From a discretization of the path integral formulation of quantum mechanics, it is possible to relate equilibrium quantum many body theory to classical statistical mechanics. In this paper, we significantly extend and analyze this well known isomorphism in terms of the equilibrium theory of classical molecular fluids composed of flexible polyatomic species. We show how quantum influence functionals are isomorphic to classical cavity distribution functions. The former describe the influence of surrounding media on the dynamics of quantal degrees of freedom, and the latter describe environmental effects for classical models of flexible molecules and chemical equilibria. The connection allows the use of classical theories to perform nonperturbative calculations of influence functionals which treat the influence functionals and many body correlation functions in a self-consistent fashion. We illustrate the computational advantages of the method by studying its predictions for a hard sphere model of liquid helium above the l transition. The nature of quantum indistinguishability of identical particles (i.e., quantum exchange) is treated in our theory in terms of an exact isomorphism with chemical equilibria. This connection allows the treatment of exchange in condensed phases in terms of the classical law of mass action, and provides a computational advance over existing methods for interacting systems. By picturing exchange in terms of classical association equilibrium, we arrive at a view (hinted at long ago by Feynman and by Penrose and Onsager) in which the Bose condensation is related to an equilibrium polymeric sol–gel transition. Thus, below the l transition, the correlations in liquid helium are equivalent to those in a classical fluid containing a finite concentration of macroscopic polymers. We stress how the path integral aspect of the isomorphism leads to useful geometrical interpretations of quantum phenomena. For example, tunneling phenomena can be viewed in terms of solitonic (or instantonic) configurations or flexible chain molecules. With this picture, we show how the isomorphism can be employed to understand both adiabatic and nonadiabatic solvent effects on chemical bonding. For concreteness, we provide a detailed analysis of a particular model of the chemical bond for which a partial summation over intermediate quantum paths leads to an Ising model problem in the isomorphism. While applications like this are presented in the form of qualitative illustrations, a variety of methods can be employed to produce quantitative results. We sketch how these calculations can be performed for various problems, making connections with methods like the renormalization group (RG) technique. The classical isomorphism together with the modern theory of classical polyatomic systems provides a powerful framework for quantitative solutions of condensed matter quantum mechanical problems.
From a discretization of the path integral formulation of quantum mechanics, it is possible to relate equilibrium quantum many body theory to classical statistical mechanics. In this paper, we significantly extend and analyze this well known isomorphism in terms of the equilibrium theory of classical molecular fluids composed of flexible polyatomic species. We show how quantum influence functionals are isomorphic to classical cavity distribution functions. The former describe the influence of surrounding media on the dynamics of quantal degrees of freedom, and the latter describe environmental effects for classical models of flexible molecules and chemical equilibria. The connection allows the use of classical theories to perform nonperturbative calculations of influence functionals which treat the influence functionals and many body correlation functions in a self-consistent fashion. We illustrate the computational advantages of the method by studying its predictions for a hard sphere model of liquid helium above the l transition. The nature of quantum indistinguishability of identical particles (i.e., quantum exchange) is treated in our theory in terms of an exact isomorphism with chemical equilibria. This connection allows the treatment of exchange in condensed phases in terms of the classical law of mass action, and provides a computational advance over existing methods for interacting systems. By picturing exchange in terms of classical association equilibrium, we arrive at a view (hinted at long ago by Feynman and by Penrose and Onsager) in which the Bose condensation is related to an equilibrium polymeric sol–gel transition. Thus, below the l transition, the correlations in liquid helium are equivalent to those in a classical fluid containing a finite concentration of macroscopic polymers. We stress how the path integral aspect of the isomorphism leads to useful geometrical interpretations of quantum phenomena. For example, tunneling phenomena can be viewed in terms of solitonic (or instantonic) configurations or flexible chain molecules. With this picture, we show how the isomorphism can be employed to understand both adiabatic and nonadiabatic solvent effects on chemical bonding. For concreteness, we provide a detailed analysis of a particular model of the chemical bond for which a partial summation over intermediate quantum paths leads to an Ising model problem in the isomorphism. While applications like this are presented in the form of qualitative illustrations, a variety of methods can be employed to produce quantitative results. We sketch how these calculations can be performed for various problems, making connections with methods like the renormalization group (RG) technique. The classical isomorphism together with the modern theory of classical polyatomic systems provides a powerful framework for quantitative solutions of condensed matter quantum mechanical problems.
