Spin-Peierls instability of the U(1) Dirac spin liquid

Abstract

Quantum fluctuations can inhibit long-range ordering in frustrated magnets and potentially lead to quantum spin liquid (QSL) phases. A prime example are gapless QSLs with emergent U(1) gauge fields, which have been understood to be described in terms of quantum electrodynamics in 2+1 dimension (QED₃). Despite several promising candidate materials, however, a complicating factor for their realisation is the presence of other degrees of freedom. In particular lattice distortions can act to relieve magnetic frustration, precipitating conventionally ordered states. In this work, we use field-theoretic arguments as well as extensive numerical simulations to show that the U(1) Dirac QSL on the triangular and kagome lattices exhibits a weak-coupling instability due to the coupling of monopoles of the emergent gauge field to lattice distortions, leading to valence-bond solid ordering. This generalises the spin-Peierls instability of one-dimensional quantum critical spin chains to two-dimensional algebraic QSLs. We study static distortions as well as quantum-mechanical phonons. Even in regimes where the QSL is stable, the singular spin-lattice coupling leads to marked temperature-dependent corrections to the phonon spectrum, which provide salient experimental signatures of spin fractionalisation. We discuss the coupling of QSLs to the lattice as a general tool for their discovery and characterisation.

Fig. 1. Spin-Peierls distortions of unstable gapless spin liquids in one and two dimensions.

a In one dimension, the spin-half antiferromagnetic Heisenberg chain is unstable to dimerisation when coupled to the lattice. The U(1) Dirac spin liquid is unstable to $\sqrt{12}\times \sqrt{12}$ and pinwheel valence-bond order on the b triangular and c kagome lattices, respectively. The blue shifted dots show the distorted lattice, the vector displacements inside the unit cell u(r) are highlighted by white arrows. The enhanced nearest-neighbour bond strengths within each plaquette are depicted in the left panel, and the short-range spin-spin correlations $⟨\overrightarrow{S}\cdot \overrightarrow{S}⟩$ are shown on the right. Structural transitions can be measured thermodynamically, and the corresponding lattice and valence-bond order can be measured with inelastic X-ray and neutron spectroscopy.

Fig. 2. Numerical observation of the spin-Peierls instability of the triangular-lattice AFM Heisenberg model.

Density matrix renormalisation group calculations of the next-nearest-neighbour frustrated triangular lattice model (J₂/J₁ = 1/8) are presented, where the undistorted lattice displays a gapless spin-liquid ground state. a Energy gain of the spin system under various simulated lattice distortions on the L = 6 circumference cylinder. Patterns considered are generated by momenta K₃, and M_1,2,3 (here the purple shaded region represents the range of responses for the three M_a, showing minimal dependence on cylinder orientation; see Supplementary Note 3), as well as the full 12-site distortion, defined by Eq. (5). b Nearest-neighbour exchanges under the 12-site unit cell distortion; red/blue bonds show enhanced/suppressed bonds. c System-size dependence of energy gain for L = 3, 6. We compare three distinct distortion patterns generated by the momenta K₃, M₃, and X₃ which are compatible with the cylinder geometry. Logarithmic axes show the quadratic energy gain for each pattern. d L-dependence of the energy gain for the data point δ = 0.002 for cylinders with circumference of up to L = 9. The colours indicate the same distortion patterns as in c. Only the distortion at X₃ shows strong L-dependence, which confirms that a weak-coupling instability of the U(1) Dirac spin liquid is realised in this numerical study of the J₁–J₂ model. e,f The spin-spin correlation function on dimers $⟨{\overrightarrow{S}}_i\cdot {\overrightarrow{S}}_j⟩$ in the presence of a 12-site lattice distortion b. Subfigure e shows the theoretical prediction from the leading order analysis of the symmetry of operators in the CFT spectrum (on an arbitrary scale); and f is the numerical calculation performed with DMRG, evaluated with distortion parameter δ  = 0.05 and L = 6 and averaged over orientations relative to the cylinder axis.

Fig. 3. Phase diagram and Kohn anomaly of the U(1) DSL with dynamical phonons.

a Scaling phase diagram for the spin-Peierls-VBS instability of the U(1) DSL, as a function of coupling g, temperature T and frequency ω₀. The shaded regions indicate the unstable parameter regimes, based on the finite-temperature calculation in the classical (ω₀ → 0) limit and the zero-temperature result for dynamical phonons (ω₀> 0). Phonon spectral function $\log S_{\mathrm{phonon}}\left(\omega ,\mathbf{k}\right)$ plotted for b zero and c finite temperature along the momentum slice between Γ and K_a. Spin-phonon coupling is set to g = 0.3. The phonon spectral function is approximated by extending the interacting phonon propagator G_a(ω) = G(ω, X_a) to momenta X_a + q given a microscopic model for the bare phonon dispersion ω₀(q) (white dashed line). We use a heuristic scaling form for the VBS correlator at finite temperatures for illustrative purposes (see “Methods”). At finite temperature T  > T_SP the phonon dispersion displays a Kohn anomaly, and at zero temperature, the quasiparticle picture breaks down.

Fig. 4. Unit cell in real and reciprocal space.

a Triangular lattice with unit vectors a_i and nearest-neighbour bonds δ_a highlighted. b Reciprocal lattice with inverse lattice vectors g_i and high-symmetry points in the Brillouin zone K_a and M_a labeled. The points X_a= −K_a/2 are the momenta eigenvalues of gapless monopole excitations^,.