Basic problems of a microscopic theory of many body quantum systems and different aspects of a new approach which can help in solving them are discussed in detail. To this effect we make a critical study of the wave mechanics of two hard core quantum particles and discover its several untouched aspects, viz.: (i) the useful details of ψ_k(r) (representing the relative motion of two particles), (ii) the expectation value of hard core (HC) repulsion (), (iii) the inconsistency of the statements, r \leσand ψ_k(r \leσ)=0 (σ=HC diameter of a particle), with uncertainty principle particularly for low k values, (iv) the lower bound of allowed values of k=2q, (v) the dominance of interparticle phase correlation in low temperature phase. For the first time this study concludes that has zero value which does not agree with its non-zero value known for the last several decades. This also finds compelling reasons for a system of interacting bosons such as liquid ^4He to have (q, -q) pair condensation with allowed q, obviously controlled by V_{HC}(r), to satisfy q \geπ/d. Several important aspects of N body quantum systems like liquids ^4He and ^3He are also concluded. Free from any error [see editor's note J. Scientific Exploration 16(1), p.1 (2002)], our approach can help in developing nearly exact microscopic theories of widely different systems of interacting bosons and fermions, as demonstrated for liquids ^4He type systems [J. Scientific Exploration, 16, 77-116 (2002)]. The paper also sums up the expert observations with our response to facilitate one to have a critical assessment and better understanding of the new approach.
Basic problems of a microscopic theory of many body quantum systems and different aspects of a new approach which can help in solving them are discussed in detail. To this effect we make a critical study of the wave mechanics of two hard core quantum particles and discover its several untouched aspects, viz.: (i) the useful details of ψ_k(r) (representing the relative motion of two particles), (ii) the expectation value of hard core (HC) repulsion (), (iii) the inconsistency of the statements, r \leσand ψ_k(r \leσ)=0 (σ=HC diameter of a particle), with uncertainty principle particularly for low k values, (iv) the lower bound of allowed values of k=2q, (v) the dominance of interparticle phase correlation in low temperature phase. For the first time this study concludes that has zero value which does not agree with its non-zero value known for the last several decades. This also finds compelling reasons for a system of interacting bosons such as liquid ^4He to have (q, -q) pair condensation with allowed q, obviously controlled by V_{HC}(r), to satisfy q \geπ/d. Several important aspects of N body quantum systems like liquids ^4He and ^3He are also concluded. Free from any error [see editor's note J. Scientific Exploration 16(1), p.1 (2002)], our approach can help in developing nearly exact microscopic theories of widely different systems of interacting bosons and fermions, as demonstrated for liquids ^4He type systems [J. Scientific Exploration, 16, 77-116 (2002)]. The paper also sums up the expert observations with our response to facilitate one to have a critical assessment and better understanding of the new approach.
Macro-orbital representation of a particle (detailed account given in cond-mat/0603784) has been used to develop the microscopic theory of a system of interacting bosons. It concludes that: (i) below certain temperature (say, Tλ), particles assume a state of (q, -q) bound pairs, (ii) the λ−transition is a consequence of inter-particle quantum correlations clubbed with zero-point repulsion and inter-particle attraction and represents an onset of the order-disorder of particles in their ϕ−space followed simultaneously by their BEC as (q, -q) bound pairs in a state of q = qo=π/d and K = 0, (iii) particles at T≤Tλ acquire collective binding which locks them at = 0, = λ/2 and Δϕ=2nπ (with n = 1, 2, 3, ...), (iv) the entire system assumes mechanical strain in inter-particle bonds and behaves like a single macroscopic molecule, (v) there exists an energy gap between the superfluid and normal fluid phases of the system, (vi) the λ−transition represents the twin phenomena of broken gauge symmetry and phase coherence, (vii) the system does not have p = 0 condensate, (viii) a new kind of quantum quasi-particle "omon" (a phononlike wave of the oscillations of the momentum coordinates of particles) exists in superfluid phase, etc. It explains the properties of He-II, including the origin of quantized vortices, critical velocities, logarithmic singularity of specific heat, etc. at quantitative level and provides microscopic foundation to two fluid theory, Ψ−theory, idea of macroscopic wave function, etc. The framework of the theory can unify the physics of interacting bosons and fermions.
