Statistical Physics and Condensed Matter
Flow Equations for Hamiltonians
Abstracts of Papers from the Institute for Theoretical Physics at the
University Heidelberg
Flow-equations are introduced in order to bring Hamiltonians closer
to diagonalization. It is characteristic for these equations that
matrix-elements between degenerate or almost degenerate states
do not decay or decay very slowly. In order to understand
different types of physical systems in this framework it is probably
necessary to classify various types of these degeneracies
and to investigate the corresponding physical behaviour.
In general these equations
generate many-particle interactions. However, for an n-orbital
model the equations for the two-particle interaction are closed
in the limit of large n. Solutions of these equations for a
one-dimensional model are considered. There appear convergency problems,
which are removed, if instead of diagonalization only a
block-diagonalization into blocks with the same number of
quasiparticles is performed.
Back
Using a continuous unitary transformation recently proposed by Wegner
together with an approximation that neglects irrelevant
contributions, we obtain flow equations for Hamiltonians. These flow
equations finally yield a diagonal or almost diagonal Hamiltonian.
As an example we investigate the Anderson Hamiltonian for dilute
magnetic alloys. We study the different fixed points
of the flow equations and the corresponding
relevant, marginal or irrelevant contributions. Our results are
comparable to results obtained with a numerical renormalization group
method, but our approach is considerably simpler.
Back
Using continuous unitary transformations recently introduced by Wegner
we obtain flow equations for the parameters of the spin-boson
Hamiltonian. Interactions not contained in the original Hamiltonian
are generated by this unitary transformation. Within an approximation
that neglects additional interactions quadratic in the bath operators,
we can close the flow equations. Applying this formalism to the case
of Ohmic dissipation at zero temperature,
we calculate the renormalized tunneling frequency.
We find a transition from an untrapped to a trapped state
at the critical coupling constant alpha=1.
We also obtain the static susceptibility via the
equilibrium spin correlation function. Our results are both consistent
with results known from the Kondo problem and those obtained from
mode coupling theories. Using this formalism at finite
temperature, we find a transition from coherent to incoherent tunneling at
T_2\approx T_1, where T_1 is the corssover temperature of the dynamics from
underdamped to overdamped motion known from the NIBA.
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S.K. Kehrein, A. Mielke:
Theory of the Anderson impurity model:
The Schrieffer-Wolff transformation re-examined
Ann. Phys. (NY) 252 (1995) 1
We apply the method of infinitesimal unitary transformations
recently introduced by Wegner
to the Anderson single impurity model. It is demonstrated
that this method provides a good approximation scheme
for all values of the on-site
interaction U, it becomes exact for U=0.
We are able to treat an arbitrary density of states, the
only restriction being that the hybridization should
not be the largest parameter in the system.
Our approach constitutes a consistent framework
to derive various results usually obtained by either
perturbative renormalization in an
expansion in the hybridization
Anderson's "poor man's" scaling approach
or the Schrieffer-Wolff unitary transformation. In contrast to the
Schrieffer-Wolff result we find the correct high-energy
cutoff and avoid singularities in the induced couplings.
An important characteristic of our method as compared to
the "poor man's" scaling approach is that we continuously
decouple modes from the impurity that have a large energy
difference from the impurity orbital energies. In the usual scaling
approach this criterion is provided by the energy difference
from the Fermi surface.
Back
We study the spin-boson model with a sub-Ohmic bath using
infinitesimal unitary transformations. Contrary to some results
reported in the literature we find a zero temperature transition
from an untrapped state for small coupling to a trapped state for
strong coupling.
We obtain an explicit expression for
the renormalized level spacing as a function of the bare
parameters of the system.
Furthermore we show that typical dynamical equilibrium correlation
functions exhibit an algebaric decay at zero temperature.
Back
A recently proposed method of continuous unitary transformations is used
to decouple the interaction between electrons and phonons. The differential
equations for the couplings represent an infinitesimal formulation of a
sequence of Fröhlich transformations. The two approaches are compared.
Our result will turn out to be less singular than Fröhlich's.
Furthermore the interaction between electrons belonging to a Cooper
pair will always be attractive in our approach. Even in the case
where Fröhlich's transformation is not defined (Fröhlich actually
excluded these regions from the transformation), we obtain an elimination
of the electron-phonon interaction. This is due to a sufficiently slow change
of the phonon energies as a function of the flow parameter.
