# MetropolisExchange

This sampler acts locally only on two local degree of freedom $s_i$ and $s_j$, and proposes a new state: $s_1 \dots s^\prime_i \dots s^\prime_j \dots s_N$, where in general $s^\prime_i \neq s_i$ and $s^\prime_j \neq s_j$ . The sites $i$ and $j$ are also chosen to be within a maximum graph distance of $d_{\mathrm{max}}$.

The transition probability associated to this sampler can be decomposed into two steps:

1. A pair of indices $i,j = 1\dots N$, and such that $\mathrm{dist}(i,j) \leq d_{\mathrm{max}}$, is chosen with uniform probability.
2. The sites are exchanged, i.e. $s^\prime_i = s_j$ and $s^\prime_j = s_i$.

Notice that this sampling method generates random permutations of the quantum numbers, thus global quantities such as the sum of the local quantum n umbers are conserved during the sampling. This scheme should be used then only when sampling in a region where $\sum_i s_i = \mathrm{constant}$ is needed, otherwise the sampling would be strongly not ergodic.

## Class Constructor

Constructs a new MetropolisExchange sampler given a machine and a graph.

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 $F(X)=|X|^2$.
graph netket._C_netket.graph.Graph A graph used to define the distances among the degrees of freedom being sampled.
d_max int=1 The maximum graph distance allowed for exchanges.

### Examples

Sampling from a RBM machine in a 1D lattice of spin 1/2, using nearest-neighbours exchanges.

>>> 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)
>>>
>>> # Construct a MetropolisExchange Sampler
>>> sa = nk.sampler.MetropolisExchange(machine=ma,graph=g,d_max=1)
>>> print(sa.hilbert.size)
100



## 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 $F(\Psi(v))$