Nonlinear Network description for many-body quantum systems in continuous space
NNonlinear Network description for many-body quantum systems in continuous space
Michele Ruggeri, Saverio Moroni, and Markus Holzmann
3, 4 Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany DEMOCRITOS National Simulation Center, Istituto Officina deiMateriali del CNR and SISSA, Via Bonomea 265, I-34136 Trieste, Italy LPMMC, UMR 5493 of CNRS, Universit´e Grenoble Alpes, F-38100 Grenoble France Institut Laue Langevin, BP 156, F-38042 Grenoble Cedex 9, France (Dated: November 7, 2017)We show that the recently introduced iterative backflow renormalization can be interpreted as ageneral neural network in continuum space with non-linear functions in the hidden units. We usethis wave function within Variational Monte Carlo for liquid He in two and three dimensions, wherewe typically find a tenfold increase in accuracy over currently used wave functions. Furthermore,subsequent stages of the iteration procedure define a set of increasingly good wave functions, eachwith its own variational energy and variance of the local energy: extrapolation of these energies tozero variance gives values in close agreement with the exact values. For two dimensional He, wealso show that the iterative backflow wave function can describe both the liquid and the solid phasewith the same functional form –a feature shared with the Shadow Wave Function, but now joinedby much higher accuracy. We also achieve significant progress for liquid He in three dimensions,improving previous variational and fixed-node energies for this very challenging fermionic system.
PACS numbers: PACS:
Explicit forms of many-body ground state wave func-tions have played an important role in the qualitative andquantitative understanding of many-body quantum sys-tems. Whereas pairing functions based on Bogoliubov’stheory [1] have provided a good description of superflu-idity and superconductivity of dilute gases, a full pair-product (Jastrow) wave function is usually the startingpoint for a microscopic description of liquid helium, theprototype of a strongly interacting, correlated quantumsystem. Starting from the first variational Monte Carlo(VMC) calculations of McMillan [2], liquid and solid he-lium – bosonic He as well as fermionic He – have trig-gered and challenged microscopic simulations to describemany-body quantum systems in two or three dimensionalcontinuous space.For systems described on a lattice, approaches basedon matrix product and tensor network states [3–7] haveprovided essentially exact description of many genericlow dimensional systems. Very recently, neural networkstates have been shown to lead to excellent results inone and two dimensional lattice models [8–11], a verypromising approach for lattice systems in two and threedimensions. However, generalization of these states tocontinuous systems [12] in two and three dimensions isdifficult or still lacking.In this work we elaborate on a recently introduced [13]class of wave functions for quantum many-body systemsin continuous space that include sets of auxiliary coor-dinates obtained with iterated backflow transformations.The wave function is viewed as a neural network wherethe hidden units of layer M are obtained iteratively asa function of the coordinates in layer M −
1, with layer M = 0 corresponding to the physical particles. In con-trast to neural networks on a lattice, all the functions involved here are in general non-linear. The network pa-rameters describing the various functions are optimizedwithin VMC simulations.We apply our description to liquid/solid He and liq-uid He, where we obtain a systematic lowering of theenergy as we increase the number of layers in the wavefunctions. For the bosonic systems we benchmark thequality of this explicit wave function with exact resultsobtained by stochastic projection Monte Carlo methods.We further show that our wave function is able to de-scribe equally well the fluid and the solid phase with thesame functional form, symmetric and translationally in-variant, qualitatively similar to the so-called shadow wavefunction (SWF) approach[14] but with over one order ofmagnitude gain in accuracy.Since the effective interaction between two heliumatoms, v ( r ), is quantitatively well known, a large quan-tity of computations exists which can be rather directlycompared to experiments. During the years several typesof wave functions have been used to simulate He. In thefirst VMC simulations [2], the wave function took into ac-count just two-body interparticle correlations; these wavefunctions were then generalized to include three-bodyand higher correlations, Ψ T ( R ) ∝ exp[ − U ( R )], where U ( R ) denotes a general, symmetric correlation function,and R ≡ ( r , r , . . . r N ) denotes the coordinate vector ofthe particles [15, 16].Exact results for bosonic He can be obtained improv-ing stochatically the wave function with Projector MonteCarlo techniques such as Diffusion Monte Carlo (DMC)[17] or Variational Path Integral methods[18–20]. Start-ing from any trial wave function, Ψ T ( R ), its propagation a r X i v : . [ c