This is a note for the reading of the 2nd yellow book by Nagaosa .
Bosonization
The example we will discuss is one of the so-called Tomonaga-Luttinger liquids, XXZ model.
For XXZ model, the Halmiltonian is:
H = J ⊥ ∑ i ( S i x S i + 1 x + S i y S i + 1 y ) + J z ∑ i S i z S i + 1 z = J ⊥ 2 ∑ i ( S i + S i + 1 − + S i − S i + 1 + ) + J z ∑ i S i z S i + 1 z H = J_\perp\sum_i(S^x_iS^x_{i+1}+S^y_iS^y_{i+1})+J_z\sum_iS_i^zS_{i+1}^z=\frac{J_\perp}{2}\sum_i(S^+_iS^-_{i+1}+S^-_iS^+_{i+1})+J_z\sum_iS_i^zS_{i+1}^z
H = J ⊥ i ∑ ( S i x S i + 1 x + S i y S i + 1 y ) + J z i ∑ S i z S i + 1 z = 2 J ⊥ i ∑ ( S i + S i + 1 − + S i − S i + 1 + ) + J z i ∑ S i z S i + 1 z For the discrete case, we have the Hamiltonian (Using the technique of Jordan-Wigner Transformation):
H = − J ⊥ 2 ∑ i = 1 N ( f i † f i + 1 + f i + 1 † f i ) + J z ∑ i = 1 N ( f i † f i − 1 2 ) ( f i + 1 † f i + 1 − 1 2 ) = H 1 + H 2 H = -\frac{J_\perp}{2}\sum_{i=1}^{N}(f_i^\dagger f_{i+1}+f_{i+1}^\dagger f_i)+J_z\sum_{i=1}^{N}(f_i^\dagger f_i-\frac{1}{2})(f_{i+1}^\dagger f_{i+1}-\frac{1}{2}) = H_1+H_2
H = − 2 J ⊥ i = 1 ∑ N ( f i † f i + 1 + f i + 1 † f i ) + J z i = 1 ∑ N ( f i † f i − 2 1 ) ( f i + 1 † f i + 1 − 2 1 ) = H 1 + H 2 If we introduce the Fourier transformation:
f n = 1 N ∑ k f k e i k n , f n † = 1 N ∑ k f k † e − i k n f_n = \frac{1}{\sqrt{N} }\sum_kf_ke^{ikn},f_n^\dagger = \frac{1}{\sqrt{N} }\sum_kf_k^\dagger e^{-ikn}
f n = N 1 k ∑ f k e i k n , f n † = N 1 k ∑ f k † e − i k n Then we have:
H 1 = ∑ k ( − J ⊥ cos k ) f k † f k H_1 = \sum_k (-J_\perp \cos k)f_k^\dagger f_k
H 1 = k ∑ ( − J ⊥ cos k ) f k † f k We treat J z J_z J z S t o t z = 0 S_{tot}^z = 0 S t o t z = 0 H 1 H_1 H 1 S i z = f i † f i − 1 2 S_i^z = f_i^\dagger f_i-\frac{1}{2} S i z = f i † f i − 2 1 H 2 H_2 H 2 H 2 H_2 H 2
From the perspective of perturbation, the low-energy particle-hole exitation near the Fermi surface will be dominant.
