We have studied the Heisenberg model on the triangular lattice using several Quantum Monte Carlo (QMC) techniques (up to 144 sites), and exact diagonalization (ED) (up to 36 sites). By studying the spin gap as a function of the system size we have obtained a robust evidence for a gapless spectrum, confirming the existence of long range Néel order. Our best estimate is that in the thermodynamic limit the order parameter m † = 0.41 ± 0.02 is reduced by about 59% from its classical value and the ground state energy per site is e0 = −0.5458 ± 0.0001 in unit of the exchange coupling. We have identified the important ground state correlations at short distance.71.10. Fd,71.10.Hf,75.10.Lp Historically the antiferromagnetic spin-1/2 Heisenberg model on the triangular lattice was the first proposed Hamiltonian for a microscopic realization of a spin liquid ground state (GS) [1]:where J is the nearest-neighbors antiferromagnetic exchange and the sum runs over spin-1/2 operators. At the classical level the minimum energy configuration is the well known 120• Néel state. The question whether the combined effect of frustration and quantum fluctuations favors disordered gapped resonating valence bonds (RVB) or long range Néel type order is still under debate. In fact, there has been a considerable effort to elucidate the nature of the GS and the results of numerical [2][3][4][5][6][7][8][9][10][11], and analytical [12][13][14][15][16] works are controversial. From the numerical point of view, ED, which is limited to small lattice sizes, provides a very important feature [6]: the spectra of the lowest energy levels order with increasing total spin, a reminiscence of the Lieb-Mattis theorem [17] for bipartite lattices, and are consistent with the symmetry of the classical order parameter [6]. However, other attempts to perform a finite size scaling study of the order parameter indicate a scenario close to a critical one or no magnetic order at all [3,8].The variational Quantum Monte Carlo (VMC) allows to extend the numerical calculations to fairly large system sizes, at the price to make some approximations, which are determined by the quality of the variational wavefunction (WF). Many WF have been proposed in the literature [2,4,10] and the lowest GS energy estimation was obtained with the long range ordered type. In particular, starting from the classical Néel state, Huse and Elser [4] introduced important two and three spin correlation factors in the WF:

Motivated by recent experiments on Bi3Mn4O12(NO3), we study a frustrated J1-J2 Heisenberg model on the two dimensional (2D) honeycomb lattice. The classical J1-J2 Heisenberg model on the 2D honeycomb lattice exhibits Néel order for J2 < J1/6. For J2 > J1/6, it has a family of degenerate incommensurate spin spiral ground states where the spiral wave vector can point in any direction. Spin wave fluctuations at leading order lift this accidental degeneracy in favor of specific wave vectors, leading to spiral order by disorder. For spin S = 1/2, quantum fluctuations are, however, likely to be strong enough to melt the spiral order parameter over a wide range of J2/J1. Over a part of this range, we argue that the resulting state is a valence bond solid (VBS) with staggered dimer order -this VBS is a lattice nematic which breaks lattice rotational symmetry. Our arguments are supported by comparing the spin wave energy with the energy of the VBS obtained using a bond operator formalism. Turning to the effect of thermal fluctuations on the spiral ordered state, any nonzero temperature destroys the magnetic order, but the discrete rotational symmetry of the lattice remains broken resulting in a thermal analogue of the nematic VBS. We present arguments, supported by classical Monte Carlo simulations, that this nematic transforms into the high temperature paramagnet via a thermal phase transition which is in the universality class of the classical 3-state Potts (clock) model in 2D. We discuss the relevance of our results for honeycomb magnets, such as Bi3M4O12(NO3) (with M=Mn,V,Cr), and bilayer triangular lattice magnets.

