Initial Configurations

sample advances N_mc Markov chains in parallel and needs a starting configuration for each. The configurations are stored in a State, whose leading axis runs over the chains; Core Concepts describes the type itself.

The starting configurations also select the symmetry sector. A move that conserves a quantum number never changes it — BondExchange conserves the magnetization of spins and the number of electrons of each spin — so the chains stay in the sector they start in.

Spins

SpinState holds spin-½ configurations as an integer array spins of shape (N_mc, Ns), with \(+1\) for \(\uparrow\) and \(-1\) for \(\downarrow\). init_config_fixed_magn draws configurations uniformly at random with total magnetization \(S^z\) equal to sz, that is Ns/2 + sz spins up and Ns/2 - sz down (Ns even):

import jax
from tachys.lattice.lattice_database import square
from tachys.lattice.spins.spin_state import SpinState, init_config_fixed_magn

lattice = square(shape=(4, 4))
key = jax.random.key(0)
N_mc = 1024

spins = init_config_fixed_magn(key, lattice.Ns, sz=0, N_mc=N_mc)  # (N_mc, Ns)
state = SpinState(spins=spins, lattice=lattice)

Other sectors are selected with sz: for a singlet ground state, for example, the lowest energy at sz=1 minus the one at sz=0 is the spin gap. With a move that does not conserve \(S^z\), such as SpinFlip, sz only sets where the chains start.

Fermions

FermionState holds occupation numbers, an integer array occupations of shape (N_mc, 2 Ns) with entries 0 and 1, together with the number of electrons Ne. The first Ns columns are the \(\uparrow\) occupations of sites \(0, \dots, N_s - 1\), the last Ns the \(\downarrow\) ones:

\[ (n_{0\uparrow}, \dots, n_{N_s-1\,\uparrow},\; n_{0\downarrow}, \dots, n_{N_s-1\,\downarrow}) . \]

The fermionic operators order their Jordan–Wigner string the same way.

init_config_spinful draws random configurations with Ne/2 + sz electrons \(\uparrow\) and Ne/2 - sz electrons \(\downarrow\) (Ne even), and returns these two numbers along with the array:

from tachys.lattice.fermions.fermion_state import (
    FermionState, init_config_spinful,
)

Ne = 14                                 # 1/8 hole doping on 16 sites
occupations, N_up, N_down = init_config_spinful(
    key, Ns=lattice.Ns, Ne=Ne, N_mc=N_mc,
)
state = FermionState(occupations=occupations, Ne=Ne, lattice=lattice)

Custom configurations

Any integer array of the right shape can seed the chains, which covers the cases the helpers do not, such as an odd number of electrons. To start every chain from the Néel state of the \(4 \times 4\) lattice, whose site in column \(x\) and row \(y\) has index \(x + 4y\) (Lattices):

import numpy as np
import jax.numpy as jnp

L = 4
neel = np.empty(L * L, dtype=np.int8)
for x in range(L):
    for y in range(L):
        neel[x + L * y] = 1 if (x + y) % 2 == 0 else -1     # spin of site (x, y)

state = SpinState(spins=jnp.tile(neel, (N_mc, 1)), lattice=lattice)

Chains that start from the same configuration stay correlated until the burn-in sweeps have separated them, see Monte Carlo Sampling.

The state returned by sample is itself a valid starting point: the training loop feeds it back at every step, and checkpoints store it.

Number of chains

With several devices, the chains are split evenly among them, so N_mc must be a multiple of the number of devices. The same N_mc sets the size of the sample for every energy estimate, and of the linear system solved by the optimizer (Optimization).