Feynman-Kac path integral approach for the energy spectrum of many boson systems

S .Datta

REVIEW

Feynman-Kac path integral approach for the energy spectrum of many boson systems

S .Datta

Statistical Mechanics · Submitted 1 Mar 2006 · 25 references

Pith's verdict: A legitimate GFK quantum Monte Carlo calculation whose main claim of being a better alternative to Gross-Pitaevskii is undercut by the authors' own calibration of the Morse potential to GP energies.

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arXiv IDcond-mat/0603016 (version 2)

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CitationS .Datta. "Feynman-Kac path integral approach for the energy spectrum of many boson systems." arXiv:cond-mat/0603016 (2006). https://pith.science/paper/KEZJDQRA

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Editor's note

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The one thing to know: this is a real quantum Monte Carlo calculation of trapped Bose gas spectra, but the core selling point is not supported by the evidence in the paper. The author takes the Generalized Feynman-Kac method seriously, computes ground and excited state energies for up to 200 Rb and Li atoms, checks the virial theorem, and reproduces known results from Blume-Greene and Dalfovo-Stringari where comparisons exist. That is useful work for a small subfield, and the reported numbers are not obviously wrong. The method itself is established, the trial functions are sensible for small N, and the paper is honest enough to warn in Section 5 that parameter choice is crucial.

The soft spot is exactly where the stress-test note lands. In Section 3.1 the author admits that with the Esry-Greene Morse parameters the GFK energies come out substantially lower than GP, so they choose new values (alpha = 0.29, r0 = 9.67) to get better agreement. That is calibration, not validation. The subsequent claim that GFK is a better alternative to GP is therefore a statement about a Hamiltonian that was tuned to match GP's energies, not an independent comparison. The sensitivity argument is even stronger than the stress-test note says: taking r0 from 11.65 to 9.67 changes the Born-approximation coupling D by roughly a factor of three hundred, so the two sets of parameters describe completely different interaction strengths. There is no independent physical constraint given for the final r0 and alpha.

I also have a quieter concern about the excited states. The method relies on fixed nodal surfaces from harmonic oscillator trial functions, and the paper does not test how much the energies depend on that choice. For the ground state the virial theorem and the comparisons with other methods give some confidence; for the m = 1, 2, 3 states the numbers are presented as exact, but there is no sensitivity study. The absence of code, even though the author says the Fortran is 270 lines, makes the central simplicity claim hard to evaluate.

On balance I do not think the central claim survives, but I would not desk reject this. There is a legitimate calculation here and a clear path to make it valuable: fix the potential parameters from independent scattering data, test the fixed-node bias, and present the comparison with GP as a consistency check rather than a victory. As it stands, it needs major revision but deserves a serious referee who can push on those points. For a reader working on trapped Bose gas QMC, there is something to take away; for a reader looking for a rigorous beyond-mean-field benchmark, this is a cautionary example of what happens when you tune the interaction to fit the result you want to surpass.

Referee report

Significance

If validated, the GFK approach would be a useful nonperturbative tool for mesoscopic trapped Bose gases, particularly because it is not restricted to the dilute/mean-field regime and can include correlations beyond GP. The paper has several genuine strengths: the Feynman-Kac formalism is well established; bosonic statistics avoid the sign problem; the random-walk implementation is simple; the ground-state energies are consistently lower than GP, which is qualitatively consistent with correlation effects; and the virial-theorem check is a reasonable internal consistency test. The central weakness is that the interaction parameters are tuned to reproduce GP energies, so the main validation is partly circular and does not establish that the computed energies are those of a physically faithful Hamiltonian.

Major objections (4)

