General update
This commit is contained in:
Jean-Sébastien Caux
vanhempi
f428a81fca
commit
c4af42cc0b
417
src/index.org
417
src/index.org
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@ -4959,12 +4959,12 @@ A_{n m} (\omega) = \coth \frac{|\omega|}{2} \left( e^{-\frac{|\omega|}{2}|n-m|}
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#+begin_summary
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#+html: Derivation
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#+end_summary
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\begin{eqnarray*}
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A_{n m} &=& e^{-\frac{|\omega|}{2} |n-m|} + 2 \sum_{j=1}^{\mbox{min}(n,m) - 1} e^{-\frac{|\omega|}{2} |n-m|+2j} + e^{-\frac{|\omega|}{2} (n+m)} - \delta_{nm} \nonumber \\
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&=& e^{-\frac{|\omega|}{2} |n-m|} + 2 e^{-\frac{|\omega|}{2} |n-m|} \frac{e^{-|\omega|} - e^{-|\omega| \mbox{min} (n,m)}}{1 - e^{-|\omega|}} + e^{-\frac{|\omega|}{2} (n+m)} - \delta_{nm} \nonumber \\
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&=& \frac{1}{1 - e^{-|\omega|}} \left( e^{-\frac{|\omega|}{2} |n-m|} + e^{-\frac{|\omega|}{2} (|n-m|+2)} - e^{-\frac{|\omega|}{2} (n+m)} - e^{-\frac{|\omega|}{2} (n+m+2)} \right) - \delta_{nm} \nonumber \\
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&=& \frac{1 + e^{-|\omega|}}{1 - e^{-|\omega|}} \left( e^{-\frac{|\omega|}{2} |n-m|} - e^{-\frac{|\omega|}{2} (n+m)} \right) - \delta_{nm}
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\end{eqnarray*}
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\begin{align*}
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A_{n m} &= e^{-\frac{|\omega|}{2} |n-m|} + 2 \sum_{j=1}^{\mbox{min}(n,m) - 1} e^{-\frac{|\omega|}{2} |n-m|+2j} + e^{-\frac{|\omega|}{2} (n+m)} - \delta_{nm} \nonumber \\
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&= e^{-\frac{|\omega|}{2} |n-m|} + 2 e^{-\frac{|\omega|}{2} |n-m|} \frac{e^{-|\omega|} - e^{-|\omega| \mbox{min} (n,m)}}{1 - e^{-|\omega|}} + e^{-\frac{|\omega|}{2} (n+m)} - \delta_{nm} \nonumber \\
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&= \frac{1}{1 - e^{-|\omega|}} \left( e^{-\frac{|\omega|}{2} |n-m|} + e^{-\frac{|\omega|}{2} (|n-m|+2)} - e^{-\frac{|\omega|}{2} (n+m)} - e^{-\frac{|\omega|}{2} (n+m+2)} \right) - \delta_{nm} \nonumber \\
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&= \frac{1 + e^{-|\omega|}}{1 - e^{-|\omega|}} \left( e^{-\frac{|\omega|}{2} |n-m|} - e^{-\frac{|\omega|}{2} (n+m)} \right) - \delta_{nm}
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\end{align*}
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#+end_details
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For future convenience, let us define a kernel inverting \(\delta_{nm} + A_{nm}\).
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One can verify by direct substitution that the following
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@ -6810,11 +6810,302 @@ P_{a b} = \varphi(\eta)\frac{\prod_{l=1}^M \varphi(\lambda_l - \lambda_b + \eta)
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:CUSTOM_ID: d
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:END:
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** Basics
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:PROPERTIES:
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:CUSTOM_ID: d_b
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:END:
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*** Schrödinger, Heisenberg and Interaction pictures
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:PROPERTIES:
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:CUSTOM_ID: d_b_p
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:END:
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For a time-independent Hamiltonian $H$, the Schrödinger equation for states $|\psi^S (t) \rangle$ and its solution can be written
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\begin{equation*}
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i \hbar \partial_t | \psi^S (t) \rangle = H | \psi^S (t) \rangle, \hspace{10mm}
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|\psi^S (t) \rangle = e^{-\frac{i}{\hbar} H t} | \psi^S (t=0) \rangle
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\end{equation*}
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A time-dependent matrix element of some operator ${\cal O}^S$ thus reads
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\begin{equation*}
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\langle \psi^S_1 (t) | {\cal O}^S | \psi^S_2 (t) \rangle
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\end{equation*}
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where states are time-dependent, and operators are time-independent. This is the **Schrödinger picture**.
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In the **Heisenberg picture**, the time dependence is shifted from the states to the operators:
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\begin{equation*}
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\langle \psi^S_1 (t) | {\cal O} | \psi^S_2 (t) \rangle = \langle \psi^S_1 (t=0)| e^{\frac{i}{\hbar} Ht} {\cal O}^S e^{-\frac{i}{\hbar} H t} | \psi^S_2 (t=0) \rangle \equiv \langle \psi^H_1 | {\cal O}^H (t) |\psi^H_2 \rangle
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\end{equation*}
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in which states are time-independent,
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\begin{equation*}
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|\psi^H \rangle \equiv |\psi^S (t=0) \rangle
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\end{equation*}
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and operators ${\cal O}^H (t) = e^{\frac{i}{\hbar} Ht} {\cal O}^S e^{-\frac{i}{\hbar} H t}$ obey the equation of motion
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\begin{equation*}
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\frac{d}{dt} {\cal O}^H (t) = \frac{i}{\hbar} \left[ H, {\cal O}^H (t) \right] + e^{\frac{i}{\hbar} Ht} \partial_t {\cal O}^S e^{-\frac{i}{\hbar} H t} \equiv \frac{i}{\hbar} \left[ H, {\cal O}^H (t) \right] + \left[ \partial_t {\cal O}\right]^H.
