Session Y24: Quantum Many-Particle Systems: DMFT & DMRG

A New Lanczos-Based Low Rank Algorithm for Inhomogeneous Dynamical Mean-Field Theory

Inhomogeneous DMFT is used to approximately solve models of ultracold atoms in optical lattices. The intensive part of the IDMFT algorithm is the solution of the Dyson equation for the local Green's function, which involves the computation of the diagonal of the inverse of a sparse matrix. Our new algorithm for finding the diagonal of the inverse of a large sparse matrix is based on domain decomposition into interior and interface points. Since the number of interface points is much less than the interior points, it is a low-rank matrix. Using this matrix allows for a much smaller number of Lanczos steps to obtain the exact solution of the diagonal of the inverse and hence reduces the need for as many re-orthogonalization steps in Lanczos. We show that the problem of finding the diagonal of the inverse is transformed into a naturally parallel GMRES solver (based on the domain decomposition) solved at each of the Lanczos iterations. We successfully implemented a coarray fortran (CAF) program code of this new algorithm for the 2D Fermionic-Bosonic Falicov-Kimball Hamiltonian (mixture of light and heavy atoms). Results of parallel performance and advantages of using a CAF implementation are discussed, in terms of a 3D implementation which is planned for the Hubbard model.   

Continuum Numerical Renormalization Group

The numerical renormalization group (NRG) has emerged as one of the most powerful techniques for calculating emergent renormalized low-temperature properties of strongly correlated systems. When applied as an impurity solver within the dynamical mean-field theory (DMFT), it allows one to directly find spectral functions and how they evolve with temperature. In spite of this success, the NRG method has a number of well-known shortcomings. It fails to properly produce the Fermi liquid state down to T=0 in DMFT and hence it does not properly calculate transport, and it produces only semiquantitative features of the spectral functions. We believe, this is mainly due to the fact that the spectral representation involves a discrete set of delta function peaks at logarithmically discretized frequency intervals which are broadened to the continuum. The broadening parameters are chosen in an {\it ad-hoc} basis. Here we formulate this problem as a discrete degree of freedom embedded in a continuum, which involves coupling the original semi-infinite NRG chain to another semi-infinite chain, arising from the neglected continuum degrees of freedom. This residual coupling can be solved perturbatively and provides an {\it ab initio} approach to constructing spectral functions from the NRG.   

Nonequilibrium dynamical mean-field study of correlated electron systems driven by a mono-cycle pulse

A few-cycle pulse can be generated and has been widely used in recent ultrafast optical experiments, but its potential application to correlated electron systems has remained unexplored in contrast to many-cycle pulses. What is characteristic of the few-cycle pulse is to induce a shift of electron's momentum dynamically. To reveal its effect on the electronic properties of the system, we study the single-band Hubbard model driven by half-cycle and mono-cycle pulses using the nonequilibrium dynamical mean-field theory. As an impurity solver, we employ the continuous-time quantum Monte Carlo method and the iterative perturbation theory. We show that when the momentum shift is nearly $\pi$ (half of the Brillouin zone) the shifted population relaxes to a negative-temperature state, where the electron-electron interaction is effectively switched from repulsive to attractive. The shift is found to deviate from the dynamical phase $\phi=\int F(t)dt$ due to electron correlation effects, which suggests that one can generate the repulsion-to-attraction transition by a mono-cycle pulse with $\phi=0$.   

PIMC study of spin-polarized 1D trapped fermions with strong attractive contact interaction

Spin imbalance in a trapped one-dimensional gas of ${}^6$Li atoms ($F=1/2$) is studied with continuous-space path-integral Monte Carlo simulation. This follows closely the experiment of Liao et al. [1], which aims to confirm the existence of the FFLO pairing predicted from the Bethe ansatz [2,3] and DMRG [4,5]. Algorithmic improvements [6] to the configuration-space sampling efficiency of a previous work [7] is made in order to explore the conditions where the attractive contact interaction between unlike-spin atoms can be stronger than in the previously accessible. Signatures of FFLO pairing is looked for in the pair momentum distribution.\\ \\ $\left[1\right]$ Y.-A. Liao et al., Nature \textbf{467}, 567 (2010).\\ $\left[2\right]$ X.-W. Guan et al., Phys. Rev. B \textbf{76}, 085120 (2007).\\ $\left[3\right]$ E. Zhao et al., Phys. Rev. Lett. \textbf{103}, 140404 (2009).\\ $\left[4\right]$ M. Rizzi et al., Phys. Rev. B \textbf{77}, 245105 (2008).\\ $\left[5\right]$ F. Heidrich-Meisner et al., Phys. Rev. A \textbf{81}, 023629 (2010).\\ $\left[6\right]$ M. Boninsegni et al. Phys. Rev. E, \textbf{74}, 036701, (2006).\\ $\left[7\right]$ M. Casula et al. Phys. Rev. A, \textbf{78}, 033607, (2008).   

