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arXiv:cond-mat/0603179 · Physical Review Letters

Quantum Criticality and Percolation in Dimer-Diluted Two-Dimensional Antiferromagnets

Anders W. Sandvik

Published May 22, 2006• 31 citations
arXivHTMLPDF

Abstract

The S = 1/2 Heisenberg model is considered on bilayer and single-layer square lattices with couplings J1, J2, with each spin belonging to one J2-coupled dimer. A transition from a Néel to disordered ground state occurs at a critical value of g = J2/J1. The systems are here studied at their dimer-dilution percolation points p*. The multicritical point (g*,p*) previously found for the bilayer is not reproduced for the single layer. Instead, there is a line of critical points (g < g*, p*) with continuously varying exponents. The uniform magnetic susceptibility diverges as T(-alpha) with alpha element of [1/2,1]. This unusual behavior is attributed to an effective free-moment density approximately T(1-alpha). The susceptibility of the bilayer is not divergent but exhibits remarkably robust quantum-critical scaling.

DOI
10.1103/physrevlett.96.207201

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The S = 1/2 Heisenberg model is considered on bilayer and single-layer square lattices with couplings J1, J2, with each spin belonging to one J2-coupled dimer. A transition from a Néel to disordered ground state occurs at a critical value of g = J2/J1. The systems are here studied at their dimer-dilution percolation points p*. The multicritical point (g*,p*) previously found for the bilayer is not reproduced for the single layer. Instead, there is a line of critical points (g < g*, p*) with continuously varying exponents. The uniform magnetic susceptibility diverges as T(-alpha) with alpha element of [1/2,1]. This unusual behavior is attributed to an effective free-moment density approximately T(1-alpha). The susceptibility of the bilayer is not divergent but exhibits remarkably robust quantum-critical scaling.
We use quantum Monte Carlo simulations to study a dimer-diluted S = 1/2 Heisenberg model on a bilayer square lattice with intralayer interaction J 1 and interlayer interaction J 2 . Below the classical percolation threshold p c , the system has three phases reachable by tuning the interaction ratio g = J 2 / J 1 : a Néel ordered phase, a gapless quantum glass phase, and a gapped quantum paramagnetic phase. We present the ground-state phase diagram in the plane of dilution p and interaction ratio g . The quantum glass phase is certified to be of the gapless Mott glass type, having a uniform susceptibility vanishing at zero temperature T and following a stretched exponential form at T > 0; χ u exp(- b / T α ) with α < 1. At the phase transition point from Neel ordered to Mott glass, we find that the critical exponents are different from those of the clean system described by the standard O (3) universality class in 2+1 dimensions.
We investigate quantum phase transitions in the spin-1∕21∕21∕2 Heisenberg antiferromagnet on square lattices with inhomogeneous bond dilution. It is shown that quantum fluctuations can be continuously tuned by inhomogeneous bond dilution, eventually leading to the destruction of long-range magnetic order on the percolating cluster. Two multicritical points are identified at which the magnetic transition separates from the percolation transition, introducing a quantum phase transition. Beyond these multicritical points a quantum-disordered phase appears, characterized by an infinite percolating cluster with short ranged antiferromagnetic order. In this phase, the low-temperature uniform susceptibility diverges algebraically with non-universal exponents. This is a signature that the quantum-disordered phase is a quantum Griffiths phase, as also directly confirmed by the statistical distribution of local gaps. This study thus presents evidence of a genuine quantum Griffiths phenomenon in a two-dimensional Heisenberg antiferromagnet.