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\end{equation*}
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Let us now consider a generic, time-dependent Hamiltonian
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\begin{equation*}
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H(t) = H_0 + V^S(t)
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\end{equation*}
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in which $H_0$ is the time-independent Hamiltonian of some exactly-solvable theory for which we know all the eigenstates, in other words for which we can provide a complete set of states $|\alpha^0 \rangle$ such that
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\begin{equation*}
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H_0 |\alpha^0\rangle = E_{\alpha^0} |\alpha^0\rangle.
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\end{equation*}
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The operator $V^S(t)$ (in the Schrödinger representation) then represents some perturbation/additional interaction which we would like to take into account. The idea of the **interaction representation** is to "Heisenbergize" using only $H_0$, meaning that we define states and operators as (here from their Schrödinger representation)
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\begin{equation*}
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|\psi^I (t) \rangle = e^{\frac{i}{\hbar} H_0 t} |\psi^S (t) \rangle, \hspace{10mm}
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{\cal O}^I (t) = e^{\frac{i}{\hbar} H_0 t} {\cal O}^S e^{-\frac{i}{\hbar} H_0 t}.
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\end{equation*}
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The time evolution of states in the interaction representation can be simply obtained from the Schrödinger equation as
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\begin{equation*}
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i\hbar \partial_t |\psi^I(t) \rangle = e^{\frac{i}{\hbar} H_0 t} \left[ -H_0 + H (t) \right] |\psi^S(t) \rangle = e^{\frac{i}{\hbar} H_0 t} V^S(t) |\psi^S (t) \rangle.
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\end{equation*}
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This can be simply rewritten as
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\begin{equation*}
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i\hbar \partial_t |\psi^I (t) \rangle = V^I (t) |\psi^I (t) \rangle.
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\label{eq:SEIR}
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\end{equation*}
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Thus, in the interaction representation, the change of the phase of a wavefunction is driven solely by the interaction term, and the time evolution of an operator is driven solely by the exactly-solvable part of the Hamiltonian.
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Formally, one can write a solution to the interaction picture Schrödinger equation as
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\begin{equation*}
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|\psi^I(t) \rangle = U^I (t, t_0) | \psi^I (t_0) \rangle
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\end{equation*}
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in terms of the propagator $U^I$ in the interaction representation. If $V^S (t)$ is in fact time-independent, we immediately have
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\begin{equation*}
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U^I (t, t_0) = e^{\frac{i}{\hbar} H_0 t} e^{-\frac{i}{\hbar} H (t - t_0)} e^{-\frac{i}{\hbar} H_0 t_0}.
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\end{equation*}
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For a generic time-dependent $V^S (t)$, we have
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\begin{equation*}
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i\hbar \partial_t U^I (t, t_0) |\psi^I(t_0) \rangle = V^I (t) U^I (t, t_0) |\psi^I (t_0) \rangle
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\end{equation*}
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so the propagator satisfies the equation (with obvious boundary condition)
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\begin{equation*}
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i\hbar \partial_t U^I (t, t_0) = V^I (t) U^I (t, t_0), \hspace{10mm}
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U^I (t_0, t_0) = 1.
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\end{equation*}
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We can write an iterative solution to this. Integrating from $t_0$ to $t$ gives
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\begin{equation*}
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U^I (t, t_0) = 1 + \frac{-i}{\hbar} \int_{t_0}^t dt' V^I (t') U^I (t', t_0)
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\end{equation*}
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so we can develop the perturbative series
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\begin{equation*}
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U^I (t, t_0) = 1 + \frac{-i}{\hbar} \int_{t_0}^t dt' V^I (t') + \left(\frac{-i}{\hbar} \right)^2 \int_{t_0}^t dt_1 V^I (t_1) \int_{t_0}^{t_1} dt_2 V^I (t_2) + ...
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\end{equation*}
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This series can be represented as
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\begin{align*}
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U^I (t, t_0) &= \sum_{n=0}^\infty \left(\frac{-i}{\hbar}\right)^n \int_{t_0}^t dt_1 \int_{t_0}^{t_1} dt_2 ... \int_{t_0}^{t_{n-1}} dt_n V^I (t_1) ... V^I(t_n) \nonumber \\
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&= \sum_{n=0}^\infty \frac{(-i/\hbar)^n}{n!} \int_{t_0}^t dt_1 ... dt_n T_t \left[ V^I(t_1) ... V^I (t_n) \right]
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\end{align*}
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in which we have introduced the time-ordering operator $T_t$ acting as (here, for ${\cal O}$ operators which are bosonic in character)
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\begin{equation*}
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T_t \left[ {\cal O}_1^I (t_1) {\cal O}_2^I (t_2) \right] \equiv \left\{ \begin{array}{ll} {\cal O}_1^I (t_1) {\cal O}_2^I (t_2), & t_1 > t_2 \\ \\ {\cal O}_2^I (t_2) {\cal O}_1^I (t_1), & t_1 < t_2 \end{array} \right.
