Path integrals are usually formulated in discrete Euclidean time using the Trotter formula. We propose a new method to study discrete quantum systems, in which we work directly in the Euclidean time continuum. The method is of general interest and can be applied to studies of quantum spin systems, lattice fermions, and in principle also lattice gauge theories. Here it is applied to the Heisenberg quantum antiferromagnet using a continuous-time version of a loop cluster algorithm. The computational advantage of this algorithm is exploited to confirm the predictions of chiral perturbation theory in the extreme low temperature regime, down to T = 0.01J. A fit of the low-energy parameters of chiral perturbation theory gives excellent agreement with previous results and with experiments.

The correlation length of the square-lattice spin-1͞2 Heisenberg antiferromagnet is studied in the lowtemperature (asymptotic-scaling) regime. Our novel approach combines a very efficient loop cluster algorithm-operating directly in the Euclidean time continuum-with finite-size scaling. This enables us to probe correlation lengths up to j ഠ 350 000 lattice spacings, more than 3 orders of magnitude larger than in any previous study. We resolve a conundrum concerning the applicability of asymptoticscaling formulas to experimentally and numerically determined correlation lengths. Our results have direct implications for the zero-temperature behavior of spin-1͞2 ladders. [S0000-0000 (98)00001-1] PACS numbers: 75.10.Jm, 02.70.Lq, 31.15.Kb, 71.10.FdSoon after the discovery of high-temperature superconductivity in doped lamellar copper oxides it was found that the undoped compounds are quasi-two-dimensional (2D) spin S 1͞2 quantum antiferromagnets. A theoretical model that captures the essential features of these materials is the nearest-neighbor quantum antiferromagnetic Heisenberg model (AFHM) on a square lattice. Through experimental, numerical, and theoretical efforts much progress has been made in the understanding of these systems. In particular, detailed neutron scattering measurements of the spin-spin correlation length in the magnet Sr 2 CuO 2 Cl 2 were found to be described quantitatively [1] by both high-temperature numerical results for the AFHM [2] and low-temperature theory for the renormalized classical regime of the ͑2 1 1͒D O(3)-symmetric nonlinear s model [3,4].The ground state of the above systems shows longrange antiferromagnetic order, thus spontaneously breaking the O(3) rotational symmetry to O(2). The low-energy excitations are two massless bosons called magnons or spin waves. These long-range excitations determine the dynamics at low energies. One can use chiral perturbation theory (CPT) to derive universal expressions for low-energy observables in terms of three material-specific parameters: the staggered magnetization M s , the spinwave velocity c, and the spin stiffness r s [5]. Both numerical data and predictions of CPT are in apparent good agreement with experimental results for S 1͞2 [1]. However, neutron scattering measurements on antiferromagnets with S . 1͞2 [1,6] reveal a striking discrepancy with CPT predictions based on three-loop asymptotic scaling. It has been suggested that for S . 1͞2, asymptotic scaling sets in only at very low temperatures-that is, for correlation lengths much larger than those accessed experimentally and numerically [7].In this Letter, we investigate the correlation length of the 2D nearest-neighbor square-lattice spin-1͞2 Heisenberg antiferromagnet at unprecedentedly low temperatures. We resolve the puzzle concerning the applicability of asymptotic scaling to experimentally and numerically determined correlation lengths [1,[6][7][8]. To effect this, we combine an efficient loop cluster algorithm operating in the Euclidean time continuum-hence with zero system...

