4HEX Improved Staggered Fermions

• The paper introduces a 4HEX staggered fermion discretization that replaces standard one-link gauge transporters with four HEX-smeared links, achieving O(a⁴) accuracy.
• It combines Naik, Lepage corrections and clover improvement to minimize taste breaking and lattice-spacing errors while preserving gauge invariance.
• Practical enhancements include improved spectral separation and reduced quantum-simulation cost through efficient local smearing and reunitarization techniques.

Searching arXiv for the cited and closely related staggered-fermion improvement papers. Looking up the two specific arXiv records and adjacent work on HEX/HISQ/ASQTAD improvement. 4HEX improved staggered fermions are staggered-fermion discretizations in which the one-link gauge transporter is replaced by a link obtained after four successive levels of hex smearing, typically combined with additional improvement terms such as the tree-level Naik correction, the Lepage correction, or, in taste-split formulations, Adams or Hoelbling mass terms together with a Symanzik clover counterterm. In the Hamiltonian formulation on a hypercubic lattice in $d$ spatial dimensions, the resulting $4{\rm HEX}$ staggered-fermion Hamiltonian is constructed to be accurate through $\mathcal O(a^4)$; in the taste-split action formulation, all gauge links in the covariant derivative and in the multi-hop taste-splitting operators are replaced by a four-level HEX-smeared link $V_\mu(x)$, with $c_{SW}=1$ at tree level (Gustafson et al., 2024, Durr, 2013).

1. Definition within staggered-fermion lattice formulations

In the Kogut–Susskind staggered-fermion formalism, each spatial site $x\in\Lambda$ carries one-component fermions $\chi(x)$, $\bar\chi(x)$, while the oriented bond $(x,x+\hat\mu)$ carries an $SU(3)$ gauge link $4{\rm HEX}$0. The staggered phases are the usual $4{\rm HEX}$1, and the alternating sign entering the mass term is $4{\rm HEX}$2. Within this setting, the $4{\rm HEX}$3 Hamiltonian is written as

$4{\rm HEX}$4

with the gauge part unchanged from the standard KS Hamiltonian, the conventional staggered mass term, and an improved kinetic operator built from a four-times hex-smeared one-link transporter together with Naik and Lepage corrections (Gustafson et al., 2024).

The taste-split action formulation uses the same one-component staggered field $4{\rm HEX}$5 and the usual staggered phases

$4{\rm HEX}$6

but augments the staggered covariant derivative by taste-non-singlet mass terms. Three such operators are specified: the Adams operator $4{\rm HEX}$7, the symmetric Hoelbling operator $4{\rm HEX}$8, and the mixed operator $4{\rm HEX}$9. In that construction, the full $\mathcal O(a^4)$0-improved action replaces all gauge links appearing in the derivative and in the multi-hop mass operators by the final fat link $\mathcal O(a^4)$1 obtained after four HEX steps (Durr, 2013).

These two formulations share the same central mechanism—repeated local link smearing followed by projection back into $\mathcal O(a^4)$2—but they target different operator algebras. The Hamiltonian construction emphasizes reduction of lattice-spacing artifacts and quantum-simulation cost, whereas the taste-split construction emphasizes spectral separation, chiral structure, and Symanzik improvement.

2. Hamiltonian structure of the 4HEX discretization

The gauge sector is kept in the standard KS form,

$\mathcal O(a^4)$3

where $\mathcal O(a^4)$4 is the left-electric field and $\mathcal O(a^4)$5 the plaquette. The mass term is

$\mathcal O(a^4)$6

The nontrivial part is the improved kinetic Hamiltonian, which contains four distinct hopping structures: a nearest-neighbor term built from $\mathcal O(a^4)$7, a three-link Naik term $\mathcal O(a^4)$8, a Lepage-corrected one-link term $\mathcal O(a^4)$9, and a one-link hex-smeared term $V_\mu(x)$0 (Gustafson et al., 2024).

The Naik operator is the three-link straight transporter

$V_\mu(x)$1

while the coefficient choices

$V_\mu(x)$2

are stated to eliminate all tree-level $V_\mu(x)$3 and $V_\mu(x)$4 errors and to push the leading discretization effects to $V_\mu(x)$5. In this sense, the Hamiltonian is not merely smeared: it is a composite improvement scheme in which straight-link, rectangle-subtraction, and repeated hex-smearing contributions are tuned against one another.

A useful distinction follows directly from the operator content. The $V_\mu(x)$6 Hamiltonian does not modify the gauge Hamiltonian itself; the improvement acts in the matter sector through the hopping structure. The leading changes therefore occur in the fermion dispersion, taste breaking, and rotational-symmetry violations induced by the lattice derivative, rather than in the plaquette term.

