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Flow equations for band-matrices

Andreas Mielke

Published Mar 17, 1998• 39 citations
OpenAlex

Abstract

Accepted for publication in European Physical Journal 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. PACS-Numbers: 03.65.-w, 02.70.Hm, 31.15.+q, 21.60.-n 1 E–mail:

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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. Supported by DFG Grant 436 RUS 113/205/1 on Russian German CooperationThe well known Foldy-Wouthuysen (FW) transformation [1] is a unitary transformation that separates big and small components in the Dirac equation for a particle interacting with an external electromagnetic field. If the external field is stationary then the FW transformation reduces to a unitary transformation of the Hamiltonian H of the Dirac particle H ′ = e iS (⃗α(⃗p − e ⃗ A(⃗x)) + βm + eA0(⃗x))e −iS (1) where S can be expanded in a sequence of hermitean operators in powers of 1/m. Recently a novel method to diagonalize Hamiltonians via continuous unitary transformations has been proposed [2], where the Hamiltonian H and the S are considered to be functions of a continuous variable, the flow parameter l. This new method has been applied to several topics like the n − orbital model [2], superconductivity and the electron phonon interaction [3, 4]. We want to apply it here to the Dirac particle in an external field in order to explore its possibilities with a well known problem. Thereby we can gain experience in solving problems like the elimination of negative energy states in near light cone QCD [5], where the solution has yet to be discovered. For infinitesimal steps exp(η(l)dl) a continuous unitary transformation of the Hamiltonian H leads to the so-called Hamiltonian flow equation dH(l) dl = [η(l), H(l)] (2) with initial condition H(0) = H. The transformation matrix η(l) is antihermitian and related to S as follows η(l) = ( d dl eiS(l))e −iS(l). The parameter l has dimension of energy −2 and characterizes the amount of diagonalization which has been achieved. In the energy representation the matrix elements of the Hamiltonian H will be grouped in a band of width

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•Jul 31, 2025•0 citations
Accepted for publication in European Physical Journal 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. PACS-Numbers: 03.65.-w, 02.70.Hm, 31.15.+q, 21.60.-n 1 E–mail:
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 zzz 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.
In the present time we observe a growing number of publications where the, so-called, flow equations are successfully used to diagonalize Hamiltonians by means of an appropriate unitary transformation. Here we discuss and compare the flow equations (FE) method (proposed in 1994) with the method of one step continuous unitary transformations (OS CUT) (proposed in 1982). It is shown that the FE method can be considered as a generalization of the OS CUT approach to the case of parameter dependent generator. The OS CUT method gives linear differential equations for the diagonalization procedure. In the FE method the system of differential equations is nonlinear. Finally we discuss the generalization of idea of continuous unitary transformations for the case of quantum equations of motion (Heisenberg picture and density matrix).
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 one 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.
Abstract The Wegner flow equation is a differential equation for a family of Hamiltonians and can be considered as a continuous unitary transformation. In essence, the transformation continuously decouples degrees of freedom and gradually converts the transformed Hamiltonian into the near-diagonal form. The Wegner flow equation has so far been applied to many areas of physics, but mainly in perturbative calculations. However, the exact solutions are known only for the simplest of two-dimensional case and this certainly limits the practical application of the method. On the other hand, the knowledge of exact solutions is essential for many physical problems as well as it can help in constructing improved approximated approaches. In this paper we present the exact solution of the Wegner flow equation with the Mielke generator for 3 × 3 Hermitian matrices, the general solutions for N × N tridiagonal Hermitian matrices and partially for 4 × 4 real symmetric matrices.
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. Supported by DFG Grant 436 RUS 113/205/1 on Russian German CooperationThe well known Foldy-Wouthuysen (FW) transformation [1] is a unitary transformation that separates big and small components in the Dirac equation for a particle interacting with an external electromagnetic field. If the external field is stationary then the FW transformation reduces to a unitary transformation of the Hamiltonian H of the Dirac particle H ′ = e iS (⃗α(⃗p − e ⃗ A(⃗x)) + βm + eA0(⃗x))e −iS (1) where S can be expanded in a sequence of hermitean operators in powers of 1/m. Recently a novel method to diagonalize Hamiltonians via continuous unitary transformations has been proposed [2], where the Hamiltonian H and the S are considered to be functions of a continuous variable, the flow parameter l. This new method has been applied to several topics like the n − orbital model [2], superconductivity and the electron phonon interaction [3, 4]. We want to apply it here to the Dirac particle in an external field in order to explore its possibilities with a well known problem. Thereby we can gain experience in solving problems like the elimination of negative energy states in near light cone QCD [5], where the solution has yet to be discovered. For infinitesimal steps exp(η(l)dl) a continuous unitary transformation of the Hamiltonian H leads to the so-called Hamiltonian flow equation dH(l) dl = [η(l), H(l)] (2) with initial condition H(0) = H. The transformation matrix η(l) is antihermitian and related to S as follows η(l) = ( d dl eiS(l))e −iS(l). The parameter l has dimension of energy −2 and characterizes the amount of diagonalization which has been achieved. In the energy representation the matrix elements of the Hamiltonian H will be grouped in a band of width
Continuous unitary transformations, or flow equations, provide a powerful non-perturbative framework for diagonalizing quantum many-body Hamiltonians. This work presents a first-principles proof that the complex, high-dimensional dynamics generated by a Wegner-type flow equation can, under specific and well-defined conditions, rigorously collapse to an exactly integrable second-order ordinary differential equation. We demonstrate that for a one-dimensional spinless fermion model in the non-interacting limit (Δ₀=0), the flow equations for the coupling constants can be derived exactly, revealing a hidden mathematical structure. We term this phenomenon "Emergent Integrability," where the diagonalization flow itself possesses an integrable submanifold, even if the initial Hamiltonian is not integrable. This discovery, established through rigorous and unabridged derivations, reveals that the process of diagonalization can be fundamentally simpler than the initial problem, shedding new light on the intrinsic mathematical structure of renormalization group-like transformations.