D-theory provides an alternative lattice regularization of the (1 + 1)-d CP (N − 1) quantum field theory. In this formulation the continuous classical CP (N − 1) fields emerge from the dimensional reduction of discrete SU (N ) quantum spins. In analogy to Haldane's conjecture, ladders consisting of an even number of transversely coupled spin chains lead to a CP (N − 1) model with vacuum angle θ = 0, while an odd number of chains yields θ = π. In contrast to Wilson's formulation of lattice field theory, in D-theory no sign problem arises at θ = π, and an efficient cluster algorithm is used to investigate the θ-vacuum effects. At θ = π there is a first order phase transition with spontaneous breaking of charge conjugation symmetry for CP (N − 1) models with N > 2.PACS numbers: 05.50.+q,75.10.Jm CP (N − 1) models are interesting (1 + 1)-d quantum field theories [1] which share a number of important features with (3 + 1)-d QCD. These include asymptotic freedom, the dynamical generation of a mass-gap, and an instanton topological charge leading to non-trivial θ-vacuum effects. Despite the fact that Nature has chosen θ = 0, it is an interesting challenge to understand the structure of θ-vacua, which in QCD may, for example, lead to 't Hooft's oblique confinement phases [2]. It has been conjectured that CP (N − 1) models have a phase transition at θ = π. In the CP (1) = O(3) case this phase transition is known to be second order with a vanishing mass-gap [3][4][5]. For N > 2, on the other hand, the transition is conjectured to be first order [6], which is consistent with large N analytic results [7]. For finite N , however, the investigation of these nonperturbative problems is highly nontrivial. In particular, in contrast to O(N ) models and other (1 + 1)-d quantum field theories, CP (N − 1) models with N > 2 cannot be solved with the Bethe ansatz. This is because in the CP (N − 1) case an infinite set of classical symmetries is anomalous and can thus not be used to solve the quantum theory analytically.The numerical investigation of lattice CP (N −1) models is also far from being straightforward, even at θ = 0. Again, in contrast to O(N ) models which can be studied with the efficient Wolff cluster algorithm [8], no efficient cluster algorithm is available for CP (N − 1) models [9]. There is even a no-go theorem that forbids the construction of an efficient Wolff-type embedding algorithm for these models [10]. Still, at θ = 0 a rather efficient multigrid algorithm was developed in [11]. However, at θ = π the situation is much worse due to a very severe sign problem: the contributions from odd topological charge sectors almost completely cancel those from even charge sectors. This makes it exponentially hard to access large lattices which is necessary for reaching reliable conclusions about the phase structure. For this reason, previous numerical studies of θ-vacua were limited to mod- erate volumes [12,13] or rely on additional assumptions [14]. In the CP (1) = O(3) case, a Wolff-type meroncluster algorithm allows...

A new non-perturbative approach to quantum field theory -D-theory -is proposed, in which continuous classical fields are replaced by discrete quantized variables which undergo dimensional reduction. The 2-d classical O(3) model emerges from the (2 + 1)-d quantum Heisenberg model formulated in terms of quantum spins. Dimensional reduction is demonstrated explicitly by simulating correlation lengths up to 350,000 lattice spacings using a loop cluster algorithm. In the framework of D-theory, gauge theories are formulated in terms of quantum links -the gauge analogs of quantum spins. Quantum links are parallel transporter matrices whose elements are non-commuting operators. They can be expressed as bilinears of anticommuting fermion constituents. In quantum link models dimensional reduction to four dimensions occurs, due to the presence of a 5-d Coulomb phase, whose existence is confirmed by detailed simulations using standard lattice gauge theory. Using Shamir's variant of Kaplan's fermion proposal, in quantum link QCD quarks appear as edge states of a 5-d slab. This naturally protects their chiral symmetries without fine-tuning. The first efficient cluster algorithm for a gauge theory with a continuous gauge group is formulated for the U (1) quantum link model. Improved estimators for Wilson loops are constructed, and dimensional reduction to ordinary lattice QED is verified numerically.

Path integrals are usually formulated in discrete Euclidean time using the Trotter formula. We propose a new method to study discrete quantum systems, in which we work directly in the Euclidean time continuum. The method is of general interest and can be applied to studies of quantum spin systems, lattice fermions, and in principle also lattice gauge theories. Here it is applied to the Heisenberg quantum antiferromagnet using a continuous-time version of a loop cluster algorithm. The computational advantage of this algorithm is exploited to confirm the predictions of chiral perturbation theory in the extreme low temperature regime, down to T = 0.01J. A fit of the low-energy parameters of chiral perturbation theory gives excellent agreement with previous results and with experiments.