=1 + V )Ψ0〉 ≈∫R2
(2π%Ψ0(x)2 + V (x)%Ψ0(x)
)dx
The Lieb–Thirring inequality: (Lieb, Thirring, 1975)
〈Ψ, (Tα=1 + V )Ψ〉 ≥∫R2
(CLT %Ψ(x)2 + V (x)%Ψ(x)
)dx
Emergence of anyons Lundholm Slide 13/37
Page 36
Compare with the ideal Fermi gas in 2D
Know: Ψ0 =∧N−1k=0 ϕk, ϕk lowest states of H1 = −∆R2 + V (x)
The free Fermi gas in a box Q ⊂ R2: (Weyl asymptotics)
E0(N,L) =
N−1∑k=0
λk ∼ 2πN2
L2= 2π%2 L2
⇒ Thomas–Fermi approximation: (Thomas, Fermi, 1927 — precursor to modern DFT)
〈Ψ0, (Tα=1 + V )Ψ0〉 ≈∫R2
(2π%Ψ0(x)2 + V (x)%Ψ0(x)
)dx
The Lieb–Thirring inequality: (Lieb, Thirring, 1975)
〈Ψ, (Tα=1 + V )Ψ〉 ≥∫R2
(CLT %Ψ(x)2 + V (x)%Ψ(x)
)dx
Emergence of anyons Lundholm Slide 13/37
Page 37
Compare with a relaxed Pauli principle
If ν particles allowed in each state: Ψ0 =⊗ν ∧N/ν ϕk,
The free Fermi gas in a box Q ⊂ R2: (Weyl asymptotics)
E0(N,L) =
N−1∑k=0
λk ∼ 2πν(N/ν)2
|Q|= 2πν−1%2 |Q|
⇒ Thomas–Fermi approximation: (Thomas, Fermi, 1927 — precursor to modern DFT)
〈Ψ0, (Tα=1 + V )Ψ0〉 ≈∫R2
(2πν−1%Ψ0(x)2 + V (x)%Ψ0(x)
)dx
The Lieb–Thirring inequality: (Lieb, Thirring, 1975)
〈Ψ, (Tα=1 + V )Ψ〉 ≥∫R2
(CLT ν
−1%Ψ(x)2 + V (x)%Ψ(x))dx
Emergence of anyons Lundholm Slide 14/37
Page 38
Average-field approximation
Huge past literature: see e.g. Wilczek 1990 review
For anyons one may consider an average-field approximation
〈Ψ0, (Tα + V )Ψ0〉 ≈∫R2
(2π|α|%Ψ0(x)2 + V (x)%Ψ0(x)
)dx,
where B = curlαAj ≈ 2πα% with LLL energy/particle ∼ |B|.
A particular almost-bosonic limit α = β/N → 0 leads to
Eaf [ψ] :=
∫R2
(∣∣(−i∇+ βA[|ψ|2])ψ(x)
∣∣2 + V (x)|ψ(x)|2)dx,
where curlA[|ψ|2] = 2π|ψ|2 and β the only parameter.DL, Rougerie, 2015; Correggi, DL, Rougerie, 2017; Chern-Simons coupled to non-rel. matter ∼1980
Emergence of anyons Lundholm Slide 15/37
Page 39
Average-field approximation
Huge past literature: see e.g. Wilczek 1990 review
For anyons one may consider an average-field approximation
〈Ψ0, (Tα + V )Ψ0〉 ≈∫R2
(2π|α|%Ψ0(x)2 + V (x)%Ψ0(x)
)dx,
where B = curlαAj ≈ 2πα% with LLL energy/particle ∼ |B|.
