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== Luttinger Liquid == | |||
Fermi liquid theory which is based on quasi-particle picture breaks down in 1-D. Low energy excitations of gapless one-dimensional spinless fermion system can be described by the Luttinger model (Luttinger 1963), which is exactly solvable. We should emphesize on gapless feature, because there are some other systems which we cannot be described by Luttinger model because their low energy spectrum is not gapless. | Fermi liquid theory which is based on quasi-particle picture breaks down in 1-D. Low energy excitations of gapless one-dimensional spinless fermion system can be described by the Luttinger model (Luttinger 1963), which is exactly solvable. We should emphesize on gapless feature, because there are some other systems which we cannot be described by Luttinger model because their low energy spectrum is not gapless. | ||
The first question is why we consider 1-D model. We know that exact solutions are nearly impossible when we treat strongly interacting electrons in 3-D, but when we are working in 1-D, there are some situations which we can solve them exactly and fortunately we need the result to compare them with experiment and in these experiments we look for correlations on energy scale which are usually small compared with Fermi energy. Luttinger model can help us in many applicable situations such as carbon nanotubes, semiconductor wires, organic crystals and edge states, i.e. as in fractional quantum hall effect. | The first question is why we consider 1-D model. We know that exact solutions are nearly impossible when we treat strongly interacting electrons in 3-D, but when we are working in 1-D, there are some situations which we can solve them exactly and fortunately we need the result to compare them with experiment and in these experiments we look for correlations on energy scale which are usually small compared with Fermi energy. Luttinger model can help us in many applicable situations such as carbon nanotubes, semiconductor wires, organic crystals and edge states, i.e. as in fractional quantum hall effect. | ||
In 1-D chains we have two Fermi points (Fermi surface in 1-D) and the Fermi surface of an array of chains is a pair of parallel sheets. So we can translate a Fermi sheet to another by a single wavevector <math>k_F</math>, this produces a singular particle-hole response at <math>2 k_F</math> which is known as Peierls which is a problem in 1-D. Also we have BCS singularity for attractive interaction as in 3-D. These feature are included in one-dimensional spinless fermion system that was proposed by Luttinger and solved exactly by Mattis and Lieb | In 1-D chains we have two Fermi points (Fermi surface in 1-D) and the Fermi surface of an array of chains is a pair of parallel sheets. So we can translate a Fermi sheet to another by a single wavevector <math>k_F</math>, this produces a singular particle-hole response at <math>2 k_F</math> which is known as Peierls which is a problem in 1-D. Also we have BCS singularity for attractive interaction as in 3-D. These feature are included in one-dimensional spinless fermion system that was proposed by Luttinger and solved exactly by Mattis and Lieb. By calculating correlation functions we can calculate all physical properties. The notion of "Luttinger liquid" was proposed by Haldane to describe low energy properties of 1-D gapless quantum systems. The basic idea had been discussed earlier by Efetov and Larkin, but it did not get enough attention. Tomonaga made a big contribution to this theory, so it is better to call it Tomonaga-Luttinger theory. If all other non-perturbative solution to a problem of 1-D Fermi liquid can not be used, the best candidate is using this theory. | ||
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We consider a system with N electrons in a system with length L with this Hamiltonian: | We consider a system with N electrons in a system with length L with this Hamiltonian: | ||
