The equilibrium statistical mechanics of classical directed polymers in two dimensions is well known to be equivalent to the imaginary-time quantum dynamics of a 1+1-dimensional many-particle system, with polymer configurations corresponding to particle world-lines. This equivalence motivates the application of techniques originally designed for one-dimensional many-particle quantum systems to the exploration of many-polymer systems, as first recognized and exploited by de Gennes [J. Chem. Phys. 48, 2257 (1968)]. In this low-dimensional setting interactions give rise to an emergent polymer fluid, and we examine how topological constraints on this polymer fluid (e.g., due to uncrossable pins or barriers) and their geometry give rise to strong, entropy-driven forces. In the limit of large polymer densities, in which a type of mean-field theory is accurate, we find that a pointlike pin causes a divergent pileup of polymer density on the high-density side of the pin and a zero-density region (or gap) of finite area on the low-density side. In addition, we find that the force acting on a pin that is only mildly displaced from its equilibrium position is sub-Hookean, growing less than linearly with the displacement, and that the gap created by the pin also grows sublinearly with the displacement. By contrast, the forces acting between multiple pins separated along the direction preferred by the polymers are super-Hookean. These nonlinear responses result from effective long-ranged interactions between polymer segments, which emerge via short-ranged interactions between distant segments of long polymer strands. In the present paper, we focus on the case of an infinitely strong, repulsive contact interaction, which ensures that the polymers completely avoid one another. In a companion paper, we consider the effects of a wider set of interpolymer interactions.
The equilibrium statistical mechanics of classical directed polymers in two dimensions is well known to be equivalent to the imaginary-time quantum dynamics of a 1+1-dimensional many-particle system, with polymer configurations corresponding to particle world-lines. This equivalence motivates the application of techniques originally designed for one-dimensional many-particle quantum systems to the exploration of many-polymer systems, as first recognized and exploited by de Gennes [J. Chem. Phys. 48, 2257 (1968)]. In this low-dimensional setting interactions give rise to an emergent polymer fluid, and we examine how topological constraints on this polymer fluid (e.g., due to uncrossable pins or barriers) and their geometry give rise to strong, entropy-driven forces. In the limit of large polymer densities, in which a type of mean-field theory is accurate, we find that a pointlike pin causes a divergent pileup of polymer density on the high-density side of the pin and a zero-density region (or gap) of finite area on the low-density side. In addition, we find that the force acting on a pin that is only mildly displaced from its equilibrium position is sub-Hookean, growing less than linearly with the displacement, and that the gap created by the pin also grows sublinearly with the displacement. By contrast, the forces acting between multiple pins separated along the direction preferred by the polymers are super-Hookean. These nonlinear responses result from effective long-ranged interactions between polymer segments, which emerge via short-ranged interactions between distant segments of long polymer strands. In the present paper, we focus on the case of an infinitely strong, repulsive contact interaction, which ensures that the polymers completely avoid one another. In a companion paper, we consider the effects of a wider set of interpolymer interactions.
The equilibrium statistical mechanics of classical directed polymers in D+1 dimensions is well known to be equivalent to the imaginary-time quantum dynamics of a quantum many-particle system in D spatial dimensions, with polymer con gurations corresponding to particle world-lines. This equivalence motivates the application of techniques originally designed for one-dimensional many-particle quantum systems to the exploration of many-polymer systems, as first recognized and exploited by P.-G. de Gennes [J. Chem. Phys. 48, 2257 (1968)]. In two dimensions, interactions give rise to an emergent polymer fluid, and I shall examine how topological constraints on this polymer fluid (e.g., due to uncrossable pins or barriers) and their geometry give rise to strong, entropy-driven forces. I shall also apply quantum techniques such as Bethe's Ansatz and bosonization to shed light on the structure of the polymer system. These techniques allow us to examine how polymer system correlations, thermodynamic properties, and response to impurities are influenced by strong polymer-polymer interactions. In three dimensions, polymers wind around one another and polymer topology may be incorporated via coupling to a Chern-Simons fi eld. As I discuss, this approach reveals the somewhat reduced role played by interactions in this higher-dimensional setting, leading to qualitatively diff erent polymer correlations, thermodynamic properties, and response to impurities.
