András Sütő

The charge flipping algorithm

This paper summarizes the current state of charge flipping, a recently developed algorithm of ab initio structure determination. Its operation is based on the perturbation of large plateaus of low electron density but not directly on atomicity. Such a working principle radically differs from that of classical direct methods and offers complementary applications. The list of successful structure-solution cases includes periodic and aperiodic crystals using single-crystal and powder diffraction data measured with X-ray and neutron radiation. Apart from counting applications, the paper mainly deals with algorithmic issues: it describes and compares new variants of the iteration scheme, helps to identify and improve solutions, discusses the required data and the use of known information. Finally, it tries to foretell the future of such an alternative among well established direct methods.

Ground State at High Density

Weak limits as the density tends to infinity of classical ground states of integrable pair potentials are shown to minimize the mean-field energy functional. By studying the latter we derive global properties of high-density ground state configurations in bounded domains and in infinite space. Our main result is a theorem stating that for interactions having a strictly positive Fourier transform the distribution of particles tends to be uniform as the density increases, while high-density ground states show some pattern if the Fourier transform is partially negative. The latter confirms the conclusion of earlier studies by Vlasov (1945), Kirzhnits and Nepomnyashchii (1971), and Likos et al. (2007). Other results include the proof that there is no Bravais lattice among high-density ground states of interactions whose Fourier transform has a negative part and the potential diverges or has a cusp at zero. We also show that in the ground state configurations of the penetrable sphere model particles are superimposed on the sites of a close-packed lattice. PACS: 61.50.Ah, 02.30.Nw, 61.50.Lt

Thermodynamic Limit and Proof of Condensation for Trapped Bosons

We study condensation of trapped bosons in the limit when the number of particles tends to infinity. For the noninteracting gas we prove that there is no phase transition in any dimension, but in any dimension, at any temperature the system is 100% condensated into the one-particle ground state. In the case of an interacting gas we show that for a family of suitably scaled pair interactions, the Gross–Pitaevskii scaling included, a less-than-100% condensation into a single-particle eigenstate, which may depend on the interaction strength, persists at all temperatures.

Galilean invariance in confined quantum systems: implications for spectral gaps, superfluid flow, and periodic order.

Galilean invariance leaves its imprint on the energy spectrum and eigenstates of N quantum particles, bosons or fermions, confined in a bounded domain. It endows the spectrum with a recurrent structure, which in capillaries or elongated traps of length L and cross-section area s⊥ leads to spectral gaps n2h2s⊥ρ/(2mL) at wave numbers 2nπs⊥ρ, where ρ is the number density and m is the particle mass. In zero temperature superfluids, in toroidal geometries, it causes the quantization of the flow velocity with the quantum h/(mL) or that of the circulation along the toroid with the known quantum h/m. Adding a ”friction” potential which breaks Galilean invariance, the Hamiltonian can have a superfluid ground state at low flow velocities but not above a critical velocity which may be different from the velocity of sound. In the limit of infinite N and L, if N/L = s⊥ρ is kept fixed, translation invariance is broken, the center of mass has a periodic distribution, while superfluidity persists at low flow velocities. This conclusion holds for the Lieb-Liniger model.

A possible mechanism of concurring diagonal and off-diagonal long-range order for soft interactions

This paper is a contribution to the theory of coherent crystals. We present arguments claiming that negative minima in the Fourier transform of a soft pair interaction may give rise to the coexistence of diagonal and off-diagonal long-range orders at high densities, and that this coexistence may be detectable due to a periodicity seen on the off-diagonal part of the one-body reduced density matrix, without breaking translation invariance. As an illustration, we study the ground state of a homogenous system of bosons in continuous space, from the interaction retaining only the Fourier modes v(k) belonging to a single nonzero wave number |k|=q. The result is a mean-field model. We prove that for v(k)>0 the ground state is asymptotically fully Bose condensed, while for v(k)<0 at densities exceeding a multiple of ℏ2q2/2m|v(k)| it exhibits both Bose–Einstein condensation and diagonal long-range order, and the latter can be seen on both the one- and the two-body density matrix.

Bose-Einstein condensation and symmetry breaking

Adding a gauge symmetry breaking field -\nu\sqrt{V}(a_0+a_0^*) to the Hamiltonian of some simplified models of an interacting Bose gas we compute the condensate density and the symmetry breaking order parameter in the limit of infinite volume and prove Bogoliubovs asymptotic hypothesis \lim_{V\to\infty}/\sqrt{V}={\rm sgn}\nu \lim_{V\to\infty}\sqrt{/V} where the averages are taken in the ground state or in thermal equilibrium states. Letting \nu tend to zero in this equation we obtain that Bose-Einstein condensation occurs if and only if the gauge symmetry is spontaneously broken. The simplification consists in dropping the off-diagonal terms in the momentum representation of the pair interaction. The models include the mean field and the imperfect (Huang-Yang-Luttinger) Bose gas. An implication of the result is that the compressibility sum rule cannot hold true in the ground state of the one-dimensional mean-field Bose gas. Our method is based on a resolution of the Hamiltonian into a family of single-mode (k=0) Hamiltonians and on the analysis of the associated microcanonical ensembles.

