One of Feynman's early applications of path integrals was to superfluid $^{4}\mathrm{He}$. He showed that the thermodynamic properties of Bose systems are exactly equivalent to those of a peculiar type of interacting classical "ring polymer." Using this mapping, one can generalize Monte Carlo simulation techniques commonly used for classical systems to simulate boson systems. In this review, the author introduces this picture of a boson superfluid and shows how superfluidity and Bose condensation manifest themselves. He shows the excellent agreement between simulations and experimental measurements on liquid and solid helium for such quantities as pair correlations, the superfluid density, the energy, and the momentum distribution. Major aspects of computational techniques developed for a boson superfluid are discussed: the construction of more accurate approximate density matrices to reduce the number of points on the path integral, sampling techniques to move through the space of exchanges and paths quickly, and the construction of estimators for various properties such as the energy, the momentum distribution, the superfluid density, and the exchange frequency in a quantum crystal. Finally the path-integral Monte Carlo method is compared to other quantum Monte Carlo methods.

Path integrals in the theory of condensed helium

Physical Content, Analytical Derivation, and Rigorous Extraction of Magnetic Hamiltonians Jean Paul Malrieu,† Rosa Caballol,‡ Carmen J. Calzado, Coen de Graaf,‡,∥ and Nathalie Guiheŕy*,† †Laboratoire de Chimie et Physique Quantiques, Universite ́ de Toulouse 3, 118 route de Narbonne, 31062 Toulouse, France ‡Departament de Química Física i Inorgaǹica, Universitat Rovira i Virgili, Marcel·lí Domingo s/n, 43007 Tarragona, Spain Departamento de Química Física, Universidad de Sevilla, Profesor Garcia Gonzalez s/n, 41012 Sevilla, Spain Institucio ́ Catalana de Recerca i Estudis Avanca̧ts (ICREA), Passeig Lluis Companys 23, 08010 Barcelona, Spain

Magnetic interactions in molecules and highly correlated materials: physical content, analytical derivation, and rigorous extraction of magnetic Hamiltonians.

The theory of magnetism in solid $^{3}\mathrm{He}$ is reviewed with emphasis on the multiple-spin-exchange theory of the magnetic interaction. Critical experiments are discussed briefly, and the most reliable present values for various quantities are chosen. Various theories of $^{3}\mathrm{He}$ magnetism are considered, with the conclusion that multiple particle exchange is the most likely mechanism. The microscopic theory of exchange is presented mainly from the standpoint of tunneling ducts. The spin Hamiltonian which results from two-, three-, and four-particle ring exchange is derived and studied in the high-temperature limit, in mean-field approximation, and with spin waves. Two reasonable parameter sets are selected which fit the experimental data, with the conclusion that the exchanges are mainly three- and four-particle rings. A discussion is given of remaining problems together with some conjectures.

Magnetism in solid He-3

We present an up-to-date survey of theoretical concepts and results in the field of one-dimensional magnetism and of their relevance to experiments and real materials. Main emphasis of the chapter is on quantum phenomena in models of localized spins with isotropic exchange and additional interactions from anisotropy and external magnetic fields.

One-dimensional magnetism

X-ray diffraction and diamond anvil techniques were used to measure the isothermal compressibility of K 3 C 60 and Rb 3 C 60 , the superconducting, binary alkali-metal intercalation compounds of solid buckminsterfullerene. These results, combined with the pressure dependence of the superconducting onset temperature T c measured by other groups, establish a universal first-order relation between T c and the lattice parameter a over a broad range, between 13.9 and 14.5 angstroms. A small secondorder intercalate-specific effect was observed that appears to rule out the participation of intercalate-fullerene optic modes in the pairing interaction.

https://science.sciencemag.org/content/sci/255/5046/833.full.pdf

Compressibility of M3C60 Fullerene Superconductors: Relation Between Tc and Lattice Parameter

Four-Spin Exchange Model and 3 He Magnetism

A Schrodinger equation is derived which allows an improvement in the wave function of a quantum crystal inside the tunneling duct for exchange. This equation is used to derive a simple estimate of the exchange rate accurate enough to be used to compare various exchange mechanisms (two-body, three-body cyclic, etc.). We find that the 3
 He bulk hcp crystal and 3
 He surface triangular crystals have their exchange dominated by three-body cyclic exchange. Therefore the dominant part of the spin Hamiltonian will be Heisenberg nearest neighbor ferromagnetic. This result explains the ferro-magnetic tendency observed in adsorbed 3
 He films. A ferromagnetic transition in hcp 3
 He at the lowest pressures is predicted at 0.05±0.01 mK, corresponding to a Curie-Weiss constant of 0.075 \+ -0.015 mK.