A particular almost-bosonic limit α = β/N → 0 leads to
Eaf [ψ] :=
∫R2
(∣∣(−i∇+ βA[|ψ|2])ψ(x)
∣∣2 + V (x)|ψ(x)|2)dx,
where curlA[|ψ|2] = 2π|ψ|2 and β the only parameter.DL, Rougerie, 2015; Correggi, DL, Rougerie, 2017; Chern-Simons coupled to non-rel. matter ∼1980
Emergence of anyons Lundholm Slide 15/37
Page 40
Average-field approximation for almost-bosonic anyons
“Less bosonic” anyons would then amount to β = αN →∞.
Theorem: As β →∞
Eaf0 (β) = ETF
0 (β) + lower order,
where ETF0 (β) is the minimum of the Thomas–Fermi functional
ETF[%] :=
∫R2
(e(1, 1)β%(x)2 + V %(x)
)dx,
∫R2
%(x)dx = 1.
Furthermore, e(1, 1) ≥ 2π, with e(β, ρ) = e(1, 1)βρ2 the energyper area of the homogeneous problem at density ρ.
Conjecture: e(1, 1) > 2πCorreggi, DL, Rougerie, 2017
Emergence of anyons Lundholm Slide 16/37
Page 41
Average-field approximation for almost-bosonic anyons
Continued study of the average-field functional Eaf [ψ] is work inprogress with M. Correggi, R. Duboscq and N. Rougerie.
Numerical simulations of |ψ0|2 at β = 318 by Romain Duboscq.
Emergence of anyons Lundholm Slide 17/37
Page 42
Universal bounds: A local exclusion principle for anyons
ei2pαπ ei(2p+1)απ
Recall: 2-particle exchange phase (2p+ 1)α times π.But anyons can also have pairwise relative angular momenta ±2q.
⇒ effective statistical repulsion DL, Solovej, 2013
Vstat(r) = |(2p+ 1)α− 2q|2 1
r2≥
α2N
r2, r = |xj − xk|
Emergence of anyons Lundholm Slide 18/37
Page 43
Universal bounds: A local exclusion principle for anyons
ei2pαπ ei(2p+1)απ
Recall: 2-particle exchange phase (2p+ 1)α times π.But anyons can also have pairwise relative angular momenta ±2q.⇒ effective statistical repulsion DL, Solovej, 2013
Vstat(r) = |(2p+ 1)α− 2q|2 1
r2≥
α2N
r2, r = |xj − xk|
Emergence of anyons Lundholm Slide 18/37
Page 44
Universal bounds: A local exclusion principle for anyons
α
α∗
αN := minp∈{0,1,...,N−2}
minq∈Z|(2p+ 1)α− 2q|
−→N→∞
α∗ :=
{1ν , if α = µ
ν is a reduced fraction with µ odd,
0, otherwise.
Emergence of anyons Lundholm Slide 19/37
Page 45
Universal bounds for the homogeneous anyon gasDL, Solovej, 2011-’13, Larson, DL, 2016-’18, DL, Seiringer, 2017Work in progress with Qvarfordt extends to non-abelian anyons.
Define the ground-state energy per particle and unit density
e(α) := lim infN,L→∞N/L2=%
E0(N,L)
N%
e(0) = 0,
e(1) = 2π
Theorem: There exist constants 0 < C1 ≤ C2 <∞ such that forany 0 ≤ α ≤ 1,
C1α ≤ e(α) ≤ C2α,
and as α→ 0,e(α) ≥ π
4α(1−O(α1/3)
)e(α) ≥ πα∗
(1−O(α
1/3∗ )
)Conjecture: optimal C1 and C2 cannot both be 2π.
Emergence of anyons Lundholm Slide 20/37
Page 46
Lieb–Thirring inequalities for anyons
Dyson, Lenard, 1967DL, Solovej, 2011-’13; LT with general local exclusion developed by Nam, Portmann, Solovej, 2013-’15;Larson, DL, 2016-’18; DL, Seiringer, 2017
Theorem (LT inequality for ideal anyons)
There exists a constant 0 < C ≤ 2π such that for any 0 ≤ α ≤ 1and any N -anyon wave function Ψ on R2,
〈Ψ, TαΨ〉 ≥ Cα
∫R2
%Ψ(x)2 dx.