We employ the stochastic series expansion quantum Monte Carlo method to investigate the S=1/2S=1/2S=1/2 antiferromagnetic Heisenberg model on a bilayer square lattice with diluted interlayer couplings. Both regular and random dilution patterns are considered. In systems with regular dilution, tuning the interlayer interaction drives a quantum phase transition from a N\'eel-ordered phase to a quantum disordered phase, consistent with the O(3)O(3)O(3) universality class. In contrast, random dilution gives rise to a two-step transition: from the N\'eel phase to an intermediate Mott glass (MG) phase, followed by a transition to the quantum disordered phase. Within the MG phase, the uniform magnetic susceptibility exhibits a stretched-exponential temperature dependence ${\ensuremath{\chi}}_{u}\ensuremath{\sim}exp(\ensuremath{-}b/{T}^{\ensuremath{\alpha}}), 0<\ensuremath{\alpha}<1$. At the N\'eel-to-glass transition, the quenched disorder modifies the critical exponents in a manner consistent with the Harris criterion. These findings provide new insights into disorder-driven quantum phase transitions and the emergence of glassy phases in diluted bilayer quantum magnets.
We use quantum Monte Carlo simulations to study a disordered S=1/2S=1/2S=1/2 Heisenberg quantum spin model with three different nearest-neighbor interactions, ${J}_{1}\ensuremath{\le}{J}_{2}\ensuremath{\le}{J}_{3}$, on the square lattice. We consider the regime in which J1{J}_{1}J1​ represents weak bonds, and J2{J}_{2}J2​ and J3{J}_{3}J3​ correspond to two kinds of stronger bonds (dimers) which are randomly distributed on columns forming coupled two-leg ladders. When increasing the average intradimer coupling (J2+J3)/2({J}_{2}+{J}_{3})/2(J2​+J3​)/2, the system undergoes a N\'eel to quantum glass transition of the ground state and later a second transition into a quantum paramagnet. The quantum glass phase is of the gapless Mott glass type (i.e., in boson language it is incompressible at temperature T=0T=0T=0), and we find that the temperature dependence of the uniform magnetic susceptibility follows the stretched exponential form $\ensuremath{\chi}\ensuremath{\sim}exp(\ensuremath{-}b/{T}^{\ensuremath{\alpha}})$, with $0<\ensuremath{\alpha}<1$. At the N\'eel-glass transition, we observe the standard O(3) critical exponents, which implies that the Harris criterion for the relevance of the disorder is violated in this system.
Series expansion studies of the J1J_1J1​-J2J_2J2​-Heisenberg bilayer
We study a bilayer of the frustrated J1J_1J1​-J2J_2J2​ Heisenberg-model on the square lattice. Starting from the dimer limit at strong interlayer coupling, we perform series expansions using the perturbative Continuous Unitary Transformation, based on the flow equation method, in order to determine the spectrum up to the two-triplon sector. From the one-triplon dispersion we obtain quantum critical lines for transitions from the dimer phase into either Néel or collinear magnetic order. For low to intermediate frustration these transitions are consistent with existing findings, based on the magnetic phases. In the region of strongest frustration, i.e. J2/J1∼0.5J_2/J_1\sim 0.5J2​/J1​∼0.5, we provide an estimate for the stability of the anticipated single-layer quantum spin-liquids against finite interlayer coupling. In the two-triplon sector we find a set of well defined (anti)bound states, which can be classified according to total spin and in-plane rotational symmetry. For vanishing frustration these states agree with previous series expansion analysis. For J2/J1≳0.5J_2/J_1\gtrsim 0.5J2​/J1​≳0.5 we provide evidence for a close-by condensation of one-triplon and two-triplon singlet bound states, suggesting that between the dimer and the collinear state, additional phases may intervene.
We study a bilayer of the frustrated ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice. Starting from the dimer limit at strong interlayer coupling, we perform series expansions using the perturbative continuous unitary transformation, based on the flow equation method, in order to determine the spectrum up to the two-triplon sector. From the one-triplon dispersion, we obtain quantum critical lines for transitions from the dimer phase into either N\'eel or collinear magnetic order. For low to intermediate frustration these transitions are consistent with existing findings, based on the magnetic phases. In the region of strongest frustration, i.e., ${J}_{2}/{J}_{1}\ensuremath{\sim}0.5$, we provide an estimate for the stability of the anticipated single-layer quantum spin-liquids against finite interlayer coupling. In the two-triplon sector, we find a set of well defined (anti)bound states, which can be classified according to total spin and in-plane rotational symmetry. For vanishing frustration, these states agree with the previous series expansion analysis. For ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.5$, we provide evidence for a close-by condensation of one-triplon and two-triplon singlet bound states, suggesting that between the dimer and the collinear state additional phases may intervene.