The input to the configuration-interaction shell model includes many dozens
or hundreds of independent two-body matrix elements. Previous studies have
shown that when fitting to experimental low-lying spectra, the greatest
sensitivity is to only a few linear combinations of matrix elements. Here we
consider interactions drawn from the two-body random ensemble, or TBRE, and
find that the low-lying spectra are also most sensitive to only a few linear
combinations of two-body matrix elements, in a fashion nearly indistinguishable
from an interaction empirically fit to data. We find in particular the spectra
for both the random and empirical interactions are sensitive to similar matrix
elements, which we analyze using monopole and contact interactions.Comment: 8 pages, 3 figure

Johnson, Calvin W.

Krastev, Plamen G.

English

arXiv

Johnson, C W

Krastev, P G

UNT Digital Library

LLNL-JRNL-423216Sensitivity Analysis of randomtwo-body interactionsCalvin W. Johnson, Plamen G. KrastevFebruary 5, 2010Physical Review CDisclaimer  This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.  INT-PUB-10-008Sensitivity analysis of random two-body interactionsCalvin W. Johnson1 and Plamen G. Krastev1, 21Department of Physics, San Diego State University,5500 Campanile Drive, San Diego, CA 92182-12332Lawrence Livermore National Laboratory,P.O. Box 808, L-414, Livermore, CA 94551AbstractThe input to the configuration-interaction shell model includes many dozens or hundreds of inde-pendent two-body matrix elements. Previous studies have shown that when fitting to experimentallow-lying spectra, the greatest sensitivity is to only a few linear combinations of matrix elements.Here we consider interactions drawn from the two-body random ensemble, or TBRE, and find thatthe low-lying spectra are also most sensitive to only a few linear combinations of two-body matrixelements, in a fashion nearly indistinguishable from an interaction empirically fit to data. We findin particular the spectra for both the random and empirical interactions are sensitive to similarmatrix elements, which we analyze using monopole and contact interactions.1I. INTRODUCTION AND MOTIVATIONThe configuration-interaction shell model is a useful framework for a detailed under-standing of low-energy nuclear structure [1–3]. The many-body basis is a large dimension(103−10) set of Slater determinants, which are antisymmeterized products of single-particlestates. One must truncate the single-particle states, corresponding to one or a few shells(typically using the harmonic oscillator as an approximation to the mean-field); the many-body basis may be further truncated. For phenomenological calculations one writes theHamiltonian in terms of single-particle energies and two-body matrix elements, while for abinitio calculations one may extend this to three-body interactions [4].The two-body matrix elements are the matrix elements of the residual interaction in thelab frame,VJT (ij, kl) = 〈ij; JT |V̂ |kl; JT 〉 (1)where |ij; JT 〉 is the normalized, antisymmeterized product of particles in orbits labeledby i and j and coupled to good angular momentum J and isospin T . If one starts from atranslationally invariant interaction between particles, one can either compute the integralin the lab frame or start in the relative frame and then transform to the lab frame; in eithercase there are correlations between the matrix elements, although they are not obvious tothe casual observer.Often for semi-phenomenological calculations, one starts from a “realistic” interaction,and then adjusts the two-body matrix elements until the rms error on a set of experimentallyknown energy levels is minimized [1]. In the 1s-0d shell, such a semi-phenomenologicalinteraction has been recently derived [5], improving on an earlier interaction[6].It has been found that the fits are empirically dominated by a few linear combinations ofmatrix elements. The physical meaning of those dominant combinations is not immediatelyobvious. One might naively guess the linear correlations are due to an underlying transla-tionally invariant interaction (although a density dependence would destroy this). Somewhatmore phenomenologically, it has been argued by appealing to mean-field properties that onecan improve fits primarily through adjusting the monopole-monopole part of the interaction,that is, interaction terms that look like na(nb − δab), where na is the number of particles inthe ath orbit. This protocol