Guide: the Heisenberg magnet¶

qsimod.usecases.heisenberg carries an anisotropic Heisenberg magnet, xxz_magnet, to a two-component optical lattice, the analogue simulator model two_component_bose_hubbard. It shares no artifact, transformation or convention with the Schwinger pipeline, whose guide introduces the attributes of a transformation, the contents of an artifact and how a transformation is read; this page states what differs.

The physics follows Jepsen et al. [1]: Eq. (1) of that article is xxz_chain, and the Methods section "Extended Hubbard model" is the superexchange reduction.

The model graph¶

flowchart TB
    subgraph LV1["Application model &nbsp;·&nbsp; <code>models.application</code>"]
        direction LR
        xxz_magnet["<b>xxz_magnet &nbsp; H<sub>XXZ</sub></b><br/>Heisenberg XXZ magnet,<br/>by its anisotropy<br/><i>Θ = {J<sub>xy</sub>, Δ}</i>"]
    end
    subgraph LV2["Intermediate representations &nbsp;·&nbsp; <code>models.intermediate</code>"]
        xxz_chain["<b>xxz_chain &nbsp; H<sub>XXZ</sub></b><br/>XXZ chain,<br/>by its two couplings<br/><i>Θ = {J<sub>xy</sub>, J<sub>z</sub>}</i>"]
        fermion_chain["<b>fermion_chain &nbsp; H<sub>tV</sub></b><br/>spinless fermions,<br/>nearest-neighbour interaction<br/><i>Θ = {J<sub>xy</sub>, J<sub>z</sub>}</i>"]
    end
    subgraph LV3["Hardware model &nbsp;·&nbsp; <code>models.hardware</code>"]
        direction LR
        two_component_bose_hubbard["<b>two_component_bose_hubbard &nbsp; H<sub>2BHM</sub></b><br/>two-component<br/>Bose–Hubbard chain<br/><i>Θ = {t, U<sub>↑↑</sub>, U<sub>↑↓</sub>, U<sub>↓↓</sub>}</i>"]
    end

    xxz_magnet ---> |"<b>anisotropy resolution</b><br/><i>exact</i>"| xxz_chain
    xxz_chain ---> |"<b>Jordan–Wigner to spinless fermions</b><br/><i>exact</i>"| fermion_chain
    xxz_chain -. "<b>second-order superexchange,<br/>solved for the knob settings</b><br/><i>approximate, regime conditions</i>" .-> two_component_bose_hubbard

    classDef ham fill:#e0f2ee,stroke:#009371,color:#003e2f
    class xxz_magnet,xxz_chain,fermion_chain,two_component_bose_hubbard ham
    style LV1 fill:#f2f7fa,stroke:#b1d0e5,color:#103c5a
    style LV2 fill:#f2f7fa,stroke:#b1d0e5,color:#103c5a
    style LV3 fill:#f2f7fa,stroke:#b1d0e5,color:#103c5a

fermion_chain is a terminal artifact of the intermediate layer: at \(J_z = 0\) it is a free-fermion model. The abstraction level is data on an artifact, not a type:

from qsimod.usecases.heisenberg import build_graph

graph = build_graph(4)
print(graph.targets())  # ('fermion_chain', 'two_component_bose_hubbard')
for node, level in graph.levels().items():
    print(f"{node:5} {level}")

The transformations at a glance¶

xxz_magnet: the XXZ magnet¶

Use case: magnet  ·  library: heisenberg_magnet

One energy \(J_{xy}\) and one dimensionless anisotropy \(\Delta = J_z / J_{xy}\): \(\Delta = 0\) is the free-fermion XX point, \(\Delta = 1\) the isotropic Heisenberg magnet, and \(\lvert \Delta \rvert > 1\) the Ising-like regimes. The structural type declares spin-1/2 degrees of freedom on a chain of \(N\) sites with a global U(1) symmetry, the conservation of the total magnetisation. Unlike the lattice gauge theory, the application model has a finite-dimensional representation:

from qsimod.realise import HilbertSpace, build_operator
from qsimod.usecases.heisenberg import ParameterNames as P, magnet