Stability of Topological Quantum Phases at Zero Temperature

We prove stability of the spectral gap for gapped, frustration-free Hamiltonians under general, quasi-local perturbations. We present a necessary and sufficient condition for stability, which we call {\it Local Topological Quantum Order} and show that this condition implies an area law for the entanglement entropy of the groundstate subspace. This result extends previous work by Bravyi \textit{et al}, on the stability of topological quantum order for the groundstate subspace of Hamiltonians composed of commuting projections with a common zero-energy subspace. Moreover, our result implies that zero-temperature topological order is robust against quasi-local perturbations, for all topologically ordered subspaces that correspond to the groundstate space of a gapped, frustration-free Hamiltonian. Finally, even in the absence of topological order, we show that symmetry-protected sectors are also stable against perturbations respecting the same symmetries.   

Recurrence relations and time evolution in the relativistic electron gas at long wavelengths

Some years ago M. Howard Lee developed the recurrence relations method to solve the Heisenberg motion equation in an exact way. The relaxation and memory functions, as other linear-response quantities, e.g., the density-density response function and the dynamic structure factor, were obtained for the two- and three-dimensional non-relativistic electronic systems at long wavelengths. In this work we study the time- and frequency-dependent behavior of the relativistic electron gas. As some applications, one can cite graphene, in two dimensions, and dwarf stars, in the three-dimensional case, since both systems have a relativistic electron gas in their compositions.   

Combining Tensor Networks with Monte Carlo: Applications to the MERA

Our recent understanding of the entanglement properties of ground states of many-body quantum systems has led to the development of a variety of variational wavefunctions based on tensor networks. The so-called \emph{bond dimension} of the tensor network, $\chi$, sets both the limit to the amount of entanglement allowed in the ansatz as well as the required computational power to contract the tensor network. Because the cost scales very strongly with $\chi$ in higher dimensions, these approaches are currently challenging in 2D and currently unusable in 3D. We present our efforts in combining Monte Carlo techniques with tensor networks to ease the computational bottleneck. Classical Monte Carlo sampling can be used to estimate the contracted value of the network, allowing one to sample expectation values and vary parameters to optimize ground states. In particular, we show a perfect sampling scheme can be efficient for tensor networks which are also unitary quantum circuits. We apply this to the Multi-scale Entanglement Renormalization Ansatz (MERA) in 1D, formally reducing the cost from $O(\chi^9)$ to $O(\chi^5)$ per sample, and demonstrate that we can optimize wavefunctions. We expect the advantages from Monte Carlo sampling will be stronger in 2D and 3D systems.   

Recent Progress In Exactly Solvable Discrete Models for Topological Phases in Two Dimensions

The study of two-dimensional topological phases in condensed matter systems is a frontier in the field of condensed matter theory as well as topological quantum computation. Discrete or lattice models, which are exactly solvble have been proposed by Kitaev and by Levin and Wen, respectively, some years ago. Here we present a summary of recent progress in studying these models and their generalizations. The topics to be covered include 1) Duality between the Kitaev and Levin-Wen models in certain special cases; 2) General procedure for computing ground state degeneracy when the models are put on a topologically non-trivial surface; 3) More detailed study of the properties (exchange and exclusion statistics etc) of topological excitations (e.g. fluxons); 4) General framework for studying constraints of topological invariance on a wide class of discrete models on more general fluctuating graphs; 5) Generalization of these models to general graphs that incorporates more general degrees of freedom. Our approach, though closely related to topological field theory and tensor category theory, could be understood by physicists.   