New address of P.L.: Max-Planck-Institut für Kolloid- und
Grenzflächenforschung, Kantstr. 55, 14513 Teltow, Germany
Back
We introduce a new theoretical approach to dissipative quantum
systems. By means of a continuous sequence of infinitesimal
unitary transformations, we decouple the small quantum system
that one is interested in from its thermodynamically large
environment. This yields a trivial final transformed Hamiltonian.
Dissipation enters through the observation that generically
observables "decay" completely under these unitary transformations,
i.e. are completely transformed into other terms. As a nontrivial
example the spin-boson model is discussed in some detail.
For the super-Ohmic bath we obtain a very satisfactory description
of short, intermediate and long time scales at small temperatures.
This can be tested from the generalized Shiba-relation that is
fulfilled within numerical errors.
Back
We study the problem of the phonon--induced electron--electron
interaction in a solid. Starting with a Hamiltonian that contains
an electron--phonon interaction, we perform a similarity renormalization
transformation to calculate an effective Hamiltonian.
Using this transformation singularities due to degeneracies
are avoided explicitely.
The effective interactions are calculated to second order
in the electron--phonon coupling. It is shown that the
effective interaction between two electrons forming a
Cooper pair is attractive in the whole parameter space.
The final result is compared with effective interactions
obtained using other approaches.
Back
A new approach to dissipative quantum systems modelled by
a system plus environment Hamiltonian is presented. Using
a continuous sequence of
infinitesimal unitary transformations the small quantum
system is decoupled from its thermodynamically large
environment. Dissipation enters through the observation
that system observables generically "decay"
completely into a different structure when the Hamiltonian
is transformed into diagonal form. The method is particularly
suited for studying low--temperature properties. This
is demonstrated explicitly for the super-Ohmic spin-boson model.
Back
Applying the method of continuous unitary transformations
to a class of Hubbard models, the derivation of the t/U-expansion
for the strong-coupling case is re-examined.
The flow equations for the coupling parameters of the higher-order effective
interactions can be solved exactly, resulting in
a systematic expansion of the Hamiltonian in powers of t/U,
valid for any lattice in arbitrary dimension
and for general band-filling. The expansion
ensures a correct treatment of the operator products generated by
the transformation, and only involves the explicit
recursive calculation of numerical coefficients.
This scheme provides a unifying framework
to study the strong-coupling expansion for the Hubbard model,
which clarifies and circumvents
several difficulties inherent to earlier approaches. Our results are
compared with those of other methods, and it is shown that the freedom
in the choice of the unitary transformation that eliminates interactions
between different Hubbard bands can affect the effective Hamiltonian only
at order t3/U2 or higher.
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It is shown that one can obtain quantitatively accurate values
for the superconducting critical temperature within a Hamiltonian framework.
This is possible if one uses a renormalized Hamiltonian that contains an
attractive electron-electron interaction and renormalized single
particle energies. It can be obtained by similarity renormalization
or using flow equations for Hamiltonians. We calculate the
critical temperature as a function of the coupling
using the standard BCS-theory. We compare
our results with Eliashberg theory and with experimental data
from various materials. The theoretical results agree with the
experimental data within 10%.
Renormalization theory of Hamiltonians provides a promising way
to investigate electron-phonon interactions in strongly correlated
systems.
Back
Flow equations for Hamiltonians are a novel method for diagonalizing
Hamilton operators. They were applied by one of the authors to a
one-dimensional SU(n)-symmetric fermionic system, solving the occuring
equations to first order of a 1/n-expansion.
In this paper, we generalize the procedure to an arbitrary number of
spatial dimensions. Although the resulting equations cannot be solved
analytically, some information can be extracted about the particle number
near the Fermi surface. The results suggest a nonuniversal behaviour
for d=1 which breaks down in favour of that of a Landau liquid in
any dimension >1.
New address of A.K.: DESY, Notkestr. 85, 22603 Hamburg, Germany
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F. Wegner:
Flow Equations for Hamiltonians.
Proceedings of the Bar-Ilan 1997 Minerva Workshop on Mesoscopics, Fractals,
and Neural Networks, Eilat
Phil. Mag. B 77 (1998) 1249
A recently developed method to diagonalize or block-diagonalize
Hamiltonians is reviewed. As an example it is applied to the elimination
of the electron-phonon-interaction. A discussion of the advantage of
this method is given.
