We introduce the concept of directed loops in stochastic series expansion and path-integral quantum Monte Carlo methods. Using the detailed balance rules for directed loops, we show that it is possible to smoothly connect generally applicable simulation schemes (in which it is necessary to include backtracking processes in the loop construction) to more restricted loop algorithms that can be constructed only for a limited range of Hamiltonians (where backtracking can be avoided). The "algorithmic discontinuities" between general and special points (or regions) in parameter space can hence be eliminated. As a specific example, we consider the anisotropic S=1/2 Heisenberg antiferromagnet in an external magnetic field. We show that directed-loop simulations are very efficient for the full range of magnetic fields (zero to the saturation point) and anisotropies. In particular, for weak fields and anisotropies, the autocorrelations are significantly reduced relative to those of previous approaches. The back-tracking probability vanishes continuously as the isotropic Heisenberg point is approached. For the XY model, we show that back tracking can be avoided for all fields extending up to the saturation field. The method is hence particularly efficient in this case. We use directed-loop simulations to study the magnetization process in the two-dimensional Heisenberg model at very low temperatures. For LxL lattices with L up to 64, we utilize the step structure in the magnetization curve to extract gaps between different spin sectors. Finite-size scaling of the gaps gives an accurate estimate of the transverse susceptibility in the thermodynamic limit: chi( perpendicular )=0.0659+/-0.0002.

The directed-loop scheme is a framework for generalized loop-type updates in quantum Monte Carlo, applicable both to world-line and stochastic series expansion methods. Here, the directed-loop equations, the solution of which gives the probabilities of the various loop-building steps, are discussed in the context of the anisotropic S = 1/2 Heisenberg model in a uniform magnetic field. This example shows how the directed-loop concept emerges as a natural generalization of the conventional loop algorithm, where the loops are selfavoiding, to cases where selfintersection must be allowed in order to satisfy detailed balance.The directed-loop algorithm

It is shown that spontaneous symmetry breaking does not modify the ground-state entanglement of two spins, as defined by the concurrence, in the XXZ-and the transverse field Ising-chain. Correlation function inequalities, valid in any dimensions for these models, are presented outlining the regimes where entanglement is unaffected by spontaneous symmetry breaking.PACS numbers: 03.67. Mn, 75.10.Jm Entanglement is a property of a quantum state shared between two or more parties. It is defined with the aim of capturing the essential quantum non-locality encoded in the state. While for a long time interests in entanglement stemmed from the opportunity to understand fundamental concepts in quantum mechanics such as the EPR-paradox and violation of Bell-inequalities, recent interest in entanglement comes from its use as a resource for performing tasks not possible by classical means.From an applied viewpoint it is thus worth quantifying the degree of entanglement in natural systems, such as solid-state materials. A number of studies have been devoted to quantifying such "natural" entanglement in states of simple models describing idealized quantum magnets, such as the XXZ-and transverse field Isingmodels [1,2,3,4,5,6]. However the possible alteration of entanglement by spontaneous symmetry breaking (SSB) were not discussed in any of these works, although the need for such a study was mentioned in Ref. [5]. SSB happens invariably in real materials described by a Hamiltonian possessing a global symmetry, thus it is important to investigate whether or not the entanglement calculated in the symmetric ground state(s) is changed by SSB. Here we find that entanglement is not modified by SSB for the XXZ-and transverse field Ising-chains.It is a fundamental requirement of entanglement that it cannot on average be created by mixing classically two quantum states. Classical mixing is incoherent, and so one should not gain more entanglement from this mixing than what is already encoded in the states that are being mixed. Thus if we consider the density matrix ρ = (ρ + + ρ − )/2 put together by an equal mixture of the broken symmetry states, ρ ± (not necessarily pure), the entanglement E(ρ) must satisfywhere we have assumed in the last equality that the entanglement in the two different symmetry-broken ground states are equal, the states being related by a global change of basis states. Thus the entanglement in the broken state cannot be smaller than in the symmetric state.Here we focus on the entanglement of two specific spins (or qubits) in the ground-state of a quantum system. The state of two spins i and j in the ground state of a quantum system is described in terms of the reduced density matrix ρ ij obtained by tracing over all spins in the ground state except the two spins i and j. Writing it out explicitly in the standard basis {| ↑↑ , | ↑↓ , | ↓↑ , | ↓↓ } one has ρ ij =

