1+1D SU(3) Lattice Gauge Theory

• 1+1D SU(3) lattice gauge theory is a framework where SU(3) gauge fields coupled with staggered fermions are reformulated using the loop-string-hadron (LSH) approach to retain only local, gauge-invariant degrees of freedom.
• The methodology employs an occupation-number basis and irreducible Schwinger bosons to reduce the dimensionality of the Hilbert space, significantly simplifying simulation and computation.
• Local Abelian Gauss-law constraints decouple the Hamiltonian into separate hopping terms, facilitating block-diagonalization and scalable classical and quantum simulation strategies.

The 1+1-dimensional SU(3) lattice gauge theory describes the dynamics of SU(3) gauge fields—central to quantum chromodynamics (QCD)—coupled to dynamical matter (typically staggered fermions) on a spatial lattice with one spatial and one temporal dimension. The loop-string-hadron (LSH) framework provides a reformulation of this theory, in which all nonabelian gauge redundancies are eliminated and only local, gauge-invariant degrees of freedom are retained. The LSH approach, rigorously constructed from a reformulation of Hamiltonian lattice gauge theory in terms of irreducible Schwinger bosons, leads to a strictly local Hamiltonian, an exponentially reduced Hilbert space, and transparent Abelian Gauss-law constraints, providing significant advantages for both classical and quantum simulation strategies (Kadam et al., 2022).

1. Hamiltonian Formulation in the LSH Basis

The LSH Hamiltonian on a one-dimensional lattice with $N$ sites (indexed by $r=1,\ldots,N$) and one staggered fermion flavor comprises purely on-site terms and nearest-neighbor links: $$H = \sum_{r=1}^{N} H_{\rm site}(r) + \sum_{r=1}^{N'} H_{\rm link}(r, r+1)$$ where $N' = N-1$ for open and $N$ for periodic boundary conditions. The on-site and link Hamiltonians are: $$H_{\rm site}(r) = \underbrace{\mu\,(-1)^r \left[\nu_1(r)+\nu_0(r)+\bar\nu_1(r)\right]}_{H_M(r)} + \underbrace{\frac{1}{3}\left[P(r)^2 + Q(r)^2 + P(r) Q(r)\right] + P(r) + Q(r)}_{H_E(r)}$$

$$H_{\rm link}(r, r+1) = x\left[J_1^\dagger(r) J_1(r+1) + J_{\bar 1}^\dagger(r) J_{\bar 1}(r+1) + J_0^\dagger(r) J_0(r+1) + \mathrm{H.c.}\right]$$

where $J_f(r)$ are hopping operators for $f=1,\,\bar 1,\,0$, corresponding to the three possible quark flavors. The parameters $\mu$ and $x$ reflect the bare fermion mass and gauge coupling, respectively. The explicit form of $J_f(r)$ in terms of LSH operators, e.g. for $J_1^\dagger(r)$,

$$J_1^\dagger(r) = \chi_1^\dagger(r)\, \ell_P^\dagger(r)\, \sqrt{1-\frac{\nu_1(r)}{n_P(r)+2}}\, \sqrt{1-\frac{\nu_1(r)}{n_P(r)+n_Q(r)+3}}$$

with analogous terms for $J_{\bar 1}$ and $J_0$, reflects the coupling between fermionic and bosonic LSH degrees of freedom.

2. LSH Local Degrees of Freedom and Operator Algebra

The LSH formulation employs an occupation-number basis at each site $r$: $$|\,n_P, n_Q ; \nu_1, \nu_0, \bar\nu_1\,\rangle_r \quad \text{with} \quad n_P, n_Q \in \{0,1,2,\ldots\},\ \nu_f \in \{0,1\}$$ where

• $n_P, n_Q$ are integer variables counting “rightward” and “leftward” pure-gauge flux loops, respectively,
• $\nu_1, \nu_0, \bar\nu_1$ are fermionic (bit-valued) operators labeling three gauge-invariant “string ends” (sources/sinks of unit flux).

On-site operators include:

• Bosonic number operators: $$\hat n_l(r) | n_P, n_Q ; \cdots \rangle_r = n_l | n_P, n_Q ; \cdots \rangle_r$$ for $l=P,Q$,
• Bosonic ladder operators: $\hat\ell_l^\dagger$, $\hat\ell_l$ obeying $$[\hat n_l, \hat\ell_l^\dagger] = \hat\ell_l^\dagger$$,
• Fermionic number operators: $\hat\nu_f(r)$ for $f\in\{1,0,\bar 1\}$,
• Fermionic creation/annihilation operators: $\hat\chi_f^\dagger(r)$, $\hat\chi_f(r)$, with canonical anticommutation relations.

