Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:hep-ph/0101065

Fixed-Point Hamiltonians in Quantum Mechanics

T. Frederico, A. Delfino, Lauro Tomio, Timoteo, V. S.

Published Jan 8, 2001• 7 citations
arXivHTMLPDF

Abstract

We show how to derive fixed-point Hamiltonians in quantum mechanics from a proposed renormalization group invariance approach that relies in a subtraction procedure at a given energy scale. The scheme is valid for arbitrary interactions that originally contain point-like singularities (as the Dirac-delta and/or its derivatives). One example of diagonalization of a fixed-point Hamiltonian illustrates the procedure for an interaction that contains regular and singular parts. In another example, the fixed point Hamiltonian is derived for the case of higher order singularities in the interaction.

DOI
10.48550/arxiv.hep-ph/0101065

Related papers

Fixed-Point Few-Body Hamiltonians in Quantum Mechanics

Lauro Tomio, T. Frederico, V. S. Timóteo +1 more

We revisited how Weinberg's ideas in Nuclear Physics influenced our own work and lead to a renormalization group invariant framework within the quantum mechanical few-body problem, and we also update the discussion on the relevant scales in the limit of short-range interactions. In this context, it is revised the formulation of the subtracted scattering equations and fixed-point Hamiltonians applied to few-body systems, in which the original interaction contains point-like singularities, such as Dirac-delta and/or its derivatives. The approach is being illustrated by considering two-nucleons described by singular interactions. This revision also includes an extension of the renormalization formalism to three-body systems, which is followed by an updated discussion on the applications to four particles.

arXiv:2111.05954•Nov 10, 2021•0 citations

Renormalization-Group Transformations on Quantum States

Frank Verstraete, J. I. Cirac, José Ignacio Latorre +2 more

We construct a general renormalization-group transformation on quantum states, independent of any Hamiltonian dynamics of the system. We illustrate this procedure for translational invariant matrix product states in one dimension and show that product, Greenberger-Horne-Zeilinger, W, and domain wall states are special cases of an emerging classification of the fixed points of this coarse-graining transformation.

arXiv:quant-ph/0410227•Physical Review Letters•Apr 14, 2005•188 citations

Mass generation at a fixed point: A Functional Renormalization Group Study of the tricritical O($N$) model in $d=3$ and $N=\infty$

Shunsuke Yabunaka, Bertrand Delamotte

Renormalization group (RG) fixed points are commonly associated with scale invariance and a divergent correlation length. We show that this connection can fail in the tricritical $O(N)$ model in three dimensions in the limit $N\to\infty$. Revisiting the line of fixed points identified by Bardeen, Moshe, and Bander, we use the functional renormalization group to clarify the mechanism leading to mass generation at its singular endpoint (the BMB fixed point). We demonstrate that the generated mass is nonuniversal and originates from the nonanalytic structure of the effective potential. We show that the critical exponent $ν$ which takes the value $ν= 1/2$ along the regular part of the BMB line, that is, for $0 \leq λ< λ_{\rm BMB}$, jumps to $ν= 1/3$ on the singular part of this line with the BMB FP, corresponding to $λ= λ_{\rm BMB}$, being the pivotal point between these two regimes. We also show how its singular potential emerges dynamically along the renormalization flow.

arXiv:2606.12269•Jun 10, 2026•0 citations

RENORMALIZATION GROUP APPROACH TO EFFECTIVE HAMILTONIANS

T.J. FIELDS, James P. Vary, K.S. GUPTA

We introduce a way of implementing renormalization within the context of the theory of effective Hamiltonians. Our renormalization scheme involves manipulations at the level of the generalized G-matrix and is independent of any specific kinematics. We show how to calculate the β-function within this context and exhibit our method using simple scale-invariant quantum mechanical systems.

arXiv:hep-ph/9506320•Modern Physics Letters A•Sep 7, 1996•7 citations

Regular Hamiltonians for non-relativistic interacting quantum field theories

Bruno Galvan

In the context of non-relativistic quantum field theory, a method is proposed for multiplying field operators at the same spatial point and obtaining regular (i.e. rigorously defined) interaction terms for the Hamiltonian. The basic idea is to modify the Lebesgue measure of configuration space of many particles by adding singular measures over the subspaces of configuration space in which the positions of two or more particles coincide.

arXiv:1305.0495•May 2, 2013•1 citation

Teaching renormalization, scaling, and universality with an example from quantum mechanics