Hence
〈Ψ, HNΨ〉 ≥∫R2
(Cα%Ψ(x)2 + V (x)%Ψ(x)
)dx
i.e. a universal lower bound of the form of average-field theory.
Emergence of anyons Lundholm Slide 21/37
Page 47
References
D. L., J.P. Solovej, Hardy and Lieb-Thirring inequalities for anyons,Commun. Math. Phys. 322 (2013) 883.
D. L., J.P. Solovej, Local exclusion principle for identical particles obeyingintermediate and fractional statistics, Phys. Rev. A 88 (2013) 062106.
D. L., J.P. Solovej, Local exclusion and Lieb-Thirring inequalities forintermediate and fractional statistics, Ann. H. Poincare 15 (2014) 1061.
D. L., N. Rougerie, The Average Field Approximation for Almost BosonicExtended Anyons, J. Stat. Phys. 161 (2015) 1236.
D. L., N. Rougerie, Emergence of fractional statistics for tracer particlesin a Laughlin liquid, Phys. Rev. Lett. 116 (2016) 170401.
S. Larson, D. L.,Exclusion bounds for extended anyons, ARMA, 2018
D. L., Many-anyon trial states, Phys. Rev. A 96 (2017) 012116.
M. Correggi, D. L., N. Rougerie, Local density approximation for thealmost-bosonic anyon gas, Analysis & PDE 10 (2017) 1169.
D. L., R. Seiringer, Fermionic behavior of ideal anyons, arXiv:1712.06218
Emergence of anyons Lundholm Slide 22/37
Page 48
Discussion
• Extended case
• Harmonic trap
• Clustering trial states
Emergence of anyons Lundholm Slide 23/37
Page 49
Hardy inequality
Statistical repulsion gives rise to the following “Hardy inequality”:
Tα ≥4α2
N
N
∑1≤j<k≤N
|xj − xk|−2
Emergence of anyons Lundholm Slide 24/37
Page 50
Extended case
We use a magnetic Hardy inequality with symmetry(cf. Laptev, Weidl, 1998; Hoffmann-Ostenhof2, Laptev, Tidblom, 2008; Balinsky...)
to consider the enclosed flux inside a two-particle exchange loop,subtracted with arbitrary pairwise angular momenta. Unwantedoscillation can be controlled by smearing (but analysis is tricky!)
Vstat(r) = ρ(r)1
r2, ρ(r) = min
q∈Z
∣∣∣∣Φ(r)
2π− 2q
∣∣∣∣2
α = 1/3 r
ρ
α2∗
1
Emergence of anyons Lundholm Slide 25/37
Page 51
Extended case (clustering)
α = 1/3 r
ρ
α = 2/3 r
ρ
Emergence of anyons Lundholm Slide 26/37
Page 52
Universal bounds for the extended anyon gas
Consider ground-state energy on a box Q ⊂ R2:
E0(N,Q, α,R) := inf{〈Ψ, TRα Ψ〉 : Ψ ∈ L2
c(QN ) , ‖Ψ‖ = 1
}In the thermodynamic limit, N, |Q| → ∞ with % = N/|Q| fixed,for dimensional reasons,
E0(N,Q, α,R)
N→ e(α, γ)%, γ := R
√%.
We define (with Dirichlet b.c.)
e(α, γ) := lim infN, |Q|→∞N/|Q|=%
E0(N,Q, α,R)
%N.
Emergence of anyons Lundholm Slide 27/37
Page 53
Universal bounds for the extended anyon gas
Consider ground-state energy on a box Q ⊂ R2:
E0(N,Q, α,R) := inf{〈Ψ, TRα Ψ〉 : Ψ ∈ L2
c(QN ) , ‖Ψ‖ = 1
}In the thermodynamic limit, N, |Q| → ∞ with % = N/|Q| fixed,for dimensional reasons,
E0(N,Q, α,R)
N→ e(α, γ)%, γ := R
√%.
