# MetropolisHamiltonian

Sampling based on the off-diagonal elements of a Hamiltonian (or a generic Operator). In this case, the transition matrix is taken to be:

where $\theta(x)$ is the Heaviside step function, and $\mathcal{N}(\mathbf{s})$ is a state-dependent normalization. The effect of this transition probability is then to connect (with uniform probability) a given state $\mathbf{s}$ to all those states $\mathbf{s}^\prime$ for which the Hamiltonian has finite matrix elements. Notice that this sampler preserves by construction all the symmetries of the Hamiltonian. This is in generally not true for the local samplers instead.

## Class Constructor

Constructs a new MetropolisHamiltonian sampler given a machine and a Hamiltonian operator (or in general an arbitrary Operator).

Argument Type Description
machine netket._C_netket.machine.Machine A machine $\Psi(s)$ used for the sampling. The probability distribution being sampled from is $F(\Psi(s))$, where the function $F(X)$, is arbitrary, by default $F(X)=|X|^2$.
hamiltonian netket._C_netket.Operator The operator used to perform off-diagonal transition.

### Examples

Sampling from a RBM machine in a 1D lattice of spin 1/2

>>> import netket as nk
>>>
>>> g=nk.graph.Hypercube(length=10,n_dim=2,pbc=True)
>>> hi=nk.hilbert.Spin(s=0.5,graph=g)
>>>
>>> # RBM Spin Machine
>>> ma = nk.machine.RbmSpin(alpha=1, hilbert=hi)
>>>
>>> # Transverse-field Ising Hamiltonian
>>> ha = nk.operator.Ising(hilbert=hi, h=1.0)
>>>
>>> # Construct a MetropolisHamiltonian Sampler
>>> sa = nk.sampler.MetropolisHamiltonian(machine=ma,hamiltonian=ha)



## Class Methods

### reset

Resets the state of the sampler, including the acceptance rate statistics and optionally initializing at random the visible units being sampled.

Argument Type Description
init_random bool=False If True the quantum numbers (visible units)

### seed

Seeds the random number generator used by the Sampler.

Argument Type Description
base_seed int The base seed for the random number generator

### sweep

Performs a sampling sweep. Typically a single sweep consists of an extensive number of local moves.

## Properties

Property Type Description
acceptance numpy.array The measured acceptance rate for the sampling. In the case of rejection-free sampling this is always equal to 1.
hilbert netket.hilbert The Hilbert space used for the sampling.
machine netket.machine The machine used for the sampling.
machine_func function(complex) The function to be used for sampling. by default $|\Psi(x)|^2$ is sampled, however in general $F(\Psi(v))$
visible numpy.array The quantum numbers being sampled, and distributed according to $F(\Psi(v))$