Back
F. Wegner:
Hamiltonian Flow in Condensed Matter Physics.
in M. Grangé et al (eds.), New Non-Perturbative Methods and Quantization
on the Light Cone, Les Houches School 1997,
Editions de Physique/Springer Vol. 8 (1998) 33
A recently developed method to diagonalize or block-diagonalize
Hamiltonians by means of an appropriate continuous unitary transformation
is reviewed. Two applications in condensed matter physics
are given as examples:
(i) the interaction of an n-orbital model of
fermions in the limit of large n is brought to block-diagonal form,
and (ii) the generation of the effective attractive two-electron interaction
due to the elimination of electron-phonon interaction is given.
The advantage of this method in particular in comparison to conventional
perturbation theory is pointed out.
Back
E.L. Gubankova, F. Wegner:
Flow equations for QED in the light front dynamics
hep-th/9710233
The method of flow equations is applied to QED on the light front. Requiring
that the particle number conserving terms in the Hamiltonian are considered to
be diagonal and the other terms off-diagonal an effective Hamiltonian is
obtained which reduces the positronium problem to a two-particle problem,
since the particle number violating contributions are eliminated. No infrared
divergencies appear. The ultraviolet renormalization can be performed
simultaneously.
Back
Continuous unitary transformations can be used to
diagonalize or approximately diagonalize a given
Hamiltonian. In the last four years, this
method has been applied to a variety of models of condensed
matter physics and field theory. With a new
generator for the continuous unitary transformation
proposed in this paper one can avoid some
of the problems of former applications. General properties
of the new generator are derived. It turns
out that the new generator is especially useful for
Hamiltonians with a banded structure.
Two examples, the Lipkin model, and the spin-boson model
are discussed in detail.
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J. Stein:
Flow equations and extended Bogoliubov transformation
for the Heisenberg antiferromagnet
near the classical limit.
Eur. Phys. J. B 5 (1998) 193
(
Postscript-File)
The Heisenberg spin-S quantum
antiferromagnet is studied near the large-spin limit, applying a new
continuous unitary transformation which extends the usual Bogoliubov
transformation to higher order in the 1/S-expansion
of the Hamiltonian. This allows to
diagonalize the bosonic Hamiltonian resulting from the Holstein-Primakoff
representation beyond the conventional spin-wave approximation.
The zero-temperature flow equations derived from the extension of the
Bogoliubov transformation to order 1/S2
for the ground-state energy, the spin-wave velocity, and
the staggered magnetization are solved exactly and yield results
which are in agreement with those obtained by
a perturbative treatment of the magnon interactions.
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M. Ragwitz and F. Wegner:
Flow Equations for Electron-Phonon Interactions:
Phonon Damping.
Eur. Phys. J. B 8 (1999) 9
A recently proposed method of a continuous sequence of unitary
transformations will be used to investigate the dynamics of phonons,
which are coupled to an electronic system.
This transformation decouples the interaction between electrons
and phonons.
Damping of the phonons enters through the observation, that the phonon
creation and annihilation operators decay under this transformation into
a superposition of electronic particle-hole excitations with a pronounced
peak, where these excitations are degenerate in energy with the
renormalized phonon frequency. This procedure allows the determination
of the phonon correlation function and the spectral function.
The width of this function
is proportional to the square of the electron-phonon coupling and
the height of the function scales inversely proportional to the square of the
coupling. The function itself is non-Lorentzian, but apart from these
scales independent of the electron-phonon coupling.
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D. Cremers and A. Mielke:
Flow equations for the Henon-Heiles Hamiltonian.
Physica D 126, 123-135 (1999),
quant-ph/9809086
The Henon-Heiles Hamiltonian was introduced in 1964 as a mathematical
model to describe the chaotic motion of stars in a galaxy.
By canonically transforming the classical Hamiltonian to a
Birkhoff-Gustavson normalform Delos and Swimm obtained a discrete
quantum mechanical energy spectrum. The aim of the present work
is to first quantize the classical Hamiltonian and to then
diagonalize it using different variants of flow equations,
a method of continuous unitary transformations introduced by
Wegner in 1994. The results of the diagonalization via flow equations
are comparable to those obtained by the classical transformation.
In the case of commensurate frequencies the transformation turns out
to be less lengthy. In addition, the dynamics of the quantum
mechanical system are analyzed on the basis of the transformed observables.
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E.L. Gubankova, H.C. Pauli, F.J. Wegner:
Light-cone Hamiltonian flow for positronium, preprint MPI-H-V33-1998.
The technique of Hamiltonian flow equations is applied to the canonical
Hamiltonian of quantum electrodynamics in the front form and 3+1 dimensions.
