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A new method, dual-space cluster expansion, is proposed to study classical phases transitions in the continuum. It relies on replacing the particle positions as integration variables by the momenta of the relative displacements of particle pairs. Due to the requirement that the particles must be static, coupling via the momenta partitions the set of particles into a set of clusters, and transforms the partition function into a sum over the different cluster decompositions. This allows us to derive a formula for the density that finite clusters can carry in the infinite system. In a simplified example, we then demonstrate that in two and higher dimensions this density has a threshold, beyond which the particles form infinite clusters. The transition is accompanied by a singularity in the free energy. We also show that infinite clusters are always present in condensed phases, most likely submacroscopic in liquids and macroscopic in crystals., Comment: 12 pages

In the first-quantized description of bosonic systems permutation cycles formed by the particles play a fundamental role. In the ideal Bose gas Bose-Enstein condensation (BEC) is signaled by the appearance of infinite cycles. When the particles interact, the two phenomena may not be simultaneous, the existence of infinite cycles is necessary but not sufficient for BEC. We demonstrate that their appearance is always accompanied by a singularity in the thermodynamic quantities which in three and four dimensions can be as strong as a one-sided divergence of the isothermal compressibility. Arguments are presented that long-range interactions can give rise to unexpected results, such as the absence of infinite cycles in three dimensions for long-range repulsion or their presence in one and two dimensions if the pair potential has a long attractive tail., Comment: 13 pages, minor changes

We prove two equilibrium properties of a system of interacting atoms in three or higher dimensional continuous space. (i) If the particles interact via pair potentials of a nonnegative Fourier transform, their self-organization into infinite permutation cycles is simultaneous with off-diagonal long-range order. If the cycle lengths tend to infinity not slower than the square of the linear extension of the system, there is also Bose-Einstein condensation. (ii) If the pair potential is also nonnegative, cycles composed of a nonzero fraction of the total number of particles do appear if the density exceeds a temperature-dependent threshold value. The two together constitute the proof that in such a system Bose-Einstein condensation takes place at high enough densities., Comment: 66 pages, combined from 2106.10032[math-ph], 2108.02659 [math-ph], 2208.08931[math-ph]. arXiv admin note: text overlap with arXiv:2208.08931, arXiv:2106.10032, arXiv:2108.02659

In this third paper of a series that started with arXiv:2106.10032 [math-ph] and continued with arXiv:2108.02659 [math-ph] we show that in $d\geq 3$ dimensions at low temperatures or high densities bosons interacting via pair potentials that are both positive and positive type form permutation cycles whose length diverges proportionally with the number of particles. Based on the second-cited paper, this implies Bose-Einstein condensation., Comment: 29 pages, end of the proof about the coupling of cycles rewritten. arXiv admin note: text overlap with arXiv:1710.04441

Based on the paper "Fourier formula for quantum partition functions", arXiv:2106.10032 [math-ph], we show that in an infinite system of identical bosons interacting via a positive-type pair potential there is off-diagonal long-range order if and only if a nonzero fraction of the particles form infinite permutation cycles. In particular, there is Bose-Einstein condensation if and only if the diverging cycle lengths increase at least as fast with $N$, the number of particles, as $N^{2/d}$ in $d\geq 3$ dimensions. This extends a similar result known for the ideal Bose gas., Comment: 27 pages, minor changes

Fourier expansion of the integrand in the path integral formula for the partition function of quantum systems leads to a deterministic expression which, though still quite complex, is easier to process than the original functional integral. It therefore can give access to problems that eluded solution so far. Here we derive the formula; applications to the problem of Bose-Einstein condensation are presented in the papers arXiv:2108.02659 [math-ph] and arXiv:2208.08931 [math-ph]., Comment: 25 pages, Appendix about merger graphs improved

We prove a property of Brownian bridges whose certain time-equidistant sequences of points are pairwise coupled by an interaction. Roughly saying, if the total time span $t$ of the bridge tends to infinity while the distance of its end points is fixed or increases slower than $\sqrt{t}$, the process asymptotically forgets this distance, just as in the absence of interaction. The conclusion remains valid if the bridge interacts in a similar way also with another set of trajectories. The main example for the interaction is the Coulomb potential., Comment: 6 pages