[Sec. 3.1, Eqs. (22) and (29)] The interaction parameters are not fixed independently: after reporting that the literature Morse parameters alpha=0.35, r0=11.65 give results 'substantially lower than GP', the authors state that they obtain 'a better agreement with GP by choosing alpha=.29 and r0=9.67'. The agreement in Table 2 is therefore, at least in part, a calibration artifact rather than evidence that GFK is a more accurate alternative to GP. An independent determination of alpha and r0 (for example from spectroscopic dimer potentials or from a direct two-body scattering calculation) or an explicit sensitivity analysis over the physically allowed parameter range is needed before the central claim can be supported.
[Sec. 3.1, Eq. (22)] The Born-approximation relation used to set the Morse strength is extremely sensitive to r0 at the chosen values: with alpha=0.29, e^{alpha r0} is about 16.5, so the denominator e^{alpha r0}(e^{alpha r0}-16) is about 8.4, and d ln D/d ln r0 is approximately -95. A 1% change in r0 changes D by roughly 60%. In addition, r0=9.67 oscillator lengths corresponds to a range far larger than the physical van der Waals range, so the claim that this is a 'realistic potential model' is not supportable, and the mapping of the experimental scattering length to D in Eq. (22) is unreliable.
[Sec. 4, Tables 3-4 with Sec. 2.2, Eq. (17)] For excited states, the GFK fixed-node calculation is exact only if the nodal surfaces of the trial function coincide with those of the true interacting wavefunction. The trial functions are noninteracting harmonic-oscillator modes, and no test of fixed-node bias is reported for the interacting system. Therefore the abstract's statement that the method is 'exact in principle' should be qualified, or the excited-state nodal surfaces should be checked, before claiming that the excited-state energies are essentially exact.
[Sec. 5] The discussion of the Lieb-Yngvason and Dyson bounds is confused. The GP energy is not generally a lower bound to the true many-body energy, and the dilute-limit results of Lieb and Yngvason are asymptotic statements about convergence to the GP energy, not a bound of the kind stated in the conclusions. This does not invalidate the numerical method, but the argument that being 'lower than GP' is consistent with exactness needs to be restated correctly.

Minor objections (6)

[Abstract and throughout] The manuscript contains numerous typographical errors, such as 'interacing', 'exacxtly', 'potetial', 'summerize', 'attaractive', and 'priniciple', which should be corrected in a revision.
[Eq. (8)] The definition of Z_m appears to have an extra minus sign inside the exponential compared with the Feynman-Kac formula in Eq. (2); this should be checked carefully.
[Sec. 3.1, Eq. (21)] The notation gamma_1, gamma_2, and gamma_3 is inconsistent, and Eq. (21) appears to be missing a factor or a definition of s; the dimensionless strengths should be defined uniformly.
[Table 2] The bracketed reference values in Table 2 are not clearly attributed; the caption should identify reference [20] explicitly, and the same applies to bracketed values in Table 4.
[Figures 4-6] The y-axis ranges in Figures 4-6 make the wavefunctions appear almost constant; the figures should be replotted on a scale that shows the actual spatial variation or replaced with density profiles.
[Sec. 3.1] The sentence 'We have checked that for these choice of parameters, Morse solution is extremely good' is vague and should be replaced with a quantitative test of the two-body and many-body properties being used to justify the parameter choice.

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The paper

The paper claims

This paper claims that a generalized Feynman-Kac path-integral method, implemented as a guided random walk, can compute ground- and excited-state energies of trapped dilute Bose gases directly from a realistic Morse interaction, without a…

The reading

This paper argues that a generalized Feynman-Kac path-integral method, implemented as a guided random walk with trial wave functions, yields accurate ground- and excited-state energies for trapped dilute Bose gases without relying on mean-field approximations. It applies the method to rubidium-87, with repulsive interactions and positive scattering length, and lithium-7, with attractive interactions and negative scattering length, interacting through a Morse potential and confined in an anisotropic harmonic trap, with up to 200 atoms. The author claims the method is exact in principle within numerical limits, includes correlations omitted by mean-field treatments, and produces lower energies than the mean-field equation for all cases studied. The paper also reports that the virial theorem is satisfied and that a low-energy scattering approximation connecting the scattering length to the potential strength works in the dilute regime. If correct, this offers a simpler computational alternative for many-boson spectra.

What carries the argument

The central object is the generalized Feynman-Kac representation of the lowest energy for a given symmetry: rewrite $H=H_0+V_p$ with $H_0=-\Delta/2+\lambda_T+\Delta\psi_T/(2\psi_T)$, so that $V_p=V-(\lambda_T+\Delta\psi_T/(2\psi_T))$; then the energy is $\lambda=\lambda_T-\lim_{t\to\infty}\frac{1}{t}\ln E_x\exp\left(-\int_0^t V_p(Y(s))\,ds\right)$, where $Y(s)$ is a diffusion with drift $\nabla\psi_T/\psi_T$. The trial function $\psi_T$ from the $3N$-dimensional harmonic oscillator keeps the walk localized, so the potential is sampled efficiently. This is paired with a Morse interaction potential, a two-body potential with a repulsive core and an attractive well, $V(r)=D e^{-\alpha(r-r_0)}(e^{-\alpha(r-r_0)}-2)$, whose strength $D$ is related to the $s$-wave scattering length through the low-energy scattering formula $D=4\hbar^2 a\alpha^3/[m e^{\alpha r_0}(e^{\alpha r_0}-16)]$. The machinery converts the many-boson Schr\"odinger equation into a guided random-walk simulation that returns energy eigenvalues.

