Session X13: Quantum Monte Carlo Methods and Strongly Correlated Systems

Variational approach for 1D antiferromagnetic Heisenberg chain with matrix-product states

In order to explore the practical applicability of variational Monte Carlo simulations based on matrix-product states (MPS) [1], we present two implementations for the one-dimensional antiferromagnetic Heisenberg model with periodic boundary conditions [2]. We compare the convergence properties of two different schemes, which use either two sets of matrices corresponding to the two sublattices, or a 2-spin block representation. It is found that the use of symmetries considerably speeds up the convergence with the matrix size D. We also present an efficient ``cooling'' schedule for the stochastic method used to optimize the matrices, which significantly reduces the computational effort. Finally, we will discuss application of the scheme to n-leg ladders with periodic boundary condition. \newline [1] A. W. Sandvik and G. Vidal, arXiv:0708.2232. \newline [2] Y.-J. Kao, L. Wang, and A. W. Sandvik (unpublished)   

Excited states from variational Monte Carlo simulations with matrix-product states

We report a further development [1] of a recently proposed variational Monte Carlo method for matrix-product states (MPS) [2]. Using the frustrated $J_1-J_2$ Heisenberg chain as a test case, we show how the matrices can be optimized not just for the ground state, but also, simultaneously, for the lowest states in several different lattice and spin symmetry sectors. This is useful in, e.g., studies of quantum phase transitions associated with crossings of excited-state energies. \newline [1] Y.-J. Kao, L. Wang, and A. W. Sandvik (unpublished) \newline [2] A. W. Sandvik and G. Vidal, arXiv:0708.2232.   

Scale-renormalized matrix-product states for correlated quantum systems

A generalization of matrix product states (MPS) is introduced which is suitable for describing interacting quantum systems in two and three dimensions. These {\it scale-renormalized matrix-product states} (SR-MPS) are based on a course-graining of the lattice in which the blocks at each level are associated with matrix products that are further transformed (scale renormalized) with other matrices before they are assembled to form blocks at the next level. Using variational Monte Carlo simulations of the two-dimensional transverse-field Ising model as a test, it is shown that the SR-MPS converge much more rapidly with the matrix size than a standard MPS. It is also shown that the use of lattice-symmetries speeds up the convergence very significantly.   

DMRG applied to quantum impurity models

Quantum impurity models are analyzed routinely and reliably at very low energies using the Numerical Renormalization Group (NRG). Its great benefit of energy scale separation, however, comes at the cost of limited resolution at finite energy. By realizing that the NRG shares the same algebraic structure as the density matrix renormalization group (DMRG) given in terms of matrix product states, several strict NRG constraints such as its rigid discretization scheme in energy space can be relaxed due to the variational principle of DMRG. Our recent work in that respect will be discussed.   

Ameliorating the sign problem for frustrated magnets using plaquette grouping

Frustrated quantum magnets are not amenable to simulation using conventional quantum Monte Carlo because of the infamous sign problem. In the overcomplete basis of singlet product states, updates have a many-to-one property that allows for grouping of updates around plaquettes in such a way that the negative sampling weights are almost entirely eliminated. Results for the J1-J2 quantun Heisenberg model on the square lattice are discussed.   

Bold Diagrammatic Monte Carlo: Generic Technique for Polaron Problems (and More?)

We introduce a Monte Carlo scheme for sampling bold-line diagrammatic series specifying an unknown function in terms of itself. The range of convergence of this bold(-line) diagrammatic Monte Carlo (BMC) is significantly broader than that of a simple iterative scheme for solving integral equations. With BMC technique, a moderate ``sign problem" turns out to be an advantage in terms of the convergence of the process. As an illustrative application, we solve the problem of fermipolaron (one spin-down particle interacting with the spin-up fermionic sea). The problem solved is prototypical for all polaron problems, and, probably, for many-particle systems as well.   

Interfacing Determinant Quantum Monte Carlo and Density Functional Theory

Over the last decade many body theory and electronic structure calculations have come together within the ``LDA+DMFT" approach in which dynamical mean field theory (DMFT) provides a frequency dependent self-energy $\Sigma(\omega)$ for electronic structure calculation within the local density approximation (LDA). Here we describe initial results with a new approach which uses the determinant Quantum Monte Carlo method to supply the self energy. This technique has the advantage of providing a momentum dependent $\Sigma({\bf k},\omega)$. However, the fermion sign problem can limit the ability to access the ground state value of the self energy. We present tests of the approach on a model of cuprate superconductors.   

Spin waves and local magnetizations on the Penrose tiling

The Penrose tiling is a perfectly ordered two dimensional structure with fivefold symmetry and scale invariance under site decimation. Quantum spin models on such a system can be expected to differ significantly from more conventional structures as a result of its special symmetries. We consider a Heisenberg antiferromagnet on the Penrose tiling, a quasiperiodic system having an inhomogeneous Neel-ordered ground state. Spin wave energies and wavefunctions are studied in the linear spin wave approximation. A linear dispersion law is found at low energies, as in other bipartite antiferromagnets, with an effective spin wave velocity lower than in the square lattice. Spatial properties of eigenmodes are characterized in several different ways. At low energies, eigenstates are relatively extended, and show multifractal scaling. At higher energies, states are more localized, and, depending on the energy, confined to sites of a specified coordination number. The ground state energy of this antiferromagnet, and local staggered magnetizations are calculated. Perpendicular space projections are shown, showing the underlying simplicity of this ``complex'' ground state. A simple analytical model, the two-tier Heisenberg star, is presented to explain the staggered magnetization distribution in this antiferromagnetic system.   

Loop Algorithm for the SU(N) Heisenberg Model

The SU(N) generalization of the Heisenberg model is studied with a new loop algorithms with non-binary loop variables.[1] The split-spin representation is used for high-dimensional representations. While we have confirmed our previous result[3] that the ground state switches from the Neel state to the VBS state around N=5 for the fundamental representation, we also find that there is an apparent U(1) symmetry in the VBS state. For higher representation, we have not observed any VBS state, although the disappearance of the Neel order parameter has been detected as we increase N. \ \\ \ \\ {}[1] N. Kawashima and K. Harada, J. Phys. Soc. Jpn. {\bf 73} 1397 (2004).\\ {}[2] N. Kawashima and Y. Tanabe, Phys. Rev. Lett. {\bf 98} 057202 (2007).\\ {}[3] K. Harada, N. Kawashima and M. Troyer, Phys. Rev. Lett. {\bf 90} 117203 (2003).   

Dynamical properties of SSH and breathing type Hamiltonians

Using a QMC method based on exact phonon integration in Fourier space and on loop updates in particle space, we study fermionic systems coupled to dynamical phonons in one dimension. Within this method it is possible to investigate Su-Schrieffer-Heeger (SSH) as well as Holstein type models, with momentum dependent couplings (e.g. breathing phonons) and arbitrary phonon dispersions. We access the dynamical properties of the systems via momentum dependent phonon spectral functions and electron Greens functions. In case of the standard Holstein model, we present precise data for the phonon spectral function in both the metallic Luttinger liquid and the insulating charge density wave phase, for a wide range of phonon frequencies.