Tangent-Fermion Discretization

• Tangent-Fermion discretization is a lattice regularization method that replaces the standard sine derivative with a tangent operator to eliminate fermion doubling and produce a single Dirac cone.
• It converts a nonlocal hopping Hamiltonian into a local generalized eigenvalue problem, enabling efficient tensor-network and quantum-circuit simulations.
• Applications span one-dimensional helical Luttinger liquids, chiral lattice models, and two-dimensional edge-state phenomena under magnetic fields.

Searching arXiv for papers on tangent-fermion discretization and related lattice Dirac methods. Tangent-fermion discretization is a lattice regularization of Dirac or Majorana fermions in which the continuum momentum is replaced by a tangent symbol, $$k_\mu \mapsto \frac{2}{a}\tan\!\bigl(\frac{a k_\mu}{2}\bigr)$$, rather than the usual sine symbol of nearest-neighbor finite differences. In the formulations developed for one- and two-dimensional Dirac problems, this replacement yields a single Dirac cone in the Brillouin zone, with the would-be doubler replaced by a pole at the zone boundary. The price is nonlocality in the literal hopping Hamiltonian, but the same spectral problem can be written as a local generalized eigenvalue problem, and in several settings as an exact low-bond-dimension MPO or an efficient LCU block encoding (Beenakker et al., 2023, Zakharov et al., 2024, Beenakker, 17 Jun 2026).

1. Definition and basic lattice construction

In one dimension, the tangent scheme studied for massless Dirac or helical fermions replaces the standard nearest-neighbor discretization of $\partial_x$ by Stacey’s nonlocal derivative. For a chiral branch this produces the dispersion

$E(k)=2t_0\tan\frac{k}{2},$

instead of the sine dispersion $E(k)=t_0\sin k$ (Zakharov et al., 2024). The tangent symbol is linear near $k=0$,

$E(k)\approx t_0 k \quad (k\to 0),$

but has poles at $k=\pm \pi$, so the Brillouin-zone edge does not host a second zero (Zakharov et al., 2024).

In real space, the corresponding Stacey derivative is nonlocal. In the one-dimensional formulation reviewed for tangent fermions, it acts as

$$\frac{df}{dx}\mapsto 2\sum_{n=1}^{\infty}(-1)^n\bigl[f(x-n)-f(x+n)\bigr],$$

and the associated hopping becomes all-to-all, with $t_{nm}=2 i t_0 (-1)^{n-m}$ in the translationally invariant case (Zakharov et al., 2024). In two dimensions, the same idea yields the single-cone dispersion

$$E(\mathbf{k})=\pm \frac{2\hbar v}{a}\sqrt{\tan^2\!\left(\frac{a k_x}{2}\right)+\tan^2\!\left(\frac{a k_y}{2}\right)},$$

which is linear near $\partial_x$0 and has no additional Dirac points in the Brillouin zone (Beenakker et al., 2023).

The central comparison throughout the literature is between three momentum symbols. The sine symbol is local but doubled; the SLAC or sawtooth symbol is nonlocal and single-cone but has a cusp at the zone boundary; the tangent symbol is nonlocal and single-cone, but can be rewritten as a ratio of local operators (Beenakker et al., 2023).

Scheme	Symbol	Characteristic consequence
Sine	$\partial_x$1 or $\partial_x$2	Local, but has doublers
SLAC / sawtooth	$\partial_x$3 modulo $\partial_x$4	Single cone, nonlocal, cusp at zone boundary
Tangent / Stacey	$\partial_x$5 or $\partial_x$6	Single cone, poles at zone boundary

2. Fermion doubling, nonlocality, and hidden locality

The standard nearest-neighbor discretization realizes the one-dimensional manifestation of the Nielsen–Ninomiya obstruction: $\partial_x$7 has zeros at $\partial_x$8 and $\partial_x$9, so it carries two Dirac points rather than one (Zakharov et al., 2024). In two dimensions, the same mechanism yields multiple cones for the naive sine discretization, while Wilson and staggered variants remove or reduce doubling at the cost of breaking symmetries or retaining more than one cone (Beenakker et al., 2023).