We find evidence for decaying magnons at strong magnetic field in the square lattice spin-1/2 Heisenberg antiferromagnet. The results are obtained using quantum Monte Carlo simulations combined with a Bayesian inference technique to obtain dynamics and are consistent with predictions from spin-wave theory.The square lattice spin-1/2 Heisenberg antiferromagnet ͑2DHAF͒ is the archetype quantum antiferromagnet and describes the magnetism of the mother compounds of the high-T c cuprate materials. 1 While the 2DHAF itself is rather well understood, 2 less is known when additional interactions are added. In this Rapid Communication we report on the nature of the excitations of the 2DHAF in an external magnetic field. Such a perturbation is highly relevant from an experimental point of view and has very interesting consequences.Adding a magnetic field introduces interactions among the magnon excitations of the pure 2DHAF. According to spin-wave theory these interactions will cause the magnons in certain regions of the Brillouin zone to be unstable at very high fields. 3 One should, however, note that spin-wave theory, which is an expansion in the parameter 1 / S, is not at all guaranteed to work for S =1/ 2. Furthermore the spinwave prediction involves certain assumptions as it is a selfconsistent calculation which neglects vertex corrections and renormalizes the spin-wave spectrum considerably. In addition there are also other theoretical predictions of what happens to the 2DHAF in a magnetic field: a scenario where the spin-1 excitation is viewed as a composite excitation of twodimensional spinons predicts additional bands of low-energy excitations at moderate values of the field. 4,5 Thus to settle the issue an independent calculation of the excitation spectrum of the 2DHAF in a magnetic field is needed. It is rather surprising that no such calculation has been performed before, given the prominence and simplicity of the model. In this Rapid Communication we use quantum Monte Carlo ͑QMC͒ simulations combined with a Bayesian inference technique to give a detailed and unbiased description of the excitation spectrum of the 2DHAF in a magnetic field. We find regions with broad spectral features at strong magnetic fields indicating decaying magnons consistent with spinwave theory, and we also show details of the spectrum in the decaying regions.In zero magnetic field the ground state of the 2DHAF is Néel ordered with a renormalized moment. Spin-wave theory explains very well the renormalized moment as well as the dispersion relation of the spin-1 excitations except for an anomaly occurring at ͑ ,0͒ ͑we use units where the lattice spacing a =1͒ where the dispersion softens and the spectrum broadens anomalously. [6][7][8] This anomaly at ͑ ,0͒ is, on the contrary, natural in a picture of the spin-1 excitation as a composite excitation of two spin-1/2 spinons that are excitations about a mean-field pi-flux state. 4 While this bound state has almost the same dispersion as predicted by spin-wave theory, it differs around ͑ ,0͒...

We show that a dilute ensemble of epoxy-bonded adatoms on graphene has a tendency to form a spatially correlated state accompanied by a gap in graphene's electron spectrum. This effect emerges from the electronmediated interaction between adatoms with a peculiar 1 / r 3 distance dependence. The partial ordering transition is described by a random bond three-state Potts model. DOI: 10.1103/PhysRevB.80.233409 PACS number͑s͒: 73.20.Hb, 68.35.Rh, 73.61.Ϫr Graphene ͑monolayer of graphite͒ is a truly twodimensional crystal, just one atom thick. 1 It is a gapless semiconductor with charge carriers mimicking relativistic dynamics of massless Dirac fermions, 2 a peculiarity dictated by the bonding of carbon atoms into a highly symmetric honeycomb lattice. Graphene can host various adsorbents, in particular atoms, retaining its own structural integrity. Such chemisorbed atoms ͑adatoms͒ may strongly affect electronic properties of graphene 3-9 introducing symmetry-breaking perturbations into the lattice. The type of symmetry breaking depends on the position of the adatom in the hexagonal unit cell of the crystal. In particular, alkali atoms position themselves over the centers of the hexagons. 10 Oxygen, nitrogen, boron, or an additional carbon 11 prefer "epoxy" bonded positions ͑e-type͒ and reside above the middle of a carboncarbon bond. Atomic hydrogen and halogens reside in the symmetric on-site position above the carbon ͑s-type͒. 12 It has also been noticed that a pair of hydrogen atoms on the neighboring sites of graphene lattice forms a stable H-H dimer which acts as an e-type adsorbent. 13 Here we predict that an ensemble of e-type adatoms ͑those perturbing C-C bonds͒ tend to order, mimicking a superlattice structure, even when graphene coverage by adsorbents is low. The underlying mechanism is a long-range electron-mediated interaction between adatoms similar to the Ruderman-Kittel-Kasuya-Yoshida ͑RKKY͒ exchange between localized spins in metals. 14 The effect is peculiar to graphene. Unlike metals, charge neutral graphene has a pointlike Fermi surface positioned in the corners K and KЈ =−K of the hexagonal Brillouin zone-called valleys. The electron density of states vanishes at the Fermi level. As a result, the Friedel oscillations in charge neutral graphene are commensurate with its honeycomb lattice and decay as the inverse cube of the distance to the adatom. 15 We show that such an interaction in a dilute ensemble of e-type adsorbents may result in their partial ordering associated with a superlattice structure with the unit cell three times larger than in graphene, as illustrated in Fig. 1. We present our results in the following order. Starting with a particular tight-binding model for an e-type adsorbent, we determine the form of a perturbation it creates for the electrons in graphene. Using group theory we classify such interactions beyond a specific microscopic model and determine the conditions under which RKKY interaction between adatoms leads to a partially ordered state with a gapful electronic sp...

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Electronic and transport properties of Graphene, a one-atom thick crystalline material, are sensitive to the presence of atoms adsorbed on its surface.An ensemble of randomly positioned adatoms, each serving as a scattering center, leads to the Bolzmann-Drude diffusion of charge determining the resistivity of the material. An important question, however, is whether the distribution of adatoms is always genuinely random. In this Article we demonstrate that a dilute adatoms on graphene may have a tendency towards a spatially correlated state with a hidden Kekulé mosaic order. This effect emerges from the interaction between the adatoms mediated by the Friedel oscillations of the electron density in graphene. The onset of the ordered state, as the system is cooled below the critical temperature, is accompanied by the opening of a gap in the electronic spectrum of the material, dramatically changing its transport properties.

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