Amplified magnetic catalysis in non-Hermitian Euclidean and hyperbolic Dirac liquids

TL;DR

This work addresses how non-Hermiticity affects magnetic catalysis in two-dimensional Dirac liquids by introducing a NH Dirac framework with non-reciprocal hopping that preserves an all-real spectrum for |α|<1. The authors show that commuting class masses (CDW and AFM) can be nucleated at subcritical interactions when a magnetic field is present, and that non-Hermiticity amplifies the resulting mass gaps via a self-consistent mean-field treatment, captured by a gap equation with Δ growing as α increases for fixed g and β. The study combines lattice models on Euclidean honeycomb and hyperbolic {10,3} and {14,3} lattices, biorthogonal quantum mechanics, and finite-size scaling to demonstrate robust NH amplification of magnetic catalysis across geometries, with scaling laws such as E_1^2 ∝ β and δ_CDW^2 ∝ β (similarly δ_AFM^2 ∝ β), and a universal relation C(α)/C(0)=√(1-α^2). The findings suggest experimental routes in designer materials and optical lattices, where non-reciprocal hopping and magnetic field control could realize amplified mass gaps, and highlight future directions in exploring commuting versus anticommuting mass classes and finite-temperature behavior.

Abstract

Due to their iconic linearly vanishing density of states near the zero-energy, half-filled two-dimensional Dirac materials in flat Euclidean and negatively-curved hyperbolic spaces exhibit dynamic mass generation only once a critical interaction strength is surpassed. Application of external magnetic fields onto these systems can, however, trigger the formation of such ordered phases yielding isotropic insulation near the band-center at arbitrarily weak coupling, a phenomenon known as magnetic catalysis. Recently, it has been proposed that a specific type of non-Hermiticity, allowing the system to feature an all-real eigenvalue spectrum otherwise squeezed toward the zero-energy, can bring down the requisite critical coupling of a specific family of ordered phases, commuting class masses, to a desired lower finite value in Dirac systems, a phenomenon known as non-Hermitian catalysis (arXiv:2501.18591). Here, we predict that a confluence of external magnetic fields and such a non-Hermiticity can amplify the magnitude of commuting class masses for subcritical strengths of interactions in Dirac liquids, an emergent phenomenon named non-Hermitian amplification of magnetic catalysis. We anchor this prediction from numerical self-consistent mean-field solutions of the commuting class mass charge-density-wave (antiferromagnetic) order displaying a staggered pattern of average electronic density (magnetization) between the nearest neighboring sites of the half-filled Euclidean honeycomb and hyperbolic {10, 3} and {14, 3} lattices, all featuring emergent non-Hermitian Dirac quasiparticles, after decomposing the nearest-neighbor Coulomb (on-site Hubbard) repulsion in the Hartree channel. We discuss the scaling behavior of these two orders with magnetic field and non-Hermiticity over a wide range of subcritical interactions.. Possible experimental setups to test our predictions are discussed.

Figures (10)