3. Smearing hierarchy and link construction

The link construction proceeds in layers. The first stage recalls the ASQTAD fat link, extended to include five-link and seven-link staples: $V_\mu(x)$7 with

$V_\mu(x)$8

After fattening, one reunitarizes,

$V_\mu(x)$9

The Lepage correction then subtracts the double-staple contamination through

$c_{SW}=1$0

where $c_{SW}=1$1 is the first-order covariant finite difference (Gustafson et al., 2024).

A single hex-smearing level is defined by

$c_{SW}=1$2

The final one-link transporter used in the Hamiltonian is obtained by iterating the hex map four times,

$c_{SW}=1$3

This repeated reunitarization is part of the definition, not a numerical afterthought.

In the taste-split action, the four HEX levels are described recursively. At level $c_{SW}=1$4, one forms a staple sum $c_{SW}=1$5 from the six two-link staples built out of the previous-level links, constructs

$c_{SW}=1$6

and projects back to $c_{SW}=1$7 via

$c_{SW}=1$8

Two weight choices are listed explicitly: $c_{SW}=1$9 and $x\in\Lambda$0 (Durr, 2013).

4. Improvement pattern and comparison with ASQTAD and HISQ

The Hamiltonian construction is presented as inspired by ASQTAD and HISQ but distinct from both. ASQTAD removes the tree-level $x\in\Lambda$1 errors through Naik, fat-link, and Lepage improvement, but leaves $x\in\Lambda$2 taste-exchange interactions and $x\in\Lambda$3 cubic anisotropy. HISQ applies a second fat/Lepage level on the one-link operator, reduces $x\in\Lambda$4 loops, and pushes rotation errors to $x\in\Lambda$5. By contrast, the 4HEX construction repeats hex smearing four times, and each level is said to remove an additional class of lattice hypercubic artifacts (Gustafson et al., 2024).

Scheme	Stated ingredients	Stated leading residual effects
ASQTAD	Naik + fat link + Lepage	$x\in\Lambda$6 taste exchange; $x\in\Lambda$7 cubic anisotropy
HISQ	second fat/Lepage level on one-link operator	$x\in\Lambda$8 at tree level; $x\in\Lambda$9 at one loop
4HEX Hamiltonian	four HEX levels + Naik + Lepage	$\chi(x)$0 at tree and one-loop level

The defining claim of the Hamiltonian analysis is that by the fourth hex level all operators up to dimension-six that violate continuum Lorentz or taste symmetry are canceled at tree level. The first nonzero discretization errors then occur at $\chi(x)$1, and the summary description states that the Hamiltonian is free of tree-level and one-loop $\chi(x)$2 errors. Numerical studies in $\chi(x)$3D toy models are reported to confirm an observed slope in the continuum extrapolation reduced by an additional factor of $\chi(x)$4 relative to HISQ, and the abstract specifies a demonstration of reduced lattice-spacing errors in the $\chi(x)$5d lattice Schwinger model (Gustafson et al., 2024).

A necessary qualification is that not every $\chi(x)$6 construction in the supplied literature carries the same formal improvement claim. The Hamiltonian version is stated to be accurate through $\chi(x)$7, while the taste-split construction is described in its closing sentence as an “only $\chi(x)$8-improved staggered discretization.” This suggests that repeated hex smearing alone does not define a unique Symanzik class; the formal order depends on the surrounding operator content.

5. Taste-split 4HEX actions, clover improvement, and spectral diagnostics

The taste-split formulation starts from the staggered kinetic operator

$\chi(x)$9

and augments it by taste-non-singlet mass terms. The Adams operator uses

$\bar\chi(x)$0

producing two light tastes and two heavy tastes. The symmetric Hoelbling operator employs the two-hop taste-tensor operator $\bar\chi(x)$1 through

$\bar\chi(x)$2

and the mixed operator is

$\bar\chi(x)$3

The full $\bar\chi(x)$4-improved action then adds the staggered version of the Wilson-clover counterterm,

$\bar\chi(x)$5

with $\bar\chi(x)$6 at tree level (Durr, 2013).

The weak-coupling/Symanzik expansion in that construction is used to motivate the clover term as the counterterm required to cancel the leading $\bar\chi(x)$7 artifacts. The full action is therefore a compound object: staggered kinetic term, ordinary mass $\bar\chi(x)$8, one of the taste-splitting operators $\bar\chi(x)$9, and the clover operator, all written with links that have already undergone four HEX steps. The role of 4HEX here is partly ultraviolet filtering and partly a preconditioner for the spectral and chiral behavior of the kernel.