The Resonating Valence Bond (RVB) theory for two-dimensional quantum antiferromagnets is shown to be the correct paradigm for large enough "quantum frustration". This scenario, proposed long time ago but never confirmed by microscopic calculations, is very strongly supported by a new type of variational wave function, which is extremely close to the exact ground state of the J1−J2 Heisenberg model for 0.4 < ∼ J2/J1 < ∼ 0.5. This wave function is proposed to represent the generic spin-half RVB ground state in spin liquids. 75.10.Jm, 71.27.+a, 74.20.Mn The question whether a frustrated spin-half system is well described by a spin-liquid ground state (GS) -with no type of crystalline order -25 years after the first proposal [1] is still controversial, mainly because of the lack of reliable analytical or numerical solutions of model systems. For unfrustrated or weakly frustrated quantum antiferromagnets a deep understanding of the nature of the GS together with a quantitative description of the ordered phase is obtained by including Gaussian quantum fluctuations over a classical Néel state. [2,3] For sizeable frustration, instead, this description is known to break down. However, the short-range RVB state [4] does not prove a good starting point for the description of frustrated models; it rather turns out to be the exact GS of ad hoc Hamiltonians. [4][5][6] As a prototype of a realistic frustrated two-dimensional system, which has been recently realized experimentally in Li 2 VOSiO 4 compounds, [7] we investigate the spinhalf Heisenberg model with nearest (J 1 ) and next-nearest neighbor (J 2 ) superexchange couplings:on an N −site square lattice with periodic boundary conditions. In the (J 2 = 0) unfrustrated case, it is well established that the GS of the Heisenberg Hamiltonian has Néel long-range order, with a sizable value of the antiferromagnetic order parameter.[8] However, variational studies [9] have shown that disordered, long-range RVB states have energies very close to the exact one. It is therefore natural to imagine that by turning on the next-nearest neighbor interaction J 2 , the combined effect of frustration and zero-point motion may eventually melt antiferromagnetism and stabilize a non-magnetic GS of purely quantum-mechanical nature. Indeed, for 0.4 < ∼ J 2 /J 1 < ∼ 0.6 there is a general consensus on the disappearance of the Néel order towards a state whose nature is still much debated. [10] In a seminal paper, [11] Anderson proposed that a physically transparent description of a RVB state can be obtained in fermionic representation by starting from a BCS-type pairing wave function (WF), of the form

We investigate the non magnetic phase of the spin-half frustrated Heisenberg antiferromagnet on the square lattice using exact diagonalization (up to 36 sites) and quantum Monte Carlo techniques (up to 144 sites). The spin gap and the susceptibilities for the most important crystal symmetry breaking operators are computed. A genuine and somehow unexpected "plaquette RVB", with spontaneously broken translation symmetry and no broken rotation symmetry, comes out from our numerical simulations as the most plausible ground state for J2/J1 ≃ 0.5.The nature of the non magnetic phases of a quantum antiferromagnet is a topic of great interest and has been a subject of intense theoretical investigation since Anderson's suggestion [1] about the possible connections with the mechanism of high-T c superconductivity.Within the Heisenberg model the simplest way in which the antiferromagnetism can be destabilized is by introducing a next-nearest-neighbor frustrating interaction leading to the so calledoperators on a square lattice. J 1 and J 2 are the (positive) antiferromagnetic superexchange couplings between nearest and next-nearestneighbor pairs of spins respectively. In the following we will consider finite clusters of N sites with periodic boundary conditions (tilted by 45 • only for N = 32) .Although there is a general consensus about the disappearance of the Néel order in the ground state (GS) of the present model for 0.38 -4], no definite conclusion has been drawn on the nature of the non magnetic phase yet. In particular an open question is whether the GS of the J 1 −J 2 Heisenberg model is a resonating valence bond (RVB) spin liquid with no broken symmetries, as it was originally suggested by Figueirido et al. [5]. The other possibility is a GS which is still SU (2) invariant, but nonetheless breaks some crystal symmetries, dimerizing in some special pattern [6][7][8][9][10][11].In this paper we address this point using exact diagonalization (ED) and a quantum Monte Carlo technique, the Green function Monte Carlo (GFMC), which allows the calculation of GS expectation values on fairly large system sizes (L ≤ 144). This is extremely important to draw reasonable conclusions on the physical thermodynamic, zero temperature, properties of the model.For frustrated spin systems as well as for fermionic models, quantum Monte Carlo methods are affected by the so called sign problem that can be controlled, at present, only at the price of introducing some kind of approximations, such as the fixed node (FN) one [12]. In this work we have also extensively used a recently developed technique, the Green function Monte Carlo with Stochastic Reconfiguration (GFMCSR), which improves systematically the accuracy of the FN approximation for GS calculations [4,[13][14][15].The FN method allows to work without any sign problem by using the following simple strategy: the exact imaginary time propagator e −τĤ -used to filter out the GS from the best variational guess |ψ G -is replaced by an approximate propagator e −τĤFN such that the node...

