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arXiv:1210.1703 · Acta Physica Polonica B

Application of a Numerical Renormalization Group Procedure to an Elementary Anharmonic Oscillator

K.P. Wójcik

Published Jan 1, 2013• 2 citations
arXivHTMLPDF

Abstract

The canonical quantum Hamiltonian eigenvalue problem for an anharmonic oscillator with a Lagrangian L = \dotϕ^2/2 - m^2 ϕ^2/2 - g m^3 ϕ^4 is numerically solved in two ways. One of the ways uses a plain cutoff on the number of basis states and the other employs a renormalization group procedure. The latter yields superior results to the former because it allows one to calculate the effective Hamiltonians. Matrices of effective Hamiltonians are quite small in comparison to the initial cutoff but nevertheless yield accurate eigenvalues thanks to the fact that just eight of their highest-energy matrix elements are proper functions of the small effective cutoff. We explain how these cutoff-dependent matrix elements emerge from the structure of the Hamiltonian and the renormalization group recursion, and we show that such small number of cutoff-dependent terms is sufficient to renormalize any band-diagonal Hamiltonian.

DOI
10.5506/aphyspolb.44.69

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The canonical quantum Hamiltonian eigenvalue problem for an anharmonic oscillator with a Lagrangian L = \dotϕ^2/2 - m^2 ϕ^2/2 - g m^3 ϕ^4 is numerically solved in two ways. One of the ways uses a plain cutoff on the number of basis states and the other employs a renormalization group procedure. The latter yields superior results to the former because it allows one to calculate the effective Hamiltonians. Matrices of effective Hamiltonians are quite small in comparison to the initial cutoff but nevertheless yield accurate eigenvalues thanks to the fact that just eight of their highest-energy matrix elements are proper functions of the small effective cutoff. We explain how these cutoff-dependent matrix elements emerge from the structure of the Hamiltonian and the renormalization group recursion, and we show that such small number of cutoff-dependent terms is sufficient to renormalize any band-diagonal Hamiltonian.
This study aims to apply the matrix method to determine the energy of anharmonic oscillators containing linear, quadratic, cubic, quartic, quintic, sextic, septic, octic, nonic, and decic anharmonicities.Energy calculations were performed analytically using the matrix method with a matrix of size 4×4.The calculations were carried out by finding the Hamiltonian matrix representation of the oscillators, which was then solved using the time-independent Schrödinger equation.As a comparison, computational calculations were also conducted with Wolfram Mathematica for large matrix sizes.Results for small coupling constants and small perturbation terms such as linear anharmonic oscillators showed that energies obtained agree with the exact solution (up to 3 digits).Meanwhile, the accuracy of the analytical matrix method was found to decrease for larger coupling constants.The application of analytical matrix methods with small matrix sizes can produce relatively accurate energies at small coupling constants, while large coupling constants require larger matrix sizes to obtain reasonably accurate results.The calculation of energy by matrix method with a large matrix size will be more effectively done computationally, such as by using Wolfram Mathematica.
The finite-element approach to lattice field theory is both highly accurate (relative errors ∼1/N2\sim1/N^2∼1/N2, where NNN is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly preserved at the lattice sites). In this talk I construct matrix elements for dynamical variables and for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian is H=p2/2+λq4/4.H=p^2/2+λq^4/4.H=p2/2+λq4/4. Construction of such matrix elements does not require solving the implicit equations of motion. Low order approximations turn out to be extremely accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator ground state gives a result for the anharmonic oscillator ground state energy accurate to better than 1\%, while a two-state approximation reduces the error to less than 0.1\%.
Theory of the quantum quartic oscillator is developed with close attention to the energy cutoff one needs to impose on the system in order to approximate the smallest eigenvalues and corresponding eigenstates of its Hamiltonian by diagonalizing matrices of limited size. The matrices are obtained by evaluating matrix elements of the Hamiltonian between the associated harmonic-oscillator eigenstates and by correcting the computed matrices to compensate for their limited dimension, using the Wilsonian renormalization-group procedure. The cutoff dependence of the corrected matrices is found to be described by a spiral motion of a three-dimensional vector. This behavior is shown to result from a combination of a limit-cycle and a floating fixed-point behaviors, a distinct feature of the foundational quantum system that warrants further study. A brief discussion of the research directions concerning renormalization of polynomial interactions of degree higher than four, spontaneous symmetry breaking and coupling of more than one oscillator through the near neighbor couplings known in condensed matter and quantum field theory, is included.
Quantum field theories require a cutoff to regulate divergences that result from local interactions, and yet physical results can not depend on the value of this cutoff. The renormalization group employs a transformation that changes the cutoff to isolate hamiltonians that produce cutoff-independent eigenvalues. The similarity renormalization group is based on similarity transformations that regulate off-diagonal matrix elements, forcing the hamiltonian towards a band-diagonal form as the cutoff is lowered. This avoids pathologies that plagued tradition transformations acting on hamiltonians, making it possible to produce a well-behaved perturbative approximation of renormalized hamiltonians in asymptotically free theories. We employ a simple two-dimensional delta function example to illustrate this new renormalization technique.
As in the usual quantum field theory, the states, and therefore also the eigenvalue spectrum of an anharmonic oscillator can be characterized by means of the so-called τ-functions, that is the matrix element of the type 〈0 | qn | j〉. For the calculation of these matrix elements, the equation of motion of the anharmonic oscillator can be used to obtain an infinite set of equations, which define an eigenvalue problem. To solve it a new set of functions, the so-called φ-functions, are introduced by means of a transformation, whose matrix corresponds formally to the WICK rule. An analysis of this infinite system of φ-equations shows that a convergent secular polynomial can be obtained, which exists as a limiting value of the polynomials for the truncated N φ -equationsystems in the limit N → ∞ . It is therefore permissible to calculate the eigenvalues of the infinite system in an approximate way from the truncated systems. Such an approximation procedure is the essential content of the so-called TAMM-DANCOFF method. The above mentioned convergence of the determinants therefore provides its justification. The convergence of the eigenvalues of the truncated systems to the exact oscillator values is numerically examined up to N= 20. The results are satisfactory.