• Figure 1: Density of states (DOS) over a small energy ($E$) window around $E=0$ in Hermitian ($\alpha=0$) and non-Hermitian (finite $\alpha$) (a) Euclidean $\{6, 3\}$ or honeycomb, (b) hyperbolic $\{10, 3\}$, and (c) hyperbolic $\{14, 3\}$ lattices at a fixed magnetic field flux per plaquette $\beta$ (measured in units of $\Phi_0/(2 \pi)$, where $\Phi_0$ is the magnetic flux quanta) and subcritical (with respect to the zero magnetic field case) nearest-neighbor (NN) Coulomb repulsion $V$ (see Table \ref{['tab:citicalvalues']}). The results are obtained from the self-consistent solutions of the charge-density-wave (CDW) order, a member of the commuting class mass family in our construction of non-Hermitian Dirac system, where non-Hermiticity stems from an imbalance of the hopping amplitudes between the NN sites in the opposite directions, see Fig. \ref{['fig:NHpeierlssub']}. Panels (d)-(f) are analogous to (a)-(c), respectively, but DOS in all these systems are obtained from the self-consistent solutions of the antiferromagnet (AFM) order, yet another member of the commuting class mass family in our specific construction of non-Hermitian Dirac systems, in the presence of subcritical on-site Hubbard repulsion $U$ (see Table \ref{['tab:citicalvalues']}). The formation of the spectral gaps for subcritical $V$ and $U$ in Hermitian systems confirms their magnetic catalysis, which remains operative in non-Hermitian systems. Results show that the spectral gap near the zero-energy increases with increasing non-Hermiticity for subcritical $V$ and $U$ in all these systems when they are subject to uniform external magnetic fields, thereby endorsing our proposed scenario of non-Hermitian amplification of magnetic catalysis. See legends for specifications of various parameter values.
• Figure 2: Schematic representations of a uniform magnetic flux attachment in non-Hermitian (NH) Dirac systems on (a) Euclidean $\{6, 3\}$ or honeycomb and (b) hyperbolic $\{10, 3\}$ lattices, in which the non-Hermiticity, quantified by $\alpha$, resulting from an imbalance in the hopping amplitudes between the nearest-neighbor (NN) sites in the opposite directions, given by $t(1 \pm \alpha)$, is represented by solid and dashed curved black arrows in their directions. Here, $t$ is the reciprocal hopping amplitude and $\alpha$ measures the non-Hermiticity. The colored arrow head represents the Peierls phase added to each NN hopping term in its direction, yielding a uniform perpendicular external magnetic field in NH systems, following Eq. \ref{['eq:NHGeneral']}. Notice that the phases are threaded around each plaquette in a way such that the total phase accumulated in the clockwise direction around each plaquette is always equal to $\beta$, yielding the magnetic flux [Eq. \ref{['eq:psmagflux']}]. The black circle in (b) corresponds to the conformal boundary of the Poincaré disk representation. With $\beta^0=\beta/14$ and $\beta^1=(\beta + \beta^0)/11$, the construction from (b) can immediately be generalized to $\{ 14,3 \}$ hyperbolic lattice to mimic a uniform magnetic field therein.
• Figure 3: Density of states (DOS) $\rho$ as a function of energy in (a) Euclidean $\{6, 3\}$ or honeycomb, (b) hyperbolic $\{10, 3\}$, and (c) hyperbolic $\{14, 3\}$ lattices with a fixed non-zero total magnetic flux ($\beta$) in Hermitian ($\alpha=0$) and non-Hermitian (finite $\alpha$) Dirac systems. For $\alpha=0$, but with a finite and uniform $\beta$, a large (finite) number of near zero energy states emerges on honeycomb (hyperbolic) lattice(s), in agreement with an index theorem. While on honeycomb lattice we observe a clear formation of a few discrete Landau levels at finite energies, any such Landau quantization is absent on hyperbolic lattices due to the background constant negative curvature. Non-Hermiticity ($\alpha$) does not change the number of exact zero-energy modes, but it squeezes all the states at finite energies closer to the zero energy. The scaling of the near zero-energy DOS, defined as $\Delta \rho(0) = \rho_{\beta}(E=0) - \rho_{0}(E=0)$, where $\rho_{\beta}(E=0)$ [$\rho_{0}(E=0)$] is the near zero-energy DOS with [without] magnetic flux, in (d) $\{ 6,3 \}$, (e) $\{ 10,3 \}$, and (f) $\{ 14,3 \}$ lattices with $\beta$ for a few choices of $\alpha$. Notice that the subtraction of such a zero-field DOS, $\rho_{0}(E=0)$, is necessary to avoid systematic error brought on by the finite-width of the zero-energy bin. On the honeycomb lattice $\Delta \rho(0)$ scales linearly with $\beta$ and the numerically computed slope of such a linear fit is insensitive to $\alpha$ due to a finite Landau gap in the spectrum, which are consistent with the predictions of the Aharonov-Casher index theorem in Hermitian and non-Hermitian Euclidean Dirac systems. On the hyperbolic lattices $\Delta \rho(0)$ scales roughly as $\beta^2$ and a slight enhancement in $\Delta \rho(0)$ with finite $\alpha$ in our numerical calculation results from the absence of any Landau gap in the spectrum, causing minor deviation from the $\alpha$-independent $\beta^2$ scaling of $\Delta \rho(0)$ in non-Hermitian systems.
• Figure 4: Density of states (DOS) $\rho$ as a function of energy in Hermitian ($\alpha=0$) (a) Euclidean $\{ 6,3 \}$ or honeycomb, (b) hyperbolic $\{10, 3\}$, and (c) hyperbolic $\{14, 3\}$ lattices for zero ($\beta=0$) and finite uniform magnetic flux (finite $\beta$) through each plaquette in the absence ($V=0$) and for subcritical strength (finite $V$) of nearest-neighbor Coulomb repulsion (see Table \ref{['tab:citicalvalues']}). See legends for specifications of the parameter values. The insets display zoomed-in pictures of the DOS near the zero-energy. With no magnetic field ($\beta=0$) and interaction ($V=0$), the DOS in all these systems vanishes near zero energy, consistent with their Dirac liquid classification (yellow curves). Introduction of a finite and strong uniform magnetic field in non-interacting systems yields a nonzero DOS near zero energy, as predicted by index theorems (cyan curves). Finally, upon turning on subcritical (with respect to the zero magnetic field case) interactions (finite $V$), a gap forms in the all these systems (magenta curves), thereby confirming that the magnetic field catalyzes spontaneous breaking of the sublattice symmetry through the formation of a charge-density-wave order for subcritical $V$. Panels (d), (e), and (f) are analogous to (a), (b), and (c), respectively, but in non-Hermitian systems with a non-zero $\alpha$, showing that the magnetic catalysis mechanism is equally operative in non-Hermitian Euclidean and hyperbolic Dirac systems for a commuting class mass (charge-density-wave in this case). For detailed discussion, see Sec. \ref{['sec:coulomb']}. The DOS in honeycomb lattice without magnetic fields and in the absence of interactions is obtained with periodic boundary conditions to suppress the imprints of topological boundary modes living on its zigzag edges.
• Figure 5: The scaling of the charge-density-wave (CDW) order parameter $\delta_{\rm CDW}$ as a function of the total magnetic flux $\Phi_B$ with varying degree of non-Hermiticity $\alpha$ and a few fixed (in a given subfigure) strengths of subcritical nearest-neighbor Coulomb repulsion $V$ in Euclidean $\{6, 3\}$ or honeycomb [(a)-(c)], hyperbolic $\{10, 3\}$ [(d)-(f)], and hyperbolic $\{14, 3\}$ [(g)-(i)] lattices. In all cases, it is evident that with increasing total magnetic flux enclosed by the system ($\Phi_B$), the amplitude of the CDW order increases. The emergence of the CDW order at subcritical strengths of nearest-neighbor Coulomb repulsion (see Table \ref{['tab:citicalvalues']}) endorses the magnetic catalysis in these systems, regardless of the underlying spatial curvature and the degree of non-Hermiticity in the system. Most importantly, with increasing non-Hermiticity (larger $\alpha$) magnitude of the CDW order amplifies monotonically when $\beta$ and $V$ are held fixed, thereby promoting the proposed mechanism of amplified magnetic catalysis of commuting class masses (CDW in this case). See legends for specific parameter values and Sec. \ref{['sec:coulomb']} for a detailed discussion.