In the path integral representation of quantum theory, a few body quantum problem becomes a classical many body problem. To exploit this isomorphism, it becomes necessary to develop methods by which degrees of freedom can be explicitly removed from consideration. The interactions among the remaining relevant variables are described by effective interactions. In this paper, we present a general methodology to carry out the reduction in numbers of degrees of freedom. Certain path integral algorithms are shown to correspond to reference systems for the full isomorphic classical many body problem. The correspondence allows one to determine systematic corrections to the algorithms by low order perturbation approximations familiar in the theory of simple classical fluids. We show how to use discretized path integrals to compute rigorous upper and lower bounds to the free energy for nontrivial quantum systems, and we discuss how to optimize the upper bounds with variational theories. Several illustrative examples are provided including treatments of a tunneling system that is strongly coupled to a nonadiabatic heat bath. The examples demonstrate that even for strongly perturbed systems in which the quantum energy scales are as large as 40kBT, the methods discussed herein provide a practical route for obtaining equilibrium properties with both high accuracy and analytical simplicity.
Often considered separated worlds, classical and quantum mechanics share numerous connections with one another. Indeed, as classical mechanics corresponds to a limiting case of quantum mechanics, certain concepts and elements of physical intuition developed in one theory can be and have been used to tackle issues in the other. Nevertheless, these connections do not only cover conceptual issues but also numerous techniques. Indeed, the inherently probabilistic nature of quantum mechanics and its close resemblance to Markovian stochastic processes opens the door to the application of a broad range of powerful methods, initially developed for quantum mechanics, in classical equilibrium and non-equilibrium statistical mechanics. In this thesis we develop progress in three subjects by taking advantage of either a conceptual or a methodological connection between quantum and classical mechanics. First, we develop the well-known mapping between systems of strongly repelling, two-dimensional directed lines and systems of one-dimensional fermions in order to extend the directed polymer problem to richer geometries. By expressing path integrals in generalized curvilinear coordinates, we successfully generalize the model to settings such as polymers anchored to curved edges, polymers constrained to uneven walls, and polymers constrained to curved surfaces. In each case, we identify the Hamiltonian of an analogous quantum system, which, because of the new geometry of each setting, develops features such as a time- and position-dependent mass and an external electromagnetic vector potential. Along the way, we perform an in-depth analysis of the approximations made and establish regimes of their validity. Finally, in order to obtain analytical results, we employ an extension of the time-dependent perturbation theory scheme of quantum mechanics to imaginary time. Complementing this with the assumption of ground-state dominance, we obtain compact expressions for universal shifts in the free energies of the various systems that allows for isolation of the effects of the distinct geometrical properties of each system. The second piece of work presented into this thesis relates to the non-equilibrium setting of active particles. Strongly interacting, self-propelled particles can form a spontaneously flowing, polar (i.e. motionally aligned), active fluid. The study of the connection between the microscopic dynamics of a single, self-propelled particle and the macroscopic dynamics of a liquid comprising such particles can yield insights into experimentally realizable active flows, but this connection is well understood in only a few select cases. Here, we introduce a model of self-propelled particles that is based on an analogy with the motion of an electron subject to strong spin-orbit coupling. We find that, within our model, self-propelled