Using the rigorous path integral formalism of Feynman and Kac we prove London's eighty years old conjecture that during the superfluid transition in liquid helium Bose-Einstein condensation (BEC) takes place. The result is obtained by proving first that at low enough temperatures macroscopic permutation cycles appear in the system, and then showing that this implies BEC. We find also that in the limit of zero temperature the infinite cycles cover the whole system, while BEC remains partial. For the Bose-condensed fluid at rest we define a macroscopic wave function. Via the equivalence of 1/2 spins and hard-core bosons the method extends to lattice models. We show that at low enough temperatures the spin-1/2 axially anisotropic Heisenberg models, including the isotropic ferro- and antiferromagnet and the XY model, undergo magnetic ordering., Comment: proof of main theorems corrected, simplified

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/\sqrt{N} is normal at all temperatures and densities, and in two dimensions the tail distribution of P/\sqrt{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 systems 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. 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. We discuss how dissipation can give rise to a critical velocity, predict the temperature dependence of the latter, and prove that Landau's criterion cannot explain superfluidity and its breakdown in a dissipative flow. We also comment on the relation between superfluidity and Bose-Einstein condensation., Comment: 19 pages

The total momentum of $N$ interacting bosons or fermions in a cube equipped with periodic boundary conditions is a conserved quantity. Its eigenvalues follow a probability distribution, determined by the thermal equilibrium state. While in non-interacting systems the distribution is normal with variance $\sim N$, interaction couples the single-particle momenta, so that the distribution of their sum is unpredictable, except for some implications of Galilean invariance. First, we present these implications which are strong in 1D, moderately strong in 2D, and weak in 3D. Then, we speculate about the possible form of the distribution in fluids, crystals, and superfluids. The existence of phonons suggests that the total momentum can remain finite when $N\to\infty$. We argue that in fluids the finite momenta distribute continuously, but their integrated probability is smaller than 1, because the momentum can also tend to infinity with $N$. In the fluid-crystal transition we expect that the total momentum becomes finite with full probability and distributed over a lattice, and that in the fluid-superfluid transition a delta peak appears only at zero total momentum. Based on this picture, we discuss the superfluid flow in both the frictionless and the dissipative cases, and derive a temperature-dependent critical velocity. Finally, we show that Landau's criterion for excitations in moving superfluids is an in some cases correct result of an erroneous derivation., Comment: revised; references updated

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 $s_\perp$ leads to spectral gaps $n^2h^2s_\perp\rho/(2mL)$ at wavenumbers $2n\pi s_\perp\rho$, where $\rho$ 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 $N/L=s_\perp\rho$ 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., Comment: Improved, final version. Equation (22) is slightly more general than in the publication. The upper bound for the critical velocity on p. 4 is corrected

We investigate the effect of a constant external velocity field on the ground state of a bosonic quasiparticle Hamiltonian. Below a critical velocity the ground state is a quasiparticle vacuum, corresponding to a pure superfluid phase at zero temperature. Beyond the critical velocity energy minimization leads to a macroscopic condensation of quasiparticles at a nonzero wave vector k_v parallel to the velocity v. Simultaneously, physical particles also undergo a condensation at k_v and, to a smaller extent, at -k_v. Together with the BEC at k=0, the three entangled condensates give rise to density modulations of wave vectors k_v and 2k_v. For larger |v| our model predicts a bifurcation of k_v with corresponding two pure condensates and no density modulation., Comment: Proof of quasiparticle condensation completed

We review the recent developments in the theory of the one-dimensional tight-binding Schr\"odinger equation for a class of deterministic ergodic potentials. In the typical examples the potentials are generated by substitutional sequences, like the Fibonacci or the Thue-Morse sequence. We concentrate on rigorous results which will be explained rather than proved. The necessary mathematical background is provided in the text., Comment: 51 pages

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 one, 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., Comment: Discussion added

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 superposed on the sites of a close-packed lattice., Comment: Note added

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 order 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 $\hbar^2 q^2/2m|v(\k)|$ 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., Comment: published version

We study variational wave functions of the product form, factorizing according to the wave vectors k, for the ground state of a system of bosons interacting via positive pair interactions with a positive Fourier transform. Our trial functions are members of different orthonormal bases in Fock space. Each basis contains a quasiparticle vacuum state and states with an arbitrary finite number of quasiparticles. One of the bases is that of Valatin and Butler (VB), introduced fifty years ago and parametrized by an infinite set of variables determining Bogoliubov's canonical transformation for each k. In another case, inspired by Nozi\`eres and Saint James the canonical transformation for k=0 is replaced by a shift in the creation/annihilation operators. For the VB basis we prove that the lowest energy is obtained in a state with ~sqrt{volume} quasiparticles in the zero mode. The number of k=0 physical particles is of the order of the volume and its fluctuation is anomalously large, resulting in an excess energy. The same fluctuation is normal in the second type of optimized bases, the minimum energy is smaller and is attained in a vacuum state. Associated quasiparticle theories and questions about the gap in their spectrum are also discussed.

