application of continuous unitary transformations...
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Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
APPLICATION OF CONTINUOUS UNITARYTRANSFORMATIONS TO IONIC HUBBARD
MODEL
S.A. Jafari1
M. Hafez2, M.R. Abolhassani2
1 Isfahan Univ. of Tech.,2 Tarbiat Modares Univ.
2008
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Outline
Introduction to CUT Method
Ionic Hubbard Model
Flow Equations for IHM
Summary and Conclusions
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Introduction to CUT MethodI Eigenvalues, eigenvectors and correlation functions of
a quantum systemI Diagonalization using continuous unitary
transformationH = Hd + H r
I Transformed Hamiltonian
H (`) = U (`) HU† (`)
I Flow equations: Define a Generator η(`)
dH (`)
dl= [η (`) , H (`)] , U(`) = eη(`)
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
The generator governs the flow, and hence determines in whichsector the Hamiltonian renormalizes itself
I Wegner Generator: Quantum fluctuations driven flow
H (`) = Hd (`) + H r (`)
η (`) = [H (`) , H r (`)] ,
I Wegner Generator ⇒ Block Diagonalization
` →∞ : hab (∞) (haa (∞)− hbb (∞)) = 0
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Band Matrices
A matrix is called band matrix, iff
Hnm = 0; for |n −m| > M
MKU (Mielke, Knetter, Uhrig) generator
ηij(`) = sgn(qi(`)− qj(`))Hij(`)
where Q is an operator counting number of some kind ofexcitations, allows for "particle number conserving" flow.
I ηMKU preservs band nature
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
PROOF
dhnm
d`= −sgn(n −m)(hnn − hmm)hnm +∑
k 6=n,m[sgn(n − k) + sgn(m − k)]hnkhkm
I First term ∝ h ⇒ Band matrixI For |n −m| > M either of knk or hkm is zeroI For m ≤ k ≤ n,
∑k over sgn’s with different sing ⇒ zero
I For k /∈ [m, n], sgn’s add up to ±2
+2∑
m−M<k<m
hnkhkm − 2∑
n<k<M
hnkhkm
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Example: Two-level Hamiltonian
H = E1− ω
2σz +
e2
σx
Using Q = (1− σz)/2 one takes Hd = E1− ω2 σz , to obtain
η11 = η22 = 0, η12 = sgn(0− 1)e/2, η21 = sgn(1− 0)e/2,which can be summarized as η = −ie/2σy . Hence:
∂`H = − ie2 [σy , E1− ω
2 σz + e2σx ] = ieω
2 iσx + ie2
2 iσz ⇒
∂`
(E − ω/2 e/2
e/2 E + ω/2
)= −e
2
(e ωω −e
)⇒
∂`E = 0, ∂`ω = e2, ∂`e = −ωe ⇒E (∞) = E , ω(∞) =
√ω2 + e2, e(∞) = 0
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Exercise
Repeat the previous procedure with the bosonic oscillator
H = E1 + ωa†a +d2
(a†2 + a2
)Hint: Take Q = a†a to obtain η = d
2 (a†2 − a2)
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Outline
Introduction to CUT Method
Ionic Hubbard Model
Flow Equations for IHM
Summary and Conclusions
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Ionic Hubbard ModelI Motivation:
I Neutral-ionic transition in organic compoundsI Ferroelectric transition in perovskite materials.
I Ionic Hubbard Hamiltonian:
H = −t∑〈j,l〉
∑σ
(c†jσclσ + c†lσcjσ
)+U
∑j
nj↑nj↓+∆
2
∑i,σ
(−1)i c†iσciσ
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Ionic Hubbard Model
I Definition of the problem:
What is the state of the system between Mott and bandinsulators?
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
I Ionic Hubbard model in the limit U = 0:
H =∑kσ
εkc†kσckσ+∑k ,σ
εk+π c†k+πσck+πσ+∆
2
∑kσ
(c†kσck+πσ + h.c.)
Where εk = −2t cos k . Using Bogolubov transformations:
H =∑kσ
Ek
(γ†kσγkσ − γ†k+πσγk+πσ
)Where:
Ek =
√4t2 cos2 k + (
∆
2)2
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Ionic Hubbard Model
Therefore in half-filling conditions IHM is band insulator Withenergy gap ∆.
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
I Ionic Hubbard model in the limit U � t :Reduces to t-J model which at half-filling describes a Mottinsulator
I Frozen charge fluctuations at half-fillingI Low-energy spin-exciations
H = J∑〈i,j〉
~Si .~Si+1, J =4t2
U(0)
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Another solvable limit: Classic LimitI Atomic limit (t = 0):
H = U∑
j
nj↑nj↓ +∆
2
∑i,σ
(−1)i ni,σ (0)
In this limit IHM is classical and line U = ∆ separates bandinsulator from Mott insulator.
