Analytic Continuation

Analytic Continuation

Maximum entropy and stochastic analytic continuation methods for extracting real-frequency spectral functions from imaginary-time data.

Overview

QMC simulations produce correlation functions in imaginary time $G(\tau)$. Extracting the real-frequency spectral function $A(\omega)$ requires solving the inverse problem:

$$G(\tau) = \int d\omega , K(\tau, \omega) , A(\omega)$$

where $K$ is a known kernel. This is an ill-conditioned problem. ALF provides both stochastic analytic continuation (SAC) and maximum entropy (MaxEnt) methods via Max_SAC.out.

Usage

Analytic continuation is performed after the standard error analysis, on the Jackknife-analyzed time-displaced data.

Parameters

Two namelists in the parameters file control the analytic continuation:

&VAR_Max_Stoch — method and frequency grid:

&VAR_Max_Stoch
OM_st       = -10.d0      ! Start of frequency grid
OM_en       =  10.d0      ! End of frequency grid
Ndis        =  501        ! Number of frequency points
Stochastic  = .true.      ! .true. = SAC, .false. = MaxEnt
Ngamma      =  500        ! Number of delta functions in SAC
NBins       =  10         ! Number of bins for SAC sampling
NSweeps     =  1000       ! MC sweeps per bin
Nwarm       =  500        ! Warmup sweeps
N_alpha     =  20         ! Number of temperatures (must be > 10)
alpha_st    =  100.d0     ! Starting alpha
R           =  1.3d0      ! Ratio for geometric temperature set: alpha_n = alpha_st * R^(n-1)
Tolerance   =  0.01d0     ! Relative-error cutoff for data rescaling
Checkpoint  = .false.     ! Keep dump files
/

&VAR_errors — data handling:

&VAR_errors
N_skip      =  0          ! Bins to skip
N_rebin     =  1          ! Rebinning factor
N_cov       =  0          ! 1 = use full covariance matrix from g_dat
N_Back      =  0          ! Background subtraction
N_auto      =  0          ! Autocorrelation handling
/

Running

The script Scripts_and_Parameters_files/Spectral.sh automates the process for k-resolved spectral functions:

1. Loops over k-points from the analyzed time-displaced data
2. Substitutes MaxEnt parameters into an input file
3. Runs Max_SAC.out for each k-point
4. Collects output into a single file suitable for plotting

# After running ana.out or ana_hdf5.out:
bash $ALF_DIR/Scripts_and_Parameters_files/Spectral.sh

Kernels

The kernel channel is a string read from the g_dat file header and dispatched automatically — you do not choose it manually.

Channel	Kernel	Used for
PH (particle-hole)	$K(\tau,\omega) = \frac{e^{-\tau\omega} + e^{-(\beta-\tau)\omega}}{\pi(1 + e^{-\beta\omega})}$	Spin correlations, density correlations
PH_C (conductivity)	Same as PH	Optical conductivity
PP (particle-particle)	$K(\tau,\omega) = \frac{e^{-\tau\omega}}{\pi(1 + e^{-\beta\omega})}$	Pairing correlations
P (particle)	$K(\tau,\omega) = \frac{e^{-\tau\omega}}{\pi(1 + e^{-\beta\omega})}$	Single-particle Green's function
P_PH (particle, p-h symmetric)	$K(\tau,\omega) = \frac{e^{-\tau\omega} + e^{-(\beta-\tau)\omega}}{\pi(1 + e^{-\beta\omega})}$	Single-particle Green's function at half filling
T0 (zero temperature)	$K(\tau,\omega) = \frac{e^{-\tau\omega}}{\pi}$	$T = 0$ (PQMC) spectral functions

For channels PH, P_PH, and PH_C, OM_st is forced to 0.

Practical Tips

• Frequency grid: Make sure OM_st and OM_en bracket the expected spectral weight. If the spectrum shows increasing spectral weight at the boundary, the window is likely too small — ideally the spectrum should decay towards both OM_st and OM_en. Too wide wastes resolution.
• Stochastic vs. MaxEnt: SAC (Stochastic=.true.) is generally more robust and is the default. MaxEnt (Stochastic=.false.) may give sharper features but is more sensitive to noise.
• Covariance matrix: Using the full covariance (N_cov=1) can improve results when correlations between time slices are significant, but requires enough bins for a reliable covariance estimate.
• Quality check: Always compare the back-transformed $G(\tau)$ from the obtained $A(\omega)$ against the original QMC data to assess the reliability of the continuation.