Introduction to quantum spin systems (original) (raw)
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Computational Studies of Quantum Spin Systems
2010
These lecture notes introduce quantum spin systems and several computational methods for studying their ground-state and finite-temperature properties. Symmetry-breaking and critical phenomena are first discussed in the simpler setting of Monte Carlo studies of classical spin systems, to illustrate finite-size scaling at continuous and first-order phase transitions. Exact diagonalization and quantum Monte Carlo (stochastic series expansion) algorithms and their computer implementations are then discussed in detail. Applications of the methods are illustrated by results for some of the most essential models in quantum magnetism, such as the S = 1/2 Heisenberg antiferromagnet in one and two dimensions, as well as extended models useful for studying quantum phase transitions between antiferromagnetic and magnetically disordered states.
Exact ground states of a spin-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">mml:mfracmml:mn1mml:mn2Ising-Heisenberg model on the Shastry-Sutherland lattice in a magnetic field
Physical Review B, 2014
Exact ground states of a spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice with Heisenberg intra-dimer and Ising inter-dimer couplings are found by two independent rigorous procedures. The first method uses a unitary transformation to establish a mapping correspondence with an effective classical spin model, while the second method relies on the derivation of an effective hard-core boson model by continuous unitary transformations. Both methods lead to equivalent effective Hamiltonians providing a convincing proof that the spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice exhibits a zero-temperature magnetization curve with just two intermediate plateaus at one-third and one-half of the saturation magnetization, which correspond to stripe and checkerboard orderings of singlets and polarized triplets, respectively. The nature of the remarkable stripe order relevant to the one-third plateau is thoroughly investigated with the help of the corresponding exact eigenvector. The rigorous results for the spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice are compared with the analogous results for the purely classical Ising and fully quantum Heisenberg models. Finally, we discuss to what extent the critical fields of SrCu2(BO3)2 and (CuCl)Ca2Nb3O10 can be described within the suggested Ising-Heisenberg model.
Quantum phase transitions in the transverse one-dimensional Ising model with four-spin interactions
Physical Review B, 2006
In this work we investigate the quantum phase transitions at zero temperature of the one-dimensional transverse Ising model with an extra term containing four-spin interactions. The competition between the energy couplings of the model leads to an interesting zero-temperature phase diagram. We use a modified Lanczos method to determine the ground state and the first excited state energies of the system, with sizes of up to 20 spins. We apply finite size scaling to the energy gap to obtain the boundary region where ferromagnetic to paramagnetic transition takes place. We also find the critical exponent associated with the correlation length. We find a degenerate ͗3,1͘ phase region. The first-order transition boundary between this phase and the paramagnetic phase is determined by analyzing the behavior of the transverse spin susceptibility as the system moves from one region to the other.
We outline how the coupled cluster method of microscopic quantum many-body theory can be utilized in practice to give highly accurate results for the ground-state properties of a wide variety of highly frustrated and strongly correlated spinlattice models of interest in quantum magnetism, including their quantum phase transitions. The method itself is described, and it is shown how it may be implemented in practice to high orders in a systematically improvable hierarchy of (socalled LSUBm) approximations, by the use of computer-algebraic techniques. The method works from the outset in the thermodynamic limit of an infinite lattice at all levels of approximation, and it is shown both how the "raw" LSUBm results are themselves generally excellent in the sense that they converge rapidly, and how they may accurately be extrapolated to the exact limit, m → ∞, of the truncation index m, which denotes the only approximation made. All of this is illustrated via a specific application to a two-dimensional, frustrated, spin-half J X X Z 1-J X X Z 2 model on a honeycomb lattice with nearestneighbor and next-nearest-neighbor interactions with exchange couplings J 1 > 0 and J 2 ≡ κJ 1 > 0, respectively, where both interactions are of the same anisotropic XXZ type. We show how the method can be used to determine the entire zerotemperature ground-state phase diagram of the model in the range 0 ≤ κ ≤ 1 of the frustration parameter and 0 ≤ ∆ ≤ 1 of the spin-space anisotropy parameter. In particular, we identify a candidate quantum spin-liquid region in the phase space.
