Efficient Basis Formulation for (1+1)-Dimensional SU(2) Lattice Gauge Theory: Spectral calculations with matrix product states

Motivation

• Integrating out gauge degrees of freedom in (1+1) dimensions addresses redundancy because only gauge-invariant states contribute to observables.The absence of transversal directions permits gauge-field integration through Gauss law.Source: §I · p. 1

Background

• Tensor networks provide wave-function access for ground states and excitations, while avoiding the sign problem and enabling real-time, finite-density, and entanglement studies.These capabilities motivate their use for nonperturbative lattice gauge-theory calculations.Source: §I · p. 1

Limitation

• Finite-dimensional gauge truncations can preserve gauge invariance but may fail to reproduce the continuum theory or possess a continuum limit.This motivates systematic studies of truncation effects rather than assuming finite-dimensional models are equivalent.Source: §I · p. 1

Approach

• The proposed SU(2) formulation integrates gauge degrees of freedom on open lattices, permits arbitrary color-flux truncation, and supports efficient tensor-network calculations.The authors use matrix product states to compute low-lying spectra and examine mass-gap and entanglement scaling toward criticality.Source: §I · pp. 1–2

Contribution

• The reduced basis supports larger maximum color-electric fluxes than previous studies and offers a systematic tool for assessing truncation effects relevant to quantum simulators.Its Abelianized Hamiltonian is nonlocal, similar to a formulation realized for the Schwinger model in trapped ions.Source: §I · p. 2

Organization

• The paper proceeds from the model and color-neutral basis to gauge-field removal, then applies MPS methods to spectra and ground-state entanglement.The final section summarizes the formulation and its numerical applications.Source: §I · p. 2

Model

• The model is a Hamiltonian (1+1)-dimensional SU(2) lattice gauge theory formulated with Kogut–Susskind staggered fermions in temporal gauge.The fermionic fields create color-labeled particles on lattice sites, while gauge links connect neighboring sites.Source: §II · p. 2

Gauge links

• Each gauge link is a quantum rigid rotor labeled by total angular momentum j and two electric-field z components in body-fixed and space-fixed frames.The color-electric energy has eigenvalues j(j+1), with j = 0, 1/2, 1, ... .Source: §II · p. 2

Physical subspace

• Physical states satisfy the non-Abelian Gauss law generated by the local color charges and left/right electric fields.The Gauss-law constraint selects gauge-invariant states from the full fermion–link Hilbert space.Source: §II · p. 2

Dimensionless formulation

• For spectral calculations, the Hamiltonian is written in dimensionless form W = xV + W0 with x = 1/(ag)^2 and µ = 2√xm/g.Here g is the coupling, a the lattice spacing, and m the bare fermion mass.Source: §II · pp. 2–3

Strong coupling

• At strong coupling, the gauge-invariant ground state has one fermion of each color on odd sites, empty even sites, and zero link flux.This state is the lattice analogue of the Dirac sea.Source: §II · p. 3

Color neutrality

• The full basis contains color information and non-singlet states, whereas restricting to color-neutral superpositions removes physically redundant degrees of freedom.The next construction exploits this reduction to formulate the physical subspace more efficiently.Source: §II · p. 3

Color-neutral basis

• Repeated application of the color-contracted hopping operator V generates color-neutral quark–antiquark or antiquark–quark excitations connected by flux strings.The resulting states have definite link flux and site occupation numbers.Source: A. Color-neutral basis · p. 3

Reduced representation

• Because the mass and electric terms depend only on occupation and flux, the color-neutral states form a reduced basis on which W0 acts identically within the corresponding full-basis states.The basis is characterized by site occupation and color-electric flux rather than full color labels.Source: A. Color-neutral basis · p. 3

Color-neutral basis

• Figure 1 illustrates strong-coupling configurations, their four-state color-neutral superpositions, and hopping transitions with matrix elements depending on flux changes j′ = j ± 1/2.The displayed transitions are organized by gauge-invariant initial and final configurations.Source: Figure 1 · p. 3, A. Color-neutral basis · p. 3

Gauss-law constraint

• In this basis, Gauss law requires neighboring fluxes to differ by at most one quantum when a site contains a single fermion.The occupation label takes values n ∈ {0, 1, 2} after color distinctions are removed.Source: A. Color-neutral basis · p. 3