ond - m a t . o t h e r] N ov FIG. 1. Schematic representation of a SWF as a non-linearnetwork. The input layer is formed by the coordinates of thewave function, R , and we have to integrate over the coordi-nates in the hidden layer, R (cid:48) . Input and hidden layer coordi-nates are connected via a gaussian, whereas the coordinatesinside each layer are connected by the many-body correlationpotentials, V ( R ) and U ( R (cid:48) ). Including several hidden layerscorrespond to the application over several projection steps. in imaginary time, τ , can be written asΨ τ ( R ) ∝ (cid:90) d R (cid:48) G ( R , R (cid:48) ; τ )Ψ T ( R (cid:48) ) (1)For small τ , the functional form of G is given by G ( R , R (cid:48) ; τ ) ∝ exp (cid:2) − λ ( R − R (cid:48) ) − V ( R ) (cid:3) (2)with λ = m/ h τ and V ( R ) is given by the interparticlepotential, V ( R ) = τ (cid:80) i 1) which are obtained via q ( n ) i = q ( n − i + (cid:88) j (cid:16) q ( n − i − q ( n − j (cid:17) η ( n ) (cid:16) q ( n − ij (cid:17) (12) ρ = 0 . − E V MC /N σ /N E SWF /N E DMC /N J -6.8593(10) 14.80BF1 -6.9936(14) 3.03 -6.765(8) -7.0243(6)BF2 -7.0076(15) 2.14Extrap. -7.033(2) ρ = 0 . − E V MC /N σ /N E SWF /N E DMC /N J -6.9137(10) 21.40BF1 -7.1204(12) 5.22BF2 -7.1367(10) 3.30 -6.937(6) -7.1691(12)BF3 -7.1458(14) 2.36Extrap. -7.169(3) ρ = 0 . − E V MC /N σ /N E SWF /N E DMC /N J -6.0220(20) 49.99BF1 -6.4656(25) 11.20BF2 -6.5230(17) 9.34 -6.350(6) -6.5921(20)BF3 -6.5402(13) 5.84BF4 -6.5502(14) 6.87Extrap. -6.615(2)TABLE I. Ground-state energy per particle, in K, of liq-uid He in three dimensions at different densities, obtainedwith VMC ( E V MC /N , E SWF /N ) and DMC ( E DMC /N ) us-ing different types of trial wave functions: Jastrow wave func-tion without backflow ( J ), with n iterated backflow trans-formations (BF n ) and Shadow Wave Function[29]. We alsoreport the variance σ of E V MC and the extrapolation of E V MC /N to zero variance[13]. Statistical uncertainties onthe last digit(s) are given in parentheses. Tail corrections arecalculated assuming g ( r ) = 1 for distances larger than halfthe side of the simulation cell. The representations of the one-dimensional functions, u ( n )2 , ζ ( n ) , and η ( n ) , establish the network parameterswhich are determined by energy minimization using thestochastic reconfiguration method [28].Although each hidden layer increases the number ofvariational parameters, the scaling of the computationaleffort for evaluation of the wave function with respect tothe number of atoms, N , does not increase [13].In Fig. 3 and table I, we show the error in the groundstate energy obtained for N = 64 He atoms in threedimensions. We have considered the liquid at three dif-ferent densities, ρ = 0 . − (negative pressure), ρ = 0 . − (equilibrium) and ρ = 0 . − (freez-ing). The error of the Jastrow or Shadow wave functionsranges in the tenths of K. The first backflow layer al-ready results in significantly better variational energies,and additional layers of backflow transformations bringthe error down to a few hundredth K.Apart from the energy, also the variance of the localenergy, σ , can be computed at each iteration level, M ,without additional computational costs. Under suitableconditions [13] the extrapolation of the energy to σ = 0with a leading linear term gives the exact ground stateenergy. The largest error in the extrapolated values is 2 3 4 5 6 7r (Å)0.00.51.01.5 2 3 4 5 6 7 −0.50.00.51.0 g (r) · D g (r) r (Å) FIG. 4. Pair correlation functions g ( r ) for liquid He inthree dimensions at equilibrium density. Dashed lines (leftscale) show variational results without backflow terms (blue)and with three backflow iterations (red), as well as DMC re-sults (black, barely visible behind the red dashes; extrapo-lated estimate[17] using the BF3 trial wave function). Solidlines (right scale) show a tenfold magnification ot the devia-tion between the VMC and DMC results. −3 −2 −1 ( E − E ) / T r (Å −2 ) FIG. 5. Error of the ground state energy of liquid and solid He in two dimension in units of the kinetic energy T acrossthe the liquid-solid coexistence region (shaded)[32], using var-ious wave functions: Shadow[30], Jastrow, Nosanow and it-erative backflow network (5 iteration layers at coexistence, 4layers otherwise). -0.02K at the highest density.A roughly tenfold increase in accuracy is also obtainedin the pair correlation function g ( r ), as shown in Fig. 4for the equilibrium density.In order to describe freezing, the liquid and the solidphase must usually be described by different functionalforms within VMC, as the usual Jastrow wave functionis in general unable to localize the atoms in a crystal(unless the pair pseudopotential is made unreasonablyhard). This bias propagates even to projector MonteCarlo methods (DMC) based on importance sampling us-ing the Jastrow function. In order to correctly describethe solid phase, usually one uses in VMC and DMC anunsymmetrized Nosanow wave function [31] where theatoms are individually tied to predetermined lattice sitesby a one-body term. In this setup, the Jastrow(Nosanow)phase describes a metastable liquid(solid) phase at den-sities higher(lower) than the coexistence region.One important conceptual progress of SWF was thepossibility to describe both liquid and solid He withinthe same wave function [14], without breaking transla-tional invariance or Bose symmetry. Remarkably, thisfeature is shared by our network wave function. InFig. 