<math>H_0=\ | <math>H_0=\sum_{k}\epsilon_k c_k^{\dagger} c_k</math> | ||
<math>H_0=\ | <math>H_0=\sum_{k}\epsilon_k c_k^{\dagger} c_k</math> | ||
In which <math>k=\frac{2\pi}{Na}n</math> and a is lattice constant, and this <math>c</math> and <math>c^{\dagger}</math> are annihilation and creation operators that obey the Fermi anti-commutation relations: | In which <math>k=\frac{2\pi}{Na}n</math> and a is lattice constant, and this <math>c</math> and <math>c^{\dagger}</math> are annihilation and creation operators that obey the Fermi anti-commutation relations: | ||
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In general when we have external magnetic field number of up and down spins are different <math>k_{F \uparrow}\neq k_{F\downarrow}</math> and when we have external electric field we have different right and left moving fermions <math>k_F^{(+)}\neq k_F^{(-)}</math> which means that in general we have four <math>k_F</math> in channel <math>(r,s)</math> which <math>r</math> can be <math>\pm</math> near each left and right <math>\pm k_F</math> and <math>s</math> indicate the spin. Next we have to determine what the excitations are. If we do not have external fields the ground state is the Fermi sea <FS> and <math>k_F=\frac{\pi N }{2L}</math>. and our Fermi sea in this 1-D are two points at <math>k_F</math> and all states between these two state are filed in <math>T=0</math>. Here we focus on low energy excitations: | In general when we have external magnetic field number of up and down spins are different <math>k_{F \uparrow}\neq k_{F\downarrow}</math> and when we have external electric field we have different right and left moving fermions <math>k_F^{(+)}\neq k_F^{(-)}</math> which means that in general we have four <math>k_F</math> in channel <math>(r,s)</math> which <math>r</math> can be <math>\pm</math> near each left and right <math>\pm k_F</math> and <math>s</math> indicate the spin. Next we have to determine what the excitations are. If we do not have external fields the ground state is the Fermi sea <FS> and <math>k_F=\frac{\pi N }{2L}</math>. and our Fermi sea in this 1-D are two points at <math>k_F</math> and all states between these two state are filed in <math>T=0</math>. Here we focus on low energy excitations: | ||
Near a Fermi surface with <math>k \approx k_F</math> we linearize the energy and we call this electron right movers. And near a Fermi surface with <math> k \approx -k_F</math> we linearize the energy and we define a range | Near a Fermi surface with <math>k \approx k_F</math> we linearize the energy and we call this electron right movers. And near a Fermi surface with <math> k \approx -k_F</math> we linearize the energy and we define a range of <math>\lambda</math>. So we can write the effective Hamiltonian in this way: | ||
<math>H\approx \ | <math>H \approx \sum_{k=k_F-\lambda}^{k_F+\lambda} \epsilon_k c_k^{\dagger} c_k +\sum_{k=-k_F-\lambda}^{-k_F+\lambda} \epsilon_k c_k^{\dagger} c_k </math> | ||
A single particle in 1-D cannot be excited alone, if one particle tries to move, it pushes the next particles to move too, thus instead of single particle excitation we have collective excitation. We can conclude that a basis for our system is not single particle excitation <math> c_k^{\dagger}</math>, but rather collective excitations like electron-hole excitation <math>c_{k+q}^{\dagger} c_k</math>. So we define a new quantum operator: | A single particle in 1-D cannot be excited alone, if one particle tries to move, it pushes the next particles to move too, thus instead of single particle excitation we have collective excitation. We can conclude that a basis for our system is not single particle excitation <math> c_k^{\dagger}</math>, but rather collective excitations like electron-hole excitation <math>c_{k+q}^{\dagger} c_k</math>. So we define a new quantum operator: | ||
