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arXiv:0808.0496 · Physical Review B

Edge and impurity response in two-dimensional quantum antiferromagnets

Max A. Metlitski, Subir Sachdev A

Published Nov 10, 2008• 12 citations
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Abstract

Motivated by recent Monte Carlo simulations of H\"oglund and Sandvik (arXiv:0808.0408), we study edge response in square lattice quantum antiferromagnets. We use the O(3) nonlinear $\ensuremath{\sigma}$ model to compute the decay asymptotics of the staggered magnetization, energy density, and local magnetic susceptibility away from the edge. We find that the total edge susceptibility is negative and diverges logarithmically as the temperature $T\ensuremath{\rightarrow}0$. We confirm the predictions of the continuum theory by performing a $1/S$ expansion of the microscopic Heisenberg model with the edge. We propose a qualitative explanation of the edge dimerization seen in Monte Carlo simulations by a theory of valence-bond-solid correlations in the N\'eel state. We also discuss the extension of the latter theory to the response of a single nonmagnetic impurity, and its connection to the theory of the deconfined critical point.

DOI
10.1103/physrevb.78.174410

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Motivated by recent Monte Carlo simulations of H"oglund and Sandvik (arXiv:0808.0408), we study edge response in square lattice quantum antiferromagnets. We use the O(3) nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ model to compute the decay asymptotics of the staggered magnetization, energy density, and local magnetic susceptibility away from the edge. We find that the total edge susceptibility is negative and diverges logarithmically as the temperature T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0. We confirm the predictions of the continuum theory by performing a 1/S1/S1/S expansion of the microscopic Heisenberg model with the edge. We propose a qualitative explanation of the edge dimerization seen in Monte Carlo simulations by a theory of valence-bond-solid correlations in the N'eel state. We also discuss the extension of the latter theory to the response of a single nonmagnetic impurity, and its connection to the theory of the deconfined critical point.
We describe the spin distribution in the vicinity of a nonmagnetic impurity in a two-dimensional antiferromagnet undergoing a transition from a magnetically ordered N\'eel state to a paramagnet with a spin gap. The quantum critical ground state in a finite system has total spin S=1∕2S=1∕2S=1∕2 (if the system without the impurity had an even number of S=1∕2S=1∕2S=1∕2 spins), and recent numerical studies in a double layer antiferromagnet [K. H. H\"oglund et al., Phys. Rev. Lett. 98, 087203 (2007)] have shown that the spin has a universal spatial form delocalized across the entire sample. We present the field theory describing the uniform and staggered magnetizations in this spin texture for two classes of antiferromagnets: (i) the transition from a N\'eel state to a paramagnet with local spin singlets, in models with an even number of S=1∕2S=1∕2S=1∕2 spins per unit cell, which are described by a O(3) Landau-Ginzburg-Wilson field theory; and (ii) the transition from a N\'eel state to a valence bond solid, in antiferromagnets with a single S=1∕2S=1∕2S=1∕2 spin per unit cell, which are described by a ``deconfined'' field theory of spinons.
The spin texture surrounding a nonmagnetic impurity in a quantum antiferromagnet is a sensitive probe of the novel physics of a class of quantum phase transitions between a N\'eel ordered phase and a valence-bond solid phase in square lattice S=1/2S=1/2S=1/2 antiferromagnets. Using a newly developed T=0T=0T=0 quantum Monte Carlo technique, we compute this spin texture at these transitions and find that it does not obey the universal scaling form expected at a scale invariant quantum critical point. We also identify the precise logarithmic form of these scaling violations. Our results are expected to yield important clues regarding the probable theory of these unconventional transitions.
Recent scanning tunneling microscopy (STM) experiments on underdoped cuprates have displayed modulations in the local electronic density of states, which are centered on a Cu-O-Cu bond [Kohsaka et al., Science 315, 1380 (2007)]. As a paradigm of the pinning of such bond-centered ordering in strongly correlated systems, we present the theory of valence bond solid (VBS) correlations near a single impurity in a square lattice antiferromagnet. The antiferromagnet is assumed to be in the vicinity of a quantum transition from a magnetically ordered N\'eel state to a spin-gap state with long-range VBS order. We identify two distinct classes of impurities: (i) local modulation in the exchange constants and (ii) a missing or additional spin, for which the impurity perturbation is represented by an uncompensated Berry phase. The ``boundary'' critical theory for these classes is developed. In the second class, we find a VBS pinwheel around the impurity, accompanied by a suppression in the VBS susceptibility. Implications for numerical studies of quantum antiferromagnets and for STM experiments on the cuprates are noted.
We use quantum Monte Carlo simulations to study effects of free edges in the two-dimensional spin-12\frac{1}{2}21​ Heisenberg antiferromagnet. We find that the magnetic response of an edge is smaller than the bulk susceptibility. This counterintuitive quantum effect can be traced to enhanced antiferromagnetic nearest-neighbor spin correlations, i.e., tendency to local singlet formation, at and close to the edge. These correlations form a comblike pattern, which can be reproduced with a simple variational valence-bond state. We also study rough edges and find that these instead significantly enhance the susceptibility due to local sublattice imbalance impeding singlet formation.
This thesis is devoted to the study of strongly correlated electron systems in two spatial dimensions. The main perspective used in this work is that of scaling, implemented by the renormalization group formalism. This thesis can be divided into three main parts. The first part, comprised of chapters 2-5, investigates various aspects of two dimensional insulating quantum antiferromagnets. In particular, chapters 2-4 deal with the phase transition between an ordered antiferromagnet and a valence bond solid on a square lattice. This transition falls outside the conventional Landau-Ginzburg-Wilson paradigm and is described by a so-called “deconfined ” quantum critical point. In chapter 2, we present an elegant calculation of the scaling dimension of the valence bond solid order parameter at this quantum critical point. Chapters 3 and 4 investigate signatures of deconfined criticality in the response of an antiferromagnet to non-magnetic impurities. Chapter 5 describes the edge response of a conventional ordered square-lattice antiferromagnet. The second part of this thesis, comprised of chapters 6 and 7, studies the subject