Superimposed particles in 1D ground states

For a class of nonnegative, range-1 pair potentials in one-dimensional continuous space we prove that any classical ground state of lower density ≥1 is a tower-lattice, i.e. a lattice formed by towers of particles the heights of which can differ only by 1, and the lattice constant is 1. The potential may be flat or may have a cusp at the origin; it can be continuous, but its derivative has a jump at 1. The result is valid on finite intervals or rings of integer length and on the whole line.

A Probabilistic Approach to the Models of Spin Glasses

Introducing the notions of quenched and annealed probability measures, a systematic study of some problems in the description of spin glasses is attempted. Inequalities and variational principles for the free energies are derived. The absence of spontaneous breakdown of the gauge symmetry is discussed and some high-temperature properties are studied. Examples of annealed models with more than one phase transition are shown.

The total momentum of quantum fluids

The probability distribution of the total momentum P is studied in N-particle interacting homogeneous quantum systems at positive temperatures. Using Galilean invariance we prove that in one dimension, the asymptotic distribution of P/N is normal at all temperatures and densities, and in two dimensions, the tail distribution of P/N is normal. We introduce the notion of the density matrix reduced to the center of mass and show that its eigenvalues are N times the probabilities of the different eigenvalues of P. A series of results is presented for the limit of sequences of positive definite atomic probability measures, relevant for the probability distribution of both the single-particle and the total momentum. The P = 0 ensemble is shown to be equivalent to the canonical ensemble. Through some conjectures we associate the properties of the asymptotic distribution of the total momentum with the characteristics of fluid, solid, and superfluid phases. Our main suggestion is that in interacting quantum system...N is normal. We introduce the notion of the density matrix reduced to the center of mass, and show that its eigenvalues are N times the probabilities of the different eigenvalues of P. A series of results is presented for the limit of sequences of positive definite atomic probability measures, relevant for the probability distribution of both the singleparticle and the total momentum. The P = 0 ensemble is shown to be equivalent to the canonical ensemble, and an argument is given why superfluidity and its breakdown cannot be explained by Landau’s criterion. Through some conjectures we associate the properties of the asymptotic distribution of the total momentum with the characteristics of fluid, solid, and superfluid phases. Our main suggestion is that above one dimension in infinite space the total momentum is finite with a nonzero probability at all temperatures and densities. In solids this probability is 1, and in a crystal it is distributed on a lattice. Since it is less than 1 in two dimensions, we conclude that a 2D system is always in a fluid phase; so if the hexatic phase existed classically, it would be destroyed by quantum fluctuations. For a superfluid we conjecture that the total momentum is zero with a nonzero probability and otherwise its distribution is continuous. We define a macroscopic wave function based on the density matrix reduced to the center of mass. Finally, we discuss how dissipation can give rise to a critical velocity, predict the temperature dependence of the latter and comment on the relation between superfluidity and Bose-Einstein condensation.

Normal and Generalized Bose Condensation in Traps: One Dimensional Examples

We prove the following results. (i) One-dimensional Bose gases which interact via unscaled inte-grable pair interactions and are confined in an external potential increasing faster than quadrat-ically undergo a complete generalized Bose-Einstein condensation (BEC) at any temperature, in the sense that a macroscopic number of particles are distributed on a o(N) number of one-particle states. (ii) In a one dimensional harmonic trap the replacement of the oscillator frequency ω by ω ln N/N gives rise to a phase transition at a ≡ ωβ = 1 in the noninteracting gas. For a < 1 the limit distribution of n0/N a is exponential and n0/N a → 1. For a > 1 there is BEC with a con-densate density n0/N → 1 − a −1. For a ≥ 1, (ln N/N)(n0 − n0) is asymptotically distributed following Gumbels law. For any a > 0 the free energy is −(π 2 /6βa)N/ ln N + o(N/ ln N), with no singularity at a = 1. (iii) In Model (ii) both above and below the critical temperature the gas undergoes a complete generalized BEC, thus providing a coexistence of ordinary and generalized condensates below the critical point. (iv) Adding an interaction UN = o(N ln N) to Model (ii) we prove that a complete generalized BEC occurs for any β > 0.