# 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 = $

$V_{ij} = $

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: options

### Monte Carlo options

Option     Default     Remark
THERMALIZATION   0   1) number of Monte Carlo configuration updates (sweeps) needed for thermalization
2) no measurements are performed in the thermalization stage
SWEEPS   1000000   total number of Monte Carlo configuration updates (sweeps) after thermalization
SKIP   1.   number of Monte Carlo configuration updates (sweeps) per measurement $t$

### ALPS lattice library options

Option     Default     Remark
LATTICE   --   which lattice do you want?
L   --   length of lattice

A first quick guide to the ALPS Lattice library can be found here.

### Boson Hubbard model options

Option     Default     Remark
MODEL   --   set as "boson Hubbard"
Nmax   --   maximum number of bosons allowed per site
t   1.   hopping strength $t$
U   0.   onsite interaction strength $U$
mu   0.   chemical potential $\mu$

Note:
The following definitions for mu are allowed:

• mu=0.5
• mu="0.5 - 0.001*((x-(L-1)/2.)*(x-(L-1)/2.) + (y-(L-1)/2.)*(y-(L-1)/2.) + (z-(L-1)/2.)*(z-(L-1)/2.)"

### Other options

Option     Default     Remark
T   0.   temperature $T$
tof_phase   0.   time-of-flight phase $\gamma$
MEASURE   true   shall we measure the common observables?
MEASURE[Simulation Speed]   true   shall we measure the simulation performance?

### More measurement options

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_

## Measurements

### 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