Emergent gauge flux in QED$_3$ with flavor chemical potential: application to magnetized U(1) Dirac spin liquids

TL;DR

The work investigates emergent gauge flux in a lattice realization of non-compact $QED_3$ with a flavor chemical potential, interpreted as a Zeeman field, using determinant quantum Monte Carlo to map out the phase diagram and gauge-field dynamics. It identifies a chiral flux (CF) phase with a finite emergent flux and relativistic Landau-level spectra for spinons, accompanied by a gapless photon mode observable in the longitudinal spin structure factor, and a coexisting CF–AFM region alongside a conventional DSL and AFM phases. The study connects continuum $QED_3$ predictions to lattice observables, showing Landau-level-inspired spectral features, Larmor modes in the transverse channel, and photon signatures in $S^{zz}$, thereby providing experimentally testable signatures for magnetized Dirac spin liquids in frustrated magnets. These results offer a robust framework for identifying deconfined gauge-field physics in condensed-matter systems under magnetic fields and guide future experimental searches for magnetized DSL states.

Abstract

We design a lattice model of a non-compact U(1) gauge field coupled to fermions with a flavor chemical potential and solve it with large-scale determinant quantum Monte Carlo simulations. For zero flavor chemical potential, the model realizes three-dimensional quantum electrodynamics (QED3) which has been argued to describe the ground state and low-energy excitations of the Dirac spin liquid phase of quantum antiferromagnets. At finite flavor chemical potential, corresponding to a Zeeman field perturbing the Dirac spin liquid, we find a "chiral flux" phase which is characterized by the generation of a finite mean emergent gauge flux and, accordingly, the formation of relativistic Landau levels for the Dirac fermions. In this state, the U(1)m magnetic symmetry is spontaneously broken, leading to a gapless free photon mode which, due to spin-flux-attachment, is observable in the longitudinal spin structure factor. We numerically compute longitudinal and transverse spin structure factors which match our continuum and lattice mean-field theory predictions. In a different region of the phase diagram, strong fluctuations of the emergent gauge field give rise to an antiferromagnetically ordered state with gapped Dirac fermions coexisting with a deconfined gauge field. We also find an interesting intermediate phase where the chiral flux phase and the antiferromagnetic phase coexist. We argue that our results pave the way to testable predictions for magnetized Dirac spin liquids in frustrated quantum antiferromagnets.

Figures (17)

• Figure 1: Lattice model and DQMC phase diagram. (a) Fermions are on the sites of cubic lattice represented by filled blue balls while the gauge fields are on the bonds of the lattice represented by filled yellow balls. The gauge field on the temporal direction is fixed to $0$ (denoted by the dashed lines). Fermions hop between nearest neighbor sites with a phase $e^{\pm i a_{ij}}$. Each plaquette is attached to a flux $\Phi_\Box$, computed from gauge field on the four bonds. Black arrows indicate spatial and temporal directions, with lattice constant $1$ for spatial and $0.1$ for temporal directions. $\tau_n$ and $\tau_{n-1}$ are adjacent temporal layers. (b) Phase diagram obtained from DQMC simulation. The red line is determined from spin correlation ratio $r_{\text{AFM}}$ while the blue line is from flux Binder cumulant $U_{\text{flux}}$. Right of red line is the AFM phase which breaks the U(1)$_f$ symmetry of spin rotation when $B\neq 0$. Left of blue line is chiral flux state (CF), in which the $\mathcal{T}$ symmetry of flux of the gauge field is broken. There are a non-magnetic (NM) phase at the top, Dirac spin liquid (DSL) phase at the bottom corner and a possible co-existing CF-AFM phase, represented by a color gradient between CF and AFM phases. Note that the finite extent of the DSL phase for $B>0$ is due to the non-zero temperature accessible in our simulations. We expect the DSL to give way to the CF phase for infinitesimal Zeeman fields $B>0$ in the zero-temperature thermodynamic limit.
• Figure 2: Lattice mean field calculations at $L=32$. (a) Eigenvalues of the fermion hopping matrix at different flux sector from $0$ to $2\pi$, known as Hofstadter's butterfly hofstadter1976. The red(green) line shows the location of Fermi level for up(down) fermion. (b) Total energy of system versus flux $\Phi_\Box$ for different values of Zeeman field $B$. With finite Zeeman field, the flux sectors with the minimal energy deviate from $\pi$ and are symmetric about $\pi$. (c) The flux sector with the minimal total energy versus Zeeman field. The data points are obtained from (b). (d) $\pi-\Phi_\Box$ versus magnetization $m_z$, shows linear relation and indicates the induced orbital magnetic field is proportional to magnetization.
• Figure 3: Determination of the phase boundary in Fig. \ref{['fig:fig1']} (b) at finite $B$. Data are at Zeeman field $B/t=2$ with system sizes $L=6,8,\cdots,16$. (a) Critical $J/t \sim 0.9$ determined from flux Binder cumulant $U_{\text{flux}}$. (b) Critical $J/t \sim 0.6$ determined from transverse spin correlation ratio $r_{\text{AFM}}$. The phase boundaries of the CF and AFM phases in Fig. \ref{['fig:fig1']} (b), at $B/t=1,2,3, \cdots, 5$, are determined in this way.
• Figure 4: Dynamical flux correlation function $C_{\chi}(\tau)$. For system size $L=12$ and inverse temperature $\beta=24$, all values of $J/t$ at Zeeman field $B/t=3$ exhibit clear deviations from exponential decay. This behavior stands in contrast to the compact case at $J/t=3, B/t=0$, where the correlation function shows well-defined exponential decay, characteristic of a confined phase xu2019monte. In the non-compact case with large $J/t=3$ and $B/t=3$, the correlators display persistent curvature in the semi-logarithmic plot, indicating a breakdown of exponential scaling and suggesting deconfined gauge field even inside AFM phase.
• Figure 5: Illustration of excitations that contribute to the transverse ($\langle S^+ S^- \rangle$ and $\langle S^- S^+ \rangle$) and longitudinal ($S^z S^z$) channels of magnetic spectra. The finite flux of the gauge field leads to Landau levels of the Dirac fermions, with energies $\pm E_n - \alpha b/2 = \pm \sqrt{2n}/\ell - \alpha b/2$, where the finite Zeeman fields lead to a spin splitting for $\alpha = \uparrow (+1), \downarrow(-1)$ spinons. For the former, spin-flip excitations can occur either within the spin-split $n=0$ Landau level, or between distinct sets of spin-split Landau levels, while the longitudinal channel only receives contributions from same-spin inter-Landau level transitions.