Truncation

• Restricting fluxes to j ≤ jmax preserves gauge invariance and gives link dimension dlink = 2jmax + 1, compared with full-basis dimensions 5, 14, 30, and 55 for jmax = 1/2, 1, 3/2, and 2.The corresponding reduced link dimensions are 2, 3, 4, and 5.Source: A. Color-neutral basis · p. 3

B. Removing the gauge fields

• Under open boundary conditions, the incoming boundary flux and site occupations recursively determine every link flux except for a ±1/2 ambiguity at singly occupied sites.The states |1+⟩ and |1−⟩ encode whether flux increases or decreases relative to the incoming link.Source: B. Removing the gauge fields · p. 4
• The resulting local site basis is {|0⟩, |1−⟩, |1+⟩, |2⟩}, with the left-boundary flux j0 fixed separately.A system of N sites is represented using these decorated fermionic states rather than explicit gauge links.Source: B. Removing the gauge fields · p. 4

Effective Hamiltonian

• The site configuration uniquely determines gauge-link content, effectively Abelianizing the model while making both electric and hopping terms nonlocal.Hopping matrix elements inherit flux dependence from the eliminated links.Source: B. Removing the gauge fields · p. 4

Physical-subspace constraint

• For j0 = 0, physical states satisfy lk,+ ≥ lk,− at every site, preventing negative reconstructed flux; their dimension is expressed through Catalan numbers.This condition removes unphysical configurations from the reduced basis.Source: B. Removing the gauge fields · p. 4

Efficiency

• The reduced formulation contains fewer link degrees of freedom than the full basis and is designed to support efficient gauge-invariant truncation.Its state-count relation is compared with the U(1) dynamical-fermion case.Source: B. Removing the gauge fields · p. 4

Mapping to the full basis

• A sequence of local isometries reconstructs the full color-singlet state from the reduced basis, acting left to right as a quantum circuit of depth equal to system size.The map takes a site and incoming flux as input and adds the outgoing link.Source: B. Removing the gauge fields · p. 4, Figure 2 · p. 4

Generality

• The construction retains the entire physical-subspace information and could in principle be generalized from SU(2) to SU(Nc) with Nc ≥ 2.After determining the relevant vertex matrix elements, gauge fields can likewise be encoded in fermionic sites.Source: B. Removing the gauge fields · p. 4

• MPS computes the ground state and vector state at j0 = 0 in the sector jN = 0.Source: §IV · p. 4
• Unlike Monte Carlo methods, MPS provides access to the entanglement entropy alongside spectral calculations.Source: §IV · p. 4
• The analysis studies von Neumann entanglement entropy in the ground state while approaching the continuum limit.Source: §IV · p. 5

Numerical methods

• Open-boundary MPS represents the N-site wavefunction through site tensors with bond dimension D controlling variational parameters and entanglement.Source: A. Numerical methods · p. 5
• Ground states are obtained by sweeping variational tensor updates until the relative energy change falls below tolerance ε.Source: A. Numerical methods · p. 5
• Excited states are computed variationally after projecting the Hamiltonian onto the subspace orthogonal to previously obtained states.Source: A. Numerical methods · p. 5
• The vector candidate is identified as the lowest-lying zero-momentum mesonic excitation with charge conjugation −1 and parity −1 in the continuum.On finite open lattices, remnants of broken momentum and charge-conjugation symmetries identify the state.Source: A. Numerical methods · p. 5
• The MPO bond dimension grows linearly with maximum color-electric flux, motivating truncation of jmax for computational feasibility.In the j0 = 0 = jN sector, jmax is bounded by N/2 × 1/2.Source: A. Numerical methods · p. 5
• The formulation reaches color-electric flux values far beyond jmax = 1/2 used in previous tensor-network studies.Source: A. Numerical methods · p. 5
• Energy penalties remove unphysical states, baryonic states, and nonzero-total-charge states from the targeted low-lying spectrum.Source: A. Numerical methods · p. 5

Numerical procedure

• For each (m/g, jmax), the calculation extrapolates first in bond dimension, then to the thermodynamic limit, and finally to vanishing lattice spacing.The study varies N ∈[100, 200], x ∈[50, 150], and D ∈[50, 200] across jmax = 1/2, 1, 3/2, 2.Source: B. Low-lying spectrum · p. 5
• The thermodynamic extrapolation gives precise ground-state energy-density and vector-gap estimates even for jmax = 2, where finite-bond-dimension effects are expected to be largest.Figures 3(c) and 3(d) illustrate this extrapolation at jmax = 2 and x = 150.Source: B. Low-lying spectrum · p. 5, Figure 3 · p. 6