This paper supplements and partly extends an earlier publication, Phys. Rev. Lett. 95, 265501 (2005). In $d$-dimensional continuous space we describe the infinite volume ground state configurations (GSCs) of pair interactions $\vfi$ and $\vfi+\psi$, where $\vfi$ is the inverse Fourier transform of a nonnegative function vanishing outside the sphere of radius $K_0$, and $\psi$ is any nonnegative finite-range interaction of range $r_0\leq\gamma_d/K_0$, where $\gamma_3=\sqrt{6}\pi$. In three dimensions the decay of $\vfi$ can be as slow as $\sim r^{-2}$, and an interaction of asymptotic form $\sim\cos(K_0r+\pi/2)/r^3$ is among the examples. At a dimension-dependent density $\rho_d$ the ground state of $\vfi$ is a unique Bravais lattice, and for higher densities it is continuously degenerate: any union of Bravais lattices whose reciprocal lattice vectors are not shorter than $K_0$ is a GSC. Adding $\psi$ decreases the ground state degeneracy which, nonetheless, remains continuous in the open interval $(\rho_d,\rho_d')$, where $\rho_d'$ is the close-packing density of hard balls of diameter $r_0$. The ground state is unique at both ends of the interval. In three dimensions this unique GSC is the bcc lattice at $\rho_3$ and the fcc lattice at $\rho_3'=\sqrt{2}/r_0^3$., Comment: Published version

Pair interactions whose Fourier transform is nonnegative and vanishes above a wave number K_0 are shown to give rise to periodic and aperiodic infinite volume ground state configurations (GSCs) in any dimension d. A typical three dimensional example is an interaction of asymptotic form cos(K_0 r)/r^4. The result is obtained for densities rho >= rho_d where rho_1=K_0/2pi, rho_2=(sqrt{3}/8)(K_0/pi)^2 and rho_3=(1/8sqrt{2})(K_0/pi)^3. At rho_d there is a unique periodic GSC which is the uniform chain, the triangular lattice and the bcc lattice for d=1,2,3, respectively. For rho>rho_d the GSC is nonunique and the degeneracy is continuous: Any periodic configuration of density rho with all reciprocal lattice vectors not smaller than K_0, and any union of such configurations, is a GSC. The fcc lattice is a GSC only for rho>=(1/6 sqrt{3})(K_0/pi)^3., Comment: final version

Based on a classic paper by Ginibre [Commun. Math. Phys. {\bf 8} 26 (1968)] it is shown that whenever Bogoliubov's approximation, that is, the replacement of a_0 and a_0^* by complex numbers in the Hamiltonian, asymptotically yields the right pressure, it also implies the asymptotic equality of |< a_0>|^2/V and < a_0^*a_0>/V in symmetry breaking fields, irrespective of the existence or absence of Bose-Einstein condensation. Because the former was proved by Ginibre to hold for absolutely integrable superstable pair interactions, the latter is equally valid in this case. Apart from Ginibre's work, our proof uses only a simple convexity inequality due to Griffiths., Comment: An error in my summary of previous results (the definition of F') is corrected. The correction is to be done also in the PRL

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 Bogoliubov's asymptotic hypothesis \lim_{V\to\infty}< a_0>/\sqrt{V}={\rm sgn}\nu \lim_{V\to\infty}\sqrt{< a_0^*a_0>/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., Comment: 8 pages; to appear in Phys. Rev. A

We prove the following results. (i) One-dimensional Bose gases which interact via unscaled integrable pair interactions and are confined in an external potential increasing faster than quadratically 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 \omega by \omega\ln N/N gives rise to a phase transition at a=\hbar\omega\beta=1 in the noninteracting gas. For a<1 the limit distribution of n_0/N^a is exponential and /N^a tends to 1. For a>1 there is BEC with a condensate density /N going to 1-1/a. For a>=1, (\ln N/N)(n_0-) is asymptotically distributed following Gumbel's law. For any a>0 the free energy is -(\pi^2/6a\beta)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 the gas undergoes a complete generalized BEC, thus providing a coexistence of ordinary and generalized condensates below the critical point. (iv) Adding an interaction =o(N\ln N) to Model (ii) we prove that a complete generalized BEC occurs at all temperatures., Comment: Published version with further improvements