The impact of polymer-polymer interactions of various types on the thermodynamics, structure, and accommodation of topological constraints is addressed for systems comprising many directed polymers in two spatial dimensions. The approach is predicated on the well-known equivalence between the classical equilibrium statistical mechanics of directed polymers in two spatial dimensions and the imaginary-time quantum dynamics of particles in one spatial dimension, originally exploited by P.-G. de Gennes [P.-G. de Gennes, J. Chem. Phys. 48, 2257 (1968)]. Known results concerning two exactly solvable microscopic models of quantum particles moving in one spatial dimension---the Lieb-Liniger model of contact interactions and the Calogero-Sutherland model of long-range interactions---are used to shed light on the behavior of the corresponding polymeric systems. In addition, the technique of bosonization is used to reveal how generic polymer interactions give rise to an emergent polymer fluid that has universal collective excitations. Additionally, the response of the system to topological constraints such as pins though which polymers cannot pass is explored. Immediately on the compressed side of a pin there is a divergent pile-up in polymer density, while on the other side there is a gap of finite area in which polymer density is negligible. Comparison of this response to that of a system of simply noncrossing (i.e., noncrossing but otherwise noninteracting) directed polymers, explored in a companion paper, reveals that generic interactions leave the structure quantitatively unchanged on the line transverse to the pin, and leave it qualitatively unchanged throughout the two dimensions of the system's extent. Furthermore, the free-energy cost associated with a pin that partitions a system having generic interactions is found to be proportional to the pin-partitioning cost for a system of simply noncrossing polymers.
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.
Anyon systems are studied in connection with several interesting applications including high TC superconductivity and topological quantum computing. In this work we show that these systems can be realized starting from directed polymers braided together to form a nontrivial link configuration belonging to the topological class of plats. The statistical sum of a such plat is related here to the partition function of a two-component anyon gas. The constraints that preserve the topological configuration of the plat are imposed on the polymer trajectories using the so-called Gauss linking number, a topological invariant that has already been well studied in polymer physics. Due to these constraints, short-range forces act on the monomers or, equivalently, on the anyon quasiparticles in a way that closely resembles the appearance of reaction forces in the constrained systems of classical mechanics. If the polymers are homogeneous, the anyon system reaches a self-dual point, in which these forces vanish exactly. A class of self-dual solutions that minimize the energy of the anyons is derived. The two anyon gas discussed here obeys an abelian statistics, while for quantum computing it is known that nonabelian anyons are necessary. However, this is a limitation due to the use of the Gauss linking invariant to impose the topological constraints, which is a poor topological invariant and is thus unable to capture the nonabelian characteristics of the braided polymer chains. A more refined treatment of the topological constraints would require more sophisticated topological invariants, but so far their application to the statistical mechanics of linked polymers is an open problem.
The growing field of topological orders has been extensively studied both form the communities of condensed matter and quantum simulation. However, very little is known about the fate of topological order in the presence of disturbing effects such as external noise or dissipation. In the first part of this thesis, we start by studying how the edge states of a topological insulator become unstable when interacting with thermal baths. Motivated by these results, we generalise the notion of Chern insulators from the well-known Hamiltonian case to Liouvillian dynamics. We achieve this goal by defining a new topological witness that is still related to the quantum Hall conductivity at finite temperature. The mixed character of edge states is also well captured by our formalism, and explicit models for topological insulators and dissipative channels are considered. Additionally, we find new topological phases that remain quantised at finite temperature. The construction is based on the Uhlmann phase, a geometric quantum phase defined for general density matrices. Using this new tool, we are able to characterise topological insulators and superconductors at finite temperature both in one and two spatial dimensions. From the experimental side, we propose a state-independent protocol to measure the topological Uhlmann phase in the context of quantum simulation. Symmetry-protected topological orders have traditionally emerged from shortrange interactions. It remains very much unknown what the role played by longrange interactions is, within the physics of these topological systems. In the second part of this thesis, we analyse how topological superconducting phases are affected by the inclusion of long-range couplings. Remarkably, we unveil new topological quasi-particles due to long-range interactions, that were absent in short-range models. We also study how topological invariants are modified by the presence of long-range effects. In the appendix section of the thesis, we explore new numerical methods for driven-dissipative phase transitions. We consider quantum systems with a dissipative term driving the system into a non-equilibrium steady state. The inclusion of short-range fluctuations out-of-equilibrium deeply modifies the shape of the phase-diagram, something never observed in equilibrium thermodynamics.
In this work, we analyze the corrections obtained on a homogeneous one-dimensional Bose gas within the high densities limit by means of the non-regular representation of quantum mechanics introduced by the polymer quantization scheme. Thus, starting from the Bogoliubov formalism, we analyze the ground expectation value of the polymer momentum operator in terms of semiclassical states in order to obtain an analytic expression for the ground state energy of the N-body system, which allows us to solve the pathological behavior commonly associated with the one-dimensional Bose–Einstein condensation through the introduction of finite size effects characterized by the contribution of the polymer corrections. We also discuss the speed of sound in our polymer version of the Bose gas and the corresponding relative shift induced by the introduction of a minimum length parameter. Finally, we investigate the emergent superfluid behavior in our polymer model by implementing an appropriate Landau’s criterion. In this case, we are able to consequently analyze the changes in the critical velocity which defines the limit between the superfluid-condensate regions, thus deducing that the polymer length acts as a kind of pseudo-potential which induces a dissipationless flow associated with the superfluid phase even in the absence of self-interactions.