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Then with the idea of collective excitation we introduce two operators: | Then with the idea of collective excitation we introduce two operators: | ||
<math>\rho ^R(q)=\ | <math>\rho ^R(q)=\sum_{k} c_{k+q}^{R\dagger} c_k^R</math> | ||
Which means that we shift right-movers by an amount <math>q</math>. And another one for left-movers: | Which means that we shift right-movers by an amount <math>q</math>. And another one for left-movers: | ||
<math>\rho ^L(q)=\ | <math>\rho ^L(q)=\sum_{k} c_{k+q}^{L\dagger} c_k^L</math> | ||
We can easily check that this new operator is not Hermitian: | We can easily check that this new operator is not Hermitian: | ||
<math> | <math> | ||
\rho^{R \dagger} (q) | \rho^{R \dagger} (q) = \sum_{k} c_{k}^{R\dagger} c_{k+q}^R | ||
= \sum_{k} c_{k-q}^{R\dagger} c_{k+q-q}^R | |||
=\sum_{k} c_{k-q}^{R\dagger} c_{k}^R | |||
=\rho ^R(-q) | |||
</math> | </math> | ||
which means that the hermitian conjugate annihilates an excitation by amount <math>k</math>. | |||
Next we consider the commutation relation between these two operator and with Hamiltonian: | Next we consider the commutation relation between these two operator and with Hamiltonian: | ||
<math>[\rho ^{R \dagger} (q),\rho ^R(q')] = [ \rho ^R(-q),\rho ^R(q')]=\delta_{qq'} \frac{qN}{2\pi}</math> | <math>[\rho ^{R \dagger} (q),\rho ^R(q')] = [ \rho ^R(-q),\rho ^R(q')]=\delta_{qq'} \frac{qN}{2\pi}</math> | ||
<math>[H,\rho ^{R} (q)]= v_F q \rho ^{R} (q)</math> | <math>[H,\rho ^{R} (q)]= v_F q \rho ^{R} (q)</math> | ||
And the next step is to just rescale the operators to have a nice commutation relation, so we introduce operator | And the next step is to just rescale the operators to have a nice commutation relation, so we introduce operator <math>b</math> according to <math>\rho</math> as: | ||
<math> | <math> | ||
b ^{R \dagger} (q) | b ^{R \dagger} (q) = i \sqrt{\frac{2\pi}{qN}} \rho ^{R} (q) | ||
b ^{R } (q) | b ^{R } (q) = -i \sqrt{\frac{2\pi}{qN}} \rho ^{R} (-q) | ||
b ^{L \dagger} (q) | b ^{L \dagger} (q) = -i \sqrt{\frac{2\pi}{qN}} \rho ^{L} (-q) | ||
b ^{L } (q) | b ^{L } (q) = i \sqrt{\frac{2\pi}{qN}} \rho ^{L} (q) | ||
</math> | </math> | ||
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<math> c_k^{\dagger}=\frac{1}{\sqrt{N}} \sum_{j=1}^{N} e^{ikx_j} \psi^{\dagger}(x_j)</math> | <math> c_k^{\dagger}=\frac{1}{\sqrt{N}} \sum_{j=1}^{N} e^{ikx_j} \psi^{\dagger}(x_j)</math> | ||
(here we assumed that | (here we assumed that x is discrete as <math>x_j=ja</math>) | ||
And we can write this fermion field operators as : | And we can write this fermion field operators as : | ||
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<math>= \frac{1}{\sqrt{N}}( \sum_{k=k_F-\Lambda}^{k_F+\Lambda} + \sum_{k=-k_F-\Lambda}^{-k_F+\Lambda}) e^{ikx_j} c_k </math> | <math>= \frac{1}{\sqrt{N}}( \sum_{k=k_F-\Lambda}^{k_F+\Lambda} + \sum_{k=-k_F-\Lambda}^{-k_F+\Lambda}) e^{ikx_j} c_k </math> | ||
< | <math>= \frac{1}{\sqrt{N}}e^{ik_F x_j} \sum_{k} e^{ik x_j} c_k ^R+ e^{-ik_F x_j}\sum_{k} e^{ikx_j} c_k^L</math> | ||
Here if we define these two operators: | Here if we define these two operators: | ||
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We will have finally: | We will have finally: | ||
<math> | |||
<math>\sqrt{a} ( e^{ik_F x_j}\psi^R(x_j)+ e^{-ik_F x_j}\psi^L(x_j))</math> | |||
Note that the anti-commutation relation is: | Note that the anti-commutation relation is: | ||
<math>\{ \psi^{\dagger R/L} (x),\psi^{R/L}(y)\} =\delta (x-y)</math> | <math>\{ \psi^{\dagger R/L} (x),\psi^{R/L}(y)\} =\delta (x-y)</math> | ||
'''Introducing Interaction to the model''' | |||
In gereneral, we can write the interaction term as | In gereneral, we can write the interaction term as | ||
<math> H_{int}=\sum_q^{'} V(q)\rho(q)\rho(-q)</math> | <math> H_{int}=\sum_q^{'} V(q)\rho(q)\rho(-q)</math> | ||
But we must exclude term corresponding to q=0, because we have uniform charge background. | But we must exclude term corresponding to q=0, because we have uniform charge background. | ||