Tangent fermions avoid doubling by violating the locality assumption of the no-go theorem. The nonlocal hopping $E(k)=2t_0\tan\frac{k}{2},$0 has no decay with $E(k)=2t_0\tan\frac{k}{2},$1, so the theorem’s locality hypothesis does not apply (Zakharov et al., 2024). The resulting lattice spectrum contains a single chiral branch with definite velocity sign throughout the Brillouin zone in one-dimensional chiral constructions, and a single Dirac cone in the two-dimensional Dirac problem (Zakharov et al., 23 Jun 2026, Beenakker et al., 2023).

A recurrent misconception is that tangent fermions are “local” because the practical algorithms are local. The underlying Hamiltonian is not local in position space. What is local is the generalized eigenvalue formulation. For rational dispersions $E(k)=2t_0\tan\frac{k}{2},$2 with finite polynomials in $E(k)=2t_0\tan\frac{k}{2},$3, one can write

$E(k)=2t_0\tan\frac{k}{2},$4

with $E(k)=2t_0\tan\frac{k}{2},$5 and $E(k)=2t_0\tan\frac{k}{2},$6 short-range difference operators. For the tangent dispersion $E(k)=2t_0\tan\frac{k}{2},$7,

$E(k)=2t_0\tan\frac{k}{2},$8

so the spectral problem is local even though the effective hopping Hamiltonian is not (Zakharov et al., 2024).

This hidden locality has direct algorithmic consequences. In one-dimensional tensor-network implementations, both sine and tangent free spinless Hamiltonians admit exact MPOs of bond dimension $E(k)=2t_0\tan\frac{k}{2},$9, while the helical tangent Hamiltonian with local Hubbard interaction admits an MPO of bond dimension $E(k)=t_0\sin k$0; with periodic boundary conditions, the tangent MPO stays at $E(k)=t_0\sin k$1, whereas the sine MPO grows from $E(k)=t_0\sin k$2 to $E(k)=t_0\sin k$3 because of the extra $E(k)=t_0\sin k$4–$E(k)=t_0\sin k$5 hopping term (Zakharov et al., 2024). In two dimensions, the generalized eigenproblem is likewise formulated with local Hermitian operators $E(k)=t_0\sin k$6 and $E(k)=t_0\sin k$7, rather than a single local Hamiltonian, and this is the mechanism by which tangent fermions evade doubling without breaking time-reversal or chiral symmetry (Vela et al., 2024).

3. Helical Luttinger liquids and tensor-network simulations

A major use of tangent-fermion discretization is the simulation of interacting one-dimensional Dirac systems with tensor networks. In the helical Luttinger-liquid setting, the lattice Hamiltonian

$E(k)=t_0\sin k$8

was studied for sine and tangent hoppings, with on-site Hubbard interaction

$E(k)=t_0\sin k$9

At half filling, the continuum bosonization prediction is the power-law decay

$k=0$0

with $k=0$1 and $k=0$2 (Zakharov et al., 2024).

The numerical comparison is sharp. For $k=0$3, both discretizations reproduce the free continuum behavior. At finite interaction, however, only the tangent discretization remains gapless: the smoothed propagator $k=0$4 and transverse spin correlator $k=0$5 follow the continuum power laws with the bosonization exponents, whereas the sine discretization shows exponential decay, indicating an interaction-induced gap opened by fermion doubling (Zakharov et al., 2024). This result was obtained with DMRG in TeNPy on $k=0$6 sites, with MPS bond dimensions up to $k=0$7 (Zakharov et al., 2024).

The same framework was extended to a helical Luttinger liquid with two-particle backscattering processes. In that model, the tangent-fermion kinetic term preserves time-reversal symmetry while avoiding the sine-dispersion obstruction. Bosonization predicts that the two-particle Umklapp interaction becomes relevant at half filling for

$k=0$8

while a distinct single-particle Umklapp term remains non-gap-opening in the regime studied (Zakharov et al., 14 Jan 2026). The tensor-network calculation confirms the expected scenario: at half filling and sufficiently strong forward interaction, the propagator departs from Luttinger-liquid scaling, a gap opens, and the transverse spin correlator saturates at long distance, indicating a gapped phase with spontaneously broken time-reversal symmetry (Zakharov et al., 14 Jan 2026).