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\end{equation*}
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(with straightforward generalization to an arbitrary product of operators at different times).
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The propagator in the interaction representation is thus compactly represented as
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\begin{equation*}
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U^I (t, t_0) = T_t \left[ e^{-\frac{i}{\hbar} \int_{t_0}^t dt' V^I (t')} \right].
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\end{equation*}
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*** Fermi's Golden Rule
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:PROPERTIES:
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:CUSTOM_ID: d_b_F
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:END:
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Introducing some perturbation into a system generates some generally very complex time-dependent behaviour. One way to picture this time dependence is to still consider the original unperturbed basis of states, but have time-dependent state amplitudes. The probability of finding the system in a given state thus becomes time-dependent. For a small perturbation, the rates at which this probability flows from one state to another is given by **Fermi's Golden rule**.
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Let us consider an exactly-solvable, time-independent Hamiltonian $H_0$ for which we know a basis of eigenstates
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\begin{equation*}
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H_0 |\alpha^0 \rangle = E_{\alpha^0} |\alpha^0 \rangle.
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\end{equation*}
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Let us consider perturbing this theory with a time-dependent operator which is adiabatically turned on from $t = -\infty$ onwards:
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\begin{equation*}
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V(t) = V e^{-i \omega t + \eta t}, \hspace{20mm} \eta \rightarrow 0^+
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\end{equation*}
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(we will evaluate this $t$ for times much less than $1/\eta$). Here, $V$ is some perturbing operator in the Schrödinger representation.
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We now address the following question. If the initial state is
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\begin{equation*}
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|\psi^S (t = t_0) \rangle = |\alpha^0_i \rangle
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\end{equation*}
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for some initial state $|\alpha^0_i\rangle$, what is the probability amplitude for finding the system in state $|\alpha^0_f \rangle$ (with $i \neq f$) at time $t$?
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In the interaction representation, we had \(|\psi^I (t) \rangle = U^I (t, t_0) |\psi^I (t_0) \rangle.\) Since
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\(|\psi^I(t) \rangle = e^{\frac{i}{\hbar} H_0 t} |\psi^S(t) \rangle\), we have \(|\psi^I(t_0)\rangle = e^{\frac{i}{\hbar} H_0 t_0} |\alpha^0_i \rangle\) and thus
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\begin{equation*}
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|\psi^S(t) \rangle = e^{-\frac{i}{\hbar} H_0 t} U^I (t, t_0) e^{\frac{i}{\hbar} H_0 t_0} |\alpha^0_i\rangle.
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\end{equation*}
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Consider now calculating the amplitude for being in state $|\alpha^0_f\rangle$, $f \neq i$, at time $t$:
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\begin{equation*}
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\langle \alpha^0_f | \psi^S(t) \rangle = \langle \alpha^0_f | e^{-\frac{i}{\hbar} H_0 t} U^I (t, t_0) e^{\frac{i}{\hbar} H_0 t_0} | \alpha^0_i \rangle
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= e^{-\frac{i}{\hbar} E_{\alpha^0_f} t + \frac{i}{\hbar} E_{\alpha^0_i} t_0} \langle \alpha^0_f | U^I (t, t_0) | \alpha^0_i \rangle.
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\end{equation*}
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Using the series expansion for the propagator and keeping only the linear (in $V$) response, the matrix element of the propagator can be written
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\begin{equation*}
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\langle \alpha^0_f | U^I (t, t_0) | \alpha^0_i \rangle
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= \langle \alpha^0_f | \alpha^0_i \rangle - \frac{i}{\hbar} \int_{t_0}^t dt' \langle \alpha^0_f | V^I(t') | \alpha^0_i \rangle + ...
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\end{equation*}
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The first term vanishes (we are looking for transition rates, so the initial and final states are different, $f \neq i$). The second term gives (substituting the explicit form of the perturbation)
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\begin{equation*}
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-\frac{i}{\hbar} \int_{t_0}^t dt' \langle \alpha^0_f| e^{\frac{i}{\hbar} H_0 t'} V (t') e^{-\frac{i}{\hbar} H_0 t'} |\alpha^0_i \rangle = -\frac{i}{\hbar} \int_{t_0}^t dt' e^{\frac{i}{\hbar} [E_{\alpha^0_f} - E_{\alpha^0_i} - \hbar \omega - i \hbar \eta] t'} \langle \alpha_f^0 | V | \alpha_i^0 \rangle.
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\end{equation*}
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Since the matrix element is time-independent, we can perform the time integral, giving us
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\begin{equation*}
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\langle \alpha^0_f | U^I (t, t_0) | \alpha^0_i \rangle
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= -\frac{\langle \alpha_f^0 | V | \alpha_i^0 \rangle}{E_{\alpha^0_f} - E_{\alpha^0_i} - \hbar \omega - i \hbar \eta} e^{\frac{i}{\hbar} [E_{\alpha^0_f} - E_{\alpha^0_i} - \hbar \omega - i \hbar \eta] t'}|_{t_0}^t + \mbox{O} (V^2).
