Emergent QED$_3$ at the bosonic Laughlin state to superfluid transition

TL;DR

This work investigates a continuous transition between the bosonic $\nu=1/2$ Laughlin state and a superfluid in (2+1)D, proposed to be governed by a $N_f=2$ $QED_3$-CS fixed point. Using infinite-cylinder DMRG on half-filled bosons in the lowest Landau level with a lattice potential, the authors observe a single continuous Laughlin–to–SF transition, with adiabatic $2\pi$ flux insertion revealing massless Dirac Dirac quasiparticles and a linear gap closure. Momentum-resolved correlation lengths exhibit an emergent $SO(3)$ symmetry among the lattice CDW channels $X,Y,M$, consistent with the $QED_3$–CS description and indicating stability against gauge fluctuations. The study also introduces flux-threading spectroscopy and symmetry-resolved scaling as practical diagnostics for Landau-forbidden criticality, with implications for fractional Chern insulators and moiré materials.

Abstract

Quantum phase transitions between topologically ordered and symmetry-broken phases lie beyond Landau theory. A prime example is the conjectured continuous transition from the bosonic $ν= 1/2$ Laughlin state to a superfluid, proposed to be governed by a QED$_3$--Chern--Simons (CS) critical point whose stability remains uncertain. We study half-filled bosons in the lowest Landau level subject to a lattice potential. Infinite-cylinder DMRG reveals a single continuous Laughlin--to--superfluid transition. Adiabatic flux insertion collapses the many-body gap and exposes massless Dirac quasiparticles, while momentum-resolved correlation lengths show that three lattice-related density modes share the same critical exponent, evidencing an emergent $SO(3)$ symmetry. The joint appearance of Dirac dispersion and symmetry enlargement provides microscopic support for a stable QED$_3$--CS fixed point. Our numerical strategy also offers a blueprint for exploring Landau-forbidden transitions in fractional Chern insulators and composite Fermi liquids realised in moire and cold-atom systems.

Figures (8)

• Figure 1: (a) Setup: bosons in the LLL on an infinite cylinder subject to a square lattice potential. (b) The unit cell of the physical boson encloses one flux quantum ($\phi_b=2\pi$). (c) Phase diagram at $L_y=8a$. The longest neutral correlation length $\xi_{\max}$ (blue) and the $M$-point correlation length $\xi_M$ (red) are shown for several bond dimensions $\chi$. Their behaviour identifies a single Laughlin–to–superfluid transition near $V_m\approx0.07$. Insets give entanglement spectra (top) and real-space correlators (bottom) on either side of the transition ($V_m = 0.02$ and $0.1$ respectively).
• Figure 2: Momentum–resolved correlation lengths $\xi(\mathbf k)$ (color scale) on a cylinder of circumference $L_y=8a$ at $\chi = 10000$. Panels (a)–(c) show the charge-neutral sector at $V_m=0.022$ (Laughlin), $0.084$ (critical), and $0.10$ (superfluid). Panel (d) shows the charge-$1$ sector at $V_m=0.10$. At criticality the correlation lengths at $X$, $Y$, and $M$ become almost equal, indicating an emergent $\mathsf{SO(3)}$ symmetry.
• Figure 3: (a) Upper: square lattice for physical bosons and $c$, $f$ partons. Each boson sees $2\pi$ flux, and each fermionic parton sees $\pi$ flux. Lower: Brillouin zone for the $\pi$-flux Hofstadter model of $f$ parton. Green dashed square marks magnetic Brillouin zone. Blue crosses are the Dirac points $f_{1,2}$. Orange horizontal lines label the quantized $k_y$ values for $f$ in the ground state on a cylinder with $L_y=4a$. After threading a $\Phi_f = \pi$ flux along the cylinder the allowed momentum shift to the blue lines, crossing all Dirac points. Red arrows indicate charge–density–wave momentum that connect the two cones. (b) Flux–insertion spectroscopy: the inverse correlation length $\Delta\equiv\ell_B/\xi_Y$ (interpreted as the excitation gap) versus inserted flux $\Phi$ on a cylinder of circumference $L_y=8a$ at $\chi = 8000$. The inset shows the relevant particle-hole excitations. The gap collapses near $\Phi=2\pi$ when $V_m$ approaches $V_m^{c}$, revealing a massless Dirac spectrum.
• Figure 4: Neutral sector correlation lengths in the flux-threaded sector ($\Phi=2\pi$). (a) Momentum-resolved correlation lengths at the critical coupling $V_m=0.07$. The correlation lengths at $X$, $Y$, and $M$ diverge together, exceeding the cylinder circumference $L_y=8a$. (b) Growth of $\xi_X$, $\xi_Y$, and $\xi_M$ on approaching the transition from the Laughlin side. Solid curves are raw DMRG data at bond dimension $\chi = 10000$; faint curves are extrapolations to $\chi \to \infty$. The inset shows the ratio $\xi_X/\xi_Y$ and $\xi_M/\xi_Y$. The three correlation lengths converge toward one another and diverge with a common exponent, consistent with an emergent $\mathsf{SO(3)}$ symmetry at criticality.
• Figure S1: Correlation-length diagnostics for cylinders with circumference $L_y/a = 8, 10, 12$. (a)-(c) Ground-state neutral sector: longest correlation length $\xi_{\max}$ and $M$-point correlation length $\xi_M$ versus potential strength $V_m$. Their contrasting trends locate a single Laughlin–to–superfluid transition whose critical coupling shifts slightly with $L_y$. (d)-(f) Flux-threaded neutral sector with $\Phi = 2\pi$: momentum-resolved correlation lengths at the respective critical couplings $V_m = 0.070, 0.072,$ and $0.084$ at bond dimension $\chi = 10000$. The color scale highlights converging peaks at $X$, $Y$, and $M$, evidencing an emergent degeneracy. (g)-(i) Correlation lengths at the three CDW momenta for the flux-threaded sector. The simultaneous divergence of $\xi_X$, $\xi_Y$, and $\xi_M$ confirms an emergent $\mathsf{SO(3)}$ symmetry at all circumferences.