We would like to write <math>H_{int}</math> similar to <math>H_0</math> in term of density operators. To do so, we first write it in terms of creation operators for left/right moving particles. There are two interaction terms corresponding to forward scattering and back scattering, as it follows: | We would like to write <math>H_{int}</math> similar to <math>H_0</math> in term of density operators. To do so, we first write it in terms of creation operators for left/right moving particles. There are two interaction terms corresponding to forward scattering and back scattering, as it follows: | ||
<math>H_{1,int}=\frac{1}{2L}\sum_{|q|<\Lambda}V(q)(\rho_R(q)+\rho_L(q))(\rho_R(-q)+\rho_L(-q))</math> | <math>H_{1,int}=\frac{1}{2L}\sum_{|q|<\Lambda}V(q)(\rho_R(q)+\rho_L(q))(\rho_R(-q)+\rho_L(-q))</math> | ||
<math>H_{2,int}=-\frac{1}{2L}\sum_{|q'|<\Lambda}V_{(2k_F)}(q)(\rho_R(-q)\rho_L(q)+\rho_R(q)\rho_L(-q)</math> | <math>H_{2,int}=-\frac{1}{2L}\sum_{|q'|<\Lambda}V_{(2k_F)}(q)(\rho_R(-q)\rho_L(q)+\rho_R(q)\rho_L(-q)</math> | ||
Now, by using the same annihilation and creation operators as defined previously, we can rewrite the Hamiltonian and the interact term as | Now, by using the same annihilation and creation operators as defined previously, we can rewrite the Hamiltonian and the interact term as | ||
<math>H=H_0+H_{int}=\sum_{q>0}|q|(v_F+V_q)(b^{L\dagger}(q)b^L(q)+b^{L\dagger}(q)b^L(q)) + \sum_{q>0}|q|(V_q-V_{2k_F})(b^{R\dagger}(q)b^{L\dagger}(q)+b^{L(q)}b^R(q)) </math> | <math>H=H_0+H_{int}=\sum_{q>0}|q|(v_F+V_q)(b^{L\dagger}(q)b^L(q)+b^{L\dagger}(q)b^L(q)) + \sum_{q>0}|q|(V_q-V_{2k_F})(b^{R\dagger}(q)b^{L\dagger}(q)+b^{L(q)}b^R(q)) </math> | ||
This Hamiltonian is quadratic and can be diagonalized. | This Hamiltonian is quadratic and can be diagonalized. | ||
'''Introducing Spin to the model''' | |||
We can consider the case of spin-<math>\frac{1}{2}</math> fields by addition of spin indices. The kinetic terms will be as follow | We can consider the case of spin-<math>\frac{1}{2}</math> fields by addition of spin indices. The kinetic terms will be as follow | ||
<math> H^0=\sum_{k}\sum_{\sigma}\epsilon_{k}c^{\dagger}_{k,\sigma}c_{k,\sigma} </math> | <math> H^0=\sum_{k}\sum_{\sigma}\epsilon_{k}c^{\dagger}_{k,\sigma}c_{k,\sigma} </math> | ||
where <math>\sigma=\uparrow,\downarrow</math>. The interaction terms arise naturally if we consider all possible diagrams pertaining to interaction of fields with different indices, namely | where <math>\sigma=\uparrow,\downarrow</math>. The interaction terms arise naturally if we consider all possible diagrams pertaining to interaction of fields with different indices, namely <math>\sigma</math> and r. This leads to the below interaction terms: | ||
<math> | <math> | ||
H_2=\int \mathrm{d}x \sum_\sigma \left(\frac{g_{2\parallel}}{2}\rho_{R,\sigma}(x)\rho_{L,-\sigma}(x)+\frac{g_{2\perp}}{2} \rho_{R,\sigma}(x) \rho_{L,-\sigma}(x)\right) | H_2=\int \mathrm{d}x \sum_\sigma \left(\frac{g_{2\parallel}}{2}\rho_{R,\sigma}(x)\rho_{L,-\sigma}(x)+\frac{g_{2\perp}}{2} \rho_{R,\sigma}(x) \rho_{L,-\sigma}(x)\right) | ||
</math> | </math> | ||
<math> | <math> | ||
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</math> | </math> | ||
where | where the g coefficients are momentum exchange coefficients between the same and opposite spins, respectively. For further discussion on spin interaction in this model refer to arXiv:cond-mat/9510014. | ||
Latest revision as of 01:38, 15 December 2012
Luttinger Liquid
Fermi liquid theory which is based on quasi-particle picture breaks down in 1-D. Low energy excitations of gapless one-dimensional spinless fermion system can be described by the Luttinger model (Luttinger 1963), which is exactly solvable. We should emphesize on gapless feature, because there are some other systems which we cannot be described by Luttinger model because their low energy spectrum is not gapless.