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\end{equation*}
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Let us now take the limit $t_0 \rightarrow -\infty$, with $\eta \rightarrow 0^+$ but $\eta t_0 \rightarrow -\infty$ while keeping $t$ finite (that is, the perturbation is turned on at a vanishingly slow rate from the infinite past). This gives us
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\begin{equation*}
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\langle \alpha^0_f | \psi^S(t) \rangle = \frac{\langle \alpha_f^0 | V | \alpha_i^0 \rangle}{\hbar \omega - (E_{\alpha^0_f} - E_{\alpha^0_i}) + i\eta \hbar} e^{-\frac{i}{\hbar} E_{\alpha_i^0} (t-t_0)} e^{-i\omega t + \eta t} + O(V^2).
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\end{equation*}
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The probability to be in state $|\alpha^0_f\rangle$ at time $t$, given that we initially started in state $|\alpha^0_i\rangle$ is thus
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\begin{equation*}
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P_{f \leftarrow i} (t) = |\langle \alpha^0_f | \psi^S(t) \rangle|^2 = \frac{|\langle \alpha_f^0 | V | \alpha_i^0 \rangle|^2 e^{2\eta t}}{(\hbar \omega - (E_{\alpha^0_f} - E_{\alpha^0_i}))^2 + \eta^2 \hbar^2}.
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\end{equation*}
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The rate at which this probability changes is thus
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\begin{equation*}
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\frac{d}{dt} P_{f \leftarrow i} (t) = |\langle \alpha_f^0 | V | \alpha_i^0 \rangle|^2 \lim_{\eta \rightarrow 0^+} \frac{2\eta}{(\hbar \omega - (E_{\alpha^0_f} - E_{\alpha^0_i}))^2 + \eta^2 \hbar^2}.
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\end{equation*}
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Using the representation of the Dirac delta function $\delta(x) = \lim_{\eta \rightarrow 0^+} \frac{1}{\pi} \frac{\eta}{x^2 + \eta^2}$ then yields **Fermi's Golden rule**
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#+begin_eqlabel
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<<Fgr>>
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#+begin_alteqlabels
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#+end_alteqlabels
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#+end_eqlabel
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\begin{equation}
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\frac{d}{dt} P_{f \leftarrow i} (t) = \frac{2\pi}{\hbar} |\langle \alpha_f^0 | V | \alpha_i^0 \rangle|^2 \delta (\hbar \omega - (E_{\alpha^0_f} - E_{\alpha^0_i}))
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\tag{Fgr}\label{Fgr}
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\end{equation}
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*** Correlators; Lehmann representation
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:PROPERTIES:
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:CUSTOM_ID: d_b_c
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:END:
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For a given specific state \(\alpha\) and (Hermitian) operator \({\cal O}\), we will define the dynamical correlator
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\begin{equation*}
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S_\alpha (j, j'; t, t') \equiv \langle \alpha | O_j (t) O_{j'} (t') | \alpha \rangle.
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\end{equation*}
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If the system exhibits translational invariance, and if the Hamiltonian is time-independent, then the correlator becomes a function of the position and time differences only,
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\begin{equation*}
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S_\alpha (j, j'; t, t') = S_\alpha (j - j', t - t').
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\end{equation*}
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Let us introduce the Fourier representation
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\begin{equation*}
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S_\alpha (k, \omega) \equiv \frac{1}{N} \sum_{j, j'} e^{-i k (j-j')} \int_{-\infty}^\infty dt e^{i \omega t} S_\alpha (j-j', t)
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\end{equation*}
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for the dynamical correlation, and
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\begin{equation*}
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{\cal O}_j = \frac{1}{N} \sum_k e^{ikj} {\cal O}_k, \hspace{5mm}
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{\cal O}_k = \sum_j e^{-ikj} {\cal O}_j
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\end{equation*}
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for the operator. Introducing a resolution of the identity \({\bf 1} = \sum_{\alpha} | \alpha \rangle \langle \alpha |\) in terms of eigenstates of the Hamiltonian, we have
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\begin{align*}
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S_\alpha (k, \omega) &= \frac{1}{N} \int_{-\infty}^\infty dt e^{i \omega t} \langle \alpha | e^{iHt} \sum_j e^{-ikj} {\cal O}_j e^{-iHt} \sum_{\alpha'} | \alpha' \rangle \langle \alpha' | \sum_{j'} e^{i k j'} {\cal O}_{j'} | \alpha \rangle \\
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&= \frac{1}{N} \sum_{\alpha'} \int_{-\infty}^\infty dt e^{i (\omega + E_\alpha - E_{\alpha'}) t}
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\langle \alpha | {\cal O}_k | \alpha' \rangle \langle \alpha' | O_{-k} | \alpha \rangle.
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\end{align*}
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This can be rewritten using the identity \(\int_{-\infty}^\infty dt e^{i (\omega - \omega') t} = 2\pi \delta (\omega - \omega')\) (and \((O_k)^\dagger = O_{-k}\) for a Hermitian operator), yielding the **Lehmann representation**
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#+begin_eqlabel
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<<Lr>>
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#+begin_alteqlabels
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#+end_alteqlabels
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#+end_eqlabel
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\begin{equation}
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S_\alpha (k, \omega) = \frac{2\pi}{N} \sum_{\alpha'} |\langle \alpha | O_k | \alpha' \rangle|^2 \delta(\omega - (E_{\alpha'} - E_\alpha)).