4. Chiral fermions, symmetric mass generation, and one-dimensional anomaly-free models

Tangent fermions have also been used to formulate strictly one-dimensional chiral lattice models without mirror partners. In the anomaly-free $k=0$9–$E(k)\approx t_0 k \quad (k\to 0),$0–$E(k)\approx t_0 k \quad (k\to 0),$1–$E(k)\approx t_0 k \quad (k\to 0),$2 model of Wang and Wen, four chiral branches carry charges $E(k)\approx t_0 k \quad (k\to 0),$3 as right-movers and $E(k)\approx t_0 k \quad (k\to 0),$4 as left-movers. The tangent discretization realizes these branches directly on a one-dimensional lattice through the free Hamiltonian

$E(k)\approx t_0 k \quad (k\to 0),$5

with

$E(k)\approx t_0 k \quad (k\to 0),$6

so that no mirror branch occurs in the Brillouin zone (Zakharov et al., 23 Jun 2026).

The symmetry-allowed gapping perturbations in that model are six-fermion interactions with interaction vectors

$E(k)\approx t_0 k \quad (k\to 0),$7

which satisfy the Haldane-null condition and therefore can gap the spectrum without frustrating one another (Zakharov et al., 23 Jun 2026). In the noninteracting theory their scaling dimension is $E(k)\approx t_0 k \quad (k\to 0),$8, so they are strongly irrelevant. A Hubbard-type density-density interaction renormalizes the Luttinger parameter to $E(k)\approx t_0 k \quad (k\to 0),$9, reducing the scaling dimension to

$k=\pm \pi$0

The interaction becomes relevant for

$k=\pm \pi$1

and DMRG shows the opening of an excitation gap in this regime without the appearance of a degenerate ground state, which the paper identifies as the hallmark of symmetric mass generation (Zakharov et al., 23 Jun 2026).

This suggests a broader role for tangent-fermion discretization in anomaly-free chiral lattice models. In the cited one-dimensional setting, it supplies the ingredient that a strictly local chiral discretization cannot provide: genuinely chiral lattice fermions without mirror partners, yet still in a tensor-network-friendly representation (Zakharov et al., 23 Jun 2026).

5. Two-dimensional boundary, edge-state, and magnetic-field formulations

In two dimensions, tangent fermions have been developed both for bulk Dirac cones and for confined geometries. The local generalized eigenproblem

$k=\pm \pi$2

with

$k=\pm \pi$3

$k=\pm \pi$4

reproduces the two-dimensional tangent dispersion and preserves time-reversal and chiral symmetries in the generalized-eigenproblem sense (Vela et al., 2024). In the broader survey of tangent fermions, this single-cone formulation was used to study Klein tunneling, strong antilocalization, gauge-invariant magnetic fields, the anomalous quantum Hall effect, and a disordered Majorana-metal phase (Beenakker et al., 2023).

For bounded geometries, the generalized eigenproblem can accommodate continuum Dirac boundary conditions. The construction proceeds by restricting $k=\pm \pi$5 and $k=\pm \pi$6 to the physical domain, rotating the local boundary projector $k=\pm \pi$7 to $k=\pm \pi$8 with a site-dependent unitary $k=\pm \pi$9, and then deleting the spin-down row and column at each boundary site in both matrices (Vela et al., 2024). In channel geometries this reproduces the continuum edge-state phenomenology: no edge states for infinite-mass boundaries, a flat zero-energy edge band for zigzag boundaries, and dispersive edge modes for intermediate boundary angles. A notable subtlety is that near the zigzag limit a spurious partner band can appear; increasing the deviation from zigzag pushes this lattice artifact out of the low-energy window (Vela et al., 2024).

