34 research outputs found
Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems
In these lecture notes, we present a pedagogical review of a number of
related {\it numerically exact} approaches to quantum many-body problems. In
particular, we focus on methods based on the exact diagonalization of the
Hamiltonian matrix and on methods extending exact diagonalization using
renormalization group ideas, i.e., Wilson's Numerical Renormalization Group
(NRG) and White's Density Matrix Renormalization Group (DMRG). These methods
are standard tools for the investigation of a variety of interacting quantum
systems, especially low-dimensional quantum lattice models. We also survey
extensions to the methods to calculate properties such as dynamical quantities
and behavior at finite temperature, and discuss generalizations of the DMRG
method to a wider variety of systems, such as classical models and quantum
chemical problems. Finally, we briefly review some recent developments for
obtaining a more general formulation of the DMRG in the context of matrix
product states as well as recent progress in calculating the time evolution of
quantum systems using the DMRG and the relationship of the foundations of the
method with quantum information theory.Comment: 51 pages; lecture notes on numerically exact methods. Pedagogical
review appearing in the proceedings of the "IX. Training Course in the
Physics of Correlated Electron Systems and High-Tc Superconductors", Vietri
sul Mare (Salerno, Italy, October 2004
Spatial noise correlations of a chain of ultracold fermions - A numerical study
We present a numerical study of noise correlations, i.e., density-density
correlations in momentum space, in the extended fermionic Hubbard model in one
dimension. In experiments with ultracold atoms, these noise correlations can be
extracted from time-of-flight images of the expanding cloud. Using the
density-matrix renormalization group method to investigate the Hubbard model at
various fillings and interactions, we confirm that the shot noise contains full
information on the correlations present in the system. We point out the
importance of the sum rules fulfilled by the noise correlations and show that
they yield nonsingular structures beyond the predictions of bosonization
approaches. Noise correlations can thus serve as a universal probe of order and
can be used to characterize the many-body states of cold atoms in optical
lattices.Comment: 12 pages, 7 figure
Time evolution of one-dimensional Quantum Many Body Systems
The level of current understanding of the physics of time-dependent strongly
correlated quantum systems is far from complete, principally due to the lack of
effective controlled approaches. Recently, there has been progress in the
development of approaches for one-dimensional systems. We describe recent
developments in the construction of numerical schemes for general
(one-dimensional) Hamiltonians: in particular, schemes based on exact
diagonalization techniques and on the density matrix renormalization group
method (DMRG). We present preliminary results for spinless fermions with
nearest-neighbor-interaction and investigate their accuracy by comparing with
exact results.Comment: Contribution for the conference proceedings of the "IX. Training
Course in the Physics of Correlated Electron Systems and High-Tc
Superconductors" held in Vietri sul Mare (Salerno, Italy) in October 200
Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks
We present a tree-tensor-network-based method to study strongly correlated
systems with nonlocal interactions in higher dimensions. Although the
momentum-space and quantum-chemistry versions of the density matrix
renormalization group (DMRG) method have long been applied to such systems, the
spatial topology of DMRG-based methods allows efficient optimizations to be
carried out with respect to one spatial dimension only. Extending the
matrix-product-state picture, we formulate a more general approach by allowing
the local sites to be coupled to more than two neighboring auxiliary subspaces.