The first question is why we consider 1-D model. We know that exact solutions are nearly impossible when we treat strongly interacting electrons in 3-D, but when we are working in 1-D, there are some situations which we can solve them exactly and fortunately we need the result to compare them with experiment and in these experiments we look for correlations on energy scale which are usually small compared with Fermi energy. Luttinger model can help us in many applicable situations such as carbon nanotubes, semiconductor wires, organic crystals and edge states, i.e. as in fractional quantum hall effect.
In 1-D chains we have two Fermi points (Fermi surface in 1-D) and the Fermi surface of an array of chains is a pair of parallel sheets. So we can translate a Fermi sheet to another by a single wavevector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_F} , this produces a singular particle-hole response at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 k_F} which is known as Peierls which is a problem in 1-D. Also we have BCS singularity for attractive interaction as in 3-D. These feature are included in one-dimensional spinless fermion system that was proposed by Luttinger and solved exactly by Mattis and Lieb. By calculating correlation functions we can calculate all physical properties. The notion of "Luttinger liquid" was proposed by Haldane to describe low energy properties of 1-D gapless quantum systems. The basic idea had been discussed earlier by Efetov and Larkin, but it did not get enough attention. Tomonaga made a big contribution to this theory, so it is better to call it Tomonaga-Luttinger theory. If all other non-perturbative solution to a problem of 1-D Fermi liquid can not be used, the best candidate is using this theory.
For describing low energy physics we need to know the ground state and elementary excitation.
We consider a system with N electrons in a system with length L with this Hamiltonian:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0=\sum_{k}\epsilon_k c_k^{\dagger} c_k}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0=\sum_{k}\epsilon_k c_k^{\dagger} c_k}
In which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=\frac{2\pi}{Na}n} and a is lattice constant, and this Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^{\dagger}} are annihilation and creation operators that obey the Fermi anti-commutation relations:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{c_k,c_{k'}^{\dagger}\}=\delta _{k k'}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{c_k,c_{k'}\}=\{c_k^{\dagger},c_{k'}^{\dagger}\}=0}
In general when we have external magnetic field number of up and down spins are different Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{F \uparrow}\neq k_{F\downarrow}} and when we have external electric field we have different right and left moving fermions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_F^{(+)}\neq k_F^{(-)}} which means that in general we have four Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_F} in channel Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (r,s)} which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} can be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm} near each left and right Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm k_F} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} indicate the spin. Next we have to determine what the excitations are. If we do not have external fields the ground state is the Fermi sea <FS> and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_F=\frac{\pi N }{2L}} . and our Fermi sea in this 1-D are two points at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_F} and all states between these two state are filed in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=0} . Here we focus on low energy excitations: Near a Fermi surface with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \approx k_F} we linearize the energy and we call this electron right movers. And near a Fermi surface with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \approx -k_F} we linearize the energy and we define a range of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} . So we can write the effective Hamiltonian in this way:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \approx \sum_{k=k_F-\lambda}^{k_F+\lambda} \epsilon_k c_k^{\dagger} c_k +\sum_{k=-k_F-\lambda}^{-k_F+\lambda} \epsilon_k c_k^{\dagger} c_k }
A single particle in 1-D cannot be excited alone, if one particle tries to move, it pushes the next particles to move too, thus instead of single particle excitation we have collective excitation. We can conclude that a basis for our system is not single particle excitation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_k^{\dagger}} , but rather collective excitations like electron-hole excitation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{k+q}^{\dagger} c_k} . So we define a new quantum operator:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_k^R=c_{k_{F}+k}} , as right moving annihilation operator
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_k^L=c_{-k_{F}+k}} , as left moving annihilation operator in which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |k|<\Lambda} .
Then with the idea of collective excitation we introduce two operators: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho ^R(q)=\sum_{k} c_{k+q}^{R\dagger} c_k^R} Which means that we shift right-movers by an amount Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} . And another one for left-movers: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho ^L(q)=\sum_{k} c_{k+q}^{L\dagger} c_k^L}
We can easily check that this new operator is not Hermitian:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^{R \dagger} (q) = \sum_{k} c_{k}^{R\dagger} c_{k+q}^R = \sum_{k} c_{k-q}^{R\dagger} c_{k+q-q}^R =\sum_{k} c_{k-q}^{R\dagger} c_{k}^R =\rho ^R(-q) } which means that the hermitian conjugate annihilates an excitation by amount Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} .
