Quantum Coulomb Liquids of Different Rank in the Breathing Pyrochlore Antiferromagnet

TL;DR

The paper tackles whether higher-rank Coulomb phases can survive quantum fluctuations in a three-dimensional quantum magnet. It studies the $S= frac{1}{2}$ Heisenberg antiferromagnet on the breathing pyrochlore with symmetry-allowed Dzyaloshinskii–Moriya interactions using the controlled pf-FRG method to map the zero-temperature phase diagram in the $D_a$-$D_b$ plane. It finds that breathing asymmetry stabilizes extended quantum analogues of both rank-1 and rank-2 $U(1)$ liquids, distinguished by distinctive pinch-point morphologies in momentum space, and that quantum fluctuations also generate an incommensurate spiral phase and an extended quantum-paramagnetic region absent in the classical model. The results establish breathing pyrochlore as a tunable, experimentally relevant platform for diagnosing emergent gauge structures in 3D quantum magnets, with clear signatures in neutron-scattering structure factors.

Abstract

Emergent gauge fields and Coulomb liquids have long been central to the physics of frustrated pyrochlore magnets, yet their realization beyond conventional, i.e. rank-1 $U(1)$, spin ice and into fully quantum higher-rank regimes has remained elusive. Here we provide a controlled demonstration of this physics in the spin-$\tfrac{1}{2}$ quantum Heisenberg antiferromagnet on the breathing pyrochlore lattice with symmetry-allowed Dzyaloshinskii--Moriya interactions, using the pseudofermion functional renormalization group. We show that tuning the breathing asymmetry stabilizes extended quantum analogues of both rank-1 and rank-2 $U(1)$ Coulomb liquids within a single microscopic model, directly distinguished by their characteristic pinch-point morphologies in momentum space. This provides the first controlled quantum realization in three dimensions where gauge theories of different rank emerge within a single microscopic spin Hamiltonian. In addition, quantum fluctuations qualitatively reshape the classical nearest-neighbor atlas of phases, causing an incommensurate spiral instability and an extended quantum-disordered regime without dipolar order, both absent from the classical model. Our results establish the breathing pyrochlore as a timely and experimentally relevant platform where higher-rank gauge constraints, conventional magnetic order, and fluctuation-driven quantum phases compete on equal footing, opening a direct route to diagnosing emergent gauge structure in three-dimensional quantum magnets.

Figures (7)

• Figure 1: Quantum phase diagram. (a) Unit cell of the breathing pyrochlore lattice with the sublattice labels indicated near every site. (b) Quantum phase diagram obtained from pf-FRG. Solid lines indicate the boundaries between regions with dipolar magnetic order (red) and quantum paramagnetic (PM) regions (blue) without dipolar order. Dashed lines serve as guides to the eye and indicate regimes where the structure factor within the quantum PM regions changes. (c) Polarized neutron scattering structure factors in the spin-flip (SF, top) and non-spin-flip (NSF, bottom) channels in the $hk0$ plane for each phase, calculated in the low-cutoff limit (see Supplemental Material for the corresponding structure factor in the $hhl$ plane).
• Figure 2: Classical phase diagram. (a) Band width of the lowest-energy band as a function of the DM couplings $D_a,D_b$. (b) Interaction matrix $J^{\alpha\beta}(\bm q)$ energy bands for a set of parameters where $D_b=0$. (c,d) Low-temperature prediction of the spin-flip (SF) and non-spin-flip (NSF) channels of the polarized neutron scattering obtained via the self-consistent Gaussian approximation (SCGA) for a set of parameters where $D_b=0$. (e,f) Specific heat obtained via classical MC simulations for three representative points marked by stars in panel (a). For two of them the system orders into a $\Gamma_5$ phase and for one the low-temperature phase is a $\bm q=\bm W$ phase.
• Figure 3: Flows and pinch-points for $\mathbf{D_b = 0}$. (a) Flow of $\chi^{zz}(\bm{k}^\mathrm{max})$ for increasing $D_a$ at lattice truncation $L=5$. Both the absolute magnitude and the system-size scaling increase sharply at $D_a = -0.1$ in the low-cutoff limit, which we interpret as the transition into the $\bm{q}=W$ ordered phase. The insets show the corresponding flows for $L = 3$ and $L = 5$ in the quantum paramagnetic ($D_a = -0.07$) and the $\bm{q}=W$$(D_b = -0.13)$ phases, highlighting the qualitative difference in their scaling behavior. (b) Polarized neutron scattering structure factors in the spin-flip channel, zoomed into the region around $[0,0,2]$ as indicated in Fig. \ref{['fig:quantum-phase-diagram']}(c), for different $D_b$ within the spin-liquid phases. The emerging four-fold pinch point is a characteristic signature of the rank-2 $U(1)$ spin liquid.
• Figure 4: Structure factors in the PM and ICS phase in the hhl-plane. (a) Neutron-scattering structure factor obtained from SCGA in the cooperative paramagnetic regimes at $T > T_c$ for characteristic points where the pf-FRG identifies the PM and ICS phases. The left (right) side show the spin-flip (SF) [non-spin-flip (NSF)] channel. (b) Corresponding neutron-scattering structure factor obtained from pf-FRG at different RG cutoff scales $\Lambda$ (see Supplemental Material for the corresponding structure factor in the $hhl$ plane).
• Figure 5: Evolution of the momentum of maximal spectral weight in the quantum spin-spin correlations. Shown is the magnitude of the momentum $\bm{k}^{\mathrm{max}}$ at which the static spin-spin correlation $\chi^{zz}(\bm{k})$ attains its maximum in the low-cutoff limit.