Computing the density operator commutation relations (Altland & Simons)

Question

I'm trying to work through Altland and Simons' example of interacting fermions in one dimension. It's in chapter 2, page 70 (you can find it here).

They define fermionic operators $$ a_{sk}^\dagger $$ where $s=L/R$. $a_{Lk}^\dagger$ is an operator that creates an electron going to the left with momentum $(-k_F+k)$, and $a_{Rk}^\dagger$ is an operator that creates an electron going to the right with momentum $(k_F+k)$. So basically, $a_{Lk}^\dagger=a_{-k_F+k}^\dagger$, $a_{Rk}^\dagger=a_{k_F+k}^\dagger$. These operators are restricted to exist only for small $k$.

Then, they define density operators

$$ \rho_{sq}=\sum_k a^\dagger_{sk+q}a_{sk} $$

They go on to show that the commutation relations for the density operators are

$$ [\rho_{sq},\rho_{s'q'}]=\delta_{s,s'}\sum_k (a^\dagger_{sk+q}a_{sk-q'}-a^\dagger_{sk+q+q'}a_{sk}) $$

Now, here's the part I don't understand. They say they want to replace the right side of the equation with its ground state expectation value. They define the ground state of the theory by $|\Omega\rangle$. Then they claim that

$$ \langle\Omega|a^\dagger_{sk}a_{sk'}|\Omega\rangle = \delta_{kk'} $$

Why should this be true? I understand that in the noninteracting theory, $a^\dagger_{sk}a_{sk'}|\Omega\rangle$ is orthogonal to $|\Omega\rangle$ unless $k=k'$. But in the interacting theory, the ground state could be in a superposition of states that means $\langle\Omega|a^\dagger_{sk}a_{sk'}|\Omega\rangle\neq0$.

They ultimately use this to prove

$$ \langle\Omega|[\rho_{sq},\rho_{s'q'}]|\Omega\rangle = \delta_{s,s'}\delta_{q,-q'}\sum_k\langle\Omega|(a^\dagger_{sk+q}a_{sk+q}-a^\dagger_{sk}a_{sk})|\Omega\rangle $$ and I don't see any other way to prove this.

What am I missing?

Answer 1

$\def\kk{\mathbf{k}} \def\ii{\mathrm{i}} \def\qq{\mathbf{q}}$

You can prove this using translation invariance, without any other assumptions on the nature of the interacting ground state. I am going to give the argument in $D$ dimensions, which obviously holds in $D=1$ as a special case.

Spatial translations of the system as a whole are generated by the centre-of-mass momentum ($\hbar = 1$) $${\bf P} = \sum_{\kk,s} \kk a_{\kk s}^\dagger a_{\kk s},$$ where boldface type denotes a vector in $D$ dimensions. The unitary operator $T_{\bf r} = \mathrm{e}^{\ii {\bf P} \cdot {\bf r}}$ describes a translation of the coordinate system by an amount ${\bf r}$. Now, the crucial assumption is that the interacting ground state satisfies the condition ${\bf P} \lvert \Omega \rangle = 0$, i.e. $T_{\bf r} \lvert \Omega \rangle = \lvert \Omega \rangle $. This holds for any system with translation-invariant interactions, since in that case 1) the Hamiltonian $H$ commutes with ${\bf P}$, so that eigenstates of $H$ are also eigenstates of ${\bf P}$, and 2) eigenstates of ${\bf P}$ gain a positive contribution ${\bf P}^2/2M$ to their energy (in non-relativistic approximation), with $M$ the total mass of the system. Therefore, the ground state $\lvert \Omega \rangle$ lies in the ${\bf P}=0$ sector. Of course, the above formal arguments prove what should already be obvious: a system in its ground state has no overall motion in the lab frame.

The rest of the argument follows from some simple algebra. One readily proves the commutation relation $$ [\mathbf{P},a_{\kk s}] = -\kk a_{\kk s}, $$ which means physically that the ladder operator $a_{\kk s}$ reduces the total momentum of the system by an amount $\kk$, and further implies the translation property $$T_{\bf r} a_{\kk s}T_{\bf r}^\dagger = \mathrm{e}^{-\ii {\bf k} \cdot {\bf r}}a_{\kk s}.$$ It follows that \begin{align} \langle \Omega \rvert a^\dagger_{\kk s}a_{\kk' s} \lvert \Omega \rangle & = \langle \Omega \rvert \left(T_{\bf r}^\dagger T_{\bf r}\right) a^\dagger_{\kk s} \left(T_{\bf r}^\dagger T_{\bf r}\right) a_{\kk' s} \left(T_{\bf r}^\dagger T_{\bf r}\right) \lvert \Omega \rangle \\ & = \langle \Omega \rvert \left(T_{\bf r}a^\dagger_{\kk s}T_{\bf r}^\dagger \right)\left(T_{\bf r} a_{\kk' s}T_{\bf r}^\dagger \right)\lvert \Omega \rangle \\ & = \mathrm{e}^{\ii (\kk - \kk')\cdot {\bf r}}\langle \Omega \rvert a^\dagger_{\kk s}a_{\kk' s} \lvert \Omega \rangle, \end{align} where the first equality follows from unitarity of $T_{\bf r}$, the second from the translation-invariance of the interacting ground state, and the third using the translation property of the ladder operators. Noting that the above equality holds for all ${\bf r}$, we conclude that both sides must vanish unless $\kk = \kk'$.

Finally, it is worth mentioning that this argument does not rely in any way on fermionic statistics, and the same holds true in a bosonic system. Indeed, a similar property holds true for any operator which increases the momentum of the system by a definite amount (thus satisfying the translation property above). This includes the density Fourier components themselves, $$ \rho_{\qq s} = \sum_\kk a^\dagger_{\kk+\qq s}a_{\kk s},$$ which satisfy $$\langle \Omega \rvert \rho_{\qq s}^\dagger \rho_{\qq' s}\lvert \Omega \rangle \propto \delta_{\qq\qq'}.$$ Moreover, the argument also holds for other eigenstates of $\mathbf{P}$ (with non-zero eigenvalue). In fact, it generalises to expectation values with respect to any translation-invariant mixed state $\varrho$, which is any mixture of $\mathbf{P}$-eigenstates (possibly with different momenta) such that $[\varrho,\mathbf{P}]=0$.