880 research outputs found
Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems
In this paper we explore the practical use of the corner transfer matrix and
its higher-dimensional generalization, the corner tensor, to develop tensor
network algorithms for the classical simulation of quantum lattice systems of
infinite size. This exploration is done mainly in one and two spatial
dimensions (1d and 2d). We describe a number of numerical algorithms based on
corner matri- ces and tensors to approximate different ground state properties
of these systems. The proposed methods make also use of matrix product
operators and projected entangled pair operators, and naturally preserve
spatial symmetries of the system such as translation invariance. In order to
assess the validity of our algorithms, we provide preliminary benchmarking
calculations for the spin-1/2 quantum Ising model in a transverse field in both
1d and 2d. Our methods are a plausible alternative to other well-established
tensor network approaches such as iDMRG and iTEBD in 1d, and iPEPS and TERG in
2d. The computational complexity of the proposed algorithms is also considered
and, in 2d, important differences are found depending on the chosen simulation
scheme. We also discuss further possibilities, such as 3d quantum lattice
systems, periodic boundary conditions, and real time evolution. This discussion
leads us to reinterpret the standard iTEBD and iPEPS algorithms in terms of
corner transfer matrices and corner tensors. Our paper also offers a
perspective on many properties of the corner transfer matrix and its
higher-dimensional generalizations in the light of novel tensor network
methods.Comment: 25 pages, 32 figures, 2 tables. Revised version. Technical details on
some of the algorithms have been moved to appendices. To appear in PR
Functional renormalization group approach to correlated fermion systems
Numerous correlated electron systems exhibit a strongly scale-dependent
behavior. Upon lowering the energy scale, collective phenomena, bound states,
and new effective degrees of freedom emerge. Typical examples include (i)
competing magnetic, charge, and pairing instabilities in two-dimensional
electron systems, (ii) the interplay of electronic excitations and order
parameter fluctuations near thermal and quantum phase transitions in metals,
(iii) correlation effects such as Luttinger liquid behavior and the Kondo
effect showing up in linear and non-equilibrium transport through quantum wires
and quantum dots. The functional renormalization group is a flexible and
unbiased tool for dealing with such scale-dependent behavior. Its starting
point is an exact functional flow equation, which yields the gradual evolution
from a microscopic model action to the final effective action as a function of
a continuously decreasing energy scale. Expanding in powers of the fields one
obtains an exact hierarchy of flow equations for vertex functions. Truncations
of this hierarchy have led to powerful new approximation schemes. This review
is a comprehensive introduction to the functional renormalization group method
for interacting Fermi systems. We present a self-contained derivation of the
exact flow equations and describe frequently used truncation schemes. Reviewing
selected applications we then show how approximations based on the functional
renormalization group can be fruitfully used to improve our understanding of
correlated fermion systems.Comment: Review article, final version, 59 pages, 28 figure
Symmetry improvement of 3PI effective actions for O(N) scalar field theory
[Abridged] n-Particle Irreducible Effective Actions (PIEA) are a powerful
tool for extracting non-perturbative and non-equilibrium physics from quantum
field theories. Unfortunately, practical truncations of PIEA can
unphysically violate symmetries. Pilaftsis and Teresi (PT) addressed this by
introducing a "symmetry improvement" scheme in the context of the 2PIEA for an
O(2) scalar theory, ensuring that the Goldstone boson is massless in the broken
symmetry phase [A. Pilaftsis and D. Teresi, Nuc.Phys. B 874, 2 (2013), pp.
594--619]. We extend this by introducing a symmetry improved 3PIEA for O(N)
theories, for which the basic variables are the 1-, 2- and 3-point correlation
functions. This requires the imposition of a Ward identity involving the
3-point function. The method leads to an infinity of physically distinct
schemes, though an analogue of d'Alembert's principle is used to single out a
unique scheme. The standard equivalence hierarchy of PIEA no longer holds
with symmetry improvement and we investigate the difference between the
symmetry improved 3PIEA and 2PIEA. We present renormalized equations of motion
and counter-terms for 2 and 3 loop truncations of the effective action, leaving
their numerical solution to future work. We solve the Hartree-Fock
approximation and find that our method achieves a middle ground between the
unimproved 2PIEA and PT methods. The phase transition predicted by our method
is weakly first order and the Goldstone theorem is satisfied. We also show
that, in contrast to PT, the symmetry improved 3PIEA at 2 loops does not
predict the correct Higgs decay rate, but does at 3 loops. These results
suggest that symmetry improvement should not be applied to PIEA truncated to
loops. We also show that symmetry improvement is compatible with the
Coleman-Mermin-Wagner theorem, a check on the consistency of the formalism.Comment: 27 pages, 15 figures, 2 supplemental Mathematica notebooks. REVTeX
4.1 with amsmath. Updated with minor corrections. Accepted for publication in
Phys. Rev.
Rotating Bose-Einstein condensates: Closing the gap between exact and mean-field solutions
When a Bose-Einstein condensed cloud of atoms is given some angular momentum,
it forms vortices arranged in structures with a discrete rotational symmetry.
For these vortex states, the Hilbert space of the exact solution separates into
a "primary" space related to the mean-field Gross-Pitaevskii solution and a
"complementary" space including the corrections beyond mean-field. Considering
a weakly-interacting Bose-Einstein condensate of harmonically-trapped atoms, we
demonstrate how this separation can be used to close the conceptual gap between
exact solutions for systems with only a few atoms and the thermodynamic limit
for which the mean-field is the correct leading-order approximation. Although
we illustrate this approach for the case of weak interactions, it is expected
to be more generally valid.Comment: 8 pages, 5 figure
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