1st step: Linearization
In case that only the vicinity of the Fermi surface that counts, we replace the dispersion relation ε k = − J ⊥ cos k \varepsilon_k = -J_\perp \cos k ε k = − J ⊥ cos k k F k_F k F
f n = 1 N ∑ k f k e i k n = R ( x n ) e i k F x n + L ( x n ) e − i k F x n f_n =\frac{1}{\sqrt{N} }\sum_kf_ke^{ikn}= R(x_n)e^{ik_Fx_n}+L(x_n)e^{-ik_Fx_n}
f n = N 1 k ∑ f k e i k n = R ( x n ) e i k F x n + L ( x n ) e − i k F x n And because R ( x n ) , L ( x n ) R(x_n),L(x_n) R ( x n ) , L ( x n ) We write:
ψ ( x ) = ( R ( x ) L ( x ) ) \psi(x) =
\begin{pmatrix}
R(x)\\
L(x)\\
\end{pmatrix}
ψ ( x ) = ( R ( x ) L ( x ) ) Then we have:
H 1 = − J ⊥ ∫ d x ψ ˉ i γ 1 ∂ x ψ H_1 = -J_\perp \int dx \bar{\psi} i \gamma_1\partial_x\psi
H 1 = − J ⊥ ∫ d x ψ ˉ i γ 1 ∂ x ψ Notes on proof:
ψ ˉ i γ 1 ∂ x ψ = i ( R † ∂ x R − L † ∂ x L ) f l † f l + 1 + f l + 1 † f l = ( R l † e − i k F l + L l † e i k F l ) ( R l + 1 e i k F ( l + 1 ) + L l + 1 e − i k F ( l + 1 ) ) + h . c . = R l † e i k F R l + 1 + R l † e − i k F ( 2 l + 1 ) L l + 1 + L l † e i k F ( 2 l + 1 ) R l + 1 + L l † e − i k F L l + 1 + h . c . = i R l † R l + 1 − i ( − 1 ) l R l † L l + 1 + i ( − 1 ) l L l † R l + 1 − i L l † L l + 1 + h . c . ≈ i R l † R l + 1 − i L l † L l + 1 + h . c . \bar{\psi} i \gamma_1\partial_x\psi = i(R^\dagger\partial_xR-L^\dagger\partial_xL)\\
\begin{aligned}
f_l^\dagger f_{l+1}+f_{l+1}^\dagger f_l &= (R^\dagger_le^{-ik_Fl}+L^\dagger_le^{ik_Fl})(R_{l+1}e^{ik_F(l+1)}+L_{l+1}e^{-ik_F(l+1)})+h.c.\\
& = R^\dagger_le^{ik_F}R_{l+1}+R^\dagger_le^{-ik_F(2l+1)}L_{l+1}+L^\dagger_le^{ik_F(2l+1)}R_{l+1}+L^\dagger_le^{-ik_F}L_{l+1}+h.c.\\
&= iR_l^\dagger R_{l+1}-i(-1)^lR_l^\dagger L_{l+1}+i(-1)^lL^\dagger_lR_{l+1}-iL^\dagger_l L_{l+1}+h.c.\\
&\approx iR_l^\dagger R_{l+1}-iL^\dagger_l L_{l+1}+h.c.\\
\end{aligned}
ψ ˉ i γ 1 ∂ x ψ = i ( R † ∂ x R − L † ∂ x L ) f l † f l + 1 + f l + 1 † f l = ( R l † e − i k F l + L l † e i k F l ) ( R l + 1 e i k F ( l + 1 ) + L l + 1 e − i k F ( l + 1 ) ) + h . c . = R l † e i k F R l + 1 + R l † e − i k F ( 2 l + 1 ) L l + 1 + L l † e i k F ( 2 l + 1 ) R l + 1 + L l † e − i k F L l + 1 + h . c . = i R l † R l + 1 − i ( − 1 ) l R l † L l + 1 + i ( − 1 ) l L l † R l + 1 − i L l † L l + 1 + h . c . ≈ i R l † R l + 1 − i L l † L l + 1 + h . c . Performing a Fourier transformation, we have:
R ( x ) = 1 2 π ∫ d k e i k x R ( k ) , ∂ x R = 1 2 π ∫ d k e i k x i k R ( k ) H 1 = ∑ k J ⊥ k ( R † ( k ) R ( k ) − L † ( k ) L ( k ) ) R(x) = \frac{1}{2\pi}\int dk e^{ikx}R(k), \partial_xR = \frac{1}{2\pi}\int dk e^{ikx}ikR(k)\\
H_1 = \sum_k J_\perp k(R^\dagger(k)R(k)-L^\dagger(k)L(k))
R ( x ) = 2 π 1 ∫ d k e i k x R ( k ) , ∂ x R = 2 π 1 ∫ d k e i k x i k R ( k ) H 1 = k ∑ J ⊥ k ( R † ( k ) R ( k ) − L † ( k ) L ( k ) ) To compute the J z J_z J z
f n † f n = R n † R n + L n † L n + e 2 i k F n ( R n † L n + L n † R n ) f^\dagger_nf_n = R_n^\dagger R_n+L^\dagger_nL_n+e^{2ik_Fn}(R_n^\dagger L_n+L^\dagger_nR_n)
f n † f n = R n † R n + L n † L n + e 2 i k F n ( R n † L n + L n † R n ) 
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