Steve Paik

We discuss the quantum mechanics of a particle restricted to the half-line >x 0 with potential energy a = V x 2 for a -< < 1 4 0. It is known that two scale-invariant theories may be defined.By regularizing the near-origin behavior of the potential by a finite square well with variable width b and depth g, it is shown how these two scale-invariant theories occupy fixed points in the resulting (b, g)space of Hamiltonians.A renormalization group (RG) flow exists in this space and scaling variables are shown to exist in a neighborhood of the fixed points.Consequently, the propagator of the regulated theory enjoys homogeneous scaling laws close to the fixed points.Using RG arguments it is possible to discern the functional form of the propagator for long distances and long imaginary times, thus demonstrating the extent to which fixed points control the behavior of the cut-off theory.By keeping the width fixed and varying only the well depth, we show how the mean position of a bound state diverges as g approaches a critical value.It is proven that the exponent characterizing the divergence is universal in the sense that its value is independent of the choice of regulator.Two classical interpretations of the results are discussed: standard Brownian motion on the real line, and the free energy of a certain one-dimensional chain of particles with prescribed boundary conditions.In the former example, V appears as part of an expectation value in the Feynman-Kac formula.In the latter example, V appears as the background potential for the chain, and the loss of extensivity is dictated by a universal power law.

arXiv:1707.04388•Journal of Physics Communications•Dec 21, 2017•5 citations
We show how to derive fixed-point Hamiltonians in quantum mechanics from a proposed renormalization group invariance approach that relies in a subtraction procedure at a given energy scale. The scheme is valid for arbitrary interactions that originally contain point-like singularities (as the Dirac-delta and/or its derivatives). One example of diagonalization of a fixed-point Hamiltonian illustrates the procedure for an interaction that contains regular and singular parts. In another example, the fixed point Hamiltonian is derived for the case of higher order singularities in the interaction.
We revisited how Weinberg's ideas in Nuclear Physics influenced our own work and lead to a renormalization group invariant framework within the quantum mechanical few-body problem, and we also update the discussion on the relevant scales in the limit of short-range interactions. In this context, it is revised the formulation of the subtracted scattering equations and fixed-point Hamiltonians applied to few-body systems, in which the original interaction contains point-like singularities, such as Dirac-delta and/or its derivatives. The approach is being illustrated by considering two-nucleons described by singular interactions. This revision also includes an extension of the renormalization formalism to three-body systems, which is followed by an updated discussion on the applications to four particles.
Mass generation at a fixed point: A Functional Renormalization Group Study of the tricritical O(NNN) model in d=3d=3d=3 and N=∞N=\inftyN=∞
Renormalization group (RG) fixed points are commonly associated with scale invariance and a divergent correlation length. We show that this connection can fail in the tricritical O(N)O(N)O(N) model in three dimensions in the limit N→∞N\to\inftyN→∞. Revisiting the line of fixed points identified by Bardeen, Moshe, and Bander, we use the functional renormalization group to clarify the mechanism leading to mass generation at its singular endpoint (the BMB fixed point). We demonstrate that the generated mass is nonuniversal and originates from the nonanalytic structure of the effective potential. We show that the critical exponent ννν which takes the value ν=1/2ν= 1/2ν=1/2 along the regular part of the BMB line, that is, for 0≤λ<λBMB0 \leq λ< λ_{\rm BMB}0≤λ<λBMB​, jumps to ν=1/3ν= 1/3ν=1/3 on the singular part of this line with the BMB FP, corresponding to λ=λBMBλ= λ_{\rm BMB}λ=λBMB​, being the pivotal point between these two regimes. We also show how its singular potential emerges dynamically along the renormalization flow.
We discuss the quantum mechanics of a particle restricted to the half-line >x 0 with potential energy a = V x 2 for a -< < 1 4 0. It is known that two scale-invariant theories may be defined.By regularizing the near-origin behavior of the potential by a finite square well with variable width b and depth g, it is shown how these two scale-invariant theories occupy fixed points in the resulting (b, g)space of Hamiltonians.A renormalization group (RG) flow exists in this space and scaling variables are shown to exist in a neighborhood of the fixed points.Consequently, the propagator of the regulated theory enjoys homogeneous scaling laws close to the fixed points.Using RG arguments it is possible to discern the functional form of the propagator for long distances and long imaginary times, thus demonstrating the extent to which fixed points control the behavior of the cut-off theory.By keeping the width fixed and varying only the well depth, we show how the mean position of a bound state diverges as g approaches a critical value.It is proven that the exponent characterizing the divergence is universal in the sense that its value is independent of the choice of regulator.Two classical interpretations of the results are discussed: standard Brownian motion on the real line, and the free energy of a certain one-dimensional chain of particles with prescribed boundary conditions.In the former example, V appears as part of an expectation value in the Feynman-Kac formula.In the latter example, V appears as the background potential for the chain, and the loss of extensivity is dictated by a universal power law.