Abstract The wave mechanics of two impenetrable hard core particles in a 1-D box is analyzed. Each particle in the box behaves like an independent entity represented by a macro-orbital (a kind of pair waveform). While the expectation value of their interaction, 〈 V HC(x) 〉, vanishes for every state of two particles, the expectation value of their relative separation, 〈 x 〉, satisfies 〈 x 〉≥λ/2 (or q ≥ π/d, with 2d=L being the size of the box). The particles in their ground state define a close-packed arrangement of their wave packets (with 〈 x 〉= λ/2, phase position separation Δϕ = 2π and momentum |q o| = π/d) and experience a mutual repulsive force (zero point repulsion) f o=h 2/2md 3 which also tries to expand the box. While the relative dynamics of two particles in their excited states represents usual collisional motion, the same in their ground state becomes collisionless. These results have great significance in determining a correct microscopic understanding of widely different many-body systems.
The collective behavior exhibited by a large number of microscopic quantum particles is at the heart of some of the most striking phenomena in condensed matter physics such as Bose-Einstein condensation and superconductivity. Physicists and mathematicians have made great progress in understanding when and how these collective phenomena emerge through the interplay of particle statistics, particle interaction and the value of thermodynamic parameters like the temperature or the chemical potential. Due to the extreme complexity of realistic many-body systems, it is natural to introduce appropriate simplifications to render their analysis feasible. Three examples of such simplifications which have proven themselves as viable starting points for a fruitful and mathematically rigorous analysis of many-body systems are the following: (a) the study of integrable models; (b) the derivation of effective theories, valid on a macroscopic scale, from more fundamental microscopic theories under appropriate coarse-graining; and (c) the use of quantum information theory to understand general connections between correlation, entanglement and particle statistics. In this thesis, we present mathematically rigorous results that were obtained in these three directions. (1) We prove anomalous quantum many-body transport in XY quantum spin chains for certain choices of the external magnetic field. The anomalous transport is described via new kinds of anomalous Lieb-Robinson bounds, including one of power-law type. We note that the XY spin chain is integrable as it can be mapped to free fermions via the non-local Jordan-Wigner transformation. (2) We derive effective macroscopic theories of Ginzburg-Landau type from the microscopic BCS theory of superconductivity in certain circumstances. We study the case of a multi-component order parameter for translation-invariant systems and the condensation of fermion pairs at zero temperature in a domain with a hard boundary. (3) We use techniques from quantum information-theory to derive bounds on the entropy of fermionic reduced density matrices, a measure of the entanglement inherent to a fermionic quantum state.
We extend to the N-body case previous techniques for solving three-body problems with repulsive interparticle potentials and periodic boundary conditions on each particle. For clarity, we begin with one-dimensional problems, although the techniques are not peculiar to them and can be generalized to three dimensions. We decompose the wavefunction ψ into N(N − 1)/2 parts, according to Ψ= ∑ i<j=1Nψij;from the Schrödinger equation we find a basic equation for a typical ψij. As a test of this equation, we apply it for N = 4 and the case of δ-function interactions, and solve it numerically to find close agreement between the energy per particle of a four-body system and a system of the same density in which N → ∞. The numerical results are consistent with the analytic one that Eg(∞), which is the ground-state energy per particle of the system with an infinite number of particles, is related to Eg(N), which is that energy for the N-body system of the same density, by Eg(N) = Eg(∞)[1 − (1/N)] for weak repulsion and by Eg(N) = Eg(∞)[1 − (1/N2)] for very strong repulsion. We begin a similar comparison for the three-dimensional hard-sphere case by working out the problem of two hard spheres with periodic boundary conditions over a length L2 such that the density is the same as for an N-body problem in volume V=LN3:2/L23=N/LN3. We find that we get the analog of the one-dimensional result, for which case Eg(2) = Eg(∞)/2 for N = 2. That is, we find Eg(2) = πa2h/mv, which is just one-half of the many-body theory result. We also calculate the two-body correlation function and find good agreement with the many-body one, as calculated by Lee, Huang, and Yang.