5, we compare the performance of network, Jastrow-Nosanow, and Shadow wave functions for N = 16 He atoms in two dimensions around the liquid-solidtransition[32]. Again, our backflow network functionachieves a roughly tenfold reduction of the variationalerror with respect to Shadow[30], Jastrow, and Nosanowwave functions –over a large density range and acrossa phase transition. For the higher density, ρ = 0 . g ( x, y ) calculated with VMC, which is hardly distinguish-able from the Nosanow (bona fide solid) result. For thelowest density g ( x, y ) turns into a radial, liquid-like pairdistribution function, while at coexistence it is interme-diate between the Nosanow and Jastrow results, muchcloser to the former.Up to now, we have demonstrated the quality of ourbackflow network to describe bosonic quantum systems,where stochastic projection Monte Carlo methods pro-vide exact results for benchmarking. Now, we showthat our approach significantly improves the descrip-tion of strongly correlated fermions in three dimensions,similar to previous results [13] obtained for two dimen-sional liquid He. In table II we list estimates of theground state energy obtained with different wave func-tions for N = 66 He atoms in three dimensions, atequilibrium and freezing density. The previous best es-timates [16] were obtained introducing explicit correla-tions up to four-particle in the Jastrow factor and three-particle in the backflow coordinates. The results fromRef. [16] included in table II refer to a spin-singlet pair-ing wave function, which performs marginally better thana Slater determinant of plane waves. They lie betweenBF1 and BF2, showing that the implicit inclusion of cor-relations at all orders through backflow iteration is moreeffective than explicit construction of successive n -orderterms (although nothing prevents the two approaches tobe combined). Furthermore, systematic improvement ismore easily obtained by adding further layers of backflowtransformations than further explicit correlations.In the lack of exact benchmark results for thisfermionic case, we compare our results to the experimen-tal equation of state [33]. The HFDHE2 pair potential ρ = 0 . − E V MC /N σ /N E FNDMC /N E EXP /N J -1.6812(17) 38.23 -2.0925(16)BF1 -2.0844(13) 16.70 -2.2760(10)BF2 -2.2278(23) 8.10 -2.3190(14) -2.481BF3 -2.2576(15) 5.60 -2.3288(14)Extrap. -2.34(1)Ref. [16] -2.168(3) 14 -2.306(4) ρ = 0 . − E V MC /N σ /N E FNDMC /N E EXP /N J 0.7582(32) 111.06 -0.2890(43)BF1 0.0952(17) 65.03 -0.5572(41)BF2 -0.3661(42) 29.43 -0.6545(15) -0.918BF3 -0.5177(32) 17.47 -0.6911(16)BF4 -0.5531(27) 14.41 -0.7013(42)Extrap. -0.723(3)Ref. [16] -0.127(5) 49 -0.6485(4)TABLE II. Ground-state energy per particle, in K, of liquid He in three dimensions at equilibrium and freezing densi-ties, obtained with VMC ( E V MC /N ) and fixed-node DMC( E FNDMC /N ) using different types of trial wave functions:Jastrow wave function without backflow ( J ), and with n it-erated backflow transformations (BF n ). We also report thevariance σ of E V MC , the extrapolation of E V MC /N to zerovariance[13], and the experimental value E EXP /N [33]. Sta-tistical uncertainties on the last digit(s) are given in paren-theses. Tail corrections are calculated assuming g ( r ) = 1 fordistances larger than half the side of the simulation cell. Theresults from entry S3BF4 of table I of Ref. [16] are correctedby a perturbative estimate of the difference due to their useof a different [34] pair potential v ( r ). adopted here is accurate within a few hundredth of aK from equilibrium to freezing density for He [35], andpresumably equally reliable also for He at slightly lowerdensities; furthermore the number of particles, N = 66, ischosen to give a small finite-size shell effect on the kineticenergy, so that the energies in the table are reasonablyclose to the thermodynamic limit. Our best estimate,the extrapolation to zero variance, is higher than the ex-perimental energy by 0.14 K at equilibrium density, andby 0.19 K at freezing. This comes to a surprise, as for He in two and three dimension and small systems of Hein two dimensions –very similar cases where exact resultsare available– the error in the zero-variance extrapolationis of order of 0.01 K. Nevertheless the improvement overthe previous variational and fixed-node energies remainssignificant.In summary, in this paper, we have put the iteratedbackflow description [13] in a more general frame. Wehave demonstrated the quality of our backflow networkfor quantitative description of bosonic and fermionic he-lium systems in two and three dimensional continuousspace.In our description, each backflow transformation cor-responds to a hidden layer, and each new layer dependson the coordinates in the previous one. 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