The same two-dimensional tangent framework has been adapted to magnetic problems where chiral protection is central. For the massless Dirac equation in a perpendicular magnetic field, a chirality-preserving tangent-fermion discretization, combined with the boundary condition

$$\frac{df}{dx}\mapsto 2\sum_{n=1}^{\infty}(-1)^n\bigl[f(x-n)-f(x+n)\bigr],$$0

produces a zero-energy sector of single chirality on a finite lattice (Vela et al., 19 May 2025). In a channel geometry, the resulting zeroth Landau level contains a spin-up polarized bulk mode centered at $$\frac{df}{dx}\mapsto 2\sum_{n=1}^{\infty}(-1)^n\bigl[f(x-n)-f(x+n)\bigr],$$1 together with a spin-up edge mode, and in strongly non-uniform magnetic fields the higher Landau levels broaden while the zero-energy mode remains exactly flat within numerical accuracy (Vela et al., 19 May 2025). By contrast, the corresponding graphene comparison broadens because the zeroth Landau level contains both chiralities (Vela et al., 19 May 2025).

6. Quantum-circuit representation and broader methodological significance

The generalized-eigenproblem formulation of tangent fermions has a direct quantum-algorithmic consequence. A direct LCU decomposition of the tangent derivative is inefficient: both the number of terms and the subnormalization factor grow with the lattice size. In one dimension, however, the tangent Dirac equation can be written as the local operator pencil

$$\frac{df}{dx}\mapsto 2\sum_{n=1}^{\infty}(-1)^n\bigl[f(x-n)-f(x+n)\bigr],$$2

with

$$\frac{df}{dx}\mapsto 2\sum_{n=1}^{\infty}(-1)^n\bigl[f(x-n)-f(x+n)\bigr],$$3

where $$\frac{df}{dx}\mapsto 2\sum_{n=1}^{\infty}(-1)^n\bigl[f(x-n)-f(x+n)\bigr],$$4 is the antiperiodic shift operator (Beenakker, 17 Jun 2026). Each member of the pencil has an exact LCU with term count independent of lattice size and with subnormalization factor of order unity. In one dimension,

$$\frac{df}{dx}\mapsto 2\sum_{n=1}^{\infty}(-1)^n\bigl[f(x-n)-f(x+n)\bigr],$$5

and in $$\frac{df}{dx}\mapsto 2\sum_{n=1}^{\infty}(-1)^n\bigl[f(x-n)-f(x+n)\bigr],$$6 dimensions the subnormalization factors remain $$\frac{df}{dx}\mapsto 2\sum_{n=1}^{\infty}(-1)^n\bigl[f(x-n)-f(x+n)\bigr],$$7 and $$\frac{df}{dx}\mapsto 2\sum_{n=1}^{\infty}(-1)^n\bigl[f(x-n)-f(x+n)\bigr],$$8, again independent of lattice size (Beenakker, 17 Jun 2026).

This makes tangent fermions unusual among nonlocal, symmetry-preserving discretizations. The direct tangent Hamiltonian is highly nonlocal, but the operator pencil is local and admits efficient block encodings for generalized eigenvalue searches and Green-function calculations (Beenakker, 17 Jun 2026). A plausible implication is that tangent-fermion discretization occupies a distinctive methodological position: it trades locality of the physical Hamiltonian for locality of the operator pencil, and that trade turns out to be compatible with tensor networks, sparse generalized eigenproblems, and quantum-circuit primitives (Zakharov et al., 2024, Beenakker, 17 Jun 2026).

Across these developments, the defining features remain stable. Tangent-fermion discretization replaces the doubled sine cone by a single tangent cone, preserves the symmetry structure of the target Dirac problem, and relocates the no-go obstruction from the spectrum to the representation. In one-dimensional interacting systems it reproduces the gapless continuum limit where sine discretizations fail under interaction; in two-dimensional boundary and magnetic problems it supports single-cone and single-chirality constructions unavailable to local standard discretizations; and in generalized-eigenproblem form it admits compact numerical and quantum-algorithmic representations (Beenakker et al., 2023, Zakharov et al., 2024, Vela et al., 2024, Vela et al., 19 May 2025, Zakharov et al., 23 Jun 2026, Beenakker, 17 Jun 2026).