Composite on-site irreps,

$$P(r) = n_P(r) + \nu_1(r)\left[1-\nu_0(r)\right],\qquad Q(r) = n_Q(r) + \nu_0(r)\left[1-\bar\nu_1(r)\right]$$

encode net “rightward” and “leftward” flux at each site. Conditional bosonic ladder operators, e.g. $$(\hat\ell_l)^{\hat\nu_f} = \hat\nu_f \hat\ell_l + (1-\hat\nu_f)$$, allow selective state transitions based on the fermionic degree of freedom.

3. Reformulation from Kogut–Susskind via Irreducible Schwinger Bosons

Starting from the 1+1D Kogut–Susskind Hamiltonian in temporal gauge,

$H = H_M + H_E + H_I$

with mass, electric, and interaction terms, Schwinger boson prepotentials, $\{a_\alpha, b^\alpha\}_{L,R}$, are introduced at each link to construct electric field and link variables that automatically satisfy the SU(3) algebra and the Casimir constraint $E^2(L) = E^2(R)$. The link operator $U^\alpha{}_\beta$ is built from the prepotentials so as to ensure unitarity and determinant-1 constraints.

Irreducible Schwinger bosons (ISB), defined by

$$A^\dagger_\alpha = a^\dagger_\alpha - \frac{1}{\hat N_a + \hat N_b + 1}(a^\dagger \cdot b^\dagger) b_\alpha$$

$$B^{\dagger\alpha} = b^{\dagger\alpha} - \frac{1}{\hat N_a + \hat N_b + 1}(a^\dagger \cdot b^\dagger) a^\alpha$$

produce traceless SU(3) irreps and address the multiplicity problem associated with naive Schwinger bosons. The minimal coupling to staggered fermions is local and manifestly gauge-invariant. Transition from the ISB and fermion variables to LSH variables solves all eight nonabelian Gauss law constraints, leaving only two Abelian ones per link.

4. Abelian Gauss Laws and Superselection Sectors

The full local SU(3) Gauss law,

$$G^a(r) = E^a(L, r) + E^a(R, r-1) + \psi^\dagger(r) T^a \psi(r) \stackrel{!}{=} 0$$

is reduced in the LSH basis to two Abelian “flux continuity” constraints per link: $P(r) = P(r+1), \qquad Q(r) = Q(r+1)$ ensuring the conservation of rightward and leftward fluxes, respectively. These constraints define superselection sectors labeled by total flux mismatches $\Delta P, \Delta Q$. Three global $U(1)$ fermion-number symmetries $\sum_r \nu_f(r)$ are typically recast as

$$\mathcal{F} = \sum_r (\nu_1 + \nu_0 + \bar\nu_1), \quad \Delta P = \sum_r [\nu_1(r) - \nu_0(r)], \quad \Delta Q = \sum_r [\bar\nu_1(r) - \nu_0(r)]$$

The physically relevant vacuum in the strong coupling regime is realized in the sector with $\mathcal{F} = 3N/2$ and $\Delta P = \Delta Q = 0$ (Kadam et al., 2022).

5. Physical and Computational Advantages of the LSH Framework

The LSH formulation presents several key benefits:

• Gauge invariance is strict and built-in, eliminating the need for nonlocal projectors or penalty terms.
• All operators are local; non-Abelian, nonlocal “Coulomb” interactions are replaced by two simple Abelian constraints per link.
• The local Hilbert space dimension scales as $\dim_{\rm LSH} \propto e^{1.932 N}$ compared to $8^N$ in colored-fermion formulations, representing a substantial reduction.
• The Hamiltonian splits into three decoupled hopping terms, each compatible with qubit or boson-qubit architectures and well-suited to efficient Trotterization or error mitigation in quantum algorithms.
• Abelian superselection sectors allow immediate block-diagonalization for both classical diagonalization and quantum circuit construction.
• The underlying structure carries over to higher dimensions, preserving the gauge-invariant flux line structure critical to QCD.

In summary, the LSH formalism for 1+1-dimensional SU(3) lattice gauge theory replaces nonlocal, nonabelian gauge redundancies with strictly local Abelian constraints, dramatically reducing computational complexity and enabling scalable simulation approaches on both classical and quantum platforms (Kadam et al., 2022).

6. Context and Implications

The loop-string-hadron approach was previously developed for SU(2) gauge theories and is here systematically extended to SU(3) coupled to dynamical staggered fermions in 1+1D. Numerical validation confirms that, for open boundary conditions, the LSH Hamiltonian reproduces the physics of the fully gauge-fixed Hamiltonian—whose nonlocal, long-range interactions do not generalize to periodic boundaries or higher dimensions—while maintaining local structure and scalability.

A plausible implication is that the LSH structure, by keeping gauge-invariant flux lines explicit and eliminating redundant gauge degrees of freedom, makes tensor network and quantum simulation algorithms significantly more tractable, not only for (1+1)D SU(3) but as a foundation for studies in higher-dimensional nonabelian lattice gauge theories.