Flow Equations For Hamiltonians Research Articles

Hamiltonian flow equations for a Dirac particle in large scalar and vector potentials

An efficient solution of the Dirac Hamiltonian flow equations has been proposed through a novel expandsion with the inverse of the Dirac effective mass. The efficiency and accuracy of this new expansion have been demonstrated by reducing a radial Dirac Hamiltonian with large scalar and vector potentials to two nonrelativistic Hamiltonians corresponding to particles and antiparticles, respectively. By solving the two nonrelativistic Hamiltonians, it is found that the exact solutions of the Dirac equation, for both particles and antiparticles, can be reproduced with a high accuracy up to only a few lowest order terms in the expansion. This could help compare and bridge the relativistic and nonrelativistic nuclear energy density functional theories in the future.

Topological quantum order: Stability under local perturbations

We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions. Given such a Hamiltonian H0, we prove that there exists a constant threshold ϵ>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions, the perturbed Hamiltonian H=H0+ϵV has well-defined spectral bands originating from low-lying eigenvalues of H0. These bands are separated from the rest of the spectra and from each other by a constant gap. The band originating from the smallest eigenvalue of H0 has exponentially small width (as a function of the lattice size). Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively bounded operators, and the Lieb–Robinson bound.

Yang-Mills Flows on Nearly Kähler Manifolds and G 2-Instantons

We consider Lie(G)-valued G-invariant connections on bundles over spaces G/H, RxG/H and R^2xG/H, where G/H is a compact nearly Kaehler six-dimensional homogeneous space, and the manifolds RxG/H and R^2xG/H carry G_2- and Spin(7)-structures, respectively. By making a G-invariant ansatz, Yang-Mills theory with torsion on RxG/H is reduced to Newtonian mechanics of a particle moving in a plane with a quartic potential. For particular values of the torsion, we find explicit particle trajectories, which obey first-order gradient or hamiltonian flow equations. In two cases, these solutions correspond to anti-self-dual instantons associated with one of two G_2-structures on RxG/H. It is shown that both G_2-instanton equations can be obtained from a single Spin(7)-instanton equation on R^2xG/H.

Superfluidity near phase separation in Bose-Fermi mixtures

We study the transition to fermion pair superfluidity in a mixture of interacting bosonic and fermionic atoms. The fermion interaction induced by the bosons and the dynamical screening of the condensate phonons due to fermions are included using the nonperturbative Hamiltonian flow equations. We determine the bosonic spectrum near the transition towards phase separation and find that the superfluid transition temperature may be increased substantially due to phonon damping.

One-dimensional conductance through an arbitrary potential

The finite-size Tomonaga-Luttinger Hamiltonian with an arbitrary potential is mapped onto a non-interacting Fermi gas with renormalized potential. This is done by means of flow equations for Hamiltonians and is valid for small electron-electron interaction. This method also yields an alternative bosonization formula for the transformed field operator which makes no use of Klein factors. The two-terminal 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 in the asymptotic regime. We recover the algebraic behavior of the conductance obtained by Kane and Fisher in the limit of low temperatures and weak electron-electron interaction, but our results remain valid for arbitrary external potential.

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. Universal 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 can be characterized by a non-local differential equation which only depends on one parameter - independent of the dissipative system or 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. Furthermore, the low energy behavior of correlation functions is determined analytically. The presented procedure can also be applied if relevant perturbations are present as is demonstrated by evaluating dynamical correlation functions for sub-Ohmic environments. It can further be generalized to other dissipative systems.

Tomonaga-Luttinger model with an impurity for a 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 fermion operators and is valid for small electron-electron interactions. In this regime, the exact expression for the anomalous dimension is recovered. 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.

Flow equations for Hamiltonians: contrasting different approaches by using a numerically solvable model

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.

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 η 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.

FLOW EQUATIONS FOR HAMILTONIANS

A method to diagonalize or block-diagonalize Hamiltonians by means of an appropriate continuous unitary transformation is reviewed. We will outline (i) the procedure for the elimination of the electron-phonon interaction and the construction of the effective attractive electron-electron interaction, and (ii) the application to some systems with electron-electron interaction (n-orbital model and Hubbard model).

Flow equations for Hamiltonians

A method to diagonalize or block-diagonalize Hamiltonians by means of an appropriate continuous unitary transformation is reviewed.

Flow equations for gluodynamics in the Coulomb gauge

A systematic procedure to consistently formulate a field theoretical, QCD bound state problem with a fixed number of constituents is outlined. The approach entails applying the Hamiltonian flow equations, which are a set of continuous unitary transformations, to a QCD motivated Hamiltonian with a confining interaction. The method is developed in detail for gluodynamics in the Coulomb gauge to obtain an effective block-diagonal Hamiltonian appropriate to a reduced Fock space with fixed number of dynamical gluons. Standard many-body techniques are used to numerically diagonalize this Hamiltonian in a constituent two gluon Fock space. The calculated gluon condensates and glueball masses are in good agreement with QCD sum rule and lattice results.

Similarity renormalization, Hamiltonian flow equations, and Dyson’s intermediate representation

A general framework is presented for the renormalization of Hamiltonians via a similarity transformation. Divergences in the similarity flow equations may be handled with dimensional regularization in this approach, and the resulting effective Hamiltonian is finite since states well-separated in energy are uncoupled. Specific schemes developed several years ago by Glazek and Wilson and contemporaneously by Wegner correspond to particular choices within this framework, and the relative merits of such choices are discussed from this vantage point. It is shown that a scheme for the transformation of Hamiltonians introduced by Dyson in the early 1950's also corresponds to a particular choice within the similarity renormalization framework, and it is argued that Dyson's scheme is preferable to the others for ease of computation. As an example, it is shown how a logarithmically confining potential arises simply at second order in light-front QCD within Dyson's scheme, a result found previously for other similarity renormalization schemes. Steps toward higher order and nonperturbative calculations are outlined. In particular, a set of equations analogous to Dyson-Schwinger equations is developed.

Hamiltonian flow equations for the Lipkin model

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.

Hamiltonian flow equations for a Dirac particle in an external potential

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.

Flow equations for Hamiltonians: crossover from Luttinger to Landau-Liquid behaviour in the n-orbital model

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 \(\tfrac{1}{n}\)-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.

Flow equations for the Anderson Hamiltonian

Using a continuous unitary transformation recently proposed by Wegner \cite{Wegner} together with an approximation that neglects irrelevant contributions, we obtain flow equations for Hamiltonians. These flow equations 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 consistent with results obtained by a numerical renormalization group method, but our approach is considerably simpler.