A qubit regularization of the $O(3)$ sigma model

Motivation

• Qubit regularization replaces infinite-dimensional local Hilbert spaces with finite-dimensional lattice-site spaces while targeting the same continuum physics.Traditional bosonic lattice fields require infinite-dimensional representations, motivating local Hamiltonians that can be represented with qubits.Source: §I · p. 1

Contribution

• The work constructs an O(3) qubit Hamiltonian in two and three spatial dimensions without requiring the dimensional reduction used in the D-theory approach.The strategy preserves important symmetries so the desired continuum theory can emerge at a quantum critical point.Source: §I · p. 1

Results

• Monte Carlo calculations reproduce O(3) Wilson-Fisher scaling in two dimensions and Gaussian-fixed-point scaling in three dimensions.Simple modifications also reproduce critical scaling for theories with O(2) and Z2 symmetries.Source: §I · p. 2

Worldline formulation

• The qubit regularization provides a simplified worldline formulation whose relativistic and continuous-time Hamiltonian viewpoints reproduce the expected critical scaling.The relativistic formulation is described as simpler than traditional dual formulations of O(N) models [27] [28] [29] [30].Source: §I · p. 2

Methods

• The paper uses worldline methods and worm algorithms to study the qubit Hamiltonian and its observables.The paper’s organization separates Hamiltonian construction, worldline formulation, algorithmic treatment, and results.Source: §I · p. 2

Target continuum theory

• The traditional O(3) lattice model has second-order critical points separating broken and symmetric phases, governed by Wilson-Fisher physics in d = 2 and Gaussian physics in d = 3.The qubit construction targets these continuum limits with a d-dimensional quantum Hamiltonian.Source: §II · p. 2

Model construction

• Two qubits per lattice site provide one singlet vacuum state and three triplet states carrying O(3) charge.The Hamiltonian combines single-site terms with nearest-neighbor hopping and pair creation or annihilation terms.Source: §II · pp. 2–3

O(3) symmetry

• At µ = 0, the model has global SU(2) symmetry acting on triplets as a spin-1 representation, while the singlet remains invariant.The chemical potential breaks O(3) symmetry and is set to zero in this work.Source: §II · p. 3

O(2) and Z2 extensions

• Adding an on-site term can suppress m = 0 states and reduce the symmetry to O(2), while a further modification yields a Z2 real-scalar theory.The O(2) reduction follows from retaining the SO(2) subgroup and including the stated Z2 transformations.Source: §II · p. 3

Phase structure

• The dimensionless coupling λ = Jt/J interpolates between a symmetric massive singlet-dominated phase at λ →∞ and a likely symmetry-broken triplet phase at λ →−∞.The construction assumes an intermediate second-order quantum critical point where the traditional scaling can emerge.Source: §II · p. 3

Worldline construction

• The partition function is expanded into bond configurations and evaluated in the singlet-triplet basis, producing worldlines for m = 0, ±1 particles in a Fock vacuum.Time discretization maps these configurations onto a space-time lattice with local bond weights.Source: §III · pp. 3–4

Worldline configurations

• Worldline configurations contain vacuum sites and closed particle loops, with m = ±1 loops oriented and m = 0 loops unoriented.Temporal bonds also encode the charge through their orientation and charge-dependent weights.Source: §III · p. 4

Limits

• The relativistic limit sets spatial and temporal weights equal, making the worldline formulation invariant under space-time lattice rotations.It uses ε = 1 and exp(−εJt) = εJ, whereas the Hamiltonian limit takes ε → 0 and tunes λ = Jt/J.Source: §III · p. 4

Worm formulation

• The worm head moves among local positions with probabilities chosen to satisfy detailed balance.Figure 4 illustrates four possible local moves from a central configuration.Source: Figure 4 · p. 5

Worm updates

• The algorithm creates a head and tail, propagates the head through local bond and monomer updates, and closes the worm when it reaches the tail.Each move satisfies detailed balance, with 2(d + 1) possible directions in general.Source: §IV · p. 6

Loop sectors

• Oriented and unoriented loops are updated in separate sectors, while a Metropolis loop flip makes the full algorithm ergodic.For O(3), worm updates are applied to oriented loops and combined with loop flips; O(2) and Z2 require only one loop type.Source: §IV · p. 6

Observables

• The algorithm measures vacuum density, average O(3) charge, current-current susceptibility ρs, and particle two-point susceptibility χ.The worm naturally samples the creation and annihilation events needed for χ.Source: §IV · p. 6

Mass measurement

• In the massive phase, the chemical potential where the average charge crosses 0.5 gives the O(3) particle mass.At µ = 0, the expected charge vanishes because of SO(3) symmetry.Source: §IV · p. 6