We define (with Dirichlet b.c.)
e(α, γ) := lim infN, |Q|→∞N/|Q|=%
E0(N,Q, α,R)
%N.
Emergence of anyons Lundholm Slide 27/37
Page 54
Universal bounds for the extended anyon gas
γ
α∗ = 0
α∗ = 1/3
α∗ = 1
γ
α = 1/3
α = 2/3
α = 1
α = 2
α = 3
Theorem ([Larson-DL’16] Bounds for the extended anyon gas)
Up to some universal constant C > 0,
e(α, γ) &
2π|ln γ| + π(j′α∗)
2 ≥ 2πα∗, γ → 0, α 6= 0
2π|α|, γ & 1.
Emergence of anyons Lundholm Slide 28/37
Page 55
Ideal anyons in a harmonic trap
Harmonic oscillator Hamiltonian:
HN = Tα + V =
N∑j=1
(1
2m(−i∇j + αAj)
2 +mω2
2|xj |2
).
Rigorous bounds for the ground-state energy E0(N):
HN |ang.mom.= L ≥ ω(N +
∣∣∣L+ αN(N−1)2
∣∣∣) (Chitra, Sen, 1992)
C1 j′αN≤ E0(N)/(ωN
32 ) ≤ C2 ∀α,N (DL, Solovej, 2013; Larson, DL, 2016)
Cp. with fermions in 2D: E0(N) ∼√
83 ωN
32 as N →∞
Average-field suggests: E0(N) ≈√
83
√|α|ωN
32 as N →∞
Emergence of anyons Lundholm Slide 29/37
Page 56
Ideal anyons in a harmonic trap
Harmonic oscillator Hamiltonian:
HN = Tα + V =
N∑j=1
(1
2m(−i∇j + αAj)
2 +mω2
2|xj |2
).
Rigorous bounds for the ground-state energy E0(N):
HN |ang.mom.= L ≥ ω(N +
∣∣∣L+ αN(N−1)2
∣∣∣) (Chitra, Sen, 1992)
C1 j′αN≤ E0(N)/(ωN
32 ) ≤ C2 ∀α,N (DL, Solovej, 2013; Larson, DL, 2016)
Cp. with fermions in 2D: E0(N) ∼√
83 ωN
32 as N →∞
Average-field suggests: E0(N) ≈√
83
√|α|ωN
32 as N →∞
Emergence of anyons Lundholm Slide 29/37
Page 57
Ideal anyons in a harmonic trap
Harmonic oscillator Hamiltonian:
HN = Tα + V =
N∑j=1
(1
2m(−i∇j + αAj)
2 +mω2
2|xj |2
).
Rigorous bounds for the ground-state energy E0(N):
HN |ang.mom.= L ≥ ω(N +
∣∣∣L+ αN(N−1)2
∣∣∣) (Chitra, Sen, 1992)
C1 j′αN≤ E0(N)/(ωN
32 ) ≤ C2 ∀α,N (DL, Solovej, 2013; Larson, DL, 2016)
Cp. with fermions in 2D: E0(N) ∼√
83 ωN
32 as N →∞
Average-field suggests: E0(N) ≈√
83
√|α|ωN
32 as N →∞
Emergence of anyons Lundholm Slide 29/37
Page 58
Anyons in a harmonic trap — exact spectrum
Exact N = 2 spectrum: Leinaas, Myrheim, 1977
Emergence of anyons Lundholm Slide 30/37
Page 59
Anyons in a harmonic trap — exact spectrum
Numerical N = 3 spectrum: Murthy, Law, Brack, Bhaduri, 1991; Sporre, Verbaarschot, Zahed, 1991
Emergence of anyons Lundholm Slide 31/37
Page 60
Anyons in a harmonic trap — exact spectrum
Numerical N = 4 spectrum: Sporre, Verbaarschot, Zahed, 1992
Emergence of anyons Lundholm Slide 32/37
Page 61
Anyons in a harmonic trap — qualitative spectrum
Schematic N →∞ spectrum: Chitra, Sen, 1992 (θ = απ)
Emergence of anyons Lundholm Slide 33/37
Page 62
Anyons in a harmonic trap — current lower bounds
j′α∗
α
E0(N)
ωN32
&
Rigorous lower bounds: DL, Solovej, 2013/’14, improved in Larson, DL, 2016, and DL, Seiringer, 2017 ...
