Model library¶

The interleaved matter-gauge structural type of an open chain.

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A structural pattern over the interleaved matter-gauge chain.

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Each link of an open chain as the indices (link, left matter, link, right matter).

The Gauss operators of an abstraction level, with their declared target eigenvalues.

On an open chain the two boundary generators omit one link and take half-integer values; their targets are the background boundary_value in the homogeneous form and its particle-hole image (-1)**(l+1) * boundary_value in the staggered form.

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The superselection sector declared by the target values of a Gauss family.

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A declared per-site occupation subspace on the interleaved chain.

The default is the doublon encoding, {0, 1} on matter positions and {0, 2} on gauge positions, for which a numerical realisation requires an occupation cutoff of at least two.

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The occupation pattern |1 0 1 0 1 ...>: matter sites occupied, gauge links empty.

The mean matter occupation <n_matter> = (1/N) sum_l n_{2l}.

The projector onto a set of occupations at one site, as a polynomial in n.

The polynomial P_k = prod_{j != k} (n - j) / (k - j) interpolates over the ladder {0, ..., cutoff}, and P_A = sum_{k in A} P_k. For A = {1} at cutoff = 2 the projector is 2n - n**2, the parity n mod 2.

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The mean weight on forbidden gauge-link occupations, (1/L_g) sum_{j in g} (1 - P_j).

This observable is the gauge violation eta of the article. Under the doublon encoding a forbidden link occupation is an odd one, so the observable is the mean link parity n mod 2.

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A plain open chain of sites degrees of freedom of a single algebra.

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The structural type a transformation requires of a plain chain of a single algebra.

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The register index of the first component at chain site j, 2*j.

The register index of the second component at chain site j, 2*j + 1.

The register size of a two-component chain of N sites, 2N.

A chain of sites positions, each carrying two bosonic modes.

The first component occupies position 2j and the second position 2j+1; the positions of the second component are declared through the link_count of the lattice.

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The structural type a transformation into a two-component bosonic chain requires.

The nearest-neighbour bond count of each site of an open chain, (1, 2, ..., 2, 1).

The nearest-neighbour XXZ terms of an open spin-1/2 chain.

sum_{j=0}^{N-2} [ (Jxy/2)( S^+_j S^-_{j+1} + h.c. ) + Jz S^z_j S^z_{j+1} ]

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The term strength * sum_j z_j S^z_j, with z_j the coordination number of site j.

Since sum_j z_j S^z_j = 2 sum_j S^z_j - S^z_0 - S^z_{N-1}, the term is a constant within a magnetisation sector plus a field on the two end spins.

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The nearest-neighbour Ising terms of an open spin-1/2 chain.

Jz sum_{j=0}^{N-2} S^z_j S^z_{j+1}  -  Gamma sum_j S^x_j  -  B sum_j S^z_j

with S^x = (S^+ + S^-)/2; the chain is antiferromagnetic for Jz > 0.

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The term strength * (S^z_0 + S^z_{N-1}), a field on the two end spins only.

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The total magnetisation sum_j S^z_j of a spin chain.

The per-site occupation operators M_j = n_{j,up} + n_{j,down}, targeted at one.

The operators select the one-atom-per-site manifold in which the superexchange derivation expands. The Hamiltonian does not commute with them, since a tunnelling event changes two of them at once, so they do not define a superselection sector; sector_projector nevertheless projects onto the manifold.

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The total particle number N = sum_j (n_{j,up} + n_{j,down}), targeted at sites.

The total particle number is a conserved quantity, so the hardware Hamiltonian is built exactly inside this sector with SectorBasis.

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The superselection sector of unit filling, N_total = N.

The Kogut-Susskind lattice Schwinger model with staggered fermions in 1+1 dimensions.

One-dimensional lattice QED, the application model H_sys of the case study:

H = (a/2) sum_l E^2_{l,l+1}
  + m sum_l (-1)**l psi^dag_l psi_l
  - (i/2a) sum_l ( psi^dag_l U_{l,l+1} psi_{l+1} - h.c. )

The parameters are the mass m, the lattice spacing a and the gauge coupling e, plus electric_gap = a e**2 / 2, the energy of one unit of electric flux, which enters no term and is read by the validity condition of the quantum-link truncation.

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The anisotropic Heisenberg (XXZ) chain, stated by an energy and a dimensionless anisotropy.

H = sum_j [ Jxy ( S^x_j S^x_{j+1} + S^y_j S^y_{j+1} ) + Delta Jxy S^z_j S^z_{j+1} ]

The anisotropy Delta = Jz / Jxy is dimensionless: 0 is the XX point, 1 the isotropic magnet.