particles experience an analog of the Heisenberg uncertainly principle that instead relates positional and rotational noise. An extension to many-component (and hence more classical) spinors, under which this uncertainty relation vanishes, contributes to the justification of this interpretation. Furthermore, by coarse-graining the microscopic model, we find expressions for the coefficients of the hydrodynamic Toner-Tu equations, established some time ago to describe an active liquid composed of “active spins.” The connection between self-propelled particles and quantum spins may possibly provide a route for realizing exotic phases of matter using active liquids, via inspiration hailing from systems composed of strongly correlated electrons. The third and final piece of work to be presented on this thesis consists of a semi-classical approach to Transitionless Quantum Driving (TQD). TQD is a method, developed by means of a reverse engineering strategy, under which non-adiabatic transitions in time-dependent quantum systems are, in M. V. Berry's words, “stifled” through the introduction of a specific auxiliary Hamiltonian. This Hamiltonian comes, however, expressed as a formal sum of outer products of the original instantaneous eigenstates and their time-derivatives. Generically, how to actually create such an operator in the laboratory is thus rarely evident. The operator may even be non-local. By following a semi-classical approach, we obtain a recipe that yields the auxiliary Hamiltonian explicitly, in terms of the fundamental operators of the system (e.g., position and momentum). By using this formalism, we are able to ascertain criteria for the locality of the auxiliary Hamiltonian, and also to determine its exact form in certain special cases. Lastly, the explicit connection between the auxiliary Hamiltonian and the observables of the system allows for a perturbation scheme in cases in which an exact solution is not easily achievable. This scheme shows that, even in situations in which an exact local auxiliary term cannot be achieved, under special circumstances it is possible to achieve an operator that is local and still approximately stifles non-adiabatic transitions.
Computational modeling of the condensed phase based on classical statistical mechanics has been rapidly developing over the last few decades and has yielded important information on various systems containing up to millions of atoms. However, if a system of interest contains important quantum effects, well-developed classical techniques cannot be used. One way of treating finite temperature quantum systems at equilibrium has been based on Feynman's imaginary time path integral approach and the ensuing quantum-classical isomorphism. This isomorphism is exact only in the limit of infinitely many classical quasiparticles representing each physical quantum particle. In this work, we present a reductionist perspective on this problem based on the emerging methodology of coarse-graining. This perspective allows for the representations of one quantum particle with only two classical-like quasiparticles and their conjugate momenta. One of these coupled quasiparticles is the centroid particle of the quantum path integral quasiparticle distribution. Only this quasiparticle feels the potential energy function. The other quasiparticle directly provides the observable averages of quantum mechanical operators. The theory offers a simplified perspective on quantum statistical mechanics, revealing its most reductionist connection to classical statistical physics. By doing so, it can facilitate a simpler representation of certain quantum effects in complex molecular environments.
In the path integral representation of the thermal density matrix, the sum over quantum paths is equivalent to the configurational partition function sum for a system composed of long polyatomic molecules. We give an iterative (renormalization group) approach which successively reduces the number of degrees of freedom in this isomorphic classical many-body problem to a more tractable few-body problem. We give as examples a two state tunneling system coupled to an adiabatic and nonadiabatic bath and an impurity with spin coupled to conduction band electrons.