Using computational techniques, it is shown that pairing is a robust property of hole doped antiferromagnetic (AF) insulators. In one dimension (1D) and for two-leg ladder systems, a BCS-like variational wave function with long-bond spin-singlets and a Jastrow factor provides an accurate representation of the ground state of the t-J model, even though strong quantum fluctuations destroy the off-diagonal superconducting (SC) long-range order in this case. However, in two dimensions (2D) it is argued -and numerically confirmed using several techniques, especially quantum Monte Carlo (QMC) -that quantum fluctuations are not strong enough to suppress superconductivity. 74.20.Mn, 71.10.Fd, 71.10.Pm, 71.27.+a The nature of high temperature superconductors remains an important unsolved problem in condensed matter physics. Strong electronic correlations are widely believed to be crucial for the understanding of these materials. Among the several proposed theories are those where antiferromagnetism induces pairing in the d x 2 −y 2 channel [1]. These approaches include the following two classes: (i) theories based on Resonant Valence Bond (RVB) wave functions, with electrons paired in long spin singlets in all possible arrangements [2,3], and (ii) theories based on two-hole d x 2 −y 2 bound states at infinitesimal doping, formed to minimize the damage of individual holes to the AF order parameter, which condense at finite pair density into a superconductor [4]. However, recent density matrix renormalization group (DMRG) calculations have seriously questioned these approaches since non-SC striped ground states were reported for realistic couplings and densities in the t-J model [5]. Clearly to make progress in the understanding of copper oxides, the 2D t-J model ground state must be fully understood, to distinguish among the many proposals.In this paper, using a variety of powerful numerical techniques, the properties of the t-J model are investigated. Our main result is that in the realistic regime of couplings the 2D t-J model supports a d x 2 −y 2 SC ground state, confirming theories of Cu-oxides based on AF correlations. The t-J model used here is. . stands for nearestneighbor sites, and n i and S i are the electron density and spin at site i, respectively. Our study focuses on the low hole-doping region of chains, two-leg ladders, and square clusters, using different numerical techniques: QMC (pure variational and fixed-node (FN) approximations), DMRG, and Lanczos. Within our QMC approach, it is possible to further improve the variational and FN accuracy by applying a few (p ≤ 2) Lanczos steps to the variational (p = 0) wave function |Ψ V ,. Non-variational estimates of energy and correlation functions can also be extracted with the variance-extrapolation method [6].Our BCS variational wave function is defined as

We describe an efficient algorithm to compute forces in quantum Monte Carlo using adjoint algorithmic differentiation. This allows us to apply the space warp coordinate transformation in differential form, and compute all the 3M force components of a system with M atoms with a computational effort comparable with the one to obtain the total energy. Few examples illustrating the method for an electronic system containing several water molecules are presented. With the present technique, the calculation of finite-temperature thermodynamic properties of materials with quantum Monte Carlo will be feasible in the near future.

We study the thermal properties of the classical antiferromagnetic Heisenberg model with both nearest (J1) and next-nearest (J2) exchange couplings on the square lattice by extensive Monte Carlo simulations. We show that, for J2/J1>1/2, thermal fluctuations give rise to an effective Z2 symmetry leading to a finite-temperature phase transition. We provide strong numerical evidence that this transition is in the 2D Ising universality class, and that T(c)-->0 with an infinite slope when J2/J1-->1/2.

In this Letter, we analyze, using scanning tunneling spectroscopy, the density of electronic states in nearly optimally doped Bi2Sr2CaCu2O(8+delta) in zero magnetic field. Focusing on the superconducting gap, we find patches of what appear to be two different phases in a background of some average gap, one with a relatively small gap and sharp large coherence peaks and one characterized by a large gap with broad weak coherence peaks. We compare these spectra with calculations of the local density of states for a simple phenomenological model in which a 2xi0 x 2xi0 patch with an enhanced or suppressed d-wave gap amplitude is embedded in a region with a uniform average d-wave gap.