Stability of charge density waves inelectron-phonon systems
Himeji Conference on Partial Differential Equations
March. 5, 2024

Tadahiro Miyao
Dept. Math. Hokkaido Univ.

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Table of Contents! Part 1: Background! Part 2: The Existence of Charge Density Waves as Long-Range Order! Part 3: Enhancement of Charge Density Waves by Electron-Phonon
Interaction

Based on:T.M. “Stability of charge density waves in electron-phonon systems”, To appear in Jour.
Stat. Phys., arXiv:2303.05667

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Part 1: Background3 / 31

Motivation! Various phases are anticipated to emerge in many-electron systems.
Eg. ferromagnetic phase, antiferromagnetic phase, charge density wave phase, Mott
insulator, Luttinger liquid, Fermi liquid, etc..
! The mathematical justification of phase diagrams has only been partially achieved.
! Systems with electron-phonon interactions are typically more challenging to handle
than ordinary many-electron systems, and there is little mathematical proof of phase
diagrams for low-temperature phases.

4 / 31History! The original meaning of CDW refers to the periodic structure of charge with a periodof 2a (where a is the lattice spacing) in one-dimensional electron systems.
In 1954, H. Fröhlich1 proposed this mechanism to theoretically explain
superconductivity (which was incorrect as a theory of superconductivity).
! In the 1970s, CDWs were discovered in several one-dimensional conductors, and their
behavior was experimentally and theoretically elucidated.2! In present times, the term “charge density wave” is commonly used to describe thetype of order depicted in the figure, even in multi-dimensional electron systems.

1 H. Fröhlich. On the Theory of Superconductivity: The One-Dimensional Case. Proceedings of the Royal
Society A 223 (1154): 296–305.
2 G. Grüner, Density Waves in Solids. Addison-Wesley, 1994.

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Mathematical investigations of charge densitywaves! D = 1: Lieb–Nachtergaele 3 , T.M. 4 , Gontier–Kouande–Sere5! D ≥ 2: Macrius–Piguet6 , Borgs–Jedrzejewski–Kotecky 7 , Borgs–Kotecky8 , T.M. 9.

.

..
'.
-.

-.

'ID'C-.

.

.
.

CDW.
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AF-.

.

.

3 E. H. Lieb and B. Nachtergaele, Phys. Rev. B 51, 4777, 1995

ID

CDW

4 T. M., Rev. Math. Phys. Vol. 23, No. 07, pp. 749-822 (2011)

5 D. Gontier, A. E. K. Kouande and E. Sere, A. H. Poincare, Vol. 24, pages 3945–3966, (2023)

6 N. Macris and C.-A. Piguet, Phys. Rev. B 60, 13484 (1999)

7 C Borgs, J Jedrzejewski, R Kotecky, J. Phys. A: Math. Gen. 29 733 (1999)

8 C. Borgs, R. Kotecky, Comm. Math. Phys., Vol. 208, pages 575–604, (2000)

9 T.M., J. Stat. Phys., Vol. 165, pages 225–245, (2016)

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Part 2: The Existence of Charge Density Waves asLong-Range Order

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Mathematical description of electrons! Here I give a mathematical framework for describing electrons on finite latticeΛ = Zd ∩ [−L, L)d . A state of a single electron is represented by a unit vector in theHilbert spaceℓ2 (Λ × {↑, ↓}).
! Since electrons are fermions, states of N electrons are expressed as vectors in theantisymmetric tensor product Hilbert space:Fe,N (Λ) =N!

ℓ2 (Λ × {↑, ↓}).

! It is more convenient to employ the fermionic Fock space to describe states withvarious numbers of electrons:Fe (Λ) =2|Λ|"Fe,N (Λ).

N =08 / 31

! The creation and annihilation operators {cx,σ , c∗x,σ : x ∈ Λ, σ = {↑, ↓}} acting on
Fe (Λ) will be needed to define various operators: 10{cxσ , cyτ } = 0 = {c∗xσ , c∗yτ },{cxσ , c∗yτ } = δxy δστ .
! Number operator of electrons:n̂xσ = c∗xσ cxσ ,n̂x = n̂x↑ + n̂x↓ .

10 A. Arai. Analysis on Fock Spaces and Mathematical Theory of Quantum Fields. WORLD SCIENTIFIC,Dec. 2016.
O. Bratteli and D. W. Robinson. Operator Algebras and Quantum Statistical Mechanics 2: Equilibrium States.
Models in Quantum Statistical Mechanics. Springer Berlin Heidelberg, 1997.