for shifting monopole strengths has been successfully appliedto several semi-empirical interactions [7–11]2A related study [12], investigating the origin of many-body forces from truncation ofthe model space, also found an empirical fit dominated by a few linear combinations ofmatrix elements. While much of the fit was dominated by the monopole interactions, evenbetter agreement was brought about using a contact interaction motivated by its usage inmean-field calculations [13, 14] and effective field theory[15].In investigating the character and origin of the dominant matrix elements, it is useful toask if there is anything special about the nuclear interaction. One way to ask this questionis to compare with interactions drawn from the two-body random ensemble (TBRE), whichdespite their arbitrary nature are known to echo some features of real nuclear spectra [16–19]. In this paper we conduct a sensitivity analysis of the low-lying spectra of randominteractions and compare against a standard empirical interaction, USDB. We find that formore measures all the interactions are nearly indistinguishable, at least on a statistical level.II. METHODOLOGY AND RESULTSOur methodology follows previous work [1, 5, 12]; we work in the 1s-0d valence spacewith an inert 16O core. Given an input set of two-body matrix elements (we leave asidesingle-particle energies and any A-dependent scaling), which we write as a vector ~v, we cancalculate the eigenvalues Eα(~v) of the many-body Hamiltonian. For this work the label αranged over all nuclides with 0 ≤ Zvalence ≤ Nvalence ≤ 10 and took the ground state bindingenergy and the first five excitation energies.If one has a target spectrum E0α, say from experiment, then the goal of the fit is tominimize∑α(Eα(~v) − E0α)2(2)(for simplicity we leave off the experimental uncertainty in each state). Expanding to firstorderEα(~v + δ~v) ≈ Eα(~v) +∑iδvi∂Eα∂vi(3)then minimizing (2) yields∑α(Eα(~v) − E0α) ∂Eα∂vi=∑j∑α∂Eα∂vi∂Eα∂vjδvj. (4)30 10 20 30 40 50 60linear combination number10-2100102104eigenvalueexcited statesground statesFIG. 1: (Color online)The spectra of eigenvalues from a singular-value decomposition of the sen-sitivity matrix A (Eq. 5) for the two-body random ensemble. The lower curve is for ground stateenergies, while the upper curve is for excitation energies.This equation is in the form ~b = A~x whereAij =∑α∂Eα∂vi∂Eα∂vj. (5)The derivatives come via the Hellmann-Feynman theorem [20]∂Eα∂vi=〈Ψα∣∣∣Ĥi∣∣∣ Ψα〉(6)where Ĥi is the Hamitonian operator whose strength is vi.We then find the eigenvalues of A (which is equivalent to finding the squares of theeigenvalues in the singular-value decomposition of ∂Eα/∂vi). We do this for both the USDBinteraction and for an ensemble of 100 sets of random two-body interactions, also called thetwo-body random ensemble (TBRE). The results are shown for the TBRE in Fig. 1, wherewe have separated out the sensitivity just for the binding energies (ground state energies)and the excitation energies. Although not shown, the equivalent SVD eigenvalues for USDBare completely within the TBRE results.The lower curve is for ground states only, while the upper curve is for excitations energiesrelative to the ground state. Clearly, and perhaps unsurprisingly, the ground state energiesare predominantly sensitive to just a few linear combinations of matrix elements–significantlyfewer than excitations energies.This leads to the next question: if the spectra are sensitive to just a few linear com-binations of matrix elements, can we place a physical interpretation on those dominant41 2 3 4 5 6linear combination number0.010.101.00eigenvalueall statesusdbrandomFIG. 2: (Color online) The SVD spectrum that measures the overlap of the subspace defined bythe six largest eigenvectors from Fig. 1, with the subspace defined by the monopole-monopoleinteraction. (Black) squares are for the TBRE, while (red) diamonds are for USDB.combinations? We specifically look at monopole and contact interactions.To answer this question, we first must introduce a method for quantifying the overlapbetween two vector subspaces [21, 22]. Consider two vectors subspaces, S1 and S2. Let V1 bea matrix whose column vectors are the (orthonormal) basis vectors of S1, and similarly withV2. From these one constructs the overlap matrix Ω = V†1V2. Note that if the subspacesare not of equal dimension then Ω is not a square matrix. In any case we do a singular valuedecomposition of Ω; the SVD