Extended reading notes

Core claim

The central discovery is that a generalized Feynman-Kac path integral, in which the random walk is guided by a trial wave function derived from the harmonic-oscillator trap, can determine the spectrum of a many-boson system with a realistic two-body Morse potential nonperturbatively. For Rb$^{87}$ with $\alpha=0.29$, $r_0=9.67$ and Li$^{7}$ with $\alpha=0.4$, $r_0=6.8$ (in oscillator units), the method yields ground- and excited-state energies per particle, chemical potentials, density profiles, and velocity-distribution aspect ratios for $N$ up to 200. The reported energies are consistently lower than the corresponding mean-field values, which the author attributes to the inclusion of interparticle correlations; for the isotropic small-$N$ case the results agree with an existing diffusion Monte Carlo calculation. The paper interprets this as evidence that the method, within numerical limitations, is exact in principle and a better alternative to mean-field theory.

Load-bearing premise

The load-bearing assumption is that a Morse-shaped two-body interaction with hand-chosen parameters ($\alpha=0.29$, $r_0=9.67$ for Rb; $\alpha=0.4$, $r_0=6.8$ for Li), linked to the measured scattering length through the first-order (Born) scattering approximation, faithfully represents the real atom-atom interaction.

Editorial extensions

If this is right

For dilute trapped bosons with up to 200 atoms, ground- and excited-state energies, chemical potentials, density profiles, and aspect ratios can be obtained from a single guided random-walk simulation rather than by solving a nonlinear mean-field equation.
Because the method treats correlations explicitly, its energies sit below mean-field values, which is consistent with the mean-field result acting as a variational upper bound in the regimes where that comparison is valid.
The low-energy scattering approximation connecting the scattering length to the Morse potential strength appears adequate for the dilute gases studied, supporting its use in similar many-body calculations.
The algorithm's structure is naturally parallel, so the same code can be extended to systems of thousands of atoms, as the paper notes is in progress.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A systematic scan over the Morse parameters $\alpha$ and $r_0$ would separate the method's intrinsic accuracy from the tuning of those parameters; the paper's own report that a different parameter choice gives substantially lower energies makes this the first test to run.
The same guided random walk could compute finite-temperature properties and correlation functions beyond energy and density by adding a temperature-dependent path-integral weight, which the energy-focused tables here do not address.
For attractive lithium, the method could locate the critical atom number for collapse directly from the many-body spectrum, rather than relying on mean-field stability arguments.
Because the trial function controls the nodal structure, adapting $\psi_T$ to non-Gaussian correlated forms could extend the approach to stronger interactions and possibly to fermionic systems, where the sign problem remains open.

Original abstract

We study the ground and excited states of weakly interacting Bose gases (with positive and negative scattering lengths) in connection with Bose Einstein Condensation to test the validity of the mean field theory and Born approximation. They behave as new quantum fluids (a gas in the weak limit and a liquid in the dense limit and we study their many body physics in the dilute limit within the realistic potential model (Morse type) by Feynman-Kac path integral technique. Within numerical limitations, this method is exact in principle and turns out to be a better alternative to GP as all the ground and excited state properties can be calculated in a much simpler way.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.85 · human participants 0.01 · controlled comparison 0.22 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.88 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We study the ground and excited states of weakly interacting Bose gases (with positive and negative scattering lengths) in connection with Bose Einstein Condensation to test the validity of the mean field theory and Born approximation. They behave as new quantum fluids (a gas in the weak limit and a liquid in the dense limit and we study their many body physics in the dilute limit within the realistic potential model (Morse type) by Feynman-Kac path integral technique. Within numerical limitations, this method is exact in principle and turns out to be a better alternative to GP as all the ground and excited state properties can be calculated in a much simpler way.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the ground and excited states of weakly interacting Bose gases (with positive and negative scattering lengths) in connection with Bose Einstein Condensation to test the validity of the mean field theory and Born approximation.	contribution 0.93	0.82	0.08	asserted 0.78
s1	They behave as new quantum fluids (a gas in the weak limit and a liquid in the dense limit and we study their many body physics in the dilute limit within the realistic potential model (Morse type) by Feynman-Kac path integral technique.	method 0.67	0.82	0.09	asserted 1.00
s2	Within numerical limitations, this method is exact in principle and turns out to be a better alternative to GP as all the ground and excited state properties can be calculated in a much simpler way.	result 0.46	0.86	0.66	asserted 0.97