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Phase diagram in classical systems

Toy model:HC,Λ = U#x∈Λn̂x,↑ n̂x,↓ + W#⟨x,y⟩∈EΛ

$Un̂x n̂y − µ + 2dW +2%#n̂x .

x∈Λ! U, W, µ: real parameters.
! The family of number operators {n̂xσ } consists of commuting operators, thus HC,Λis a classical Hamiltonian.
! The ground state of HC,Λ varies depending on the values of the parameters.
" Phase diagram

10 / 31µ/2dSe,+H2W
W/2Se,0H0

W/2−W/2−W

H1U/4dSe,−

Figure: Phase diagram of the ground states for HC,Λ .

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Order parameter and phase diagram(P)! Let HC,Λbe the Hamiltonian with the periodic boundary conditions imposed.

! Ae is the ∗-algebra generated by {cx,σ : x ∈ Zd , σ ∈ {↑, ↓}}.

! Given an observable Ψ ∈ Ae , the thermal expectation value of Ψ with respect to(P)HC,Λ is defined as(P)⟨Ψ ⟩β,C,Λ =&'(P)Tr Ψ e−βHC,Λ(P)ZC,Λ

,&'(P)(P)
ZC,Λ = Tr e−βHC,Λ .

! For any state ⟨·⟩ on Ae , we define the staggered density as
1 #(−1)x ⟨n̂x ⟩.
L→∞ |Λ|∆C = limx∈ΛWhen ∆C ̸= 0, a CDW emerges.
12 / 31µ/2dSe,+H2W
W/2

Se,0H0−W/2−W

$W/2

H1

U/4d

Se,−! ∆C = 1 (Se,0 ), ∆C = 1/2 (Se,± )! ∆C = 0 (Hi , i = 0, 1, 2)
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CDW in itinerant electron systems

The extended Hubbard model:
HH,Λ = TΛ + HC,Λ .
Here,TΛ = −t##(c∗x,σ cy,σ + c∗y,σ cx,σ ).

⟨x,y⟩∈EΛ σ=↑,↓! Consider the electron hopping term as a quantum perturbation.
! It is logical to ask whether charge density waves are stable in the case t ̸= 0.

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Theorem 2.1 (Borgs and Kotecky(2000))

The charge density waves are stable, i.e., ∆C ̸= 0 in the parameter region:(())
USe,ε = (U, µ) ∈ R2 : U < 2d(W − ε), |µ| < 2d min W − ε, W − ε −,4dif the temperature is low enough and ε > 0 and |t| are sufficiently small.
µ/2dWSie.

Se,0a

W/2−W/2

W/2

U/4d−W
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! The proof is based on the Pirogov–Sinai theory.
Remark: The method of reflection positivity can cover the case where µ = 0(half-filling system) only.
! The Pirogov–Sinai theory has emerged as a potent approach in classical statisticalmechanics for delineating first-order phase transitions and coexistence of phases atlow temperatures.11! Noteworthy research works that utilize the Pirogov–Sinai theory to many-electronsystems include Borgs-Kotecký-Ueltschi(1996), Datta-Fernández-Fröhlich(1996),Borgs-Kotecký(2000). 1211 S. A. Pirogov and Y. G. Sinai, Theoretical and Mathematical Physics, 25(3):1185–1192, Dec. 1975.

12 Borgs, C.; Kotecký, R.; Ueltschi, D. Comm. Math. Phys. 181 (1996), 409–446,Borgs, C.; Kotecký, R. Comm. Math. Phys. 208 (2000), 575–604,Datta, Nilanjana; Fernández, Roberto; Fröhlich, Jürg, J. Statist. Phys. 84 (1996), 455–534.

16 / 31Part 3: Enhancement of Charge Density Waves byElectron-Phonon Interaction

17 / 31The Holstein–Hubbard model
The Holstein–Hubbard model:HΛ = HH,Λ + g#n̂x (bx + b∗x ) + ω0x∈Λ#b∗x bx .

x∈Λ! HΛ acts on the following Hilbert space:HΛ = Fe (Λ) ⊗ Fp (Λ),

where Fp (Λ) is the bosonic Fock space over ℓ2 (Λ):
Fp (Λ) =∞"n=0⊗ns ℓ2 (Λ);⊗ns ℓ2 (Λ) stands for the n-fold symmetric tensor product of ℓ2 (Λ), with
⊗0s ℓ2 (Λ) = C.