The reported spectral diagnostics are specific. Without smearing or clover, the complex eigenvalues of the massless staggered operator lie in two “bellies” that barely separate physical from doubler branches. Adding 4-level HEX alone opens the bellies somewhat but adds large additive mass renormalization. Including the clover term with $(x,x+\hat\mu)$0 plus $(x,x+\hat\mu)$1 makes the low-lying physical branch very thin, almost circular, with exactly real topological modes cleanly detached from doublers. In the “chiral-sandwich” expectation values $(x,x+\hat\mu)$2, one sees $(x,x+\hat\mu)$3 signals for the $(x,x+\hat\mu)$4 physical zero-modes and near zero for all others, while taste-singlet chiralities $(x,x+\hat\mu)$5 remain suppressed. A diagnostic proposed explicitly is to plot the 10–20 lowest eigenvalues $(x,x+\hat\mu)$6 of the massless improved operator and the corresponding chiralities $(x,x+\hat\mu)$7; after full improvement, for $(x,x+\hat\mu)$8, one should see two real zero-modes with $(x,x+\hat\mu)$9, a tight cluster of near-zero modes in the imaginary direction, and a gap $SU(3)$0–$SU(3)$1 before the first doubler eigenvalue (Durr, 2013).

6. Quantum-simulation decomposition and practical tuning

For quantum simulation, the $SU(3)$2 Hamiltonian is Trotter-stepped in blocks corresponding to the gauge sector, nearest-neighbor hopping with $SU(3)$3, the Naik term, the Lepage plus one-hex block, and the mass term. The primitive operations are defined explicitly: $SU(3)$4 for basic gauge–matter evolution on one link, $SU(3)$5 for $SU(3)$6 matrix multiplication on one link register, $SU(3)$7 for gauge-link inversion, $SU(3)$8 and $SU(3)$9 for diagonal phase rotations in the electric-field basis, and $4{\rm HEX}$00 for reunitarization of one link. Per spatial link and per Trotter step, the tabulated counts assign one $4{\rm HEX}$01 each to the nearest-neighbor and Naik blocks, two $4{\rm HEX}$02 to the Naik block, two $4{\rm HEX}$03 and one $4{\rm HEX}$04 to the Lepage block, and for the 4HEX block “$4{\rm HEX}$05” $4{\rm HEX}$06, four $4{\rm HEX}$07, and four $4{\rm HEX}$08 (Gustafson et al., 2024).

For a $4{\rm HEX}$09-dimensional lattice of volume $4{\rm HEX}$10, assuming two color-parity Trotter-colors per direction, the resulting estimate is

$4{\rm HEX}$11

per slice, plus $4{\rm HEX}$12 multi-controlled rotations for the gauge sector. The parallelization structure follows from checkerboard decomposition: all even links may be updated simultaneously in one color and all odd links in the next. Because the fat-link and hex-link constructions are purely local operations on each link’s neighborhood, they can be scheduled in two further sub-rounds, one for staple generation and one for reunitarization. The stated consequence is a reduction of circuit depth from $4{\rm HEX}$13 to $4{\rm HEX}$14, up to a constant proportional to the number of smearing levels rather than to the lattice size (Gustafson et al., 2024).

The taste-split formulation gives concrete tuning recommendations. One starts at $4{\rm HEX}$15 and measures the pion taste splitting

$4{\rm HEX}$16

If $4{\rm HEX}$17 stays $4{\rm HEX}$18, no further tuning of $4{\rm HEX}$19 is needed; otherwise $4{\rm HEX}$20 is adjusted in increments of $4{\rm HEX}$21 until the slope $4{\rm HEX}$22 vanishes. The smearing weights $4{\rm HEX}$23 are to be chosen so as to minimize $4{\rm HEX}$24 at the target lattice spacing, and the typical values $4{\rm HEX}$25 are stated to give a $4{\rm HEX}$26–$4{\rm HEX}$27 reduction in taste splitting relative to one-level HEX. In dynamical RHMC runs, stout or nHYP variants may be used by replacing $4{\rm HEX}$28 with the exponential map to maintain reversibility. If the $4{\rm HEX}$29 kernel is used in an overlap construction, the spectral bounds are reported to be greatly tightened, reducing the degree of the sign-function polynomial by $4{\rm HEX}$30–$4{\rm HEX}$31 (Durr, 2013).

Taken together, these constructions suggest that “4HEX improved staggered fermions” is best understood as a family of staggered discretizations organized around four successive HEX-smearing levels and supplemented by additional improvement operators chosen for the target problem. In the Hamiltonian setting, the emphasis is on $4{\rm HEX}$32 scaling and quantum-circuit structure; in the taste-split setting, the emphasis is on Symanzik improvement, spectral separation, and chirality diagnostics.