What kind of paper the classifier reports this is: simulation study 0.53 · evidence from simulation or computation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** A plot for the Condensate Energy/Particle versus Numbe

**Figure 2.** A plot for the Condensate Energy/Particle versus Numbe

**Figure 3.** A plot for the Condensate Energy/Particle versus Numbe

**Figure 4.** Ground state wave function of Rb along the x axis for N = 10

**Figure 5.** Ground state wave function of Rb along the x axis for N = 40

**Figure 6.** Ground state wave function of Rb along the x axis for N = 10

**Figure 7.** Aspect ratio in Rb87 as a function of N. The horizontal line corresponds to a number close to √ λ [this work] 17

**Figure 8.** Ground state wave function of Li along the x axis for N = 10

**Figure 9.** Ground state wave function of Li along the x axis for N = 40

**Figure 10.** Ground state wave function of Li along the x axis for N = 1

**Figure 11.** Aspect ratio in li7 as a function of N. 5 Conclusions: Numerical work with bare Feynman-Kac procedure employing modern computers was reported[6] for the first time for few electron systems after forty years of original work[5] and seemed to be real useful for calculating atomic ground and excited states[7]. Tremendous success[9] in atomic physics motivated us to apply it to condensed matter physics. We have performed our calculations only with 200 particles. Nonetheless we have been able to demonstrate some of the ’Holy Grails’ of Bose Condensation, namely we have been able to…

Figures reproduced from arXiv:cond-mat/0603016 by the authors.

References

Works cited (25)

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A. Korzeniowski, J. L. Fry, D.E. Orr and N. G. Fazleev, Phys. Rev . Lett. 69, 893,(1992)

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J. Madox, Nature 358 707 (1992)

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M.Caﬀerel and P. Claverie, J. Chem Phys. 88 , 1088 (1988), 88, 1100 (1988)

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S. Datta, J. L Fry, N. G. Fazleev, S. A. Alexander and R. L. Coldw ell, Phys Rev A 61 (2000) R030502, Ph. D dissertation, The University of Texas at Arlington,(1996)

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A. Korzeniowski, J Comp and App Math, 66 333 (1996)

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J. L. duBois, Ph D dissertation, University of Delaware,(2003)

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W. Krauth, Phys. Rev Lett. 77 3695 (1995)

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M. D. Donsker and S. R. Varadhan, in Proc. of the Internation al Con- ference on Function space Integration ( Oxford Univ. Press 1975 )pp. 15-33. 24

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B. Simon, Functional Integrals and Quantum Mechanics ( Acade mic Press, New York, 1979 )

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P. B. Billingsley, Convergence of Probability measures,( Wiley, New York,1968)

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R. J. Dodd, J Res. Natl Inst. Stand. Technol 101,545(1996)

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[17]

W. Ketterle, D. S. Durpee and D. M. Stamper-Kurn in the Proce edings of international School of Physics edited by M. Inguscio, S. String rai, C. E. Wieman (1998)

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[18]

B. D. Esry, Phys Rev A 55 1147 1997, dissertation submitted to JILA 1996

View on Pith 1997
[19]

D. Blume, C. H. Green, Phys. Rev A 63 063601(2001)

View on Pith 2001
[20]

F. Dalfovo and S. Stringari, Phys. Rev A 53, 2477(1996)

View on Pith 1996
[21]

J. Tempere,F. Brosens, L. F. Lemmens, J. T. Devreese, Phys . Rev A 61 043605, 1,(2000)

View on Pith 2000
[22]

M. P. Singh and A. L. Satheesha Euro. Phys. J D 7 391 (1999

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[23]

E. H. Lieb and J. Yngvason, Phys. Rev. Lett 80 1998

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[24]

F. J. Dyson, Phys. Rev. 106,20, 1957

View on Pith 1957
[25]

E. A. Cornell and C. E. Wieman, Reviews in Modern Physics, 74 (2002) 25 Acknowledgements: Financial help from DST ( under Young Scientist Scheme (award no. SR/FTP/-76/2001 )) is gratefully acknowledged. The author would like to thank Prof J. K . Bhattacharjee, Indian Association for the Cultiv ation of Science, India for suggesting the problem and many st...