Coupled cluster method calculations of quantum magnets with spins of general spin quantum number
Journal of Statistical Physics, 2002
We present a new high-order coupled cluster method (CCM) formalism for the ground states of lattice quantum spin systems for general spin quantum number, s. This new "general-s" formalism is found to be highly suitable for a computational implementation, and the technical details of this implementation are given. To illustrate our new formalism we perform high-order CCM calculations for the one-dimensional spin-half and spin-one antiferromagnetic XXZ models and for the one-dimensional spin-half/spin-one ferrimagnetic XXZ model. The results for the ground-state properties of the isotropic points of these systems are seen to be in excellent quantitative agreement with exact results for the special case of the spin-half antiferromagnet and results of density matrix renormalisation group (DMRG) calculations for the other systems. Extrapolated CCM results for the sublattice magnetisation of the spin-half antiferromagnet closely follow the exact Bethe Ansatz solution, which contains an infinite-order phase transition at ∆ = 1. By contrast, extrapolated CCM results for the sublattice magnetisation of the spin-one antiferromagnet using this same scheme are seen to go to zero at ∆ ≈ 1.2, which is in excellent agreement with the value for the onset of the Haldane phase for this model. Results for sublattice magnetisations of the ferrimagnet for both the spin-half and spin-one spins are non-zero and finite across a wide range of ∆, up to and including the Heisenberg point at ∆ = 1.
The coupled cluster method applied to quantum magnetism
Lecture Notes in Physics, 2004
The Coupled Cluster Method (CC\I) is one of the most powerful and universally applied techniques of quanturn many-body theory. In particular, it has been used extensively in order to investigate many types of lattice quantum spin system at zero temperature. The ground-and excited-state properties of these systerns may now be determined routinely to great accuracy. In this Chapter we present an overview of the CCM formalism and we describe how the CC1\1 is applied in detaiL \Ve illustrate the power and versatility of the method by presenting results for four diH'erent spin models. These are, namely, the XXZ model, a Heisenberg model with bonds of differing strengths on the square lattice. a model which interpolates between the Kagome-and triangular-lattice antiferromagnets. and a frustrated ferrimagnetic spin system on the square lattice. vVe consider the ground-state properties of all of these systems and we present accurate results for the excitation energies of the spin-half square-lattice XXZ model. vVe utilise an "extcnded" SUB2 approximation scheme. and we demonstrate how this approximation Illay be solved exactly by using Fourier transform methods or, alternatively, by determining and solving the SUB2-m problem. \Ve also present the rcsults of "localised" approximation schemes called the LSUBm or SUBm-m schemes. \Ve note t hat we must utilise computational techniques in order to solvc these localised approximation schemes to "high order" vVe show that we are able to determine the positions of quantum phase transitions with much accuracy, and we demonstrate that we are able to determine their quantum criticality by using the CClvi in conjunction with the coherent anomaly method (CAM). Also. we illustrate that the CCM lnay be used in order to determine the "nodal surfaces" of lattice quantum spin systems, Finally. we show how connections to cumulant series expansions ma:-' be made by determining the perturbation series of a spin-half triangular-lattice antiferromagnet using the CCM at various levels of LSUBm approximation, 7.1 Introduction Key experimental observations in fields such as supcrfluidity, superconductivity, nuclear structure, quantum chemistry, quantum magnetism and strOllgly correlated electronic systems have often implied that the strong quantum correlations inherent in these systems should be fully included, at least conceptually, in any theoretical calculations that aim fully to describe their basic D.J.J. Farnc~ll and fl.P. T1ishop, The Coupled Cluster I\1f't.hud Applied to Quant HIll I\:1agrlPtisIll.
Acta Physica Polonica A, 2017
The effect of uniaxial single-ion anisotropy on magnetic and critical properties of the mixed spin-1/2 and spinS (S > 1/2) Ising model on a three-coordinated Bethe lattice is rigorously examined with the help of star-triangle transformation and exact recursion relations. In particular, our attention is focused on the ferrimagnetic version of the model, which exhibits diverse temperature dependences of the total and both sublattice magnetizations. It is shown that the critical behavior of the mixed-spin Ising model on the Bethe lattice basically depends on whether the quantum spin number S is integer or half-odd-integer.