Ground-state energy

• Ground-state energy densities deviate by at most the percentage level from the full-theory continuum result, even for jmax = 1/2, with larger truncations closer to the analytic solution.For jmax = 3/2 and 2, the largest-mass curves are nearly indistinguishable, while jmax = 1/2 and 1 remain consistently above the exact continuum solution.Source: B. Low-lying spectrum · p. 6, Figure 5 · p. 6

Vector mass gap

• Vector-gap extrapolations are substantially less controlled at small jmax: higher-order lattice-spacing corrections remain relevant for jmax = 1/2, while systematic errors are much larger for jmax = 1/2 and 1.For jmax = 3/2 and 2, the data are well described by linear fits in ag over the studied range; for jmax = 1, the continuum error remains large.Source: B. Low-lying spectrum · p. 6, Figure 6 · p. 7
• Only jmax ≥ 3/2 agrees well with the strong-coupling results, and at m/g = 1.6 these values approach the nonrelativistic prediction ∆vec = 2m/g.The jmax = 1/2 and 1 results severely differ from that prediction.Source: B. Low-lying spectrum · p. 7, Figure 7 · p. 7

Critical scaling

• Fits of the small-mass vector gap to γ(m/g)^ν yield exponents compatible with 2/3 for jmax = 3/2 and 2, while the jmax = 1/2 estimate 0.639 is unreliable because of large lattice effects.The fits use m/g ≤ 0.4; the jmax = 1 result remains compatible within relatively large errors.Source: B. Low-lying spectrum · p. 7, Table I · p. 7

C. Entanglement entropy

• The reduced formulation computes distillable, classical, representation, and total entanglement contributions from sector-resolved reduced density matrices.Flux sectors remain identifiable from fermionic content, and the reduced basis preserves the sector weights and distillable entropy while omitting the representation contribution in its effective Abelianized description.Source: C. Entanglement entropy · p. 8
• Averaging four central bipartitions and extrapolating in ag = 1/√x controls finite-size effects before fitting the entropy to its continuum form.The data are essentially converged at bond dimension D = 200, with differences from D = 150 used for error estimates.Source: C. Entanglement entropy · pp. 8–9, Figure 8 · p. 9
• The continuum extrapolation shows curvature from a logarithmic term, indicating that the ground-state entropy is ultraviolet divergent.This behavior is assessed from the total entropy obtained by summing the distinct entropy contributions.Source: C. Entanglement entropy · pp. 8–9
• For jmax = 3/2 and 2, the extracted central charges remain close to c = 2 across the studied bare-mass range, with little difference between these truncations.The full theory is expected to have c = 2 for its two Dirac fermions, and the higher truncations follow the continuum fit well.Source: C. Entanglement entropy · p. 9, Figure 9 · p. 9
• For jmax = 1/2 and 1, the entropy data are poorly compatible with logarithmic divergence at m/g ≥ 0.2, and their small-mass central charges differ from 2.Thus, ground-state entanglement is more sensitive to truncation than the ground-state energy density.Source: C. Entanglement entropy · p. 9

V. CONCLUSION

• The integrated-out basis supports gauge-invariant truncation at finite jmax and approaches the full SU(2) theory as jmax →∞.The truncation is more efficient than general finite-dimensional gauge-model truncations.Source: §V · pp. 9–10
• The formulation reaches larger flux cutoffs than previous tensor-network studies and could simplify quantum-simulator implementations.Ground-state energies may be estimated well with the simplest truncation, whereas excitations and entanglement require larger jmax.Source: §V · p. 10
• Ground-state energy density differs from the full-theory continuum result by only a percentage level at jmax = 1/2 and changes little between jmax = 3/2 and 2.This demonstrates rapid convergence for the ground-state energy observable.Source: §V · p. 10
• For the vector mass gap, jmax = 3/2 and 2 yield precise continuum extrapolations agreeing with Ref., while jmax = 1/2 and 1 retain pronounced lattice effects.The critical exponents are essentially consistent with a large-Nc calculation.Source: §V · p. 10
• Entanglement scaling likewise requires jmax = 3/2 or 2, which produce logarithmic divergence and central charges close to 2, unlike smaller truncations.The lower truncations therefore do not recover the proper continuum theory despite their accurate ground-state energy densities.Source: §V · p. 10
• The complete elimination of gauge degrees of freedom is specific to (1+1) dimensions, while higher-dimensional extensions would only reduce them maximally.The authors also target only the vector state besides the ground state, although other masses and time evolution are accessible in principle.Source: §V · p. 10