Exchange and magnetic order in HCP 3He and adsorbed 3He with triangular lattice

2014 A two parameter model including only four and three spin exchange fits the entropy curve remarkably well down to 1 mK, agrees with the unusual experimental results on the susceptibility and specific heat and gives the main features of the observed phase diagram. J. Physique LETTRES 41 (1980) L-139 L-142 15 MARS 1980, Classification Physics Abstracts 67.80 The experimental magnetic properties of bcc solid 3He in the millikelvin range, near the melting curve are in contradiction with the usual antiferromagnetic Heisenberg model [1]. i) An abrupt drop of entropy suggesting a first order transition is observed at 7~ = 1 mK ~ 0/3 (9 ^~ 3 mK : Curie Weiss constant) [2]. In a magnetic field, the transition remains apparently first order with 7~ slightly decreasing with H up to 77~ 4 kG. Above 4 kG, a second order transition is observed with T c2 increasing rapidly with H [2, 3]. The experimental phase diagram (see Fig. 4 of Ref. [1]) suggests a competition between two phases : one occurring at low field is antiferromagnetic with a low susceptibility, the second arising at high field has a high susceptibility and consequently some ferromagnetic tendency. The smallness of the critical field 7~77~/2 kB 0 ~ 0.1 indicates that these two phases are almost degenerate and suggests the proximity of a bicritical point. ii) At 7~ T;;5 10 mK, the susceptibility /(77 -~ 0), in the paramagnetic (P) phase is larger than the extrapolated Curie Weiss value [4]. The curve Z 1 (T) has been fitted with a high temperature series expansion limited to the first three terms [5] : x-1(T) = C-1(T 0+ BIT); (C = N(y~C/2)2), with () = 2.6 mK and B = 2.7 mK. This behaviour is the opposite of what is observed in usual antiferromagnets described by a Heisenberg model where (*) Michigan State University, East Lansing, MI 48824, U.S.A. (./~ : interaction between the nth neighbours and zn : number of nth neighbours) is positive. iii) The specific heat measurements [6, 7] give C, = 0.25~2 T -2 j3 T -3) with e2 ~ 7 iL 0.5 mK2 at the melting molar volume V = 24.25 cm3 and e3 less accurate but positive. A Heisenberg model with nearest (or nearest and next nearest) neighbour exchange gives e3 0. These unusual properties have led to different theoretical models. Some of them add to the nearest neighbour antiferromagnetic Heisenberg (HNNA) model a coupling to another system : a) phonons (i.e. magnetoelastic constraints [8]), b) ground state vacancies giving spin polarons [9, 10]. The models (b) lead to anomalies in the high temperature range (30 to 200 mK) [10], for which there is at present no experimental confirmation [6, 7]. These models introduce ferromagnetic tendencies and reproduce some features of the system : in particular ii). Except for reference [9] which gives, in contradiction with experiment, a second order transition at low field and a first order transition at high field, there are few detailed theoretical studies of the phase diagram in the H, T plane for these models. Another type of model generalizes the Heisenberg Hamiltonian by adding multiple (three, four spin...) exchange interactions [11, 12, 13] ; it gives no anomalies for T &#x3E; 0 and fits quantitatively the largest number of experimental data. Starting here with the idea that some kinds of permutations : four spin exchange (with possibly three spin exchange) can be preponderant and thus Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:01980004106013900 L-140 JOURNAL DE PHYSIQUE LETTRES neglecting pair exchange, we put forward a two parameter simpler multiple exchange model which fits all available data better than the previous models. We cannot trust the exchange calculations based on the one particle gaussian wave function [11] which in 3He does not satisfy the conditions defined by Herring for a reliable Home based function [14]. (The decrease of the real wave function in the exchange configuration has no reason to be gaussian [15].) So this approximation cannot even give the hierarchy of different kinds of exchange. We now put forward simple geometrical arguments based on the hard sphere model which [15] is valid for calculating the energy of solid 3He [16]. For the problem of n exchanging hard spheres, two essential parameters are to be considered : a) the free space b available in the box shaped by the surrounding atoms in the exchange configuration; b) the total displacement of the particles in the 3 N dimensional space of configuration during the exchange process. The exchange increases with 6 and decreases with d. There is an optimum between these two conditions which can favour one type of exchange. In bcc 3 He, the free space 5 is much larger for three or four spin exchange than for pair exchange [15]. Although d increases with the number of exchanging atoms, four and three spin exchanges are probably larger than pair exchange. These conjectures have been corroborated by quantitative calculations [15]. Higher order (five, six spin...) exchanges have comparable free space 6 with four spin exchange but they are disfavoured by the increasing d. (In hcp 3He, we believe that three spin exchange is most likely the dominant contribution [17].) We suppose here first that folded four spin exchange involving the most compact four atom tetrahedron (F) [12] dominates and we neglect all other exchanges. The Hamiltonian is : 5’ijkl being the four particle cyclic permutation operator. Expressing ~~ in terms of spin operators, we obtain : with Contrary to the other models, we can have exact results in the high temperature region r ~&#x3E; 0. A high temperature series expansion of the partition function Z = Tr exp(~i~), (JC is the sum of the exchange and Zeeman Hamiltonians) gives [13] :

/pdf/most-experimental-magnetic-properties-of-solid-bcc-3he-are-1r06mx22nn.pdf

Most experimental magnetic properties of solid bcc 3He are described by a two parameter model : three and four spin exchange

The zero point motion of quantum solids gives a finite frequency of exchange for the atoms. Calculations based on two-body Jastrow functions giving essentially two-atom exchange, cannot describe correctly the true geometrical hard core correlations. A simple model of four hard spheres in a box shows that four-atom cyclic exchange can be much larger than two-atom exchange. We propose a new method taking into account the hard core correlations. This model applied to solid 3He leads to preponderant four-spin exchange in the b.c.c. phase as shown by the experimental results and to a ferromagnetic three-spin exchange in h.c.p. 3He or adsorbed 3He or cyclic three-atom permutation in h.c.p. solid H2.

https://hal.archives-ouvertes.fr/jpa-00220175/document

J. M. Delrieu

Papers

Magnetism in solid He-3

Four-Spin Exchange Model and 3 He Magnetism

Exchange and magnetic order in HCP 3He and adsorbed 3He with triangular lattice

Most experimental magnetic properties of solid bcc 3He are described by a two parameter model : three and four spin exchange

Physical origin of multiple three and four spin (or atoms) exchange in quantum solids with some consequences