A quantum spin system can be modelled by an equivalent classical system, with
an effective Hamiltonian obtained by integrating all non-zero frequency modes
out of the path integral. The effective Hamiltonian H_eff(S_i) derived from the
coherent-state integral is highly singular: the quasiprobability density
exp(-beta H_eff), a Wigner function, imposes quantisation through derivatives
of delta functions. This quasiprobability is the distribution of the
time-averaged lower symbol of the spin in the coherent-state integral. We
relate the quantum Monte Carlo minus-sign problem to the non-positivity of this
quasiprobability, both analytically and by Monte Carlo integration.Comment: 4 page

Samson, J H

English

arXiv

A quantum spin system can be modelled by an equivalent classical system, with an effective Hamiltonian obtained by integrating all non-zero frequency modes out of the path integral. The effective Hamiltonian H_eff(S_i) derived from the coherent-state integral is highly singular: the quasiprobability density exp(-beta H_eff), a Wigner function, imposes quantisation through derivatives of delta functions. This quasiprobability is the distribution of the time-averaged lower symbol of the spin in the coherent-state integral. We relate the quantum Monte Carlo minus-sign problem to the non-positivity of this quasiprobability, both analytically and by Monte Carlo integration

John Samson (1251294)

Loughborough University Institutional Repository

arXiv:quant-ph/9905089 v1   26 May 1999EXACT CLASSICAL EFFECTIVE POTENTIALSJ. H. SAMSONDepartment of Physics, Loughborough University, Loughborough, Leics LE11 3TU UKE-mail: j.h.samson@lboro.ac.ukPresented at Sixth International Conference on Path Integrals from peV toTeV, Firenze, 26 August 1998A quantum spin system can be modelled by an equivalent classical system, with an effectiveHamiltonian obtained by integrating all non-zero frequency modes out of the path integral.The effective Hamiltonian Heff ({Si}) derived from the coherent-state integral is highly sin-gular: the quasiprobability density exp(−βHeff ), a Wigner function, imposes quantisationthrough derivatives of delta functions. This quasiprobability is the distribution of the time-averaged lower symbol of the spin in the coherent-state integral. We relate the quantum MonteCarlo minus-sign problem to the non-positivity of this quasiprobability, both analytically andby Monte Carlo integration.1 IntroductionIt is possible, therefore, that a closer study of the relation of classical andquantum theory might involve us in negative probabilities, and so it does.R P Feynman1One may distinguish two broad numerical approaches to quantum statistical me-chanics. One can evaluate a path integral by direct Monte Carlo methods. Herethe notorious minus-sign problem (or phase problem) often hinders direct evalua-tion of the path integral: this refers to a rapid oscillation of the integrand in sign orphase, and the resulting intolerably slow convergence of integrals evaluated by ran-dom sampling. Alternatively, one can integrate out the quantum fluctuations, leavingan effective Hamiltonian Heff(x) with c-number variables x to be used in a classicalsimulation2. A similar non-positivity arises here if the observables described by xare incompatible: the distribution exp(−βHeff(x)) is then a Wigner function, whichis not in general positive-definite.A free spin s serves here as a useful toy model to relate the path integral andthe Wigner function3. The spin coherent-state path integral, a poor starting point fornumerical simulations, presents the worst case of the sign problem: for continuouspaths the integrand is a pure Berry phase factor. The spin Wigner function is highlysingular, consisting of derivatives of delta functions supported on concentric spheresof quantised radius3,4. We demonstrate the common origin of these non-positivitiesboth analytically and numerically.Firenze: submitted to World Scientific on March 3, 2006 12 Classical effective HamiltonianWe consider a system with density matrix ρˆ ≡ exp(−βHˆ), and assign c-numbervariables x ∈ RN to operators xˆ. We define the Wigner function W (x) and classicaleffective Hamiltonian Heff(x) of these variables asW (x) ≡ exp(−βHeff(x)) = Tr (ρˆδN (x− xˆ)) , (1)where δN is the N -dimensional delta function. To remove operator-ordering ambi-guity, we defineδN (x− xˆ) ≡∫dNλ(2pi)Nexp(iλ · (x − xˆ)). (2)If the components of xˆ do not commute, the delta operator need not be positive.Here we restrict consideration to the spin operator xˆ = Sˆ in the spin-s represen-tation. The distribution is the spin Wigner function Ws(S). For a single spin withvanishing Hamiltonian