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Physical review. B./Physical review. B

Local entanglement and confinement transitions in the random transverse-field Ising model on the pyrochlore lattice

T. Pardini, Anirudha Menon, Stefan P. Hau‐Riege, Rajiv R. P. Singh

Published Oct 28, 2019• 11 citations
Publisher

Abstract

We use numerical linked cluster expansions (NLC) and exact diagonalization to study confinement transitions out of the quantum spin liquid phase in the pyrochlore-lattice Ising antiferromagnet with random transverse fields. We calculate entanglement entropies associated with local regions defined by single tetrahedron to observe these transitions. The randomness-induced confinement transition is marked by a sharp reduction in the local entanglement and a concomitant increase in Ising correlations. In NLC, it is studied through the destruction of loop resonances due to random transverse-fields. The confining phase is characterized by a distribution of local entanglement entropies, which persists to large random fields.

DOI
10.1103/physrevb.100.144437

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We use numerical linked cluster expansions (NLC) and exact diagonalization to study confinement transitions out of the quantum spin liquid phase in the pyrochlore-lattice Ising antiferromagnet with random transverse fields. We calculate entanglement entropies associated with local regions defined by single tetrahedron to observe these transitions. The randomness-induced confinement transition is marked by a sharp reduction in the local entanglement and a concomitant increase in Ising correlations. In NLC, it is studied through the destruction of loop resonances due to random transverse-fields. The confining phase is characterized by a distribution of local entanglement entropies, which persists to large random fields.
We study an effective spin model derived perturbatively from a random transverse-field Ising model on the pyrochlore lattice. The model consists of spin configurations on the pyrochlore lattice, restricted to the spin-ice subspace, with spins interacting with random Ising exchange couplings as well as ring exchanges along the hexagons of the lattice. This model is studied by exact diagonalization up to N=64N=64N=64 site systems. We calculate spin-glass correlation functions and local entanglement entropy ST{S}_{T}ST​ between spins in a single tetrahedron and the rest of the system. We find that the model undergoes two phase transitions. At weak randomness the model is in a quantum spin-ice phase where ST=ln6{S}_{T}=ln6ST​=ln6. Increasing randomness, at low transverse fields, first leads to a frozen phase, with long-range spin-glass order and ST=ln2{S}_{T}=ln2ST​=ln2 corresponding to the Cat states associated with Ising order. Further increase in randomness leads to a random resonating-hexagon phase with a frozen backbone of spins and a broad distribution of entanglement entropies. The implications of these studies for non-Kramers rare-earth pyrochlores are discussed.
The antiferromagnetic Ising model on the pyrochlore lattice exhibits a quantum phase transition in an applied transverse field from the low-field quantum spin-ice phase to the high-field polarized regime. Recent field-theoretical analysis and series expansion results indicate this to be a discontinuous, first-order transition. Here, we explore this transition using quantum Monte Carlo simulations in order to assess this scenario and study the thermodynamic properties in the vicinity of the quantum phase transition. For this purpose, we also consider several variants of extended cluster-update schemes for the transverse-field Ising antiferromagnet on frustrated lattices and compare their performance to the conventional bond-based algorithm for the transverse-field Ising model on the pyrochlore lattice.
At zero temperature, the classical antiferromagnetic Ising model on the pyrochlore lattice is a spin disorder phase of the critical spin correlation. It is a deconfined phase in that the binding energy of the monopole-anti-monopole pair is independent of their distance of separation. We show that turning on a transverse magnetic field turns it into the cooperative paramagnet, and the spin correlation becomes exponential decay. Furthermore, it introduces the quantum confinement (of magnetic monopoles), where the binding energy of the pair is proportional to their distance of separation. This disorder state undergoes adiabatic transition to the paramagnetic state in the large field limit. The effective Hamiltonian (without magnetic monopoles) in small field is the Ising Hamiltonian plus ring exchange interaction.
We study the ground state and thermodynamic properties of the spin-half XXZ model, with an Ising interaction Jz{J}_{z}Jz​ and a transverse exchange interaction Jx{J}_{x}Jx​, on a pyrochlore tube obtained by joining together elementary cubes in a one-dimensional array. Periodic boundary conditions in the transverse directions ensure that the bulk of the system consists of corner-sharing tetrahedra, with the same local geometry as the pyrochlore lattice. We use exact diagonalization, the density matrix renormalization group (DMRG), and minimally entangled typical thermal states (METTS) methods to study the system. When Jz{J}_{z}Jz​ is antiferromagnetic (Jz>0{J}_{z}>0Jz​>0) and Jx{J}_{x}Jx​ is ferromagnetic (Jx<0{J}_{x}<0Jx​<0), we find a transition from a spin liquid to an XY ferromagnet, which has power-law correlations at T=0T=0T=0. For Jz<0{J}_{z}<0Jz​<0 and Jx>0{J}_{x}>0Jx​>0, spin-two excitations are found to have lower energy than spin-one at the transition away from the fully polarized state, showing evidence for incipient spin-nematic order. When both interactions are antiferromagnetic, we find a nondegenerate ground state with no broken symmetries and a robust energy gap. The low-energy spectra evolve smoothly from predominantly Ising to predominantly XY interactions. In the spin-liquid regime of small ∣Jx∣|{J}_{x}|∣Jx​∣, we study the confinement of monopole-antimonopole pairs and find that the confinement length scale is larger for Jx<0{J}_{x}<0Jx​<0 than for Jx>0{J}_{x}>0Jx​>0, although both length scales are very short. These results are consistent with a local spin-liquid phase for the Heisenberg antiferromagnet with no broken symmetries.