Statistical Physics and Condensed Matter

Flow Equations for Hamiltonians

Abstracts of Papers from the Institute for Theoretical Physics at the University Heidelberg

F. Wegner:
Flow equations for Hamiltonians.
Ann. Physik, Leipzig 3, 77 (1994)

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.

S.K. Kehrein, A. Mielke:
Flow equations for the Anderson Hamiltonian.
J. Phys. A - Math. Gen. 27, 4259-4279 (1994); corrigendum 27, 5705 (1994)

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.

S.K. Kehrein, A. Mielke, and P. Neu:
Flow equations for the spin-boson problem.
Z. Phys. B 99, 269 (1996)

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.

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.

S.K. Kehrein, A. Mielke:
On the spin-boson model with a sub-Ohmic bath.
Phys. Lett. A 219, 313 (1996).

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.

P. Lenz, F. Wegner:
Flow equations for electron-phonon interactions.
Nucl. Phys. B482 [FS] (1996) 693-712 .

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

S.K. Kehrein, A. Mielke:
Low temperature equilibrium correlation functions in dissipative quantum systems.
Ann. Physik (Leipzig) 6, 90 (1997)

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.

A. Mielke:
Similarity renormalization of the electron-phonon coupling.
Ann. Physik (Leizpzig) 6, 215-233 (1997)

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.

S.K. Kehrein, A. Mielke:
Diagonalization of system plus environment Hamiltonians.
J. Stat. Phys. 90, 889-898 (1998), cond-mat/9701123

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.

J. Stein:
Flow equations and the strong-coupling expansion for the Hubbard model.
J. Stat. Phys. 88 (1997) 487 ( Postscript-File)

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.

A. Mielke:
Calculating critical temperatures of superconductivity from a renormalized Hamiltonian.
Europhys. Lett. 40, 195-200 (1997)

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.

A. Kabel, F. Wegner:
Flow Equations for Hamiltonians: Crossover from Luttinger to Landau-Liquid Behaviour in the n-Orbital Model
Z. Phys. B 103 (1997) 555

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

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.

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.

E.L. Gubankova, F. Wegner:
Flow equations for QED in the light front dynamics

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.

A. Mielke
Flow equations for band-matrices
Euro. Phys. J. B 5, 605-611 (1998),

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.

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.

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.

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.

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.

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

A. B. Bylev, H. J. Pirner:
Hamiltonian Flow Equations for a Dirac Particle in an external Potential
Phys. Lett. B428, 329 (1998),

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

J. Stein:
Flow equations and the Ruderman-Kittel-Kasuya-Yosida interaction.
Eur. Phys. J. B 12 (1999) 5 ( Postscript-File)

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.

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.

J. Stein:
Unitary flow of the bosonized large-N Lipkin model.
J. Phys. G 26 (2000) 377 ( Postscript-File)

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.

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.

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.

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).

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.

I. Grote, E. Körding, F. Wegner:
Stability Analysis of the Hubbard Model.
J. Low Temp. Phys. 126 (2002) 1385.

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.

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.

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. Back

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.

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. Back

T. Stauber, A. Mielke:
Equilibrium Correlation Functions of the Spin-Boson Model with Sub-Ohmic Bath.

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. Back

V. Hankevych and F. Wegner:
Possible Phases of the Two-Dimensional t-t' Hubbard Model.
Eur. Phys. Journal B 31 (2003) 497.

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. Back

T. Stauber, A. Mielke:
Contrasting Different Flow Equations for a Numerically Solvable Model.

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. Back

T. Stauber:
Universal Asymptotic Behavior in Flow Equations of Dissipative Systems.

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. Back

T. Stauber:
Tomonaga-Luttinger model with impurity at weak two-body interaction.

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. Back

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. Back

E. Koerding, F. Wegner:
Flow Equations and Normal Ordering
J. Phys. A 39 (2006) 1231-1237 cond-mat/0509801.

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. Back

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. Back
Back to main page Flow equations.

Nov. 2007