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\tag{Lr}\label{Lr}
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\end{equation}
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The physical content of the Lehmann representation is made clear by Fermi's Golden rule [[Fgr][Fgr]]: starting from the state \(\alpha\), one entry of the operator induces a transition to the excited state \(\alpha'\) which lives at momentum/energy \(k, \omega\) above \(\alpha\), and acts as a mediator of correlations. The full correlator is given by the sum over all accessible intermediate states \(\alpha'\).
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Note the very important fact that through the Lehmann representation, the dynamical correlator is given by a sum of strictly non-negative terms. This is of immense importance for the practical utility of [[#d_sr][sum rules]].
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*** Detailed balance
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:PROPERTIES:
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:CUSTOM_ID: d_b_db
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:END:
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Let's consider the following correlator
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\begin{equation*}
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S (k, \omega) = \frac{1}{N} \sum_{j, j'} e^{-i k (j-j')} \int_{-\infty}^\infty dt e^{i \omega t} \langle {\cal O}_j (t) {\cal O}_{j'} (0) \rangle_T
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\end{equation*}
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in which the average is a thermal trace in the eigenstates basis (with \(\beta \equiv k_B T\)),
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\begin{equation*}
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\langle \cdots \rangle_T \equiv \frac{1}{Z} \sum_{\alpha} \langle \alpha | \cdots | \alpha \rangle e^{-\beta E_\alpha},
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\hspace{10mm}
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Z \equiv \sum_\alpha e^{-\beta E_\alpha}
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\end{equation*}
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The Lehmann representation is easily obtained by using the single-state version [[Lr][Lr]] and performing the Gibbs sum:
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\begin{align*}
|
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S(k, \omega) = \sum_{\alpha} \frac{e^{-\beta E_\alpha}}{Z} S_\alpha (k, \omega) = \frac{2\pi}{N} \sum_{\alpha, \alpha'} \frac{e^{-\beta E_\alpha}}{Z} |\langle \alpha | O_k | \alpha' \rangle|^2 ~\delta(\omega - (E_{\alpha'} -E_\alpha))
|
||||
\end{align*}
|
||||
|
||||
Simple manipulations (interchanging the \(\alpha, \alpha'\) summations and using the delta function) then gives the **detailed balance** condition
|
||||
#+begin_eqlabel
|
||||
<<db>>
|
||||
#+begin_alteqlabels
|
||||
#+end_alteqlabels
|
||||
#+end_eqlabel
|
||||
\begin{equation}
|
||||
S(-k, -\omega) = e^{-\beta \omega} S(k, \omega)
|
||||
\tag{db}\label{db}
|
||||
\end{equation}
|
||||
which is valid for any dynamical correlator at thermal equlibrium.
|
||||
|
||||
|
||||
** Sum rules
|
||||
:PROPERTIES:
|
||||
:CUSTOM_ID: d_sr
|
||||
:END:
|
||||
|
||||
|
||||
*** Integrated intensity
|
||||
:PROPERTIES:
|
||||
:CUSTOM_ID: d_sr_ii
|
||||
:END:
|
||||
|
||||
The simplest sum rules which one can find relate to simple integrals of the dynamical correlators. First, the **static correlator** is obtained by setting the time difference to zero:
|
||||
\begin{equation*}
|
||||
S_\alpha (j - j') \equiv S_\alpha (j-j', 0)
|
||||
\end{equation*}
|
||||
or in Fourier space,
|
||||
\begin{equation*}
|
||||
S_\alpha (k) \equiv \int_{-\infty}^\infty \frac{d\omega}{2\pi} S_\alpha (k, \omega)
|
||||
= \sum_{\alpha'} | \langle \alpha | O_k | \alpha' \rangle|^2 \delta_{k, K_{\alpha'} - K_\alpha}
|
||||
\end{equation*}
|
||||
where in the last equality we have made use of the Lehmann representation [[Lr][Lr]] and explicitly enforced the requirement that the momentum difference between the excited state and the averaging state is equal to \(k\).
|
||||
|
||||
Separately, we can also consider the **autocorrelator**
|
||||
\begin{equation*}
|
||||
S_\alpha (t) \equiv S_\alpha (0, t)
|
||||
\end{equation*}
|
||||
which in Fourier space reads
|
||||
\begin{equation*}
|
||||
S_\alpha (\omega) = \frac{1}{N} \sum_k S_\alpha (k, \omega).
|
||||
\end{equation*}
|
||||
|
||||
Combining these two ideas, we get the **static autocorrelator** or **static moment**
|
||||
\begin{equation*}
|
||||
S_\alpha \equiv S_\alpha (0, 0) = \langle \alpha | {\cal O}_j {\cal O}_j | \alpha \rangle
|
||||
\end{equation*}
|
||||
The idea is that if the equal-time product \({\cal O} {\cal O}\) simplifies to a known or easy to evaluate quantity, then we obtain a constraint on the (integral of the) dynamical correlator.
|
||||
|
||||
**Spin chain**
|
||||
|
||||
For example, if we are studying a spin-\(1/2\) chain, since we know that \(S^a_j S^a_j = 1/4\) for any component or site, we obtain the sum rule
|
||||
\begin{equation*}
|
||||
S^{aa}_\alpha = \frac{1}{4} = \frac{1}{N} \sum_k \int_{-\infty}^\infty \frac{d\omega}{2\pi} S_\alpha^{aa} (k, \omega)
|
||||
\end{equation*}
|
||||
|
||||
|
||||
*** The f-sumrule
|
||||
:PROPERTIES:
|
||||
:CUSTOM_ID: d_sr_f
|
||||
|
|
@ -6824,7 +7115,7 @@ Consider a model with lattice sites \(j = 1, ..., N\).