Time Evolution of Density Matrices Using BBGKY Hierarchy

Our work starts with the BBGKY hierarchy equations which can be straightforwardly derived from the time-dependent Schr\"odinger equation for each $n$-body reduced density matrices ($n$-RDM). The equations of the BBGKY hierarchy in each level, couple an $n$-RDM to the $(n+1)$-RDM. In order to make this set of equations tractable we need to truncate the hierarchy. While people usually truncate the hierarchy at the first level, one can also truncate it at the level of second equation by approximating $3$-RDM in terms of $2$-RDM and $1$-RDM. Regardless of approximations that we choose, the total energy and momentum will be conserved if we solve the first and second equation together. However, we will show that most of the existing approximations are unstable and even diverging in time and ponder the reasons behind it.   

Optimizing Matrix- and Tensor-Product Algorithms for Momentum-Space Hamiltonians using Quantum Entropy

Momentum-space formulations of local models such as the Hubbard model are hard to treat using matrix- and tensor-product-based algorithms because they contain contain non-local interactions. Quantum entropy-based measures such as the single-site and block entropies and the mutual information can be used to map the entanglement structure in order to gain physical information and to optimize algorithms. In this contribution, we will discuss the optimization of density-matrix-renormalization-group and tree-tensor-network algorithms and their application to the two-dimensional Hubbard model.   

Topological liquid nucleation induced by vortex-vortex interactions in Kitaev's honeycomb model

We provide a microscopic understanding of the nucleation of topological quantum liquids for interacting non-Abelian anyons by making an explicit connection between the microscopics of the pairwise interactions - typically showing oscillations in sign, but decaying exponentially with distance - and the nature of the collective many-anyon state. We investigate this issue in the context of Kitaev's honeycomb lattice model, where non-Abelian vortex excitations can be arranged on superlattices. Depending on microscopic parameters such as the vortex-spacing, we observe the nucleation of several distinct Abelian topological phases. By reformulating the collective behavior of the interacting vortex superlattice in terms of an effective lattice model of tunneling Majorana fermion zero modes, we show that the pairwise interactions fully determine the phase diagram of the nucleated phases. We find that due to the oscillations longer-range interactions beyond nearest neighbor can influence the nature of the collective state and thus need to be included for a comprehensive microscopic picture. Correspondind results should hold for vortices forming an Abrikosov lattice in a p-wave superconductor or quasiholes forming a Wigner crystal in non-Abelian quantum Hall states.   

Optimizing the Hartree-Fock orbitals by the DMRG

We have proposed a density matrix renormalization group (DMRG) scheme to optimize the one-electron basis states of molecules. It improves significantly the accuracy and efficiency of the DMRG in the study of quantum chemistry or other many-fermion system with nonlocal interactions. For a water molecule, we find that the ground state energy obtained by the DMRG with only 61 optimized orbitals already reaches the accuracy of best quantum Monte Carlo calculation with 92 orbitals.   

Determination of Boundary Scattering, Intermagnon Scattering, and the Haldane Gap in Heisenberg Chains

Low-lying magnon dispersion in a $S=1$ Heisenberg antiferromagnetic (AF) chain with boundary $S/2$ spins coupling antiferromagnetically ($J_{\rm end} > 0$) is analyzed by use of the non-Abelian DMRG method. The Haldane gap $\Delta$, the magnon velocity $v$, the inter-magnon scattering length $a$, and the scattering length $a_{\rm b}$ of the boundary coupling are evaluated. The length $a_{\rm b}$, which represents the contribution of boundary effects, depends on $J_{\rm end}$ drastically, while $\Delta$, $v$, and $a$ are constant irrespective of $J_{\rm end}$. Our method estimates the gap of the $S=2$ AF chain as $\Delta = 0.0891623(9)$ using a chain length up to 2048, which is longer than the correlation length.   

Fractional exclusion statistics: the paradigm to describe interacting particle systems

The thermodynamics and statistical mechanics calculations for systems of interacting particles represent in general a difficult task. Even in relatively simple cases, like systems described in the Fermi liquid theory or in the Hartree or Hartree-Fock approximations, the dependence of the quasiparticle energies on the population of all the quasiparticle energy levels makes it impossible to apply the standard formalism. This is because either the sum of quasiparticle energies is different from the total energy of the system or the typical Bose and Fermi populations do not maximize the partition function. The solution to this problem is provided by the application of the fractional exclusion statistics. In this presentation I will compare the standard treatment of systems of interacting particles, given in terms of the Bose or Fermi populations of the quasiparticle energy levels, with a method based on the fractional exclusion statistics. This method is the only paradigm for describing rigorously the interacting particle systems in terms of quasiparticles.