! The annihilation- and creation operators of phonons are denoted by bx and b∗x ,respectively.
[bx , b∗y ] = δx,y , [bx , by ] = 0.
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! The phonons are assumed to be dispersionless with energy ω0 > 0.
! The parameter g is the strength of the electron-phonon interaction.
! Given Ψ ∈ Ae , the thermal expectation value of Ψ with respect to HΛ(P) is defined as(P)⟨Ψ ⟩β,Λ =&'(P)Tr Ψ e−βHΛ
(P)ZΛ,(P)ZΛ

&'(P)= Tr e−βHΛ .

! As before, we define the staggered density as
1 #(−1)x ⟨n̂x ⟩.
L→∞ |Λ|∆ = lim(1)x∈Λ

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! We can prove the existence of charge density waves in the following region:
Sep,ε =*(U, µ) ∈ R2 :U < 2d(W − ε) +2g 2,
ω0(Ug2|µ| < 2d min W − ε, W − ε −+4d 2dω0)+,

provided that |t| is small enough, ω0 is large enough and at sufficiently lowtemperatures.
! Se,ε ⊂ Sep,ε " The electron-phonon interaction has a significant stabilizing effecton the charge density waves.
! Such effects have been anticipated in theoretical physics based on numericalcomputations and discussions relying on certain approximation theories.

20 / 31(1)(2)(3)

The spin operators, (Sx , Sx , Sx ), at site x are defined to beSx(i) =

12#c∗x,σ (s(i) )σ,σ′ cx,σ ,i = 1, 2, 3,σ,σ ′ =↑,↓

where s(i) (i = 1, 2, 3) are the Pauli matrices:
$%
$%
$
0 1
0 −i
1s(1) =, s(2) =, s(3) =1 0i 00%
0.
−1(s(i) )σ,σ′ represents the matrix elements of s(i) , with the correspondence ↑= 1, ↓= 213 .

13 Under this convention, for example, (s(1) )↑,↑ = (s(1) )1,1 = 0 and (s(1) )↑,↓ = (s(1) )1,2 = 1.

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Theorem 3.1 (T.M., 2023)
Suppose that 0 < ε < W and (U, µ) ∈ Sep,ε . There exist certain constants
0 < β0 < ∞, 0 < ω∗ < ∞ and 0 < t0 < ∞, such that, if β ≥ β0 , ω0 ≥ ω∗ and|t| ≤ t0 , then the following (i)-(iii) hold:(i) Given an arbitrary local observable Ψ ∈ Ae , the inﬁnite volume limit:(P)⟨Ψ ⟩β(P)= lim ⟨Ψ ⟩β,ΛL→∞(P)

converges. The state ⟨·⟩β on Ae deﬁned in this way can be represented by theconvex combination of two pure states:
11(P)
(+)
(−)
⟨Ψ ⟩β = ⟨Ψ ⟩β + ⟨Ψ ⟩β .
22(±)
The states ⟨·⟩β describe charge density waves:(+)⟨n̂x ⟩β= ρ + (−1)x ∆,(−)⟨n̂x ⟩β= ρ − (−1)x ∆.

Here, ∆(+) = −∆(−) = ∆ > 0, where ∆(+) and ∆(−) are staggered(±)
densities deﬁned with respect to states ⟨·⟩β in equation (1). Additionally, ρ isgiven as
1 #(+)
ρ = lim⟨n̂x ⟩βL→∞ |Λ|x∈Λ(−)

and coincides with the density associated with ⟨·⟩β .

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Theorem 3.2 (Cont’d)(i) (±)(ii) No magnetic order exists. Namely, ⟨Sx ⟩βx ∈ Zd .

= 0 (i = 1, 2, 3) holds for every(iii) The two-point correlation function concerning arbitrary local observablesΨ, Φ ∈ Ae decays exponentially. Namely, there are constants CΨ,Φ > 0 andξℓ > 0 such that(),,dist(supp Ψ, supp Φ),(±)(±)(±) ,,,⟨Ψ Φ⟩β − ⟨Ψ ⟩β ⟨Φ⟩β , ≤ CΨ,Φ exp −ξℓwhere, for any two ﬁnite subsets A and B of Zd , the distance between A and B
is deﬁned as dist(A, B) = min{∥x − y∥ : x ∈ A, y ∈ B}.