|
|||
Let's look at the following generic correlation function of some operator \({\cal O}_j^a\), where \(a\) is some label
|
||||
(for example, for a lattice Bose gas, we could have \({\cal O}_j = \Psi_j\) or \(\Psi^\dagger_j\) or \(\rho_j \equiv \Psi_j^\dagger \Psi_j\)):
|
||||
\begin{equation*}
|
||||
{\cal S}^{a \bar{a}} (k, \omega) \equiv \frac{1}{N} \sum_{j, j'} e^{-i k(j-j')}
|
||||
{\boldsymbol S}^{a \bar{a}} (k, \omega) \equiv \frac{1}{N} \sum_{j, j'} e^{-i k(j-j')}
|
||||
\int_{-\infty}^\infty dt e^{i\omega t} \langle \frac{1}{2} \left[ {\cal O}^a_j (t), ({\cal O}^a_{j'} (0))^\dagger \right] \rangle
|
||||
\end{equation*}
|
||||
The expectation value \(\langle ... \rangle\) can be any expectation value: with respect to a specific state,
|
||||
|
|
@ -6832,7 +7123,7 @@ The expectation value \(\langle ... \rangle\) can be any expectation value: wit
|
|||
\frac{1}{\cal Z} \sum_{\gamma} \langle \gamma | ... | \gamma \rangle e^{-\beta (E_{\gamma} - \mu N_{\gamma})}\), it doesn't matter.
|
||||
In the case of a (grand-canonical) thermal average, we can write
|
||||
\begin{equation*}
|
||||
{\cal S}^{a \bar{a}} (k, \omega) \equiv \frac{1}{N} \sum_{j, j'} e^{-i k(j-j')}
|
||||
{\boldsymbol S}^{a \bar{a}} (k, \omega) \equiv \frac{1}{N} \sum_{j, j'} e^{-i k(j-j')}
|
||||
\int_{-\infty}^\infty dt e^{i\omega t} \frac{1}{\cal Z} \sum_\gamma \langle \gamma | \frac{1}{2} \left[ {\cal O}^a_j (t), ({\cal O}^a_{j'} (0))^\dagger \right] | \gamma \rangle e^{-\beta (E_\gamma - \mu N_\gamma)}
|
||||
\end{equation*}
|
||||
in which \(\mu\) is the chemical potential, \({\cal Z}\) is the grand-canonical partition function,
|
||||
|
|
@ -6840,15 +7131,15 @@ and \(\sum_{\gamma}\) represents a sum over eigenstates.
|
|||
|
||||
Consider integrating this correlation over all frequencies, calculating the first moment in frequency
|
||||
\begin{equation*}
|
||||
I_k^{(1)} \equiv \int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\cal S}^{a \bar{a}} (k, \omega)
|
||||
I_k^{(1)} \equiv \int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\boldsymbol S}^{a \bar{a}} (k, \omega)
|
||||
\end{equation*}
|
||||
This can be manipulated as follows:
|
||||
\begin{align*}
|
||||
I_k^{(1)} &=& \frac{1}{2N} \int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega \int_{-\infty}^\infty dt e^{i\omega t}
|
||||
I_k^{(1)} &= \frac{1}{2N} \int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega \int_{-\infty}^\infty dt e^{i\omega t}
|
||||
\frac{1}{\cal Z} \sum_\gamma \langle \gamma | \left[ {\cal O}^a_k (t), ({\cal O}^a_{k'} (0))^\dagger \right] | \gamma \rangle e^{-\beta (E_\gamma - \mu N_\gamma)} \nonumber \\
|
||||
&=& \frac{1}{2N} \int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega \int_{-\infty}^\infty dt e^{i\omega t}
|
||||
&= \frac{1}{2N} \int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega \int_{-\infty}^\infty dt e^{i\omega t}
|
||||
\frac{1}{\cal Z} \times \nonumber \\
|
||||
&&\times \sum_\gamma \sum_{\alpha} \left\{ e^{-i\omega_{\alpha \gamma}t} \langle \gamma | {\cal O}^a_k | \alpha \rangle
|
||||
&\times \sum_\gamma \sum_{\alpha} \left\{ e^{-i\omega_{\alpha \gamma}t} \langle \gamma | {\cal O}^a_k | \alpha \rangle
|
||||
\langle \alpha | ({\cal O}^a_k)^\dagger | \gamma \rangle - e^{i\omega_{\alpha \gamma}t} \langle \gamma | ({\cal O}^a_k)^\dagger | \alpha \rangle
|
||||
\langle \alpha | {\cal O}^a_k | \gamma \rangle \right\} e^{-\beta (E_\gamma - \mu N_\gamma)}
|
||||
\end{align*}
|
||||
|
|
@ -6890,10 +7181,16 @@ e^{-\beta (E_\gamma - \mu N_\gamma)}
|
|||
in which we have used the fact that we're working in a basis of energy eigenstates (so \(H | \alpha \rangle = (E_\alpha - \mu N_\alpha) | \alpha \rangle\)) and the resolution of the identity \(\sum_\alpha | \alpha \rangle \langle \alpha | = {\bf 1}\).