The aim is to generate a bound state equation in a quantum field theory,
particularly to derive an effective Hamiltonian which is practically solvable
in Fock-spaces with reduced particle number, such that the approach can
ultimately be used to address to the same problem for quantum
chromodynamics.
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H. J. Pirner, B. Friman:
Hamiltonian Flow Equations for the Lipkin model.
Phys. Lett. B434, 231 (1998),
nucl-th/9804039
We derive Hamiltonian flow equations giving the evolution of the Lipkin
Hamiltonian to a diagonal form using continuous unitary transformations. To
close the system of flow equations, we present two different schemes. First we
linearize an operator with three pairs of creation and destruction operators
by reducing it to the z component of the quasi spin. We obtain the well
known RPA-result in the limit of large particle number. In the second scheme
we introduce a new operator which improves the resulting spectrum considerably
especially for few particles.
Back
We derive and solve the Hamiltonian flow equations for a Dirac particle in
an external static potential. The method shows a general procedure for the set
up of continuous unitary transformations to reduce the Hamiltonian to a
quasidiagonal form.
Back
The method of continuous unitary transformations is applied to
obtain the indirect exchange coupling between
local magnetic moments in an electron gas.
The derivation of the exact analytical expression for the resulting
Ruderman-Kittel-Kasuya-Yosida interaction is presented for general
dimensionality. In odd dimensions, the result can be shown explicitly
to exhibit universal 2kF oscillatory behaviour on all
length scales.
Back
J. Stein:
Flow equations and new weak-coupling solution
for the spin-polaron in a quantum antiferromagnet.
Europhys. Lett. 50 (2000) 68
(
Postscript-File)
The t-J model for the doped two-dimensional Heisenberg quantum
antiferromagnet is studied
in the generalized Dyson-Maleev representation, applying a new
continuous unitary transformation which eliminates the coupling of spin
and charge degrees of freedom.
The analytical solutions of the resulting flow equations are derived
in the weak-coupling regime where t/J is small.
This continuous transformation yields a new weak-coupling result for the
dispersion of the spin-polaron, if the elimination
of both the nondiagonal spin-wave contributions and the terms coupling
holes and spin-waves is performed simultaneously.
The associated one-particle ground state is
lower in energy than the corresponding perturbative result, which is
reproduced upon application of subsequent transformations.
Back
The flow equations describing the continuous unitary transformation
which brings the Hamiltonian closer to diagonality are derived and solved
exactly for the Lipkin model in the Holstein-Primakoff boson representation and
for a large particle number N. The transformed Hamiltonian is diagonal
in order 1/N^3, extending known linear transformations to next-higher
orders in the inverse particle number. This approach quite naturally allows
to preserve the tridiagonal structure of the original Lipkin Hamiltonian in
the course of the transformation.
Exact analytical results for the coupling functions and
explicit expressions for the ground-state energy and for the
energy gap to the first excited state in order 1/N^2
are presented and are compared with the accurate numerical values.
Back
F. Wegner:
Flow Equations for Hamiltonians.
Physics Reports 348 (2001) 77.
Proceedings of the RG 2000 in Taxco, Mexico
A recently developed method to diagonalize or block-diagonalize
Hamiltonians by means of an appropriate continuous unitary transformation is
reviewed. The main aspects will be discussed: (i) Elimination of off-diagonal
matrix elements at different energy scales and (ii) problems and advantageous
of this method. Two applications in condensed matter physics are given as
examples: The interaction of an n-orbital model of fermions in the limit of
large n is brought to block-diagonal form, and the generation of the effective
attractive two-electron interaction due to the elimination of electron-phonon
interaction is given. The advantage of this method in particular in comparison
with conventional perturbation theory is pointed out.
Back
F. Wegner:
Flow Equations for Hamiltonians.
Nucl. Phys. B (Proc. Suppl.) 90 (2000) 141
A method to diagonalize or block-diagonalize Hamiltonians by means of
an appropriate continuous unitary transformation is reviewed.
Back
F. Wegner:
Flow Equations for Hamiltonians.
Advances in Solid State Physics 40 (2000) 113
A method to diagonalize or block-diagonalize
Hamiltonians by means of an appropriate continuous unitary transformation
is reviewed. Main advantages among others are:
(i) In perturbation theory one obtains new results for effective interactions
which are less singular than those obtained by conventional perturbation
theory, eg. for the effective pair interaction by eliminating the
electron-phonon interaction. (P. Lenz and F.W.)