model = magnet(4)
print(model.parameters)  # xxz_magnet.Jxy [rad/ms], xxz_magnet.Delta
space = HilbertSpace.of(model.structure)
print(space.dimension)  # 16
operator = build_operator(
    model.hamiltonian, space, {P.TRANSVERSE_XXZ_MAGNET: 0.5, P.ANISOTROPY: 1.0}
)
print(operator.shape)  # (16, 16)

xxz_magnet -> xxz_chain: the anisotropy resolution¶

Exactness: EXACT, a reparametrisation that leaves the operators unchanged. The relation is not an identity relation, since one parameter is the product of two others, but it is invertible:

from qsimod.usecases.heisenberg import ParameterNames as P, anisotropy

step = anisotropy()
print(step.relation)
print(step.forward_relation_kind)  # CLOSED_FORM
print(step.inverse_relation_kind)  # CLOSED_FORM
print(
    step.relation.is_invertible(
        {P.TRANSVERSE_XXZ_MAGNET, P.ANISOTROPY}, {P.TRANSVERSE_XXZ_CHAIN, P.LONGITUDINAL_XXZ_CHAIN}
    )
)  # True

xxz_chain: the XXZ chain by two couplings¶

Use case: spin_chain  ·  library: xxz_spin_chain

The Hamiltonian of xxz_magnet, parametrised by the two coupling energies \(J_{xy}\) and \(J_z\); the transverse term is stored in the ladder basis of Algebra.SPIN_HALF.

xxz_chain -> fermion_chain: the Jordan-Wigner transformation to spinless fermions¶

Exactness: EXACT. The structural type changes from spin-1/2 to fermions on the same \(N\) positions; the transverse term becomes a hopping, the longitudinal term a nearest-neighbour interaction, and the parity strings cancel because the couplings join adjacent sites. With the alternating-sign convention of the numerical realisation the two models are the same matrix:

import numpy as np

from qsimod.realise import HilbertSpace, build_operator
from qsimod.usecases.heisenberg import ParameterNames as P, fermion_chain, spin_chain

values = {"Jxy": 0.5, "Jz": 0.35}
spin, fermions = spin_chain(6), fermion_chain(6)
left = build_operator(
    spin.hamiltonian,
    HilbertSpace.of(spin.structure),
    {P.TRANSVERSE_XXZ_CHAIN: values["Jxy"], P.LONGITUDINAL_XXZ_CHAIN: values["Jz"]},
)
right = build_operator(
    fermions.hamiltonian,
    HilbertSpace.of(fermions.structure),
    {P.TRANSVERSE_FERMION_CHAIN: values["Jxy"], P.LONGITUDINAL_FERMION_CHAIN: values["Jz"]},
)
print(float(np.max(np.abs(np.asarray(left) - np.asarray(right)))) < 1e-12)  # True

fermion_chain: the Jordan-Wigner image¶

Use case: fermion_chain  ·  library: interacting_fermion_chain

\(z_j\) is the coordination number, 1 at the ends and 2 in the bulk; the last two terms are the expansion of \(J_z \sum_{j} (\hat n_j - 1/2)(\hat n_{j+1} - 1/2)\), and the constant is retained. At \(J_z = 0\) the model is free, with the band \(-J_{xy} \cos(qa)\). The leading minus sign is the alternating-sign Jordan-Wigner convention of qsimod.realise.build; the plain convention differs by \(\hat c_j \to (-1)^j \hat c_j\), which changes neither the spectrum nor the Fermi velocity.

xxz_chain -> two_component_bose_hubbard: the second-order superexchange¶

At one atom per site, virtual tunnelling onto a neighbouring site and back yields a spin-spin coupling. Only the interspecies channel exchanges two spins, so it alone sets \(J_{xy}\), whereas all three channels contribute to \(J_z\).

Exactness: APPROXIMATE; the leading error is of fourth order in \(t\), a relative error scaling like \((t/U)^2\).