Emergence of anyons Lundholm Slide 34/37
Page 63
Upper bounds: many-anyon trial states
V1 V2
V3
j
⇒
xj
αAj
Jj
V∗1
N = νK particles arranged into ν complete graphs (Vq, Eq)
α = µν even:
ψα(z) :=∏j<k
|zjk|−α S
ν∏q=1
∏(j,k)∈Eq
(zjk)µ
N∏k=1
ϕ0(zk)
(cf. Moore–Read (Pfaffian), Read–Rezayi)
Emergence of anyons Lundholm Slide 35/37
Page 64
Upper bounds: many-anyon trial states
V1 V2
V3
j
⇒
xj
αAj
Jj
V∗1
N = νK particles arranged into ν complete graphs (Vq, Eq)α = µ
ν even:
ψα(z) :=∏j<k
|zjk|−α S
ν∏q=1
∏(j,k)∈Eq
(zjk)µ
N∏k=1
ϕ0(zk)
(cf. Moore–Read (Pfaffian), Read–Rezayi)
Emergence of anyons Lundholm Slide 35/37
Page 65
Upper bounds: many-anyon trial states
V1 V2
V3
j
⇒
xj
αAj
Jj
V∗1
N = νK particles arranged into ν complete graphs (Vq, Eq)α = µ
ν even:
ψα(z) :=∏j<k
|zjk|−α S
ν∏q=1
∏(j,k)∈Eq
(zjk)µ
N∏k=1
ϕ0(zk)
(cf. Moore–Read (Pfaffian), Read–Rezayi)
Emergence of anyons Lundholm Slide 35/37
Page 66
Upper bounds: many-anyon trial states
V1 V2
V3
j
⇒
xj
αAj
Jj
V∗1
N = νK particles arranged into ν complete graphs (Vq, Eq)α = µ
ν odd:
ψα(z) :=∏j<k
|zjk|−α S
ν∏q=1
∏(j,k)∈Eq
(zjk)µK−1∧k=0
ϕk (zj∈Vq)
(cf. Moore–Read (Pfaffian), Read–Rezayi)
Emergence of anyons Lundholm Slide 35/37
Page 67
Upper bounds: many-anyon trial states
Proposition: For Ψ = Φψα, Φ ∈ H1loc(R2N ;R), α = µ
ν even,
〈Ψ, HNΨ〉 =
(1− αν − 1
2
)ωN
∫R2N
|Ψ|2 dx
+
∫R2N
N∑j=1
|∇jΦ|2|ψα|2 dx.
Emergence of anyons Lundholm Slide 36/37
Page 68
Upper bounds: many-anyon trial states
R-extended case: Replace∏j<k |zjk|−α with e−α
∑j<k wR(xj−xk).
Proposition: For the free gas on a box Q ⊂ R2, α even
TRα ψα = αWR ψα ,
WR(x) :=
N∑j 6=k=1
∆wR(xj − xk) = 2π
N∑j 6=k=1
1BR(0)
πR2(xj − xk).
Proposition: For Ψ = Φψα, Φ ∈ H10 (QN ;R), α even
〈Ψ, TRα Ψ〉 =
∫QN
N∑j=1
|∇jΦ|2 + αWR|Φ|2 |ψα|2 dx.
Emergence of anyons Lundholm Slide 37/37