Appendix A: Dimension of the physical subspace

• Physical-subspace states are counted as nonnegative-flux paths on a quaternary tree, with Catalan numbers enumerating admissible paths and yielding the dimension formula.Negative flux first appears after a |1−⟩ edge from a zero-flux vertex, so paths containing such vertices are unphysical.Source: Appendix A · p. 11
• The proposed basis is exponentially smaller than the color-neutral basis from Ref. for OBC, already at jmax = 1/2, even though its physical-state fraction decreases with system size.The comparison uses dN,Hamer = 3N(2jmax + 1)N−1 and the scaling shown in Fig. 11.Source: Appendix A · p. 11, Figure 11 · p. 12

Appendix B: Hamiltonian

• After integrating out gauge fields, the color-electric term becomes nonlocal because flux is reconstructed from fermionic occupations.The hopping term also acquires flux-dependent matrix elements and long-range interactions.Source: Appendix B · p. 12

Appendix C: Penalty terms

• In MPS calculations, appropriate penalty terms shift negative-j states, baryon states, and states outside the j0 = 0 = jN sector high enough to prevent mixing with the targeted low-lying spectrum.Source: Appendix C · pp. 12–13

Appendix D: Distinguishing vector and scalar states

• Charge-conjugation phases near π distinguish vector candidates from scalar candidates with phases near 0.The appendix combines this phase information with the dispersion obtained from the pseudomomentum operator.Source: Appendix D · p. 13
• The dispersion plots distinguish the ground state, vector states, and scalar candidate for jmax = 1/2, 1, 3/2, and 2.The ground state is marked by a blue dot, vector states by red triangles, and the scalar candidate by a green square.Source: Figure 12 · p. 13
• ⟨C(2)⟩≈1 identifies the vector candidate above the ground state, while decreasing values characterize its momentum excitations.The diagnostic uses a cyclic shift by two lattice sites as an approximate translation operator for the finite open-boundary system.Source: Appendix D · p. 13

Appendix E: Extrapolation procedure and error estimation

• The extrapolation proceeds successively through D →∞, the thermodynamic limit, and the continuum limit ag = 1/√x →0.Bond-dimension data are fitted versus 1/D, finite-size behavior is extrapolated using the asymptotic form, and continuum data use polynomial fits in 1/√x.Source: Appendix E · pp. 13–14
• The continuum central value uses the lowest-order statistically significant fit with χ2 d.o.f. < 1, while systematic errors compare higher-order fits or omit the largest lattice spacing.For vector mass gaps, small jmax can produce enhanced lattice effects, and jmax = 1/2 quadratic fits have high χ2.Source: Appendix E · p. 14
• The critical exponent is obtained by fitting γ(m/g)ν over intervals [0.1, (m/g)max] with 0.25 ≤(m/g)max ≤0.4, selecting the smallest χ2 fit.Systematic uncertainty is estimated from the most outlying fit, with statistical uncertainty given by the fitting error.Source: Appendix E · p. 14
• Finite-precision entropy uncertainty is estimated as 10^-5 by comparing simulations with ε = 10^-6 and ε = 10^-10.The comparison is motivated by the largest expected differences at the highest truncation and is applied as a common systematic error.Source: Appendix E · p. 14, Figure 13 · p. 15

1. Mapping to the full basis

• Sequential local isometries reconstruct the full-basis state from the reduced basis through a quantum circuit of depth N.The map introduces link degrees of freedom while respecting SU(2) composition through Clebsch-Gordan coefficients; for j0 = 0, the left input link is |00⟩.Source: §1 · pp. 15–16, Figure 14 · p. 15

2. Classical part of the entropy

• Flux-sector projectors commute with the basis-changing isometry, so the same sector weights pj are obtained in the reduced and full bases.In the reduced basis, the target flux is represented by the cumulative qα values of vertices to the left of the link.Source: §2 · pp. 16–17, Figure 15 · p. 17

3. Distillable part of the entropy

• The full-basis reduced density matrix in a fixed flux sector is obtained from the reduced-basis density matrix by applying the isometry, projecting onto j, and tracing out the added degrees of freedom.The resulting operator is isometrically related to the reduced-sector RDM and therefore has the same spectrum.Source: §3 · p. 17, Figure 16 · p. 17