Several microscopic theories point out that Bose-Einstein condensation (BEC), i.e., a macroscopic occupation of the lowest energy single particle state in many-boson systems, may appear also in quantum fluids and solids and that it is at the origin of the phenomenon of superfluidity. Nevertheless, the connection between BEC and superfluidity is still matter of debate, since the experimental evidences indicating a non zero condensate fraction in superfluid helium are only indirect. In the theoretical study of BEC in quantum fluids and solids, perturbative approaches are useless because of the strong correlations between the atoms, arising both from the interatomic potential and from the quantum nature of the system. Microscopic Quantum Monte Carlo simulations provide a reliable description of these systems. In particular, the Path Integral Monte Carlo (PIMC) method is very suitable for this purpose. This method is able to provide exact results for the properties of the quantum system, both at zero and finite temperature, only with the definition of the Hamiltonian and of the symmetry properties of the system, giving an easy picture for superfluidity and BEC in many-boson systems. In this thesis, we apply PIMC methods to the study of several quantum fluids and solids. We describe in detail all the features of PIMC, from the sampling methods to the estimators of the physical properties. We present also the most recent techniques, such as the high-order approximations for the thermal density matrix and the worm algorithm, used in PIMC to provide reliable simulations. We study the liquid phase of condensed 4He, providing unbiased estimations of the one-body density matrix g1(r). We analyze the model for g1(r) used to fit the experimental data, highlighting its merits and its faults. In particular we see that, even if it presents some difficulties in the description of the overall behavior of g1(r), it can provide an accurate estimation of the kinetic energy K and of the condensate fraction n0 of the system. Furthermore, we show that our results for n0 as a function of the pressure are in a good agreement with the most recent experimental results. The study of the solid phase of 4He is the most significant part of this thesis. The recent observation of non classical rotational inertia (NCRI) effects in solid helium has generated big interest in the study of an eventual supersolid phase, characterized at the same time by crystalline order and superfluidity. Nevertheless, until now it has been impossible to give a theoretical model able to describe all the experimental evidences. In this work, we perform PIMC simulations of 4He at high densities, according to different microscopic configurations of the atoms. In commensurate crystals we see that BEC does not appear, our model being able to reproduce the momentum distribution obtained form neutron scattering experiments. In a crystal with vacancies, we have been able to see a transition to a superfluid phase at temperatures in agreement with experimental results if the vacancy concentration is low enough. In amorphous solids, superfluid effects are enhanced but appear at temperatures higher than the experimental estimation for the transition temperature. Finally, we study also metastable disordered configurations in molecular para-hydrogen at low temperature. The aim of this study is to investigate if a Bose liquid other than helium can display superfluidity. Choosing accurately a ¿quantum liquid¿ initial configuration and the dimensions of the simulation box, we have been able to frustrate the formation of the crystal and to calculate the temperature dependence of the superfluid density, showing a transition to a superfluid phase at temperatures close to 1 K.
Quantum and classical results are often presented as being dependent upon separate postulates as if the two are distinct and unrelated, and there is little attempt to show how the quantum implies the classical. The transformation to classical phase space gives researchers access to a range of algorithms derived from classical statistical mechanics that promise results on much more favourable numerical terms. Quantum Statistical Mechanics in Classical Phase Space offers not just a new computational approach to condensed matter systems, but also a unique conceptual framework for understanding the quantum world and collective molecular behaviour. A formally exact transformation, this revolutionary approach goes beyond the quantum perturbation of classical condensed matter to applications that lie deep in the quantum regime. It offers scalable computational algorithms and tractable approximations tailored to specific systems. Concrete examples serve to validate the general approach and demonstrate new insights. For example, the computer simulations and analysis of the λ-transition in liquid helium provide a new molecular-level explanation of Bose–Einstein condensation and a quantitative theory for superfluid flow. The intriguing classical phase space formulation in this book offers students and researchers a range of new computational algorithms and analytic approaches. It offers not just an efficient computational approach to quantum condensed matter systems, but also an exciting perspective on how the classical world that we observe emerges from the quantum mechanics that govern the behaviour of atoms and molecules. The applications, examples, and physical insights foreshadow new discoveries in quantum condensed matter systems. Key features • A computationally efficient approach to quantum condensed-matter many-particle systems based on a formally exact transformation of quantum statistical mechanics to classical phase space. • Shows how the observed classical world emerges from the underlying quantum mechanics of atoms and molecules. • Describes successive quantum corrections to classical statistical mechanics. • Derives a number of expansions that account for the non-commutativity of quantum position and momentum operators when transformed to classical phase space. • Tests various computer simulation algorithms against benchmark results for model condensed matter systems.