Strained hyperbolic Dirac fermions: Zero modes, flat bands, and competing orders

TL;DR

This work shows that symmetry-preserving strain patterns on hyperbolic Dirac lattices {10,3} and {14,3} generate time-reversal symmetric axial magnetic fields, producing a zero-energy flat band that catalyzes ordered phases at weak couplings, such as CDW, HO, and a Hubbard-driven edge-compensated AFM. By framing these effects in a mean-field formalism and connecting them to known phenomena on strained Euclidean lattices, the authors unify axial magnetic catalysis as a general mechanism for Dirac fermions on curved and flat spaces. They further demonstrate that non-Hermiticity, via a sublattice-hopping imbalance, amplifies these orders while keeping eigenvalues real, broadening the parameter regime for observing such phases. The results illuminate how geometry, interactions, and non-Hermiticity jointly shape emergent mass orders and offer pathways to realize and probe these phases in designer quantum platforms and metamaterials.

Abstract

Starting from the nearest-neighbor tight-binding model on {10,3} and {14,3} hyperbolic lattices that, for a uniform hopping amplitude, gives rise to emergent Dirac fermions on a curved space with a constant negative curvature, displaying a vanishing density of states, we propose spatially modulated hopping pattern therein that preserve the underlying 5- and 7-fold rotational symmetries, respectively, and effectively couples fermions to time-reversal symmetric axial magnetic fields. Such strain-induced axial fields produce a flat band near zero-energy, triggering nucleation of a charge density-wave, featuring a staggered pattern of fermionic density between two sublattices, and the Haldane phase fostering intra-sublattice circulating currents with a net zero magnetic flux for weak nearest- and next-nearest-neighbor Coulomb repulsions, respectively. Sufficiently weak on-site Hubbard repulsion destabilizes such flat bands toward the formation of a magnetic phase that simultaneously supports antiferromagnetic and ferromagnetic orders in the whole system. While the magnetization in the bulk and boundary cancel each other, the Neél order is of the same sign everywhere, thereby yielding a global antiferromagnet. Throughout, we draw parallels between these findings and the well-studied qualitatively similar results on a 3-fold rotational symmetric strained honeycomb lattice, thereby unifying the phenomenon of axial magnetic catalysis for Dirac fermions, encompassing the ones living on the Euclidean plane. Finally, we show that with a specific class of non-Hermiticity, manifesting via an imbalance in the hopping amplitudes between two sublattices in the opposite directions, magnitudes of all these orders can be boosted substantially when all the eigenvalues in the noninteracting systems are real, staging a non-Hermitian amplification of axial magnetic catalysis.

Figures (11)