(ii) In systems with impurities as for example in the spin-boson problem
large parameter regions can be treated in a consistent way
(S. Kehrein and A. Mielke).
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A. Mielke:
Diagonalization of Dissipative Quantum Systems I: Exact Solution of the
Spin-Boson Model with an Ohmic bath at alpha=1/2.
preprint (2000).
This paper shows how flow equations can be used
to diagonalize dissipative quantum systems. Applying a continuous
unitary transformation to the spin-boson model, one obtains exact
flow equations for the Hamiltonian and for an observable. They are
solved exactly for the case of an Ohmic bath with a coupling
alpha=1/2. Using the explicite expression for the transformed
observable one obtains dynamical correlation functions. This yields
some new insight to the exactly solvable case alpha=1/2.
The main motivation of this work is to demonstrate, how the method
of flow equations can be used to treat dissipative quantum systems
in a new way. The approach can be used to construct controllable
approximation schemes for other environments.
Back
I. Grote, E. Körding, F. Wegner:
Stability Analysis of the Hubbard Model.
J. Low Temp. Phys. 126 (2002) 1385.
cond-mat/0106604
An effective Hartree-Fock-Bogoliubov-type interaction is calculated for the
Hubbard model in second order in the coupling by means of flow equations. A
stability analysis is performed in order to obtain the transition into
various possible phases. We find, that the second order contribution
weakens the tendency for the antiferromagnetic transition. Apart from a
possible antiferromagnetic transition the d-wave Pomeranchuk instability
recently reported by Halboth and Metzner is usually the strongest instability.
A newly found instability is of p-wave character and yields
band-splitting. In the BCS-channel one obtains the strongest contribution for
dx2-y2-waves. Other types of instabilities of comparable
strength are also reported.
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V. Hankevych, I. Grote and F. Wegner:
Pomeranchuk and other Instabilities in the t-t'
Hubbard model at the Van Hove Filling.
Phys. Rev. B66 (2002) 094516.
cond-mat/0205213
We present a stability analysis of the two-dimensional t-t' Hubbard
model for various values of the next-nearest-neighbor hopping t', and electron
concentrations close to the Van Hove filling by means of the flow equation
method. For t' > -t/3 a dx2-y2-wave Pomeranchuk
instability dominates (apart from antiferromagnetism at small t'). At
t' <-t/3 the leading instabilities are a g-wave Pomeranchuk instability and p-wave
particle-hole instability in the triplet channel at temperatures T < 0.15t,
and an s*-magnetic phase for T > 0.15t; upon increasing the electron
concentration the triplet analog of the flux phase occurs at low temperatures.
Other weaker instabilities are found also.
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V. Hankevych and F. Wegner:
Superconductivity and Instabilities in the t-t' Hubbard Model.
Acta Phys. Pol. B 34 (2003) 497, Erratum 34 (2003) 1591
Contributed paper to the International
Conference on Strongly Correlated Electron Systems SCES'02 in Cracov.
cond-mat/0205597.
We present a stability analysis of the 2D t-t' Hubbard model on a square
lattice for t' = -t/6. We find possible phases of the model (d-wave
Pomeranchuk and superconducting states, band splitting, singlet and
triplet flux phases), and study the interplay of them.
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T. Stauber, A. Mielke:
Equilibrium Correlation Functions of the Spin-Boson Model with Sub-Ohmic
Bath.
cond-mat/0207414.
The spin-boson model is studied by means of flow equations for
Hamiltonians. Our truncation scheme includes all coupling terms which are
linear in the bosonic operators. Starting with the canonical generator
ηc=[H0,H] with H0 resembling the
non-interacting bosonic bath, the flow equations exhibit a universal
attractor for the Hamiltonian flow. This allows to calculate equilibrium
correlation functions for super-Ohmic, Ohmic and sub-Ohmic baths within a
uniform framework including finite bias.
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V. Hankevych and F. Wegner:
Possible Phases of the Two-Dimensional t-t' Hubbard Model.
Eur. Phys. Journal B 31 (2003) 497.
cond-mat/0207612.