For a noninteracting Bose gas with a fixed one-body Hamiltonian H^0 independent of the number of particles we derive the inequalities _N < _{N+1}, _N < _N _N for i\neq j, \partial _N/\partial \beta >0 and ^+_N < _N. Here N_i is the occupation number of the ith eigenstate of H^0, \beta is the inverse temperature and the superscript + refers to adding an extra level to those of H^0. The results follow from the convexity of the N-particle free energy as a function of N., Comment: a further inequality added

In this paper we present an extremely simple structure solution method termed charge flipping. It works ab initio on high resolution x-ray diffraction data in the manner of Fourier recycling. The real space modification simply changes the sign of charge density below a threshold, while in reciprocal space the modification is the Fobs map without any weighting. We test the algorithm using synthetic data for a wide range of structures, analyse the solution statistics and check the quality of reconstruction. Finally, we reconsider mathematical aspects of the algorithm in detail, showing that in this chaotic iteration process the solution is a limit cycle and not a fixed point., Comment: 17 pages, 4 figures

We establish quantum dynamical lower bounds for a number of discrete one-dimensional Schr\"odinger operators. These dynamical bounds are derived from power-law upper bounds on the norms of transfer matrices. We develop further the approach from part I and study many examples. Particular focus is put on models with finitely or at most countably many exceptional energies for which one can prove power-law bounds on transfer matrices. The models discussed in this paper include substitution models, Sturmian models, a hierarchical model, the prime model, and a class of moderately sparse potentials., Comment: 20 pages

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 one-particle eigenstate, which may depend on the interaction strength, persists at all temperatures., Comment: Published version

In an earlier paper (J. Phys. A: Math. Gen. 26 (1993) 4689) we introduced the notion of cycle percolation in the Bose gas and conjectured that it occurs if and only if there is Bose-Einstein condensation. Here we give a complete proof of this statement for the perfect and the imperfect (mean-field) Bose gas and also show that in the condensate there is an infinite number of macroscopic cycles., Comment: The paper is completed with the proof of Eq. (34) and with a remark after Eq. (44) on the existence for any m>ln(rho/rho_0) of an infinite cycle which contains a fraction between exp{-(m+1)} and exp{-m} of the total number of particles

We show that in d>1 dimensions the N-particle kinetic energy operator with periodic boundary conditions has symmetric eigenfunctions which vanish at particle encounters, and give a full description of these functions. In two and three dimensions they represent common eigenstates of bosonic Hamiltonians with any kind of contact interactions, and illustrate a partial `multi-dimensional Bethe Ansatz' or `quantum-KAM theorem'. The lattice analogs of these functions exist for N<=L^[d/2] where L is the linear size of the box, and are common eigenstates of Bose-Hubbard Hamiltonians and spin-1/2 XXZ Heisenberg models., Comment: 23 pages. More references, enlarged introduction, remark 2.9 corrected. To appear in Journal of Statistical Physics

We study the ground state, sum a_X |X>, of N hard-core bosons on a finite lattice in configuration space, X={x_1,...,x_N}. All a_X being positive, the ratios a_X / sum a_Y can be interpreted as probabilities P_a (X). Let E denote the energy of the ground state and B_X the number of nearest-neighbor particle-hole pairs in the configuration X. We prove the concentration of P_a to X's with B_X in a sqrt(|E|)-neighborhood of |E|, show that the average of a_X over configurations with B_X=n increases exponentially with n, discuss fluctuations about this average, derive upper and lower bounds on E and give an argument for off-diagonal long-range order in the ground state., Comment: Equation (77) corrected

We analyze the onset of incommensurabilities around the VBS point of the S=1 bilinear-biquadratic model. We propose a simple effective field theory which is capable of reproducing all known properties of the commensurate-incommensurate transition at the disorder point $\theta_{\rm vbs}$. Moreover, the theory predicts another special point $\theta_{\rm disp}$, distinct from the VBS point, where the Haldane gap behaves singularly. The ground state energy density is an analytic function of the model parameters everywhere, thus we do not have phase transitions in the conventional sense., Comment: 8 pages, 2 figures, to appear in PRB

We investigate the properties of the ground state of a system of interacting bosons on regular lattices with coordination number $k\geq 2$. The interaction is a pure, infinite, on-site repulsion. Our concern is to give an improved upper bound on the ground state energy per site. For a density $\rho$ a trivial upper bound is known to be $-k\rho(1-\rho)$. We obtain a smaller variational bound within a reasonably large family of trial functions. The estimates make use of a large deviation principle for the energy of the Ising model on the same lattice., Comment: 17 pages, 4 tables