The line U = ∆ is metallic transition point.S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Outline
Introduction to CUT Method
Ionic Hubbard Model
Flow Equations for IHM
Summary and Conclusions
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Warm up exercise
I Flow equations for IHM in the limit U = 0:
Split H0 (`) as:
H0 (`) =∑kσ
εk (`) c†kσckσ
+∑kσ
εk+π (`) c†k+πσck+πσ
+∑
k ,σ∆k (`)
2 (c†kσck+πσ + h.c.)
Wegner generator becomes:η0 (`) =
∑k ,σ
∆k (`)2 (εk (`)− εk+π (`))
(c†kσck+πσ − h.c.
)
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Flow Equation
[η0 (`) , H0 (`)] =
−∑k ,σ
∆k (`)2 (εk (`)− εk+π (`))2
(c†kσck+πσ + h.c.
)+∑k ,σ
∆2k (`)2 (εk (`)− εk+π (`))
(c†k ,σck ,σ − c†k+πσck+πσ
)Which gives the following flow equations:
dεk (`)d` = ∆2
k (`) εk (`)
d∆k (`)d` = −4∆k (`) ε2
k (`)
εk (`) = −εk+π (`)
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Solution of flow equations
In the limit ` →∞∆k (∞) = 0
εk (∞) = ±
√(∆
2
)2
+ 4 t2 cos2 k
{+ k ∈ (−π
2 , 0]
- k ∈ (−π, π2 ]
I Result is identical to Bogolubov transformation.
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Flow Equations for IHM
I Effective Hamiltonian For IHM
H (`) is considered as:
H (`) = −t (`)∑iσ
(c†i,σci+1σ + h.c.
)+ ∆(`)
2∑iσ
(−1)i c†iσciσ
+U(`)2∑iσσ′
c†iσc†iσ′ciσ′ciσ + V (`)∑iσσ′
c†iσc†i+1σ′ci+1σ′ciσ
With initial conditions t (0) = 1, ∆ (0) = ∆, U (0) = U, andV (0) = 0
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Wegner generator for IHM
η (`) = t (`) ∆ (`)∑i,σ
(−1)i(
c†i+1,σci,σ − h.c.)
−t (`) U (`)∑
i,σσ′
(c†i,σc†i,σ′ci,σ′ci+1,σ − c†i−1,σc†i,σ′ci,σ′ci,σ − h.c.
)−t (`) V (`)
∑i,σσ′
(c†i,σc†i+1,σ′ci+1,σ′ci+1,σ + c†i,σc†i+1,σ′ci+2,σ′ci,σ
− c†i,σc†i,σ′ci+1,σ′ci,σ − c†i−1,σc†i+1,σ′ci+1,σ′ci,σ − h.c.)
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Some Algebra
With definitions: η (`) ≡ η1 (`) + η2 (`) + η3 (`)H (`) = H1 (`) + H2 (`) + H3 (`) + H4 (`)Various commutators can be calculated:[η (`) , H1 (`) + H2 (`)] =
2t2 (`) ∆ (`)∑i,σ
(−1)i(
c†i,σci,σ + c†i,σci+2,σ + h.c.)
+t (`) ∆2 (`)∑i,σ
(c†i+1,σci,σ + h.c.
)[η2 (`) , H1 (`)] = 2t2 (`) U (`)
∑i,σσ′
(c†i,σc†i,σ′ci,σ′ci,σ
−c†i+1,σc†i,σ′ci,σ′ci+1,σ + h.c.)
+ irrelevant terms
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Some More Algebra
[η3 (`) , H1 (`)] = 2t2 (`) V (`)∑
i,σσ′
(2c†i,σc†i+1,σ′ci+1,σ′ci,σ
−c†i,σc†i,σ′ci,σ′ci,σ + h.c.)