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The antiferromagnetic Ising chain in longitudinal and transverse fields.

H = Jz sum_j ( S^z_j S^z_{j+1} - hz S^z_j - hx S^x_j )

One energy Jz and two dimensionless fields (hz, hx), as in Simon et al. (2011). The transverse field breaks the magnetisation symmetry, so the structural type declares none.

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A spin-1/2 U(1) quantum-link model, in any of the four conventions.

H = sum_l [ (kappa/2)( <coupling> + h.c. ) + m <mass sign>_l psi^dag_l psi_l ] + constant

The staggered convention is H_IR1 of the case study and the homogeneous convention H_IR2, its branch point.

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The boson encoding of a pair-coupling quantum-link model.

H = sum_{j even, 0 <= j <= 2N-4} [ (kappa / 2 sqrt(2)) b_j b_{j+2} (b^dag_{j+1})^2 + h.c. ]
  + sum_{j even, 0 <= j <= 2N-2}   m n_j
  + constant

H_IR3 of the case study. The encoding is exact on the declared occupation subspace, {0, 1} on matter positions and {0, 2} on links, as the projected operator P H P (sandwich).

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The qubit-register form of a quantum-link model, a 3-local Pauli Hamiltonian.

H = sum_l (kappa/2) ( sigma^-_{2l} sigma^+_{2l+1} sigma^-_{2l+2} + h.c. )
  + sum_l m n_{2l} + constant                          (pair convention)

H_IR4 of the case study, the Jordan-Wigner image of H_IR2. Each coupling term expands into four mutually commuting Pauli strings; in the hopping convention the Jordan-Wigner string contributes a factor -1 to the coupling.

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The XXZ chain, stated by two coupling energies.

H = sum_{j=0}^{N-2} [ (Jxy/2)( S^+_j S^-_{j+1} + h.c. ) + Jz S^z_j S^z_{j+1} ]

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The Jordan-Wigner image of an XXZ chain: spinless fermions with a nearest-neighbour V.

H = - (Jxy/2) sum_j ( c^dag_j c_{j+1} + h.c. )
  + Jz sum_j n_j n_{j+1}
  - (Jz/2) sum_j z_j n_j
  + (N-1) Jz/4

z_j is the coordination number of site j; the last two terms expand Jz sum_j (n_j - 1/2)(n_{j+1} - 1/2). The hopping sign is the alternating-sign Jordan-Wigner gauge of qsimod.realise.build, under which this model and the spin chain realise to the same matrix.

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The Ising chain, stated by three energies.

H = Jz sum_{j=0}^{N-2} S^z_j S^z_{j+1} - Gamma sum_j S^x_j - B sum_j S^z_j

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The on-site interaction (U/2) n (n - 1) at one register position, as two terms.

The admissible knob set of an optical-superlattice simulator.

J = 0 and Delta = 0 are excluded strictly. Two coupled constraints of origin DERIVATION bound the tilt by tilt_fraction of the superlattice depth and of the resonance gap U - delta; a second theory on the same lattice drops them with apparatus_only.

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A tilted, staggered Bose-Hubbard chain on an optical superlattice.

The analogue simulator model H_sim of the case study:

H = sign * J sum_{j=0}^{2N-3} ( b^dag_j b_{j+1} + h.c. )
  + sum_{j=0}^{2N-2} [ (U/2) n_j (n_j - 1) + eps_j n_j ],
eps_j = (-1)**j delta/2 + j Delta

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The admissible knob set of a transverse-field Ising simulator.

A transverse-field Ising chain of qubits, Jz sum ZZ + h sum X.

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The admissible knob set of a two-component optical-lattice simulator.

t = 0 is excluded strictly. U_uu and U_ud are attractive, so that Jxy > 0; U_dd may take either sign, so its box contains the pole U_dd = 0 and a solve needs an initial point on the intended side. Three coupled constraints U**2 >= (mott_ratio * t)**2 keep the Mott gap of every channel above the tunnelling.

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A two-component Bose-Hubbard chain: two hyperfine states in one optical lattice.

H = -t sum_{sigma} sum_{j=0}^{N-2} ( b^dag_{j,sigma} b_{j+1,sigma} + h.c. )
  + (U_uu/2) sum_j n_{j,up} (n_{j,up} - 1)
  + (U_dd/2) sum_j n_{j,down} (n_{j,down} - 1)
  + U_ud     sum_j n_{j,up} n_{j,down}

Component up sits at register position 2j and down at 2j+1 (two_component_structure). The model declares the conserved total particle number, N at unit filling.

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