Institut for Theoretical Physics
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Flow equations for Hamiltonians
Using continuous unitary transformations one obtains
flow equations for Hamiltonians. This new, non-perturbative
method can be used to diagonalize or renormalize a given
Hamiltonian. The method is useful for problems with
different energy scales or strong interactions. The method
has been developed 25 years ago by Franz Wegner for
problems of many particle physics and independently by
Stan Glazek and Kenneth Wilson for bound state problems
in QCD. Since that time we use this method to treat
various kinds of problems including
One of our main interest for the future is to study correlated
electron systems using this method. We expect that the method
can be applied successfully to such problems, since it directly
renormalizes the Hamiltonian and therefore allows the description
of bound states or correlated states.
- Dissipative quantum systems.
- Electron-phonon coupling, superconductivity.
- Quantum mechanics of classically chaotic systems.
Selected publications in this field
Albert Verdeny, Andreas Mielke, Florian Mintert: Accurate effective Hamiltonians via unitary flow in Floquet space
Phys. Rev. Lett.111,
We present a systematic construction of effective Hamiltonians of periodically driven quantum systems. Due to an equivalence between the time dependence of a Hamiltonian and an interaction in its Floquet operator, flow equations, that permit to decouple interacting quantum systems, allow us to identify time-independent Hamiltonians for driven systems. With this approach, we explain the experimentally observed deviation of expected suppression of tunneling in ultra-cold atoms.
Tobias Stauber, Andreas Mielke: Flow equations for Hamiltonians: Contrasting different approaches by using a numerically solvable model
J. Phys. A36,
To contrast different generators for flow equations for Hamiltonians 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. By this we discuss the question of optimization for the first time. 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. For this, an explicit analysis of the operator flow is given for the first time. We observe that 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.
Tobias Stauber, Andreas Mielke: Equilibrium Correlation Functions of the Spin-Boson Model with Sub-Ohmic Bath
Phys. Lett. A64,
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 eta_c=[H_0,H] with H_0 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. Results for sub-Ohmic baths might be relevant for the assessment of dissipation due to 1/f-related noise, recently found in solid-state qubits.
Andreas Mielke: Diagonalization of Dissipative Quantum Systems I: Exact solution of the Spin-Boson Model with an Ohmic bath at $\alpha =1/2$
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.
Daniel Cremers, Andreas Mielke: Flow equations for the Hénon-Heiles Hamiltonian
The Hénon--Heiles Hamiltonian was introduced in 1964 [M. Hénon, C. Heiles: Astron. J. 69, 73 (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 [Ann. Physik (Leipzig) 3, 77 (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.
Stefan Kehrein, Andreas Mielke: Diagonalization of system plus environment Hamiltonians
J. Stat. Phys.90,
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.
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.
Stefan Kehrein, Andreas Mielke: Low temperature equilibrium correlation function in dissipative quantum systems
Ann. Physik (Leipzig)6,
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.
Andreas Mielke: Similarity renormalization of the electron--phonon coupling
Ann. Physik (Leipzig)6,
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.
Andreas Mielke: Calculating superconducting transition temperatures in a renormalized Hamiltonian framework
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. For small coupling we rederive the McMillan formula for $T_c$. 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.
Stefan Kehrein, Andreas Mielke, Peter Neu: Flow equations for the spin-boson problem
Z. Phys. B99,
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.
Stefan Kehrein, Andreas Mielke: Theory of the Anderson impurity model: The Schrieffer-Wolff transformation re-examined
Ann. Physics (NY)252,
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.
Stefan Kehrein, Andreas Mielke: On the spin-boson model with a sub-Ohmic bath
Phys. Lett. A219,
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.
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.
Last changes: 20.3.2022.