Gapless state of interacting Majorana fermions in a strain-induced Landau level

Mechanical strain can generate a pseudomagnetic field, and hence Landau levels (LL), for low-energy excitations of quantum matter in two dimensions. We study the collective state of the fractionalized Majorana fermions arising from residual generic spin interactions in the central LL, where the projected Hamiltonian reflects the spin symmetries in intricate ways: emergent U(1) and particle-hole symmetries forbid any bilinear couplings, leading to an intrinsically strongly interacting system; also, they allow the definition of a filling fraction, which is fixed at 1/2. We argue that the resulting many-body state is gapless within our numerical accuracy, implying ultra-short-ranged spin correlations, while chirality correlators decay algebraically. This amounts to a Kitaev ‘non-Fermi’ spin liquid and shows that interacting Majorana Fermions can exhibit intricate behavior akin to fractional quantum Hall physics in an insulating magnet.

Figure 1

Effective Hamiltonian: (a) Each hexagon has three chirality terms, $F_x,F_y,\text{and}{F}_z$, defined in terms of three-spin operators that couple two Majoranas on $A$ sites in the zero flux sector. Triaxial strain $\mathbb{C}$ acts in the directions shown. (b) Each hexagon has three rhombic plaquettes (yellow, blue, red) that signify a 4-Majorana interaction which can arise due to product of $F_{\alpha }$ operators on neighboring hexagons.

Figure 2

Ground state: (a) Ground state $\text{energy}/{\left(m_{\max }\right)}^{3/2} \equiv {\epsilon }_{gs}$ and total angular momentum eigenvalue for the ground state $\equiv M_{GS}$ vs $m_{\max }$ ($\beta =1$). $M_{GS}$ follows the time-reversal symmetric value ($=M_o$) (dashed line: mean-field energy ${\epsilon }_{\mathrm{var}}$). (b) Real-space density in ground state for $m_{\max }=24,28$.

Figure 3

Excitations: (a) Displaced low-energy spectra (from ground state energy) vs shifted angular momentum $\tilde{m}=M-M_o$ (from time-reversal symmetric sector $M_o$) for the three different system sizes $m_{\max }=12,16$ and $m_{\max }=24$, for $\beta =1$ exact system. (b) Gap to first and second excited state (${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$) vs $1/m_{\max }$. (c) $\rho \left(r\right)$ for the first excited state for $m_{\max }=24$ for $\beta =0,1$.

Figure 4

SR and LR models: (a) Gap to first excited state for both SR and LR goes to $zero$ with increasing $m_{\max }$ (for LR, the value in the $y$ axis should be multiplied by a factor 5). (b) Density oscillations in the first excited state (DMRG for SR; ED for LR).

Figure 5

Central charge: (a) Variation of ground state (SR system) entanglement entropy ($S$) of a subregion $m/m_{\max }$ with the rest showing a characteristic behavior $S=\frac{c}{6}ln[\frac{m_{\max }}{\pi }sin\left(\frac{\pi m}{m_{\max }}\right)]$ ($m_{\max }=240$, bond dimension $\chi \sim 800$). (b) Variation of $c$ showing that it goes to 1 with increasing system size for the SR model [see Eq. (10)].

Figure 6

Density correlator: Behavior of the density correlator in $m$ space for the ground state for SR, LR and exact system ($\beta =1$, average density $n=1/2$). The behavior in intermediate $m$ ($2\lesssim m\lesssim 12$) goes as $1/m^2$. The exact and LR saturate ($m_{\max }=24$, ED results) while the SR system shows a clear $1/m^2$ behavior ($m_{\max }={10}^2$, DMRG results).

Figure 7

Lattice notations: Three vectors $\vec{x}=\{\frac{\sqrt{3}}{2},-\frac{1}{2}\},\vec{y}=\{-\frac{\sqrt{3}}{2},-\frac{1}{2}\},\vec{z}=\{0,1\}$ point toward $A$ sites starting from the hexagonal plaquette centers. ${\delta }_1=\{\frac{\sqrt{3}}{2},\frac{3}{2}\}$ and ${\delta }_2=\{-\frac{\sqrt{3}}{2},\frac{3}{2}\}$ are the lattice vectors. $\mathbf{K}=\{\frac{4\pi }{3\sqrt{3}},0\}$ and ${\mathbf{K}}^{\prime }=\{-\frac{4\pi }{3\sqrt{3}},0\}$ are the two Dirac cones for the free Majorana dispersion on the honeycomb lattice. These lengths are measured in units of bond-length $a$ which is set to 1.

Figure 8

Strain profile: (a) The three kinds of plaquette terms for quartic Majoranas interaction ($v_1,v_2,v_3$) are seen in form of a vector $\mathit{V}$. (b) The magnitude of $\mathit{V}$ increases linearly away from the center of the flake and is linearly proportional to $\mathbb{C}$.

Figure 9

Density variations: (a) The ${\langle n_m\rangle }_{gs}$ for the ground state for two different values of $m_{\max }=24,28$ for $\beta =1$ is shown. (b) The ${\langle n_m\rangle }_{ex}$ for the first excited state in the half-filled sector for $m_{\max }=24$ and values of $\beta =0,1$ is shown. The corresponding real-space density profiles are shown in the main text.

Figure 10

$\beta =0$ case: (a) The total angular momentum of the ground state continues to remain in the time-reversal symmetric value $M_o$. (b) Gap to first excited and second excited states goes to zero for exact model $\beta =0$ with increasing $m_{\max }$.