Documentation:dwa

Directed worm algorithm

Theory

Quantum statistical mechanics at finite temperature

At finite temperature T, the physics is essentially captured by the partition function

$Z = \mathrm{Tr} \, \exp \left(-\beta \hat{H} \right)$

and physical quantities such as the local density

$\langle n_i \rangle = \frac{1}{Z} \, \mathrm{Tr} \, \hat{n}_i \exp \left(-\beta \hat{H} \right) = \frac{1}{Z} \sum_{\mathcal{C}} n_i (\mathcal{C}) Z(\mathcal{C})$

for some configuration $\mathcal{C}$ in the complete configuration space, with inverse temperature $\beta = 1/T$. Here, the units will be cleverly normalized later on.

Feynmann perturbation in the path-integral representation

Decompose $\hat{H} = \hat{H}_0 - \hat{V}$, where $\hat{H}_0$ is purely diagonal in the basis of choice, and $\hat{V}$ being off-diagonal.

Feynmann perturbation in the path-integral representation defines the configuration weight:

$Z(\mathcal{C}) = e^{-\beta\epsilon_1} \left( e^{-\tau_1 \epsilon_1} V_{i_1i_2} e^{\tau_1 \epsilon_2} \right) \cdots \left( e^{-\tau_m \epsilon_m} V_{i_mi_1} e^{\tau_m \epsilon_1} \right)$

for configuration

$\mathcal{C} = \left\{ m ; i_1 \cdots i_m ; \tau_1 \cdots \tau_m \, | \, m \in \mathbf{N} ; 0 < \tau_1 < \cdots < \tau_m < \beta \right\}$

where

$\epsilon_i = <i| \hat{H}_0 |i>$

$V_{ij} = <i| \hat{V} |j>$

The derivation can be found in chapter 2.1-2.2 of my thesis.

Quantum Monte Carlo (Directed Worm Algorithm)

The Quantum Monte Carlo simulation is in fact a Markov chain random walk in the (worldlines) configuration space, importance sampled by the configuration weight $Z(\mathcal{C})$ which is just a positive number assigned to some particular configuration $\mathcal{C}$ for instance shown here. How $Z(\mathcal{C})$ is being assigned depends on the model Hamiltonian as well as the ergodic algorithm that satisfies detailed balance.

For the directed worm algorithm, the configuration is updated with the worm transversing to and from the extended configuration space to ensure ergodicty. In addition, $n_i(\mathcal{C})$ is the number of particles (or state) at site i with time 0.

Each configuration update is known as a Monte Carlo sweep.

The complete step-by-step description of the directed worm algorithm can be found in chapter 2.3 of my thesis, and the code implementation in the following.

The dwa code

Monte Carlo options

Option	Default	Remark
THERMALIZATION	0	number of Monte Carlo configuration updates (sweeps) needed for thermalization, no measurements are performed
SWEEPS	1000000	total number of Monte Carlo configuration updates (sweeps)
t	1.	hopping strength $t$
U	0.	onsite interaction strength $U$
mu	0.	chemical potential $\mu$
K	0.	parabolic trapping strength $K$
T	—	temperature $T$

Options

Option	Default	Remark
SWEEPS	1000000	total number of Monte Carlo configuration updates (sweeps)
t	1.	hopping strength $t$
U	0.	onsite interaction strength $U$
mu	0.	chemical potential $\mu$
K	0.	parabolic trapping strength $K$
T	—	temperature $T$

Measurements

Basic options

The following table summarizes the basic options for measurements available to the user of our DWA code. When unspecified in the parameter list, they assume the default values.

Option	Default	Remark
SKIP	1	number of Monte Carlo configuration updates (sweeps) per measurement
MEASURE	true	turns on/ off measurements
MEASURE[Simulation Speed]	true	observe time taken per checkpoint

Measurement observables

When the measurement mode is turned on, the following is a list of common observables available to the user.

Observable		Boolean control	Binning analysis	Remark
Total Particle Number	$\langle N \rangle$	—	detailed	measure always
Energy	$\langle E \rangle$	—	detailed	measure always
Energy:Vertex	$\langle E_v \rangle$	—	detailed	measure always
Energy:Onsite	$\langle E_o \rangle$	—	detailed	measure always
Density	$\langle n \rangle$	—	detailed	measure if lattice is homoogeneous
Energy Density	$\langle \epsilon \rangle$	—	detailed	measure if lattice is homoogeneous
Energy Density:Vertex	$\langle E_v \rangle$	—	detailed	measure if lattice is homoogeneous
Energy Density:Onsite	$\langle E_o \rangle$	—	detailed	measure if lattice is homoogeneous
Total Particle Number^2	$\langle N^2 \rangle$	measure_number2_	detailed	—
Energy^2	$\langle E^2 \rangle$	measure_energy2_	detailed	—
Density^2	$\langle N^2 \rangle$	measure_density2_	detailed	measure if lattice is homogeneous
Energy Density^2	$\langle E^2 \rangle$	measure_energy_density2_	detailed	measure if lattice is homogeneous
Winding Number^2	$\langle W_\alpha^2 \rangle$	measure_winding_number2_	simple	measure if lattice is periodic: $\alpha=x,y,z$
Local Kink:Number	$\langle n_i^r \rangle$	measure_local_num_kinks_	simple	—
Local Density	$\langle n_i \rangle$	measure_local_density_	simple	—
Local Density^2	$\langle n_i^2 \rangle$	measure_local_density2_	simple	—
Green Function:0	$g_f \left(\alpha=0\right)$	—	detailed	measure always
Green Function:1	$\sum_{i=x,y,z} g_f \left(\alpha_i =1\right)$	—	detailed	measure always
Green Function	$g_f \left(\alpha ; \gamma = 0 \right)$	measure_green_function_	simple	—
Green Function:TOF	$g_f \left(\alpha \right)$	measure_green_function_	simple	measure if tof_phase != 0
Momentum Distribution:0	$\langle n_k \left( 0 ; \gamma = 0 \right) \rangle$	—	detailed	measure if tof_phase == 0
Momentum Distribution:TOF:0	$\langle n_k \left( 0 \right) \rangle$	—	detailed	measure if tof_phase != 0

More options

The following table summarizes more options for measurements available to the user of our DWA code. When unspecified in the parameter list, they assume the default values.

Option	Default	Boolean control
MEASURE[Total Particle Number^2]	false	measure_number2_
MEASURE[Energy^2]	false	measure_energy2_
MEASURE[Density^2]	false	measure_density2_
MEASURE[Energy Density^2]	false	measure_energy_density2_
MEASURE[Local Kink: Number]	false	measure_local_num_kinks_
MEASURE[Winding Number]	false	measure_winding_number_
MEASURE[Local Density]	false	measure_local_density_
MEASURE[Local Density^2]	false	measure_local_density2_
MEASURE[Green Function]	false	measure_green_function_

Option	Default	Remark
tof_phase	0.	time-of-flight phase $\gamma$