application of continuous unitary transformations...

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


Outline

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(`)




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




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




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




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




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)




Outline


Ionic Hubbard Model






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




Ionic Hubbard Model

I Definition of the problem:

What is the state of the system between Mott and bandinsulators?




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




Ionic Hubbard Model

Therefore in half-filling conditions IHM is band insulator Withenergy gap ∆.




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)




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.



Outline


Ionic Hubbard Model






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.

)




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 (`)




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.





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




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.)




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




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




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




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′




IONICITY

nB =1N

∑σ,i∈B

〈niσ〉, nA =1N

∑σ,i∈A

〈niσ〉




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

))




Phase Diagram




Outline


Ionic Hubbard Model






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




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




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] + · · ·




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)




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.



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.



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



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