|
||||
|
||||
We thus get
|
||||
\begin{equation*}
|
||||
\int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\cal S}^{a \bar{a}} (k, \omega) = \frac{-1}{2N} \langle \left[ \left[ H, {\cal O}^a_k \right], ({\cal O}^a_k)^\dagger \right] \rangle
|
||||
\end{equation*}
|
||||
(note: this holds irrespective of how the average \(\langle ... \rangle\) is defined, provided \(S(k,\omega)\) and the right-hand side of this equation are averaged in precisely the same way). This relation holds for any correlator of any model.
|
||||
#+begin_eqlabel
|
||||
<<fsr>>
|
||||
#+begin_alteqlabels
|
||||
#+end_alteqlabels
|
||||
#+end_eqlabel
|
||||
\begin{equation}
|
||||
\int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\boldsymbol S}^{a \bar{a}} (k, \omega) = \frac{-1}{2N} \langle \left[ \left[ H, {\cal O}^a_k \right], ({\cal O}^a_k)^\dagger \right] \rangle
|
||||
\tag{fsr}\label{fsr}
|
||||
\end{equation}
|
||||
(note: this holds irrespective of how the average \(\langle ... \rangle\) is defined, provided \({\boldsymbol S}(k,\omega)\) and the right-hand side of this equation are averaged in precisely the same way). This relation holds for any correlator of any model.
|
||||
|
||||
Now: the magic trick comes from the fact that for specific cases, the right-hand side can be calculated exactly.
|
||||
|
||||
|
|
@ -6924,7 +7221,7 @@ H_{LL} = \frac{1}{L} \sum_k (k^2 - \mu) \Psi^\dagger_k \Psi_k + \frac{c}{L^3}
|
|||
|
||||
The dynamical structure factor in our notations is then
|
||||
\begin{equation*}
|
||||
{\cal S}^{\rho \rho} (k, \omega) = \frac{1}{L} \int_0^L dx dx' e^{-ik (x - x')} \int_{-\infty}^\infty dt e^{i\omega t}
|
||||
{\boldsymbol S}^{\rho \rho} (k, \omega) = \frac{1}{L} \int_0^L dx dx' e^{-ik (x - x')} \int_{-\infty}^\infty dt e^{i\omega t}
|
||||
\langle \frac{1}{2}\left[ \rho(x,t), \rho(x',0) \right] \rangle
|
||||
\end{equation*}
|
||||
For the density operator, we write
|
||||
|
|
@ -6934,7 +7231,7 @@ For the density operator, we write
|
|||
\end{equation*}
|
||||
For the f-sumrule applied to the dynamical structure factor, we get
|
||||
\begin{equation*}
|
||||
\int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\cal S}^{\rho \rho} (k, \omega) = \frac{-1}{2L} \langle \left[ \left[ H, \rho_k \right], \rho_{-k} \right] \rangle.
|
||||
\int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\boldsymbol S}^{\rho \rho} (k, \omega) = \frac{-1}{2L} \langle \left[ \left[ H, \rho_k \right], \rho_{-k} \right] \rangle.
|
||||
\end{equation*}
|
||||
Let's calculate the right-hand side of the f-sumrule. Start with
|
||||
\begin{equation*}
|
||||
|
|
@ -6982,13 +7279,63 @@ We thus obtain the explicit f-sumrule for the dynamical structure factor,
|
|||
#+end_alteqlabels
|
||||
#+end_eqlabel
|
||||
\begin{equation}
|
||||
\int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\cal S}^{\rho \rho} (k, \omega) =
|
||||
\int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\boldsymbol S}^{\rho \rho} (k, \omega) =
|
||||
\frac{N}{L} k^2.
|
||||
\tag{l.dsf.fsr}\label{l.dsf.fsr}
|
||||
\end{equation}
|
||||
This allows to check the overall intensity for fixed values of \(k\).
|
||||
|
||||
|
||||
**** \(XXZ\) chain
|
||||
:PROPERTIES:
|
||||
:CUSTOM_ID: d_sr_f_xxz
|
||||
:END:
|
||||
|
||||
Let us, for convenience, temporarily consider a fully anisotropic chain with Hamiltonian
|
||||
\begin{equation*}
|
||||
H = \sum_j \left[ J_x S^x_j S^x_{j+1} + J_y S^y_j S^y_{j+1} + J_z S^z_j S^z_{j+1} - h S^z \right]
|
||||
\end{equation*}
|
||||
|
||||
We will consider the dynamical spin-spin correlations (with \(a, b = x, y, z\))
|
||||
\begin{equation*}
|
||||
{\boldsymbol S}^{ab} (k, \omega) = \frac{1}{N} \sum_{j, j'} e^{-i k (j - j')} \int_{-\infty}^\infty dt e^{i \omega (t-t')} \frac{1}{2} \langle \left[ S^a_j (t), S^b_{j'} (t') \right] \rangle
|
||||
\end{equation*}
|
||||
|
||||
The f-sumrule [[fsr][fsr]] here takes the form
|
||||
\begin{equation*}
|
||||
\int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\boldsymbol S}^{ab} (k, \omega)
|
||||
= \sum_{j, j'} e^{-i k (j - j')} \frac{-1}{2N}
|
||||
\langle \left[ \left[H, S^a_j \right], S^b_{j'} \right] \rangle
|
||||
\end{equation*}
|
||||
|
||||
Defining the first frequency moments
|
||||
\begin{equation*}
|
||||
S^{ab}_1 (k) \equiv \int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\boldsymbol S}^{ab} (k, \omega)
|
||||
\end{equation*}
|
||||
and the exchange operators
|
||||
\begin{equation*}
|
||||
X^a \equiv \sum_j S^a_j S^a_{j+1}
|
||||
\end{equation*}
|
||||
we obtain
|
||||
\begin{align*}
|
||||
S^{xx}_1 (k) &= \frac{-1}{N} \left( (J_y - J_z \cos k) \langle X^y \rangle + (J_z - J_y \cos k) \langle X^z \rangle - \frac{h}{2} S^z_{tot} \right), \\
|
||||
S^{yy}_1 (k) &= \frac{-1}{N} \left( (J_z - J_x \cos k) \langle X^z \rangle + (J_x - J_z \cos k) \langle X^x \rangle - \frac{h}{2} S^z_{tot} \right), \\
|
||||
S^{zz}_1 (k) &= \frac{-1}{N} \left( (J_x - J_y \cos k) \langle X^x \rangle + (J_y - J_x \cos k) \langle X^y \rangle \right).