• Figure 1: Realizations of a strained (a) Euclidean honeycomb lattice, and strained (b) $\{10,3\}$ and (c) $\{14,3\}$ hyperbolic lattices producing time-reversal symmetric axial magnetic fields. The modified hopping amplitudes along the nearest-neighbor (NN) bonds are color coded, showing the preserved three-fold, five-fold, and seven-fold rotational symmetries, respectively. In a non-Hermitian setup, parameterized by a real $\alpha$, the hopping amplitude ($t$) along each NN bond becomes $t(1-\alpha)$ (solid arrow) and $t(1+\alpha)$ (dashed arrow) in the opposite directions marked by the arrow heads between two sublattices ($A$ and $B$), yielding all-real eigenvalues when $|\alpha|<1$. Local density of states (LDOS) associated with the symmetric combination of two particle-hole symmetric closest to zero-energy modes on (d) honeycomb, and (e) $\{ 10, 3 \}$ and (f) $\{ 14, 3\}$ hyperbolic lattices, showing its strong bulk localization on the sites of one sublattice, namely $A$. LDOS associated with the anti-symmetric combination of two particle-hole symmetric closest to zero-energy modes on (g) honeycomb, and (h) $\{ 10, 3 \}$ and (i) $\{ 14, 3\}$ hyperbolic lattices, showing its strong boundary localization on the sites of the other sublattice, namely $B$. Two hyperbolic lattices are always shown on the Poincaré disk (golden dashed ring). For details see Secs. \ref{['sec:strainconstruct']}, \ref{['sec:zeromodes']}, and \ref{['sec:NHconstruct']}.
• Figure 2: Density of states (DOS) $\rho$ as a function of energy $E$ in the absence ($q=0$) and presence (finite $q$) of axial magnetic fields on a (a) honeycomb lattice, and on (b) $\{10,3\}$ and (c) $\{ 14, 3\}$ hyperbolic lattices. Notice that for $q=0$, in all these systems the DOS near zero-energy vanishes linearly with $E$ confirming the presence of massless Dirac fermions therein. With finite $q$, a peak in the DOS develops near $E=0$, yielding a flat band. These results are shown in Hermitian systems ($\alpha=0$). Now introducing the non-Hermitian degree of freedom, the peak in the DOS near $E=0$ gets taller with increasing non-Hermiticity $\alpha$ (see Fig. \ref{['fig:Geometry']}) as shown for the (d) honeycomb lattice, and (e) $\{ 10, 3\}$ and (f) $\{ 14,3 \}$ hyperbolic lattices with a fixed nonzero $q$ (axial field) in each system. For details see Sec. \ref{['sec:zeromodes']} and Sec. \ref{['sec:NHconstruct']}. DOS in pristine honeycomb lattice ($q=0$) is computed with periodic boundary conditions in all directions to suppress the signatures of zero energy topological edge modes, localized near the zigzag boundaries, yielding a finite DOS near $E=0$.
• Figure 3: Scaling of the self-consistent solutions of the charge density wave (CDW) order, averaged over the entire system ($\delta_{\rm CDW}$), as a function of the nearest-neighbor Coulomb repulsion ($V_1$) among spinless fermions on a honeycomb lattice in a (a) Hermitian setup ($\alpha=0$) and in non-Hermitian setups with (b) $\alpha=0.4$ and (c) $\alpha=0.8$ (see Fig. \ref{['fig:Geometry']}) in the absence of any axial magnetic field ($q=0$) and for a few choices of finite $q$ (producing axial magnetic fields). While for $q=0$, $\delta_{\rm CDW}$ strictly develops beyond an $\alpha$-dependent critical strength of $V_1$, denoted by $V_{1,c}$, see Table \ref{['tab:criticalvalues']}, finite axial magnetic fields catalyze their nucleation for subcritical $V_1$ due to the presence of a flat band near zero energy (see Fig. \ref{['fig:DOS']}). As finite $q$ also increases the bandwidth of the free fermion system, we observe a crossover between a strain-dominated regime (red shade) for weaker $V_1$ where a larger $q$ yields a bigger $\delta_{\rm CDW}$ and an interaction-dominated regime (light blue shade) for stronger $V_1$ where a larger $q$ yields a smaller $\delta_{\rm CDW}$. Although this is a crossover phenomenon, all the data for $\delta_{\rm CDW}$ for various finite $q$ seem to cross at a specific $V_1 > V_{1, {\rm c}}$. Amplification of the axial catalysis of the CDW order by the non-Hermiticity in the system for any $V_1$ is shown in (d) by comparing $\delta_{\rm CDW}$ for a fixed $q$ (axial field) but for $\alpha=0.0$, $0.4$, and $0.8$ over a wide range of $V_1$. Notice that for a fixed $q$ and $V_1$, $\delta_{\rm CDW}$ increases monotonically with $\alpha$. Vertical lines in (d), color coded according to the $\alpha$ values, mark the $\alpha$-dependent critical $V_1$ for the CDW ordering when $q=0$. Panels (e)-(h) [(i)-(l)] are analogous to panels (a)-(d), respectively, but for the $\{10,3 \}$ [$\{14,3 \}$] hyperbolic Dirac lattice. See Sec. \ref{['sec:coulomb']} and Sec. \ref{['sec:NHinteract']} for detailed discussions.
• Figure 4: Analogous to Fig. \ref{['fig:CDW']}, but showing the scaling of the self-consistent solutions of the Haldane order ($\delta_{\rm HO}$), averaged over the entire system, with the next-nearest-neighbor Coulomb repulsion ($V_2$) among spinless fermions. Notice that the anticipated crossing of all data for $\delta_{\rm HO}$ for various $q$ values at the crossover point between the strain-dominated (red shade) and interaction-dominated (light blue shade) regimes at a finite $V_2$ is not perfect for the honeycomb lattice (top row), while such crossing seems to be perfect at a specific $V_2$ on $\{ 10,3\}$ (middle row) and $\{ 14,3\}$ (bottom row) hyperbolic lattices. Such data crossings between the two regions occurs above critical $V_2\; (\equiv V_{2,c})$, which depends on $\alpha$ (see Table \ref{['tab:criticalvalues']}). For details see Secs. \ref{['sec:haldane']} and \ref{['sec:NHinteract']}.
• Figure 5: Analogous to Figs. \ref{['fig:CDW']} and \ref{['fig:Haldane']}, but showing the scaling of the self-consistent solutions of the antiferromagnetic order ($\delta_{\rm AFM}$), averaged over the entire system, with the on-site Hubbard repulsion ($U$) among spinful fermions. The anticipated crossing of all data for $\delta_{\rm AFM}$ for various $q$ values at the crossover point between the strain-dominated (red shade) and interaction-dominated (light blue shade) regimes at a finite $U$ is not perfect for the honeycomb lattice (top row), while such crossing seems to be perfect at a specific $U$ on $\{ 10,3\}$ (middle row) and $\{ 14,3\}$ (bottom row) hyperbolic lattices. This data crossing between the two regimes occurs above the critical $U \; (\equiv U_c)$, which depends on $\alpha$ (see Table \ref{['tab:criticalvalues']}). For details see Secs. \ref{['sec:hubbard']} and \ref{['sec:NHinteract']}.