Next we consider the commutation relation between these two operator and with Hamiltonian:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\rho ^{R \dagger} (q),\rho ^R(q')] = [ \rho ^R(-q),\rho ^R(q')]=\delta_{qq'} \frac{qN}{2\pi}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [H,\rho ^{R} (q)]= v_F q \rho ^{R} (q)}
And the next step is to just rescale the operators to have a nice commutation relation, so we introduce operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} according to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} as:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b ^{R \dagger} (q) = i \sqrt{\frac{2\pi}{qN}} \rho ^{R} (q) b ^{R } (q) = -i \sqrt{\frac{2\pi}{qN}} \rho ^{R} (-q) b ^{L \dagger} (q) = -i \sqrt{\frac{2\pi}{qN}} \rho ^{L} (-q) b ^{L } (q) = i \sqrt{\frac{2\pi}{qN}} \rho ^{L} (q) }
Notice that we only change them by a factor, so that we can write the commutation relations in this way: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [b ^{R} (q),b ^{R \dagger} (q')]=\delta_{qq'}} Which we see that it is a relation like for Bosonic operator. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [H,b ^{R\dagger} (q)]= v_F k b^{R\dagger} (q)} And by these we can write the Hamiltonian according to these Bosonic operators as: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=\sum_{q>0} v_F q ( b ^{R \dagger} (q) b ^{R} (q) +b ^{L \dagger} (q) b ^{R} (q))+ \frac{\pi v_F}{N } (n_R^2+n_L^2)}
In which the first term is the sums correspond to particle-hole excitation, oscillation mode, and the second term corresponds to initial addition of electrons ans is called zero mode.
We can also introduce the fermion field operators corresponding to annihilation and creation operators:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x_j)=\frac{1}{\sqrt{N}} \sum_{k=-\pi/a}^{\pi/a} e^{ikx_j} c_k} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi^{\dagger}(x_j)=\frac{1}{\sqrt{N}} \sum_{k=-\pi/a}^{\pi/a} e^{-ikx_j} c_k^{\dagger}}
And:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_k=\frac{1}{\sqrt{N}} \sum_{j=1}^{N} e^{-ikx_j} \psi(x_j)} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_k^{\dagger}=\frac{1}{\sqrt{N}} \sum_{j=1}^{N} e^{ikx_j} \psi^{\dagger}(x_j)}
(here we assumed that x is discrete as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j=ja} ) And we can write this fermion field operators as :
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x_j) = \frac{1}{\sqrt{N}} \sum_{k} e^{ikx_j} c_k } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{1}{\sqrt{N}}( \sum_{k=k_F-\Lambda}^{k_F+\Lambda} + \sum_{k=-k_F-\Lambda}^{-k_F+\Lambda}) e^{ikx_j} c_k }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{1}{\sqrt{N}}e^{ik_F x_j} \sum_{k} e^{ik x_j} c_k ^R+ e^{-ik_F x_j}\sum_{k} e^{ikx_j} c_k^L}
Here if we define these two operators:
We will have finally:
Note that the anti-commutation relation is:
Introducing Interaction to the model
In gereneral, we can write the interaction term as
But we must exclude term corresponding to q=0, because we have uniform charge background. We would like to write similar to in term of density operators. To do so, we first write it in terms of creation operators for left/right moving particles. There are two interaction terms corresponding to forward scattering and back scattering, as it follows:
Now, by using the same annihilation and creation operators as defined previously, we can rewrite the Hamiltonian and the interact term as
This Hamiltonian is quadratic and can be diagonalized.
Introducing Spin to the model
We can consider the case of spin- fields by addition of spin indices. The kinetic terms will be as follow
where . The interaction terms arise naturally if we consider all possible diagrams pertaining to interaction of fields with different indices, namely and r. This leads to the below interaction terms:
where the g coefficients are momentum exchange coefficients between the same and opposite spins, respectively. For further discussion on spin interaction in this model refer to arXiv:cond-mat/9510014.