We study the path-integral treatment of the quantum mechanics of a fully frustrated cluster of spins: a cluster in which every pair of spins is coupled equally by antiferromagnetic Heisenberg interactions. Such clusters are interesting partly because they are the building blocks of geometrically frustrated spin systems. Using a Hubbard–Stratonovich transformation to decouple the interactions, the Boltzmann factor for the spin cluster is written in terms of the time-evolution operator for a single spin in a stochastically varying magnetic field. The time-evolution operator follows a random walk in SU (2): by switching from a Langevin to a Fokker–Planck description of this walk and computing the probability distribution of its end-point, we arrive at an expression for the partition function as a finite sum of single integrals. The calculation provides an analytically tractable illustration of the auxiliary field approach, as used in quantum Monte Carlo calculations, and may potentially be extended to treat more complex frustrated spin systems.
We study the path-integral treatment of the quantum mechanics of a fully frustrated cluster of spins: a cluster in which every pair of spins is coupled equally by antiferromagnetic Heisenberg interactions. Such clusters are interesting partly because they are the building blocks of geometrically frustrated spin systems. Using a Hubbard–Stratonovich transformation to decouple the interactions, the Boltzmann factor for the spin cluster is written in terms of the time-evolution operator for a single spin in a stochastically varying magnetic field. The time-evolution operator follows a random walk in SU (2): by switching from a Langevin to a Fokker–Planck description of this walk and computing the probability distribution of its end-point, we arrive at an expression for the partition function as a finite sum of single integrals. The calculation provides an analytically tractable illustration of the auxiliary field approach, as used in quantum Monte Carlo calculations, and may potentially be extended to treat more complex frustrated spin systems.
Abstract We extend a semiclassical numerical method, bosonic auxiliary-field Monte Carlo, to quantum spin systems. This method breaks the lattice into clusters, solves each cluster precisely and couples them with classical auxiliary fields through classical Monte Carlo simulation. We test the method with antiferromagnetic spin models in one-dimensional chains, square lattices and triangular lattices, and obtain reasonable results at finite temperatures. This algorithm builds a bridge between classical Monte Carlo method and quantum methods. The algorithm can be improved with either progress in classical Monte Carlo sampling or the development of quantum solvers, and can also be further applied to systems with different lattices or interactions.
The auxiliary-field quantum Monte Carlo method is reviewed. The Hubbard-Stratonovich transformation converts an interacting Hamiltonian into a non-interacting Hamiltonian in a time-dependent stochastic field, allowing calculation of the resulting functional integral by Monte Carlo methods. The method is presented in a sufficiently general form to be applicable to any Hamiltonian with one- and two-body terms, with special reference to the Heisenberg model and one- and many-band Hubbard models. Many physical correlation functions can be related to correlation functions of the auxiliary field; general results are given here. Issues relating to the choice of auxiliary fields are addressed; operator product identities change the relative dimensionalities of the attractive and repulsive parts of the interaction. Frequently the integrand is not positive-definite, rendering numerical evaluation unstable. If the auxiliary field violates time-reversal invariance, the integrand is complex and this sign problem becomes a phase problem. The origin of this sign or phase is examined from a number of geometrical and other viewpoints and illustrated by simple examples: the phase problem by the spin (1/2) Heisenberg model, and the sign problem by the attractive SU(N) Hubbard model on a triangular molecule with negative hopping integrals. In the latter case, widely studied in the Jahn-Teller literature, the sign is due neither to fermions nor spin, but to frustration. This system is used to illustrate a number of suggested interpretations of the sign problem.
