273 research outputs found
Topological nature of spinons and holons: Elementary excitations from matrix product states with conserved symmetries
We develop variational matrix product state (MPS) methods with symmetries to
determine dispersion relations of one dimensional quantum lattices as a
function of momentum and preset quantum number. We test our methods on the XXZ
spin chain, the Hubbard model and a non-integrable extended Hubbard model, and
determine the excitation spectra with a precision similar to the one of the
ground state. The formulation in terms of quantum numbers makes the topological
nature of spinons and holons very explicit. In addition, the method also
enables an easy and efficient direct calculation of the necessary magnetic
field or chemical potential required for a certain ground state magnetization
or particle density.Comment: 13 pages, 4 pages appendix, 8 figure
Faster Methods for Contracting Infinite 2D Tensor Networks
We revisit the corner transfer matrix renormalization group (CTMRG) method of
Nishino and Okunishi for contracting two-dimensional (2D) tensor networks and
demonstrate that its performance can be substantially improved by determining
the tensors using an eigenvalue solver as opposed to the power method used in
CTMRG. We also generalize the variational uniform matrix product state (VUMPS)
ansatz for diagonalizing 1D quantum Hamiltonians to the case of 2D transfer
matrices and discuss similarities with the corner methods. These two new
algorithms will be crucial to improving the performance of variational infinite
projected entangled pair state (PEPS) methods.Comment: 20 pages, 5 figures, V. Zauner-Stauber previously also published
under the name V. Zaune
Transfer Matrices and Excitations with Matrix Product States
We investigate the relation between static correlation functions in the
ground state of local quantum many-body Hamiltonians and the dispersion
relations of the corresponding low energy excitations using the formalism of
tensor network states. In particular, we show that the Matrix Product State
Transfer Matrix (MPS-TM) - a central object in the computation of static
correlation functions - provides important information about the location and
magnitude of the minima of the low energy dispersion relation(s) and present
supporting numerical data for one-dimensional lattice and continuum models as
well as two-dimensional lattice models on a cylinder. We elaborate on the
peculiar structure of the MPS-TM's eigenspectrum and give several arguments for
the close relation between the structure of the low energy spectrum of the
system and the form of static correlation functions. Finally, we discuss how
the MPS-TM connects to the exact Quantum Transfer Matrix (QTM) of the model at
zero temperature. We present a renormalization group argument for obtaining
finite bond dimension approximations of MPS, which allows to reinterpret
variational MPS techniques (such as the Density Matrix Renormalization Group)
as an application of Wilson's Numerical Renormalization Group along the virtual
(imaginary time) dimension of the system.Comment: 39 pages (+8 pages appendix), 14 figure
Symmetry Breaking and the Geometry of Reduced Density Matrices
The concept of symmetry breaking and the emergence of corresponding local
order parameters constitute the pillars of modern day many body physics. The
theory of quantum entanglement is currently leading to a paradigm shift in
understanding quantum correlations in many body systems and in this work we
show how symmetry breaking can be understood from this wavefunction centered
point of view. We demonstrate that the existence of symmetry breaking is a
consequence of the geometric structure of the convex set of reduced density
matrices of all possible many body wavefunctions. The surfaces of those convex
bodies exhibit non-analytic behavior in the form of ruled surfaces, which turn
out to be the defining signatures for the emergence of symmetry breaking and of
an associated order parameter.
We illustrate this by plotting the convex sets arising in the context of
three paradigmatic examples of many body systems exhibiting symmetry breaking:
the quantum Ising model in transverse magnetic field, exhibiting a second order
quantum phase transition; the classical Ising model at finite temperature in
two dimensions, which orders below a critical temperature ; and a system
of free bosons at finite temperature in three dimensions, exhibiting the
phenomenon of Bose-Einstein condensation together with an associated order
parameter . Remarkably, these convex sets look all very
much alike. We believe that this wavefunction based way of looking at phase
transitions demystifies the emergence of order parameters and provides a unique
novel tool for studying exotic quantum phenomena.Comment: 5 pages, 3 figures, Appendix with 2 pages, 3 figure
Variational optimization algorithms for uniform matrix product states
We combine the density matrix renormalization group (DMRG) with matrix product state tangent space concepts to construct a variational algorithm for finding ground states of one-dimensional quantum lattices in the thermodynamic limit. A careful comparison of this variational uniform matrix product state algorithm (VUMPS) with infinite density matrix renormalization group (IDMRG) and with infinite time evolving block decimation (ITEBD) reveals substantial gains in convergence speed and precision. We also demonstrate that VUMPS works very efficiently for Hamiltonians with long-range interactions and also for the simulation of two-dimensional models on infinite cylinders. The new algorithm can be conveniently implemented as an extension of an already existing DMRG implementation
Time Evolution within a Comoving Window: Scaling of signal fronts and magnetization plateaus after a local quench in quantum spin chains
We present a modification of Matrix Product State time evolution to simulate
the propagation of signal fronts on infinite one-dimensional systems. We
restrict the calculation to a window moving along with a signal, which by the
Lieb-Robinson bound is contained within a light cone. Signal fronts can be
studied unperturbed and with high precision for much longer times than on
finite systems. Entanglement inside the window is naturally small, greatly
lowering computational effort. We investigate the time evolution of the
transverse field Ising (TFI) model and of the S=1/2 XXZ antiferromagnet in
their symmetry broken phases after several different local quantum quenches.
In both models, we observe distinct magnetization plateaus at the signal
front for very large times, resembling those previously observed for the
particle density of tight binding (TB) fermions. We show that the normalized
difference to the magnetization of the ground state exhibits similar scaling
behaviour as the density of TB fermions. In the XXZ model there is an
additional internal structure of the signal front due to pairing, and wider
plateaus with tight binding scaling exponents for the normalized excess
magnetization. We also observe parameter dependent interaction effects between
individual plateaus, resulting in a slight spatial compression of the plateau
widths.
In the TFI model, we additionally find that for an initial Jordan-Wigner
domain wall state, the complete time evolution of the normalized excess
longitudinal magnetization agrees exactly with the particle density of TB
fermions.Comment: 10 pages with 5 figures. Appendix with 23 pages, 13 figures and 4
tables. Largely extended and improved versio
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