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Outline of the proof! HH Hamiltonian:###HΛ = −tc∗x,σ cy,σ + HC,Λ + gn̂x (bx + b∗x ) + ω0 Np,Λσ ⟨x;y⟩x∈Λ" Lang–Firsov transformation:##(eﬀ)
HΛ = −teiΦx,y c∗x,σ cy,σ + HC,Λ + ω0 Np,Λσ ⟨x;y⟩where√

2g1(qx − qy ), qx = √ (bx + b∗x ),ω02$%##
Ueﬀ #(eﬀ)HC,Λ = Ueﬀnx,↑ nx,↓ + W
nx ny − µ + 2dW +nx ,2Φx,y = −x∈Λ2

Ueﬀ = U −⟨x,y⟩x∈Λ

2g.
ω024 / 31

" The partition function can be represented as a contour model on the(d + 1)-dimensional spacetime TΛ := [−L, L − 1]d × [0, β]:14n#!
−β̃ ℓ eℓ |Vℓ |Zℓ,Λ =eρ(Yi ){Y1 ,...,Yn }i=1" The Pirogov–Sinai theory can be applied.
B

14 In the case of classical systems such as the Ising model, the partition function can be represented by a contour
model in d-dimensional space.

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Difficulties:! The construction of the contour model is complex.
! Existing methods cannot be applied due to the non-conservation of the boson particle
number.
" It is challenging to control ρ(Y ).
" By utilizing explicit formulas for correlation functions in quantum field theory, it
becomes possible to evaluate ρ(Y ).

Theorem 3.3 (T.M. (2023))
Suppose that 0 < ε < W and (U, µ) ∈ Sep,ε . There exist certain constants
0 < β0 < ∞, 0 < ω∗ < ∞ and 0 < t0 < ∞, such that, if β ≥ β0 , ω0 ≥ ω∗ and|t| ≤ t0 , then the following holds:|ρ(Y )| ≤ e−(β̃ee +β̃c+γ)|supp Y | ,,, $%, ∂,e5−(β̃ee +β̃c+γ† )|supp Y |,,,, ∂ν i ρ(Y ), ≤ 2β̃C0 + e − 1 + α e1/2 |supp Y | ewhere, γ, γ† and c are some positive numbers, and the symbols ν i represent theparameters U, µ, W, g.

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! Define the weight of the contour Y as follows:
Wℓ (Y ) = ρ(Y ) eβ̃eℓ |Y |rZm,IntmYZℓ,Int m Y
m=1.

" The partition function can be expressed as follows:#Zℓ,Λ = e−β̃eℓ |Λ|n-Wℓ (Yi ).

i=1{Y1 ,...,Yn }⊂Tn
Λ

! The set X = {Y1 , . . . , Yn } is called a cluster, if Y1 , . . . , Yn are “connected”.
Pclustert^

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! For each cluster X = {Y1 , . . . , Yn }, set*+n1
Ψ(X) =ϕ(Y1 , . . . , Yn )Wℓ (Yi ).
n(Y )!
i=1Y ∈X" Cluster expansion:log Zℓ,Λ =#Ψ(X).

X:supp X⊂ΛProposition 3.4

11log Zℓ,Λ = eℓ −L→∞ |Λ|βfℓ = − lim#X:o∈supp X

1
Ψ(X).
|supp X|

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Proposition 3.5
The limit ⟨Ψ ⟩ℓ = lim ⟨Ψ ⟩ℓ,Λ exists, and the following holds:
L→∞. n/∞###1⟨Ψ ⟩ℓ =Wℓ,Ψ (YΨ )Wℓ (Yi ) ϕ(YΨ , Y1 , . . . , Yn ).
n!
n=0i=1YΨ{Y1 ,...,Yn }Proposition 3.6⟨Ψ Φ⟩ℓ − ⟨Ψ ⟩ℓ ⟨Φ⟩ℓ=!

Wℓ,Ψ (YΨ )Wℓ,Φ (YΦ )YΨ ,YΦ∞!

1(n−2)!
n=2!

{Y3 ,...,Yn }" n#i=3$Wℓ (Yi ) ϕ(YΨ , YΦ , Y3 , . . . , Yn ).

From this, it follows thatɿ%&dist(supp Ψ, supp Φ)|⟨Ψ Φ⟩ℓ − ⟨Ψ ⟩ℓ ⟨Φ⟩ℓ | ≤ CΨ,Φ exp −.
ξℓ29 / 31

Summary! We provided a mathematical justification for the physicists’ prediction that CDW
stabilizes due to the electron-phonon interaction.
! The proof relied on the quantum Pirogov–Sinai theory.
! Consideration was limited to on-site electron-phonon interactions, but extension tofinite range is possible (though the proof becomes more complex).
! There are many aspects of magnetic ordering that are not yet fully understoodmathematically.

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Thank you!

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