Numerical Correlation Functions

A correlation function requires operator ordering, a state or ensemble, and a time convention. Its numerical evaluation can use eigenstates, time propagation, or Krylov resolvents. These methods should agree on small systems, but they expose different errors and resolution limits.

Helpful background. Many-body correlation definitions explains the physical distinctions; exact diagonalization supplies finite-system states.

Operator-applied states

Use $A(t)=e^{iHt/\hbar}Ae^{-iHt/\hbar}$ and a normalized energy eigenstate $|0\rangle$ with energy $E_0$. Define

$$C_{AB}(t)=\langle0|A(t)B|0\rangle =e^{iE_0t/\hbar}\langle0|A e^{-iHt/\hbar}B|0\rangle.$$

Construct $|b\rangle=B|0\rangle$ once and propagate it. It is generally not normalized; its norm contains the total spectral weight. Normalizing it for a propagator is permissible only if that norm is restored in the final contraction.

For general $A,B$, replacing only the ket by $|\widehat b\rangle=|b\rangle/\|b\|$ requires multiplying the final contraction by $\|b\|$. For the autocorrelation with $A=B^\dagger$, it is often convenient to normalize both sides:

$$C_{B^\dagger B}(t)= \|b\|^2\langle\widehat b| e^{-i(H-E_0)t/\hbar}|\widehat b\rangle.$$

Here the factor is $\|b\|^2$. If $b=0$, the correlation vanishes and there is no state to normalize. Distinguishing these two implementations prevents a missing square root in otherwise plausible spectra.

If $B$ changes a conserved quantum number, propagate in the target sector. Keeping the ground-state sector can make a nonzero correlation vanish. For connected correlations subtract $\langle A\rangle\langle B\rangle$ explicitly.

Lehmann representation and sum rules

Inserting energy eigenstates gives

$$C_{AB}(t)=\sum_n \langle0|A|n\rangle\langle n|B|0\rangle e^{-i(E_n-E_0)t/\hbar}.$$

For $A=B^\dagger$, each weight is nonnegative. With Fourier convention $S(\omega)=\int_{-\infty}^{\infty}dt\,e^{i\omega t}C(t)$,

$$S(\omega)=2\pi\sum_n|\langle n|B|0\rangle|^2 \delta\!\left(\omega-\frac{E_n-E_0}{\hbar}\right).$$

Thus $\int d\omega\,S(\omega)/(2\pi)=C(0)=\|B|0\rangle\|^2$. Test this sum rule after any broadening or Fourier transform. Missing factors of $\hbar$ arise when switching between energy and angular-frequency axes.

At finite temperature, with $p_m=e^{-\beta E_m}/Z$, the corresponding spectrum is

$$S_{B^\dagger B}(\omega)= 2\pi\sum_{m,n}p_m|\langle n|B|m\rangle|^2 \delta\!\left(\omega-\frac{E_n-E_m}{\hbar}\right).$$

For Hermitian $B$, exchanging $m,n$ gives detailed balance, $S(-\omega)=e^{-\beta\hbar\omega}S(\omega)$, interpreted as a relation between line weights. For non-Hermitian $B$, the reversed process involves the $BB^\dagger$ spectrum instead. Include all sectors needed by both the thermal ensemble and the operator action.

A single-spin frequency test

Take $H=-\hbar\omega_0Z/2$, $\omega_0>0$, ground state $|0\rangle$, and $B=X$. Since $X|0\rangle=|1\rangle$,

$$C_{XX}(t)=e^{-i\omega_0t},\qquad S_{XX}(\omega)=2\pi\delta(\omega-\omega_0).$$

The peak is at positive frequency under the stated transform. A peak at negative frequency usually indicates a Fourier-sign or operator-time convention mismatch, not an unexpected physical excitation.

For the same system, the connected $ZZ$ correlation vanishes in the ground state. This second test detects a missing disconnected subtraction.

At finite temperature the $XX$ spectrum has weights $p_0=1/(1+e^{-\beta\hbar\omega_0})$ at $+\omega_0$ and $p_1=1-p_0$ at $-\omega_0$. Their sum is one and their ratio is $p_1/p_0=e^{-\beta\hbar\omega_0}$. Thus one small model tests Fourier sign, Boltzmann weighting, negative-frequency response, and the equal-time sum rule together. A broadened spectrum need not obey the pointwise detailed-balance formula exactly if both lines are independently convolved with the same finite-width kernel; test the underlying line weights before attributing that mismatch to dynamics.

Time windows and spectral resolution

A record of duration $T$ has frequency resolution of order $2\pi/T$. The sample interval $\Delta t$ limits the nonaliased angular-frequency range to approximately $|\omega|<\pi/\Delta t$. Increasing zero padding interpolates the discrete spectrum but does not improve physical resolution.

Window functions reduce ringing while broadening peaks and changing normalization. Record the window and verify its effect on a known line. Exponential damping $e^{-\eta|t|}$ yields Lorentzian broadening of half-width $\eta$ in angular frequency.

For energy-domain resolvents, Green-function numerics develops Lanczos continued fractions. The Krylov dimension and broadening must be converged separately.

Exercise

If an autocorrelation code replaces both occurrences of $B|0\rangle$ by its normalized vector and omits the prefactor, which spectral quantity is wrong?

Solution

All spectral weights are divided by $\|B|0\rangle\|^2$, so the integrated spectrum becomes one instead of $\langle0|B^\dagger B|0\rangle$. Peak positions may remain correct. The equal-time sum rule detects the lost normalization.

References

• Dagotto, E., “Correlated electrons in high-temperature superconductors,” Reviews of Modern Physics 66, 763–840 (1994), doi:10.1103/RevModPhys.66.763 — exact-diagonalization dynamical correlations.
• Weiße, A., Wellein, G., Alvermann, A., and Fehske, H., “The kernel polynomial method,” Reviews of Modern Physics 78, 275–306 (2006), arXiv:cond-mat/0504627 — spectral resolution and sum rules.