In this thesis, I use a quantum Monte Carlo (QMC) method based on the stochastic series expansion (SSE) to study transport, quantum phase diagrams, and dynamics of several one- and quasi-one-dimensional spin systems, including spin-1/2 and spin-1 chains, a frustrated four-leg spin tube, and N-leg spin-1/2 ladders. I establish a diffusive channel for the finite temperature spin transport of the spin-1/2 antiferromagnetic Heisenberg chain. Extensive finite size studies are detailed, allowing to extract the diffusion kernel in the thermodynamic limit. The diffusion is found to exhibit no anomalous exponent, moreover it is shown to survive for finite values of exchange anisotropy and magnetic fields, however with a strong suppression away from the isotropic point and for increasing magnetic fields. In the case of the spin-1 chain, I obtain the quantum phase diagram and spin dynamics in the combined presence of finite single-ion anisotropies and external magnetic fields. I also prove the existence of magnon modes and multi-particle continua and determine their evolutions as a function of anisotropies and magnetic fields. Moreover, I calculate the magnetic field dependence of the nuclear magnetic relaxation as a function of the single ion anisotropy. For the quasi-one-dimensional spin system of a frustrated four-leg tube, I investigate the regime of large rung coupling, where QMC is applicable. I evaluate the uniform spin susceptibility and the dynamic structure factor and show that in this regime the spin-tube exhibts a crossover from a gapped effective antiferromagnetic spin-2 chain behavior with massive magnons to a gapless Luttinger liquid phase with spinon excitations. Finally, I investigate the spin dynamics of N-leg spin-1/2 ladders with spacially isotropic interactions for N=1,..,5. I obtain the low-lying excitations of the spin ladders as a function of the number of legs and show that they exhibt an even-odd oscillation which is consistent with Haldane's conjecture and other known results for the thermodynamics of spin ladders.
We review various ways to express the partition function of the single-band Hubard model as a path integral. The emphasis is made on the derivation of the action in the integrand of the path integral and the results obtained from this approach are discussed only briefly. Since the single-band Hubbard model is a pure fermionic model on the lattice and its Hamiltonian is a polynomial in creation and annihilation fermionic operators, with the help of the fermionic coherent states of holomorphic representions the partition function of the model can be straightforwardly expressed as a path integral with respect to Grassmann fields. Then the question arises how to calculate this path integral specially in the most interesting case of the strong coupling, which is described by the term quartic in the fermion fields. The standard way to get rid of this quartic term is to use so-called Hubbard-Stratonovich transformation by introducing auxilary boson field (this procedure is usually called bosonization). Since there are several equivalent forms of the interaction term in the single-band Hubbard model the bosonization procedure is not unique. In the first section of the article we discuss two types of bosoniztions one with two real scalar fields (charge and spin fields) and another with a real vector field. The second section is devoted to a spin-rotation-invariant bosonization procedure invented by Schulz and Weng, Ting and Lee, where an arbitary quantization axis is introduced at each point in space and time and an integration over all possible orientations enters the path integral measure. Spin rotation invariant formulation is necessary to study e.g. spiral magnetization. In the third section we present the contstruction of a path integral for the partition function when the single-band Hubbard Hamiltonian is written in terms of the Hubbard operators, which are elements of the doubly graded semi-simple algebra. In this case one should use generalized coherent states for superalgebra in order to get a path integral formulation. We follow the procedure proposed recently by T ngler and Kopp, to investigate the case of strong coupling.
An introduction is given to the methodology of the path-integral formulation of quantum Monte Carlo simulation. This allows us to reformulate a d −dimensional partition function of a quantum mechanical system into the form of a d +1−dimensional path-integral written in classical mechanical variables. In addition, the quantum mechanical partition function is shown to be related by a process of analytical continuation to the time-development operator of the quantum mechanical system. The evaluation of the path-integral representation is discussed for the Maxwell-Boltzmann ideal gas, a chain of atoms, and liquid 4 He . The properties of liquid Helium I and Helium II are discussed in terms of the two-fluid model, the Ginzburg theory, and recent quantum Monte Carlo Simulation studies. The properties of liquid 3 He are discussed in the context of recent analytical p-wave pairing theories which are compared to the theory of s-wave pairing in the theory of superconducting electron.
We describe the computational ingredients for an approach to treat interacting fermion systems with pairing fields, based on path integrals in the space of Hartree-Fock-Bogoliubov (HFB) wave functions. The path integrals can be evaluated by Monte Carlo, via random walks of HFB wave functions whose orbitals evolve stochastically. The approach combines the advantage of HFB theory in paired fermion systems and many-body quantum Monte Carlo techniques. The properties of HFB states, written in the form of either product states or Thouless states, are discussed. The states preserve forms when propagated by generalized one-body operators. They can be stabilized for numerical iteration. Overlaps and one-body Green's functions between two such states can be computed. A constrained-path or phaseless approximation can be applied to the random walks of the HFB states if a sign problem or phase problem is present. The method is illustrated with an exact numerical projection in the Kitaev model, and in the Hubbard model with attractive interaction under an external pairing field.