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arXiv:cond-mat/0005185 · Physical review. B, Condensed matter

Disorder from disorder in a strongly frustrated transverse-field Ising chain

Donald Priour, Martin P. Gelfand, Shivaji L. Sondhi

Published Sep 13, 2001• 10 citations
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Abstract

We study a one-dimensional chain of corner-sharing triangles with antiferromagnetic Ising interactions along its bonds. Classically, this system is highly frustrated with an extensive entropy at $T=0$ and exponentially decaying spin correlations. We show that the introduction of a quantum dynamics via a transverse magnetic field removes the entropy and opens a gap, but leaves the ground state disordered at all values of the transverse field, thereby providing a particularly simple realization of the ``disorder by disorder'' scenario first proposed by Fazekas and Anderson in their search for resonating valence bond states. Our conclusion relies on exact diagonalization calculations as well as on the analysis of a 14th-order perturbation expansion about the large transverse field limit. This test of the perturbation expansion method suggests that it could be applied to studies of frustrated transverse-field magnets in higher dimensions.

DOI
10.1103/physrevb.64.134424

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We study a one-dimensional chain of corner-sharing triangles with antiferromagnetic Ising interactions along its bonds. Classically, this system is highly frustrated with an extensive entropy at T=0T=0T=0 and exponentially decaying spin correlations. We show that the introduction of a quantum dynamics via a transverse magnetic field removes the entropy and opens a gap, but leaves the ground state disordered at all values of the transverse field, thereby providing a particularly simple realization of the ``disorder by disorder'' scenario first proposed by Fazekas and Anderson in their search for resonating valence bond states. Our conclusion relies on exact diagonalization calculations as well as on the analysis of a 14th-order perturbation expansion about the large transverse field limit. This test of the perturbation expansion method suggests that it could be applied to studies of frustrated transverse-field magnets in higher dimensions.
An antiferromagnetic Ising model on the Δ chain, with a macroscopic ground state degeneracy and exponentially decaying spin correlation functions, is studied. By introducing transverse magnetic fields Γ on spins at tips of the triangles and λΓ on spins on the bottom chain, we study the ground state as a function of λ and Γ. The λ=0 problem is exactly solved and the ground state has a staggered magnetization on the bottom chain for any Γ>0. Because the transverse field on the bottom chain with finite λ destroys the staggered order, the order–disorder transition occurs as λ is increased. We calculate the transition point λ c (Γ/ J ) as a function of Γ/ J , where J denotes the magnitude of the exchange interaction, by using the series expansion method around the λ=0 limit and conclude that λ c (Γ/ J =+0)≃0.9459. It is shown that states in the degenerate ground state manifold at Γ=0 are crucial in getting a finite value of λ c (+0).
We investigate the transverse field Ising model on a diamond chain using series expansions about the high-field limit and exact diagonalizations. For the unfrustrated case we accurately determine the quantum critical point and its expected 2d Ising universality separating the polarized and the Z2{\mathcal{Z}}_{2}Z2​ symmetry broken phase. In contrast, we find strong evidence for a disorder by disorder scenario for the fully-frustrated transverse field Ising model, i.e., except for the pure Ising model, having an extensive number of ground states, the system is always in a quantum disordered polarized phase. The low-energy excitations in this polarized phase are understood in terms of exact local modes of the model. Furthermore, an effective low-energy description for an infinitesimal transverse field allows us to pinpoint the quantum disordered nature of the ground state via mapping to an effective transverse field Ising chain and to determine the induced gap to the elementary effective domain wall excitation very accurately.
We report on a systematic study of two-dimensional, periodic, frustrated Ising models with quantum dynamics introduced via a transverse magnetic field. The systems studied are the triangular and kagom\'e lattice antiferromagnets, fully frustrated models on the square and hexagonal (honeycomb) lattices, a planar analog of the pyrochlore antiferromagnet, a pentagonal lattice antiferromagnet, as well as two quasi-one-dimensional lattices that have considerable pedagogical value. All of these exhibit a macroscopic degeneracy at T=0T=0T=0 in the absence of the transverse field, which enters as a singular perturbation. We analyze these systems with a combination of a variational method at weak fields, a perturbative Landau-Ginzburg-Wilson approach from large fields, as well as quantum Monte Carlo simulations utilizing a cluster algorithm. Our results include instances of quantum order arising from classical criticality (triangular lattice) or classical disorder (pentagonal and probably hexagonal) as well as notable instances of quantum disorder arising from classical disorder (kagom\'e). We also discuss the effect of finite temperature, as well as the interplay between longitudinal and transverse fields---in the kagom\'e problem the latter gives rise to a nontrivial phase diagram with bond-ordered and bond-critical phases in addition to the disordered phase. We also note connections to quantum-dimer models and thereby to the physics of Heisenberg antiferromagnets in short-ranged resonating valence-bond phases that have been invoked in discussions of high-temperature superconductivity.
In this paper, using high order perturbative series expansion method, the critical exponents of the order parameter and susceptibility in transition from ferromagnetic to disordered phases for 1D quantum Ising model in transverse field, with ferromagnetic nearest neighbor and anti-ferromagnetic next to nearest neighbor interactions, are calculated. It is found that for small value of the frustrating second neighbor coupling ( k<0.4 ), the exponents are the same as those of the classical 2D Ising model (B=1/8 and Y=7/8), but a deviation from 2D classical Ising universality class is observed for larger values of frustration (0.4< K<0.5). The reason for this deviation is the “disorder line” which becomes tangent to the Ferromagnetic phase boundary line at K=0.5
We study the low-energy properties of the long-range random transverse-field Ising chain with ferromagnetic interactions decaying as a power of the distance. Using variants of the strong-disorder renormalization group method, the critical behavior is found to be controlled by a strong-disorder fixed point with a finite dynamical exponent . Approaching the critical point, the correlation length diverges exponentially. In the critical point, the magnetization shows an -independent logarithmic finite-size scaling and the entanglement entropy satisfies the area law. These observations are argued to hold for other systems with long-range interactions, even in higher dimensions.
We consider a long-range Ising antiferromagnet (LRIAF) put in a transverse field. Applying quantum Monte Carlo method, we study the variation of order parameter (spin correlation in Trotter time direction), susceptibility and average energy of the system for various values of the transverse field at different temperatures. The antiferromagnetic order is seen to get immediately broken as soon as the thermal or quantum fluctuations are added. We also discuss the phase diagram for the Sherrington-Kirkpatrick (SK) model with the same LRIAF bias, also in presence of a transverse field. We find that while the antiferromagnetic order is immediately broken as one adds an infinitesimal transverse field or thermal fluctuation to the system, an infinitesimal SK spin glass disorder is enough to induce a stable glass order in the antiferromagnet. This glass order eventually gets destroyed as the thermal or quantum fluctuations increased beyond their threshold values and the transition to para phase occurs. Indications of this novel phase transition are discussed. Because of the presence of full frustration, this surrogate property of the LRIAF for incubation of stable spin glass phase in it (induced by addition of a small disorder) should enable eventually the study of classical and quantum spin glass phases by using some perturbation theory with respect to the disorder.