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Communications in Theoretical Physics

A Renormalized-Hamiltonian-Flow Approach to Eigenenergies and Eigenfunctions*

Wen-ge Wang

Published Jul 1, 2019• 2 citations
Publisher

Abstract

Abstract We introduce a decimation scheme of constructing renormalized Hamiltonian flows, which is useful in the study of properties of energy eigenfunctions, such as localization, as well as in approximate calculation of eigenenergies. The method is based on a generalized Brillouin-Wigner perturbation theory. Each flow is specific for a given energy and, at each step of the flow, a finite subspace of the Hilbert space is decimated in order to obtain a renormalized Hamiltonian for the next step. Eigenenergies of the original Hamiltonian appear as unstable fixed points of renormalized flows. Numerical illustration of the method is given in the Wigner-band random-matrix model .

DOI
10.1088/0253-6102/71/7/861

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Abstract We introduce a decimation scheme of constructing renormalized Hamiltonian flows, which is useful in the study of properties of energy eigenfunctions, such as localization, as well as in approximate calculation of eigenenergies. The method is based on a generalized Brillouin-Wigner perturbation theory. Each flow is specific for a given energy and, at each step of the flow, a finite subspace of the Hilbert space is decimated in order to obtain a renormalized Hamiltonian for the next step. Eigenenergies of the original Hamiltonian appear as unstable fixed points of renormalized flows. Numerical illustration of the method is given in the Wigner-band random-matrix model .
This paper presents a renormalization approach to many-particle systems. By starting from a bare Hamiltonian H=H0+H1\mathcal{H}={\mathcal{H}}_{0}+{\mathcal{H}}_{1}H=H0​+H1​ with an unperturbed part H0{\mathcal{H}}_{0}H0​ and a perturbation H1,{\mathcal{H}}_{1},H1​, we define an effective Hamiltonian which has a band-diagonal shape with respect to the eigenbasis of H0.{\mathcal{H}}_{0}.H0​. This means that all transition matrix elements are suppressed which have energy differences larger than a given cutoff $\ensuremath{\lambda}$ that is smaller than the cutoff $\ensuremath{\Lambda}$ of the original Hamiltonian. This property resembles a recent flow equation approach on the basis of continuous unitary transformations. For demonstration of the method we discuss an exact solvable model, as well as the Anderson-lattice model where the well-known quasiparticle behavior of heavy fermions is derived.
A renormalization group flow of Hamiltonians for two-dimensional classical partition functions is constructed using tensor networks. Similar to tensor network renormalization [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.180405; S. Yang, Z.-C. Gu, and X.-G. Wen, Phys. Rev. Lett. 118, 110504 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.110504], we obtain approximate fixed point tensor networks at criticality. Our formalism, however, preserves positivity of the tensors at every step and hence yields an interpretation in terms of Hamiltonian flows. We emphasize that the key difference between tensor network approaches and Kadanoff's spin blocking method can be understood in terms of a change of the local basis at every decimation step, a property which is crucial to overcome the area law of mutual information. We derive algebraic relations for fixed point tensors, calculate critical exponents, and benchmark our method on the Ising model and the six-vertex model.
Perturbation methods are generally used for solving wave operator equations associated with the determination of effective Hamiltonians.In many cases the standard Rayleigh-Schrödinger and Brillouin-Wigner series either converge slowly or diverge.Therefore it is necessary to modify or to renormalize the standard wave equations.For that purpose derivative and convergence superoperators within the Ralyeigh-Schrödinger and Brillouin-Wigner formalisms were introduced.A new efficient approximation for convergence superoperators is investigated in this paper.Its application to a model system of N non-interacting molecules shows that this approximation can overcome convergence difficulties.
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Abstract The Truncated conformal space approach (TCSA) is a numerical technique for finding finite size spectrum of Hamiltonians in quantum field theory described as perturbations of conformal field theories. The truncation errors of the method have been systematically studied near the UV fixed point (when the characteristic energy related to the coupling is less than the truncation cutoff) where a good theoretical understanding has been achieved. However numerically the method demonstrated a good agreement with other methods for much larger values of the coupling when the RG flow approaches a new fixed point in the infrared. In the present paper we investigate this regime for a number of boundary RG flows testing the leading exponent and truncation errors. We also study the flows beyond the first fixed point which have been observed numerically but yet lack a theoretical understanding. We show that while in some models such flows approximate reversed physical RG flows, in other models the spectrum approaches a stable regime that does not correspond to any local boundary condition. Furthermore we find that in general the flows beyond the first fixed point are very sensitive to modifications of the truncation scheme.