+ irrelevant terms
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Differential Equations
dt(`)d` = −t (`) ∆2 (`)
d∆(`)d` = 8t2 (`) ∆ (`)
dU(`)d` = 8t2 (`) (U (`)− V (`))
dV (`)d` = 4t2 (`) (2V (`)− U (`))
I Hopping term flows to zero!I Quantum fluctuations are being renormalized to zeroI Attractive longer reange Coulomb interaction induced
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Solutions at ` →∞:
t (∞) = 0
∆ (∞) =(8 + ∆2) 1
2
U (∞) = U2
(8+∆2)12
(8+∆2)
√2
4 +(8+∆2)−√
24 ∆
√2
!∆
1+
√2
2
V (∞) =√
2U4
(8+∆2)12
−(8+∆2)
√2
4 +(8+∆2)−√
24 ∆
√2
!∆
1+
√2
2
Renormalized "Classical" Hamiltonian:Heff ≡ H (∞) =∆(∞)
2∑iσ
(−1)i niσ + U (∞)∑
ini↑ni↓ + V (∞)
∑iσσ′
niσni+1σ′
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
IONICITY
nB =1N
∑σ,i∈B
〈niσ〉, nA =1N
∑σ,i∈A
〈niσ〉
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Definitions of spin and charge gapsI Phase Transitions
∆s = E0(N
2 + 1, N2 − 1
)− E0
(N2 , N
2
)∆c = 1
2
(E0(N
2 + 1, N2 + 1
)+ E0
(N2 − 1, N
2 − 1)− 2E0
(N2 , N
2
))
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Phase Diagram
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Outline
Introduction to CUT Method
Ionic Hubbard Model
Flow Equations for IHM
Summary and Conclusions
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
For a fixed ∆
I At small U, both charge and spin gaps are identicalI In the intermediate region, charge gap vanishes ⇒ Metallic
regionI For large U, charge gap develops once more ⇒ InsulatorI Low energy spin-excitations ⇒ Mott Insulator
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
IFaculty position @ IUT physicsApplications should be addressed to:Prof. Ahmad Shirani,Head of the physics department,Isfahan University of Technology,Isfahan 84156, Iran.Fax: 0311-391 2376email: [email protected]
I Thank you for your attention
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Example 2: Electron-Phonon Interactions
I Aim: replacement of electron-phonon interaction with anelectron-electron interaction:
I Main Hamiltonian:
H =∑
q
ωq : a†qaq : +∑
k
εk : c†kck :
+∑k ,q
Mq
(a†−q + aq
)c†k+qck + E ≡ H0 + He−p(1)
I Review on Fröhlich methods:
HF = e−SHeS = H + [H, S] +12
[[H, S] , S] + · · ·
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Flow Equations for Electron-Phonon Interactions
HF =∑
k ,k ′,q
V Fk ,k ′,q : c†k+qc†k ′−qck ′ck :
+∑
k
(εF
k − 2∑
q
nk+qVk ,k+q,q
): c†kck :
+∑
q
ωFq : a†qaq : +EF + irrelevant terms (2)
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Flow Equations for Electron-Phonon InteractionsI Flow equations approach:
H (`) is approximated as:
H (`) = E (`) +∑
q
ωq (`) : a†qaq :
+∑
k
(εk (`)− 2
∑q
nk+qVk ,k+q,q (`)
): c†kck :
+∑
k ,k ′,q
Vk ,k ′,q (`) : c†k+qc†k ′−qck ′ck :
+∑k ,q
(Mk ,q (`) a†−q + M∗
k+q,−q (`) aq
)c†k+qck (3)
Last term is off-diagonal and other terms are diagonal.S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Flow Equations for Electron-Phonon InteractionsFlow equations are obtained as:
dMk ,q (`)
d`= −α2
k ,q (`) Mk ,q (`)
dVk ,k ′,q (`)
d`= Mk ,q (`) M∗
k ′−q,q (`) βk ′,−q (`)
−Mk ′,−q (`) M∗k+q,−q (`) αk ′,−q (`) (4)
dεk (`)
d`= −2
∑q
((nq + 1)
∣∣Mk ,q (`)∣∣2 αk ,q (`) + nq
∣∣Mk+q,−q (`)∣∣2 βk ,q (`)
)dωq (`)
d`= 2
∑k
∣∣Mk+q,−q (`)∣∣2βk ,q (`)
(nk+q − nk
)S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL
Introduction to CUT Method Ionic Hubbard Model Flow Equations for IHM Summary and Conclusions
Flow Equations for Electron-Phonon Interactions
dE (`)
d`=∑
k
nkdεk (`)
d`−∑k ,q
nknk+qdVk ,k+q,q (`)
d`
Where αk ,q (`) = εk+q (`)− εk (`) + ωq (`) andβk ,q (`) = εk+q (`)− εk (`)− ωq (`) are defined.Solutions in infinity:εk (∞) = εF
k , E (∞) = EF , ωq (∞) = ωFq and
Vk ,k ′,q (∞) = |Mq|2(
βk ′,−q
α2k , q + β2
k ′,−q−
αk ′,−q
α2k ′,−q + β2
k , q
)
V Fk ,k ′,q = V F
k ,−k ,q = |Mq|2ωq(
εk+q − εk)2 − ω2
q
S.A. Jafari1 M. Hafez2, M.R. Abolhassani2 1 Isfahan Univ. of Tech., 2 Tarbiat Modares Univ.
APPLICATION OF CONTINUOUS UNITARY TRANSFORMATIONS TO IONIC HUBBARD MODEL