|
||||
\end{align*}
|
||||
|
||||
Specializing to the \(XXZ\) case with \(J_x = J_y = J\) and \(J_z = J\Delta\), we get
|
||||
\begin{align*}
|
||||
S^{xx}_1 (k) &= S^{yy}_1 (k) = \frac{-1}{N} \left( (1 - \Delta \cos k) J \langle X^x \rangle + (\Delta - \cos k) J \langle X^z \rangle - \frac{h}{2} S^z_{tot} \right), \\
|
||||
S^{zz}_1 (k) &= \frac{-2}{N} \sin^2 \frac{k}{2} J \langle X^x \rangle
|
||||
\end{align*}
|
||||
|
||||
In the special case of ground-state correlations, the expectation values of the exchange operators can be obtained from the anisotropy dependence of the energy (using the explicit form [[xxz.h][xxz.h]] for the Hamiltonian):
|
||||
\begin{align*}
|
||||
\langle X^x \rangle &= \langle X^y \rangle = \frac{1}{2J} \left( E_0 - \Delta \frac{\partial E_0}{\partial \Delta} \right), \\
|
||||
\langle X^z \rangle &= \frac{1}{J} \left(\frac{\partial E_0}{\partial \Delta} + \frac{N}{4} \right).
|
||||
\end{align*}
|
||||
|
||||
|
||||
** Lieb-Liniger
|
||||
:PROPERTIES:
|
||||
:CUSTOM_ID: d_l
|
||||
|
|
@ -7399,23 +7746,23 @@ Zeit. für Physik
|
|||
:CUSTOM_ID: 1938.Hulthen.AMAF.26A
|
||||
:END:
|
||||
|
||||
L. Hulth{\'e}n,
|
||||
/\"U}ber das Austauschproblem eines Kristalles/,
|
||||
L. Hulthén,
|
||||
/Über das Austauschproblem eines Kristalles/,
|
||||
Arkiv Mat. Astron. Fysik **26A**, 1 (1938), doi:[[https://doi.org/][]].
|
||||
|
||||
| <20> | <60> |
|
||||
| Extended data | |
|
||||
|------------------+------|
|
||||
| Author | L. Hulth{\'e}n |
|
||||
| Title | \"U}ber das Austauschproblem eines Kristalles |
|
||||
| Journal | Arkiv Mat. Astron. Fysik |
|
||||
| Volume | 26A |
|
||||
| Pages | 1 |
|
||||
| Year | 1938 |
|
||||
| doi | [[https://doi.org/][]] |
|
||||
| Publication date | |
|
||||
|------------------+------|
|
||||
| Submission date | |
|
||||
| <20> | <60> |
|
||||
| Extended data | |
|
||||
|------------------+------------------------------------------------|
|
||||
| Author | L. Hulthén |
|
||||
| Title | Über das Austauschproblem eines Kristalles |
|
||||
| Journal | Arkiv Mat. Astron. Fysik |
|
||||
| Volume | 26A |
|
||||
| Pages | 1 |
|
||||
| Year | 1938 |
|
||||
| doi | [[https://doi.org/][]] |
|
||||
| Publication date | |
|
||||
|------------------+------------------------------------------------|
|
||||
| Submission date | |
|
||||
|
||||
#+begin_details
|
||||
#+begin_summary
|
||||
|
|
@ -9256,7 +9603,7 @@ Commun. Math. Phys. **86**, 391-418 (1982), doi:[[https://doi.org/10.1007/BF0121
|
|||
issue = {3},
|
||||
doi = {10.1007/BF01212176},
|
||||
OPTnote = {10.1007/BF01212176},
|
||||
abstract = {A class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered. Eigenfunctions of the Hamiltonians are known for these models. Norms of these eigenfunctions in the finite box are calculated in the present paper. These models include in particular the quantum nonlinear Schr{\"o}dinger equation and the Heisenberg XXZ model.},
|
||||
abstract = {A class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered. Eigenfunctions of the Hamiltonians are known for these models. Norms of these eigenfunctions in the finite box are calculated in the present paper. These models include in particular the quantum nonlinear Schrödinger equation and the Heisenberg XXZ model.},
|
||||
year = {1982}
|
||||
}
|
||||
|
||||
|
|
|
|||
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