Following Shi. et. al. [Phys. Rev. A, 74, 022320 (2006)], we treat a tree-like
network ansatz with arbitrary coordination number z, where the z=2 case
corresponds to the one-dimensional scheme. For this ansatz, the long-range
correlation deviates from the mean-field value polynomially with distance, in
contrast to the matrix-product ansatz, which deviates exponentially. The
computational cost of the tree-tensor-network method is significantly smaller
than that of previous DMRG-based attempts, which renormalize several blocks
into a single block. In addition, we investigate the effect of unitary
transformations on the local basis states and present a method for optimizing
such transformations. For the 1-d interacting spinless fermion model, the
optimized transformation interpolates smoothly between real space and momentum
space. Calculations carried out on small quantum chemical systems support our
approach
Time evolution of correlations in strongly interacting fermions after a quantum quench
Using the adaptive time-dependent density matrix renormalization group, we
study the time evolution of density correlations of interacting spinless
fermions on a one-dimensional lattice after a sudden change in the interaction
strength. Over a broad range of model parameters, the correlation function
exhibits a characteristic light-cone-like time evolution representative of a
ballistic transport of information. Such behavior is observed both when
quenching an insulator into the metallic region and also when quenching within
the insulating region. However, when a metallic state beyond the quantum
critical point is quenched deep into the insulating regime, no indication for
ballistic transport is observed. Instead, stable domain walls in the density
correlations emerge during the time evolution, consistent with the predictions
of the Kibble-Zurek mechanism.Comment: Published version; minor changes, references adde
Finite Projected Entangled Pair States for the Hubbard model
We adapt and optimize the projected-pair-entangled-state (PEPS) algorithm on
finite lattices (fPEPS) for two-dimensional Hubbard models and apply the
algorithm to the Hubbard model with nearest-neighbor hopping on a square
lattice. In particular, we formulate the PEPS algorithm using projected
entangled pair operators, incorporate SU(2) symmetry in all tensor indices, and
optimize the PEPS using both iterative-diagonalization-based local bond
optimization and gradient-based optimization of the PEPS. We discuss the
performance and convergence of the algorithm for the Hubbard model on lattice
sizes of up to 8x8 for PEPS states with U(1) symmetric bond dimensions of up to
D = 8 and SU(2) symmetric bond dimensions of up to D = 6. Finally, we comment
on the relative and overall efficiency of schemes for optimizing fPEPS
Fourth-Order Perturbation Theory for the Half-Filled Hubbard Model in Infinite Dimensions
We calculate the zero-temperature self-energy to fourth-order perturbation
theory in the Hubbard interaction for the half-filled Hubbard model in
infinite dimensions. For the Bethe lattice with bare bandwidth , we compare
our perturbative results for the self-energy, the single-particle density of
states, and the momentum distribution to those from approximate analytical and
numerical studies of the model. Results for the density of states from
perturbation theory at agree very well with those from the Dynamical
Mean-Field Theory treated with the Fixed-Energy Exact Diagonalization and with
the Dynamical Density-Matrix Renormalization Group. In contrast, our results
reveal the limited resolution of the Numerical Renormalization Group approach
in treating the Hubbard bands. The momentum distributions from all approximate
studies of the model are very similar in the regime where perturbation theory
is applicable, . Iterated Perturbation Theory overestimates the
quasiparticle weight above such moderate interaction strengths.Comment: 19 pages, 17 figures, submitted to EPJ
Analytical and Numerical Treatment of the Mott--Hubbard Insulator in Infinite Dimensions
We calculate the density of states in the half-filled Hubbard model on a
Bethe lattice with infinite connectivity. Based on our analytical results to
second order in , we propose a new `Fixed-Energy Exact Diagonalization'
scheme for the numerical study of the Dynamical Mean-Field Theory. Corroborated
by results from the Random Dispersion Approximation, we find that the gap opens
at . Moreover, the density of states near the gap
increases algebraically as a function of frequency with an exponent
in the insulating phase. We critically examine other analytical
and numerical approaches and specify their merits and limitations when applied
to the Mott--Hubbard insulator.Comment: 22 pages, 16 figures; minor changes (one reference added, included
comparison with Falicov-Kimball model
Spectral Density of the Two-Impurity Anderson Model
We investigate static and dynamical ground-state properties of the
two-impurity Anderson model at half filling in the limit of vanishing impurity
separation using the dynamical density-matrix renormalization group method. In
the weak-coupling regime, we find a quantum phase transition as function of
inter-impurity hopping driven by the charge degrees of freedom. For large
values of the local Coulomb repulsion, the transition is driven instead by a
competition between local and non-local magnetic correlations. We find evidence
that, in contrast to the usual phenomenological picture, it seems to be the
bare effective exchange interactions which trigger the observed transition.Comment: 18 pages, 6 figures, submitted to J. Phys.:Condens. Matte
Quantum criticality of dipolar spin chains
We show that a chain of Heisenberg spins interacting with long-range dipolar
forces in a magnetic field h perpendicular to the chain exhibits a quantum
critical point belonging to the two-dimensional Ising universality class.
Within linear spin-wave theory the magnon dispersion for small momenta k is
[Delta^2 + v_k^2 k^2]^{1/2}, where Delta^2 \propto |h - h_c| and v_k^2 \propto
|ln k|. For fields close to h_c linear spin-wave theory breaks down and we
investigate the system using density-matrix and functional renormalization
group methods. The Ginzburg regime where non-Gaussian fluctuations are
important is found to be rather narrow on the ordered side of the transition,
and very broad on the disordered side.Comment: 6 pages, 5 figure