Relativistic limit

• The relativistic analysis tunes J near Jc and jointly fits ρs and χ to scaling forms to extract ν, η, and Jc.The universal functions are approximated by fourth-order polynomials.Source: §V · p. 7

Hamiltonian limit

• The Hamiltonian analysis similarly tunes λ = Jt/J near λc while keeping J fixed.The Hamiltonian results use the same scaling strategy for the measured observables.Source: §V · p. 7

Comparison caveat

• Scaling functions from the two limits are difficult to compare because they also depend on the lattice aspect ratio.The paper provides fit results for both limits across d = 2 and d = 3 in Table I.Source: §V · p. 7

d = 2 O(3)

• ν = 0.7113(11) and η = 0.0378(6) are reproduced for the d = 2 O(3) Wilson-Fisher fixed point in the relativistic limit.Fourth-order polynomial fits describe both ρs and χ well.Source: A. Results for the O(3) model · p. 7
• The Hamiltonian-limit analysis with ε = 0.1 and β = L also gives excellent fits, supporting recovery of the O(3) Wilson-Fisher fixed point.The temporal lattice extent is ten times larger than the spatial extent in these computations.Source: A. Results for the O(3) model · p. 7

d = 3 O(3)

• In d = 3, results in both relativistic and Hamiltonian limits are consistent with the mean-field critical exponents expected for the nearly free O(3) theory.The scalar field theory is free up to logarithmic corrections in three spatial dimensions.Source: A. Results for the O(3) model · p. 7

O(2) model

• The O(2) model's d = 2 exponents agree with XY-universality values in both relativistic and Hamiltonian limits.The reference values are ν = 0.6717(1) and η = 0.0381(2).Source: B. Extension to O(2) and Z2 models · p. 7
• In d = 3, the O(2) model yields mean-field-consistent exponents ν = 0.5 and η = 0.0.Source: B. Extension to O(2) and Z2 models · p. 7

Z2 model

• The Z2 model is evaluated using susceptibility χ alone because it has no analog of ρs.The fit is performed after tuning the coupling near the critical point.Source: B. Extension to O(2) and Z2 models · p. 7
• Z2 results agree with d = 2 Ising exponents ν = 0.629 971(4) and η = 0.036 298(2), while d = 3 results are mean-field-consistent.The agreement holds in both relativistic and Hamiltonian limits for d = 2.Source: B. Extension to O(2) and Z2 models · p. 7

Monomer density

• Monomer density v is extracted at the critical point by extrapolating data from the largest lattice sizes.The reported values appear in the last column of Table I, including the β → ∞ limit.Source: C. Monomer density · pp. 7, 10
• v is the probability that a space-time site occupies the trivial Fock vacuum state.Source: C. Monomer density · p. 10
• Values of v near 1 indicate a more perturbative ground state.Source: C. Monomer density · p. 10
• The d = 3 models have v values closer to 1 than d = 2 models, consistent with free-field behavior near the d = 3 critical point.Source: C. Monomer density · p. 10

Conclusions

• The results support matching traditional lattice-regularized quantum field theories in the continuum limits for the studied dimensions.Source: §VI · p. 10
• Two qubits per lattice site reproduce Wilson-Fisher and Gaussian fixed-point scaling in the O(3) qubit model.Source: §VI · p. 10
• Modified O(3) qubit regularizations reproduce the correct critical exponents for O(2) and Z2 theories.Source: §VI · p. 10

Limitations

• The study does not determine whether the two-qubit regularization reproduces the traditional O(3) model in d = 1.The authors defer this question because possible critical behavior, including a Kosterlitz-Thouless phase or another fixed point, remains unresolved.Source: §VI · pp. 10–11

Conclusions

• Table I compares relativistic and Hamiltonian results with Monte Carlo, conformal-bootstrap, and mean-field references.Source: Table I · p. 11

• The algorithm is tested by enumerating all configurations on a 2 × 2 lattice and evaluating four observables from the lattice partition function.Source: Appendix A · p. 11
• The Monte Carlo method is compared with exact results for O(3), O(2), and Z2 models on a 2 × 2 space-time lattice.The comparison includes observables for all three models.Source: Appendix A · p. 11, Table II · p. 12
• The formulation is checked in the εJ →0 limit by comparing charge and monomer-density calculations with exact Hamiltonian diagonalization.Finite-ε results are linearly extrapolated to ε →0.Source: Appendix A · p. 11, Table III · p. 12
• The appendix presents Table II for the exact-versus-Monte-Carlo comparison and Table III for the Hamiltonian-diagonalization results.Source: Table II · p. 12, Table III · p. 12