View on Pith 2002

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

M. H. Anderson, J.R. Ensher, M.R. Matthews, C. E. Wieman E. A. Cornell, Science 269,198 (1995)

View on Pith 1995

[2] [2]

V. L. Ginzburg and L.P. Pitaevski, Zh. Eksp. Teor Fiz, 34 1240(1958)[Sov. Phys. JETP 7, 858 (1958)], E.P. Gross, J. Math P hys.4, 195(1963)

View on Pith 1958

[3] [3]

S. Giorgini, J. Boronat and J. Casulleras, Phys. Rev A 60 5129 (1999)

View on Pith 1999

[4] [4]

B. D. Esry and C. H. Green, Phys. Rev A 60 1451, (1999)

View on Pith 1999

[5] [5]

M. D. Donsker and M. Kac, J. Res. Natl. Bur. Stand, 44 581 (1950), see also, M.Kac in Proceedings of the Second Berkley Symposium (Berkley Press, California, 1951 )

View on Pith 1950

[6] [6]

A. Korzeniowski, J. L. Fry, D.E. Orr and N. G. Fazleev, Phys. Rev . Lett. 69, 893,(1992)

View on Pith 1992

[7] [7]

J. Madox, Nature 358 707 (1992)

View on Pith 1992

[8] [8]

M.Caﬀerel and P. Claverie, J. Chem Phys. 88 , 1088 (1988), 88, 1100 (1988)

View on Pith 1988

[9] [9]

S. Datta, J. L Fry, N. G. Fazleev, S. A. Alexander and R. L. Coldw ell, Phys Rev A 61 (2000) R030502, Ph. D dissertation, The University of Texas at Arlington,(1996)

View on Pith 2000

[10] [10]

A. Korzeniowski, J Comp and App Math, 66 333 (1996)

View on Pith 1996

[11] [11]

J. L. duBois, Ph D dissertation, University of Delaware,(2003)

View on Pith 2003

[12] [12]

W. Krauth, Phys. Rev Lett. 77 3695 (1995)

View on Pith 1995

[13] [13]

M. D. Donsker and S. R. Varadhan, in Proc. of the Internation al Con- ference on Function space Integration ( Oxford Univ. Press 1975 )pp. 15-33. 24

View on Pith 1975

[14] [14]

B. Simon, Functional Integrals and Quantum Mechanics ( Acade mic Press, New York, 1979 )

View on Pith 1979

[15] [15]

P. B. Billingsley, Convergence of Probability measures,( Wiley, New York,1968)

View on Pith 1968

[16] [16]

R. J. Dodd, J Res. Natl Inst. Stand. Technol 101,545(1996)

View on Pith 1996

[17] [17]

W. Ketterle, D. S. Durpee and D. M. Stamper-Kurn in the Proce edings of international School of Physics edited by M. Inguscio, S. String rai, C. E. Wieman (1998)

View on Pith 1998

[18] [18]

B. D. Esry, Phys Rev A 55 1147 1997, dissertation submitted to JILA 1996

View on Pith 1997

[19] [19]

D. Blume, C. H. Green, Phys. Rev A 63 063601(2001)

View on Pith 2001

[20] [20]

F. Dalfovo and S. Stringari, Phys. Rev A 53, 2477(1996)

View on Pith 1996

[21] [21]

J. Tempere,F. Brosens, L. F. Lemmens, J. T. Devreese, Phys . Rev A 61 043605, 1,(2000)

View on Pith 2000

[22] [22]

M. P. Singh and A. L. Satheesha Euro. Phys. J D 7 391 (1999

View on Pith 1999

[23] [23]

E. H. Lieb and J. Yngvason, Phys. Rev. Lett 80 1998

View on Pith 1998

[24] [24]

F. J. Dyson, Phys. Rev. 106,20, 1957

View on Pith 1957

[25] [25]

E. A. Cornell and C. E. Wieman, Reviews in Modern Physics, 74 (2002) 25 Acknowledgements: Financial help from DST ( under Young Scientist Scheme (award no. SR/FTP/-76/2001 )) is gratefully acknowledged. The author would like to thank Prof J. K . Bhattacharjee, Indian Association for the Cultiv ation of Science, India for suggesting the problem and many st...

View on Pith 2002