We present a stability analysis of the 2D t-t' Hubbard model on a square
lattice for various values of the next-nearest-neighbor hopping t' and
electron concentration. Using the free energy expression, derived by means
of the flow equations method, we have performed numerical calculation for the
various representations under the point group C4mm in order to determine
the phase diagram. A surprising large number of phases has been observed. Some
of them have an order parameter with many nodes in k-space. Commonly
discussed types of order found by us are antiferromagnetism,
dx2-y2-wave singlet superconductivity,
d-wave Pomeranchuk instability and flux phase. A few instabilities newly
observed are a triplet analog of the flux phase, a particle-hole
instability of p-type symmetry in the triplet channel which gives rise to a
phase of magnetic currents, an s*-magnetic phase, a g-wave Pomeranchuk
instability and the band splitting phase with p-wave character. Other weaker
instabilities are found also. We study the interplay of these phases and
favorable situations of their occurrences. A comparison with experiments
is made.
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T. Stauber, A. Mielke:
Contrasting Different Flow
Equations for a Numerically Solvable Model.
cond-mat/0209643.
To contrast different generators for flow equations and to discuss the
dependence of physical quantities on unitarily equivalent, but effectively
different initial Hamiltonians, a numerically solvable model is considered
which is structurally similar to impurity models. A general truncation
scheme is established that produces good results for the Hamiltonian flow as
well as for the operator flow. Nevertheless, it is also pointed out that a
systematic and feasible scheme for the operator flow on the operator level is
missing. More explicitly, truncation of the series of the observable flow
after the linear or bilinear terms does not yield satisfactory results for the
entire parameter regime as - especially close to resonances - even high orders
of the exact series expansion carry considerable weight.
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T. Stauber:
Universal Asymptotic Behavior in Flow Equations of Dissipative Systems.
cond-mat/0211596.
Based on two dissipative models, universal asymptotic behavior of flow
equations for Hamiltonians is found and discussed. The asymptotic behavior
only depends on fundamental bath properties but not on initial system
parameters and the integro-differential equations possess an universal
attractor. The asymptotic flow of the Hamiltonian is characterized by a
non-local differential equation which only depends on one parameter -
independent of the dissipative system nor of the truncation scheme. Since
the fixed point Hamiltonian is trivial, the physical information is completely
transferred to the transformation of the observables. This yields a more
stable flow which is crucial for the numerical evaluation of correlation
functions. The presented procedure also works if relevant perturbations
are present as is demonstrated by evaluating the Shiba relation for sub-Ohmic
baths. It can further be generalized to other dissipative systems.
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T. Stauber:
Tomonaga-Luttinger model with impurity at weak two-body interaction.
cond-mat/0211598.
The Tomonaga-Luttinger model with impurity is studied by means of flow
equations for Hamiltonians. The system is formulated within collective density
fluctuations but no use of the bosonization formula is made. The
truncation scheme includes operators consisting of up to four fermionic
operators and is valid for small electron-electron interactions. In this
regime, the algebraic behavior of correlation functions close to the Fermi
point is recovered involving the exact exponents. Furthermore, we
verify the phase diagram of Kane and Fisher also for intermediate impurity
strength. The approach can be extended to more general one-body
potentials.
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T. Stauber:
One-dimensional conductance through an arbitrary delta impurity
cond-mat/0301586.
The finite-size Tomonaga-Luttinger Hamiltonian with an arbitrary delta
impurity at weak electron-electron interaction is mapped onto a non-interacting
Fermi gas with renormalized impurity potential by means of flow equations for
Hamiltonians. The conductance can then be evaluated using the Landauer formula.
We obtain similar results for infinite systems at finite temperature by
identifying the flow parameter with the inverse squared temperature. This also
yields the finite-size scaling relations of a free electron gas. We recover the
algebraic behavior of the conductance obtained by Kane and Fisher in the limit
of low temperatures but conclude that this limit might be hard to reach for
certain impurity strengths.
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In this paper we consider flow-equations where we allow a normal ordering which is adjusted to the one-particle energy of the Hamiltonian. We show that this flow converges nearly always to the stable phase. Starting out from the symmetric Hamiltonian and symmetry-broken normal ordering nearly always yields symmetry breaking below the critical temperature.
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F. Wegner:
Flow Equations and Normal Ordering: A Survey
J. Phys. A 39 (2006) 8221-8230
cond-mat/0511660.
First we give an introduction to the method of diagonalizing or block-diagonalizing continuously a Hamiltonian and explain how this procedure can be used to analyze the two-dimensional Hubbard model. Then we give a short survey on applications of this flow equation on other models. Finally we outline, how symmetry breaking can be introduced by means of a symmetry breaking of the normal ordering, not of the Hamiltonian.
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Nov. 2007