eigenvalue spectrum then is a measure of the overlap of thetwo spaces. If the two spaces perfectly overlap then all eigenvalues are 1, if just N of thedimensions perfectly overlap than N eigenvalues will be 1 and the rest zero. Note that thismethod is invariant under arbitrary choice of orthonormal bases.We begin with the monopole-monopole interaction of the form na(nb − δab), which hassix unique terms, and thus six vectors or linear combinations of matrix elements, in thesd-shell. These we combine with the k most dominant linear combinations that arise fromthe previous analysis; somewhat arbitrarily we chose k = 6 (our results do not changequalitatively for other small values of k). The results, the SVD eigenvalues of Ω, are shownin Fig. 2The results for ground states and for excited states are similar, so we combine all statesinto a single calculation. The eigenvalues for USDB are roughly 50% higher than for theTBRE, but otherwise qualitatively very similar.51 2 3 4 5 6linear combination number0.11.0eigenvalueall statesFIG. 3: SVD spectrum from the overlap of the dominant eigenvectors from USDB and the TBRE.We also compared for contact interactions; we took only two terms, the S = 0, T =1 channel and S = 1, T = 0 channel (there being only the s-wave channel in relativecoordinates). These results we summarize in Table 1.TABLE I: Leading eigenvalues from SVD of subspace overlaps of two-term contact interactionInteraction ground states excited states all statesUSDB 0.60 0.58 0.62TBRE 0.55 ± 0.04 0.41 ± 0.06 0.55 ± 0.04For comparison, the leading eigenvalue for the overlaps of USDB versus the six-termmonopole is 0.94, while that of the TBRE versus monopole is 0.66 ± 0.03. There is some-what more sensitivity to the monopole interaction than the contact interaction; however,the reader should keep in mind that is not the whole story. Recall that when fitting aninteraction, the linearized equations are cast in the form A~x = ~B. Our analysis in thispaper is entirely with the eigenvalues of A, but in any fit one must also look to ~b (which inpractice is the deviation of the theoretical spectra from experiment). For example, in [12] itwas found that using a contact interaction brought better agreement than a monopole inter-action. One can understand this in terms of conjugate gradient methods [23]: the directionof local steepest gradient may not in fact point towards the global minimum.By our measures so far, both the TBRE and the empirically-fit USDB look qualitativelysimilar. Therefore we take a final analysis by comparing the dominant linear combinationsof the USDB with those from the TBRE. This is show in Fig. 3, using the same analysis as6for Fig. 2. For comparison with the previous results, the leading eigenvalue is 0.72 ± 0.06.III. CONCLUSIONWe have analyzed the sensitivity of the low-lying spectra of the random two-body ensem-ble of interactions to variations of the Hamiltonian matrix elements; by using singular valuedecomposition, we find the dominant linear combinations, which would be important in anyfit to experimental data. We found the SVD eigenvalues follow a pattern remarkably simi-lar to that shown by semi-realistic/semi-phenomenological interactions such as USDB. Wealso analyzed the most dominant linear combinations of matrix elements by computing theoverlap with monopole and contact interactions. Overall, both the TBRE and the empiricalUSDB had qualitatively similar results.The U.S. Department of Energy supported this investigation through contracts DE-FG02-96ER40985 and DE-FC02-09ER41587, and through subcontract B576152 by Lawrence Liv-ermore National Laboratory under contract DE-AC52-07NA27344.[1] P.J. Brussard and P.W.M. Glaudemans, Shell-model applications in nuclear spectroscopy(North-Holland Publishing Company, Amsterdam, 1977).[2] B. A. Brown and B. H. Wildenthal, Annu. Rev. Nucl. Part. Sci. 38, 29 (1988).[3] E. Caurier, G. Mart́ınez-Pinedo, F. Nowacki, A. Poves, and A. P. Zuker, Rev. Mod. Phys. 77,427 (2005).[4] P. Navrátil and W. E. Ormand, Phys. Rev. C 68, 034305 (2003).[5] B.A. Brown and W.A. Richter, Phys. Rev. C 74 034315 (2006).[6] B.H. Wildenthal, Prog. Part. Nucl. Phys. 11, 5 (1984).[7] A. Poves and A. Zuker: Phys. 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Sensitivity Analysis of random two-body interactions

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Sensitivity analysis of random two-body interactions

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