Validity conditions. Three domain and six regime conditions:

Since \(\Delta + 1 = U_{\uparrow\downarrow}/U_{\uparrow\uparrow} + U_{\uparrow\downarrow}/U_{\downarrow\downarrow}\), a large anisotropy requires a small intraspecies channel, where \(\lvert J_z \rvert \ll U_{\uparrow\uparrow}\) binds independently of \(t \ll U_{\uparrow\uparrow}\).

from qsimod.usecases.heisenberg import ParameterNames as P, superexchange, validity

step = superexchange()
print(step.inverse_relation_kind)  # UNDER_DETERMINED
report = validity().report(
    {
        P.HOPPING: 3.0,
        P.INTERACTION_UP: -51.6,
        P.INTERACTION_MIXED: -72.1,
        P.INTERACTION_DOWN: -125.4,
        P.TRANSVERSE_XXZ_CHAIN: 0.4975,
        P.LONGITUDINAL_XXZ_CHAIN: 0.4841,
    }
)
print(report)

The forward maps take the four knobs to the two couplings; the transformation poses the inverse direction, two equations in four unknowns, as a solve, and returns the target with its parameters unbound.

What the superexchange also produces¶

The derivation also yields a longitudinal field

for which the XXZ Hamiltonian has no term; at the settings of the reference experiment it amounts to about 40 % of \(J_{xy}\). superexchange_field returns its strength and weighted_magnetisation_term builds the operator:

from qsimod.usecases.heisenberg import ParameterNames as P, field_map, transverse_map

knobs = {
    P.HOPPING: 3.0,
    P.INTERACTION_UP: -51.6,
    P.INTERACTION_MIXED: -72.1,
    P.INTERACTION_DOWN: -125.4,
}
print(round(field_map().evaluate_real(knobs) / transverse_map().evaluate_real(knobs), 3))
# 0.411

Since \(\sum_{j} z_j \hat S^z_j = 2 \sum_{j} \hat S^z_j - \hat S^z_0 - \hat S^z_{N-1}\) and the total magnetisation is conserved, the term is a constant inside a magnetisation sector plus a field on the two end spins. It vanishes at \(U_{\uparrow\uparrow} = U_{\downarrow\downarrow}\), where the default objective places the solve. examples/heisenberg_end_to_end.py reports the deviation of the spectra with and without it.

two_component_bose_hubbard: the two-component optical lattice¶

Use case: lattice  ·  library: two_component_bose_hubbard_chain

Two hyperfine states share one lattice on an interleaved register: component up occupies position \(2j\) and component down position \(2j+1\), so \(N\) chain sites occupy \(2N\) positions. The knobs are the tunnelling \(t\) and the three interaction channels:

from qsimod.usecases.heisenberg import device_limits

for line in str(device_limits()).split("; "):
    print(line)

\(U_{\downarrow\downarrow}\) may take either sign, since an anisotropy below -1 requires a negative value; the excluded value \(U_{\downarrow\downarrow} = 0\) is the midpoint of the box, so device_start chooses the branch from the requested anisotropy. The coupled constraints are the Mott lobe, |U| >= 3.4 t per channel, an operating limit of the apparatus that is weaker than the regime condition \(t \ll U\).

The artifact declares its conserved sector, the total particle number at unit filling, and no term leaves it, so a realisation inside the sector is exact:

from qsimod.realise import RealisationRequest, SectorBasis
from qsimod.usecases.heisenberg import lattice

model = lattice(4)
print(model.sector)  # unit filling [N_0=+4]
basis = SectorBasis.of(
    model.structure,
    constraints=model.constraint_operators,
    request=RealisationRequest(boson_cutoff=2),
)
print(basis.dimension, "of", basis.space.dimension)  # 266 of 6561

The one-atom-per-site manifold is neither the ground state nor the middle of the spectrum on the attractive branch; it is identified by its weight on the configurations that mott_manifold_operators declares.

References¶

1. P. N. Jepsen, J. Amato-Grill, I. Dimitrova, W. W. Ho, E. Demler and W. Ketterle, Spin transport in a tunable Heisenberg model realized with ultracold atoms, Nature 588, 403-407 (2020). doi:10.1038/s41586-020-3033-y