The ground state of a system of N impenetrable hard core quantum particles in a 1-D box is analyzed by using a new scheme applied recently to study a similar system of two such particles {\it [Centl. Eur. J. Phys., 2(4), 709 (2004)]}. Accordingly, each particle of the system behaves like an independent entity represented by a {\it macro-orbital}, -a kind of pair waveform identical to that of a pair of particles moving with (q, −q) momenta at their {\it center of mass} which may have any momentum K in the laboratory frame. It concludes: (i) $ = 0$, (ii) $ \ge λ/2$ and (iii) q≥qo(=π/d) (with d=L/N being the average nearest neighbour distance), {\it etc.} While all bosons in their ground state have q=qo and K=0, fermions have q=qo with different K ranging between 0 and K=KF (the Fermi wave vector). Independent of their bosonic or fermionic nature, all particles in the ground state define a close packed arrangement of their equal size wave packets representing an ordered state in phase (ϕ−)space with Δϕ=2nπ (with n = 1,2,3, ...), $ = λ/2 = d$, and q=qo. As such our approach uses greatly simplified mathematical formulation and renders a visibly clear picture of the low energy states of the systems and its results supplement earlier studies in providing their complete understanding.
Envisioned by Richard Feynman in the early 1980s, quantum simulation has received dramatic impetus thanks to the development of a variety of plateforms able to emulate a wide class of quantum Hamiltonians.During the past decade, most of the quantum simulators have implemented rather well-known models, hence permitting a direct comparison with theoretical calculations and a precise benchmarking of their reliability.The field has now reached a maturity such that one can address difficult problems, which cannot be solved efficiently using classical algorithms.These advances provide unprecedented opportunities to explore previously unreachable fields, test theoretical predictions, and inspire novel approaches.This contribution is an elementary introduction to quantum simulation.We discuss the challenges, define both digital and analog quantum simulators, and list the demanding conditions they require.We also provide a brief account of the contributions gathered in the dossier on Quantum simulation in the Compte rendus Physique of the French Academy of Sciences [1][2][3][4][5][6].The latter completes excellent reviews that appeared previously, see for instance . Universal models and the role of simulations in many-body physicsUnderstanding the behavior of macroscopic quantum systems is a major challenge of modern physics.The basic laws of low-energy physics are by now quite well known at the microscopic, say atomic, scale.Conversely, many fundamental questions remain open, and even debated, about the collective dynamics at the macroscopic scale.By "macroscopic scale", here we mean systems made up of a huge number of constituents, or degrees of freedom, say 10 6 , 10 23 , or even more, as relevant in condensed matter physics or in astrophysics, for instance.Such huge systems cannot be treated exactly, be it a the classical level and, even worse, at the quantum level.Yet, it is the main outcome of the thermodynamic approach that the collective behavior of a macroscopic system can drastically differ from that of its elementary constituents.For instance, the elementary interactions between the H 2 O molecules are fundamentally unchanged when a water bucket turns from the liquid phase to the solid phase at zero degree Celsius.Similarly, the interactions between the microscopic magnetic moments do not show any brutal change when a magnetic material gets magnetized underneath the Curie temperature.Hence, the dramatic effects observed in macroscopic systems are governed by large-scale instabilities, without obvious counterparts at the microscopic level.This observation takes a universal character, summarized in the celebrated motto "More is Different" [15].Such so-called emerging phenomena also appear in quantum systems, where new effects that are impossible in the classical world show up below some critical temperature or at zero temperature when some interaction parameter passes through a critical value.Celebrated examples include the superfluid transition in helium or super-conducting transitions and other metal-insulator transitions in electronic systems.Strikingly enough at first sight, while emerging phenomena only appear in very large scale macroscopic systems, their germs are contained in the mutual interactions of their elementary constituents.Hence, local two-body interactions are sufficient to explain a huge class of phase transitions, such as the liquid-solid and magnetic transitions mentioned above.The Boltzmann statistical approach proved particularly successful in describing this connection between the microscopic and macroscopic scales.To understand emerging phenomena on grounds as fundamental as possible, the programme is now well established: One tries to identify the basic microscopic terms that seem to be relevant and disregards all the other microscopic details.One then elaborates a model, as generic as possible, likely to reproduce the main experimental observations.The most usual examples are the Ising and Heisenberg models for magnetic transitions, or the Hubbard model for metal-insulator transitions [16][17][18].Then all is left to do is to check that the phenomenon of interest indeed emerges from the dynamics of the simplified model.The realization of this programme is nothing but a simulation.It consists in building up a simplified system that mimics the main properties of a real system.