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arXiv:1911.10243 · Physical review. B./Physical review. B

Confinement, reduced entanglement, and spin-glass order in a random quantum spin-ice model

Anirudha Menon, Tom Pardini, Rajiv R. P. Singh

Published May 21, 2020• 2 citations
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Abstract

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 $N=64$ site systems. We calculate spin-glass correlation functions and local entanglement entropy ${S}_{T}$ 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 ${S}_{T}=ln6$. Increasing randomness, at low transverse fields, first leads to a frozen phase, with long-range spin-glass order and ${S}_{T}=ln2$ 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.

DOI
10.1103/physrevb.101.184423

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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.
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 consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.
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.
We show that the highly frustrated transverse-field Ising model on the three-dimensional pyrochlore lattice realizes a first-order phase transition without symmetry breaking between the low-field Coulomb quantum spin liquid and the high-field polarized phase. The quantum phase transition is located quantitatively by comparing low- and high-field series expansions. Furthermore, the intriguing properties of the elementary excitations in the polarized phase are investigated. We argue that this model can be achieved experimentally by applying mechanical strain to a classical spin-ice material composed of non-Kramers spins such as Ho2Ti2O7{\mathrm{Ho}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}Ho2​Ti2​O7​. Taken together with our results, this provides an experimental platform to study quantum spin liquid physics.
A model of the quantum (transverse) Ising spin glass with a long-range random interaction is proposed and studied here. In this model the random interaction between two spins at a distance rij{r}_{\mathrm{ij}}rij​ apart falls algebraically as ${1/r}_{\mathrm{ij}}^{(d+\ensuremath{\sigma})/2}.$ Here, apart from the strength of the quantum fluctuations, the interaction range $\ensuremath{\sigma}$ is also tunable. We have studied the ground-state properties of the model, extending the ``Droplet model'' of the short-range quantum spin glass. The model is also studied using a modified form of the effective Landau action describing the transition of the short-range quantum spin glass. The important features which are due to the long-range interaction are clearly mentioned. Field theoretical renormalization group calculations fail to locate any stable weak-coupling fixed point in the non-mean-field region. The simplest conjecture is that beyond the mean-field region, the critical behavior is governed by the infinite randomness fixed point. We extend the phenomenological scaling relation for the short-range quantum spin glass (based on the assumption of the existence of a dangerously irrelevant operator) to the present long-range interacting case. Most of the possible interesting aspects associated with the quantum transition in the present model are elaborately discussed.