the trace of Eq. (2) is easily evaluated3,4 to give derivativesof delta functions supported on concentric spheres of quantised radius:Ws(S) = 〈δ3(S− Sˆ)〉 =−12s+ 1s∑m=−s12piSddSδ(S −m), (3)where S = |S| and Sˆ2 = s(s + 1). The spheres with m < 0 do not contribute andfor integer spin there is a positive delta function at the origin of weight−δ′(S)2piS(2s+ 1)=δ3(S)(2s+ 1). (4)To motivate this form, we need to verify that the correct marginal distribu-tions are obtained. Integrating Ws over a plane yields the expected distribution∑sm=−s δ(e ·S−m)/(2s+ 1) for the normal component of spin3. For two coupledspins 1/2, with Hamiltonian −JS1 ·S2, a convolution integral of the two single-spinfunctions W1/2 gives (after some manipulations)W (S1 + S2) =3eβJ/4W1(S1 + S2) + e−3βJ/4W0(S1 + S2)3eβJ/4 + e−3βJ/4. (5)This is the correct thermal average of triplet and singlet distributions, in line withthe effective-Hamiltonian interpretation. Eq. (5) can be further interpreted as a lo-cal hidden variable distribution, which must give the correct correlations for spinmeasurements on two electrons. No positive definite distribution exists for a singletstate4,5, and the distribution must therefore be non-definite.Firenze: submitted to World Scientific on March 3, 2006 23 Coherent-state path integralsSpin coherent states |n〉 for spin s are labelled by a unit vector n, such thatn · Sˆ|n〉 = s|n〉. To relate the Wigner functions just described to the coherent statepath integral, we split the exponent of Eq. (2) into L time slices,Ws(S) = 〈δ3(S− Sˆ)〉 =∫d3λ8pi3eiλ·S〈(1− iλ · Sˆ/L+O(L−2))L〉. (6)We now insert a resolution of unity in the form 2s+14pi∫dn|n〉〈n| between each slice6,represent the spin by its lower symbol (s+ 1)n7,8,Sˆ =2s+ 14pi∫dn(s+ 1)n|n〉〈n|, (7)and re-exponentiate. The lower symbol is used as the paths (here and in our MonteCarlo calculation) are only piecewise continuous. This givesWs(S) = limL→∞WLs (S), (8)the limit defined in the sense of a distribution, with approximantsWLs (S) =12s+ 1∫d3λ8pi3eiλ·S∫ L∏i=1(2s+ 14pidnie−i(s+1)λ·ni)××〈n1 | n2〉 〈n2 |· · ·|nL〉 〈nL | n1〉 (9)=12s+ 1∫ L∏i=1(2s+ 14pidni)δ3(S− (s+ 1)n¯) 〈n1 | n2〉 · · · 〈nL | n1〉 ,where n¯ =∑Li=1 ni/L is the time-averaged coherent-state label. We therefore havea sequence of well-defined distributions (9) convergent on the Wigner function. Forlarge L the discretised distribution WLs (S) can be shown to be asymptotically asmeared Wigner function; for spin 1/2 we findWL1/2(S) ≈(Lpi)3/2 ∫d3xe−L(x−S)2W1/2(S). (10)In order to verify this result and investigate the importance of the sign prob-lem, we have computed the discretised Wigner function (9) by direct Monte Carlosampling of the histogram for n¯. Each path comprises L independent pointsni, i = 1 . . . L, taken from a uniform distribution on the sphere. Since the prod-uct of overlaps in Eq. 9 is complex for L > 2, a phase problem arises. For theresults shown in Fig. 1, 107 independent paths were generated for each L from 2to 15. Further analysis shows the numerical distributions converging to the WignerFirenze: submitted to World Scientific on March 3, 2006 3-606120 0.2 0.4 0.6 0.8 1WL1/2(S) s=1⁄2S⁄(s+1)L-6-3036912150 0.25 0.5 0.75 1WL1(S) s=1S⁄(s+1)LFigure 1. Discretised Wigner function for spin 1/2 and 1 with L = 2 . . . 15 time slices and 107 MonteCarlo steps.function (3) according to Eq. (10). We also see how the phase fluctuations lead tonumerical instabilities. The correct distribution would be obtained if the the numberof Monte Carlo steps were taken to infinity before the number of time slices. It isclear from the figure that 107 steps are insufficient for convergence for spin 1; resultsfor higher spin show still slower convergence.We have thus demonstrated the emergence of a non-positive quasiprobability(the spin Wigner function) from the Berry phases of the integral. Numerical calcula-tions are however rapidly overwhelmed by the sign problem.References1. R. P. Feynman, in Quantum Implications, ed. B. J. Hiley and F. D. Peat (Rout-ledge, London, 1987), 2352. R. Giachetti and V. Tognetti, Phys. Rev. Lett. 55, 912 (1985)3. J. H. Samson, Phys. Rev. B51, 223 (1995)4. C. Chandler et al Found. Phys. 22, 867 (1992); M. O. Scully and K.Wo´dkiewicz, Found. Phys. 24, 85 (1993)5. J. S. Bell, Physics 1, 195 (1964)6. J. R. Klauder, J. Math. Phys. 23, 1797 (1982)7. E. H. Lieb, Commun. Math. Phys. 31, 327 (1973)8. I. Daubechies and J. R. Klauder, J. Math. Phys. 26, 2239 (1985)Firenze: submitted to World Scientific on March 3, 2006 4

Exact classical effective potentials

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Exact Classical Effective Potential

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