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arXiv:1409.4048

Low Energy Continuum and Lattice Effective Field Theories

Serdar Elhatisari

Published Sep 14, 2014• 0 citations
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Abstract

In the first part of the thesis we consider the constraints of causality and unitarity for particles interacting via strictly finite-range interactions. We generalize Wigner's causality bound to the case of non-vanishing partial-wave mixing. Specifically we analyze the system of the low-energy interactions between protons and neutrons. We also analyze low-energy scattering for systems with arbitrary short-range interactions plus an attractive $1/r^α$ tail for $α\geq2$. In particular, we focus on the case of $α=6$ and we derive the constraints of causality and unitarity also for these systems and find that the van der Waals length scale dominates over parameters characterizing the short-distance physics of the interaction. This separation of scales suggests a separate universality class for physics characterizing interactions with an attractive $1/r^{6}$ tail. We argue that a similar universality class exists for any attractive potential $1/r^α$ for $α\geq2$. In the second part of the thesis we present lattice Monte Carlo calculations of fermion-dimer scattering in the limit of zero-range interactions using the adiabatic projection method. The adiabatic projection method uses a set of initial cluster states and Euclidean time projection to give a systematically improvable description of the low-lying scattering cluster states in a finite volume. We use Lüscher's finite-volume relations to determine the $s$-wave, $p$-wave, and $d$-wave phase shifts. For comparison, we also compute exact lattice results using Lanczos iteration and continuum results using the Skorniakov-Ter-Martirosian equation. For our Monte Carlo calculations we use a new lattice algorithm called impurity lattice Monte Carlo. This algorithm can be viewed as a hybrid technique which incorporates elements of both worldline and auxiliary-field Monte Carlo simulations.

DOI
10.48550/arxiv.1409.4048

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In the first part of the thesis we consider the constraints of causality and unitarity for particles interacting via strictly finite-range interactions. We generalize Wigner's causality bound to the case of non-vanishing partial-wave mixing. Specifically we analyze the system of the low-energy interactions between protons and neutrons. We also analyze low-energy scattering for systems with arbitrary short-range interactions plus an attractive 1/rα1/r^α1/rα tail for α≥2α\geq2α≥2. In particular, we focus on the case of α=6α=6α=6 and we derive the constraints of causality and unitarity also for these systems and find that the van der Waals length scale dominates over parameters characterizing the short-distance physics of the interaction. This separation of scales suggests a separate universality class for physics characterizing interactions with an attractive 1/r61/r^{6}1/r6 tail. We argue that a similar universality class exists for any attractive potential 1/rα1/r^α1/rα for α≥2α\geq2α≥2. In the second part of the thesis we present lattice Monte Carlo calculations of fermion-dimer scattering in the limit of zero-range interactions using the adiabatic projection method. The adiabatic projection method uses a set of initial cluster states and Euclidean time projection to give a systematically improvable description of the low-lying scattering cluster states in a finite volume. We use Lüscher's finite-volume relations to determine the sss-wave, ppp-wave, and ddd-wave phase shifts. For comparison, we also compute exact lattice results using Lanczos iteration and continuum results using the Skorniakov-Ter-Martirosian equation. For our Monte Carlo calculations we use a new lattice algorithm called impurity lattice Monte Carlo. This algorithm can be viewed as a hybrid technique which incorporates elements of both worldline and auxiliary-field Monte Carlo simulations.
We consider two-component fermions with short-range interactions and large scattering length. This system has universal properties that are realized in several different fields of physics. In the limit of large fermion–fermion scattering length aff and zero-range interaction, all properties of the system scale proportionally with aff. For the case with shallow bound dimers, we calculate the dimer–dimer scattering phase shifts using lattice effective field theory. We extract the universal dimer–dimer scattering length add/aff=0.618(30) and effective range rdd/aff=−0.431(48). This result for the effective range is the first calculation with quantified and controlled systematic errors. We also benchmark our methods by computing the fermion–dimer scattering parameters and testing some predictions of conformal scaling of irrelevant operators near the unitarity limit.
In this work we introduce a worldline-based fermion Monte Carlo algorithm for studying few body quantum mechanics of self-interacting fermions in the Hamiltonian lattice formulation. Our motivation to construct the method comes from our interest in studying renormalization of chiral nuclear effective-field theory with lattice regularization. In particular we wish to apply our method to compute the lattice spacing dependence of local lattice interactions as we take the continuum limit of the lattice theory. Our algorithm can compute matrix elements of the operator $exp(\ensuremath{-}\ensuremath{\beta}H)$, where HHH is the lattice Hamiltonian and $\ensuremath{\beta}$ is a free real parameter. These elements help us compute deep bound states that are well separated from scattering states even at values of $\ensuremath{\beta}$ which are not very large. Computing these bound-state energies accurately can help us study renormalization of the lattice theory. In addition to developing the algorithm, in this work we also introduce a finite-volume renormalization scheme for the lattice Hamiltonian of the leading pionless effective-field theory and show how it would work in the one- and two-body sectors.
The control afforded by Feshbach resonance phenomena has enabled the exploration of strongly interacting degenerate regimes in dilute ultracold atomic alkali-gases. In these dilute systems, interactions are characterized by a single parameter, the s-wave scattering length. In this dissertation, we review the physics of quantum degenerate atomic gases from a theoretical perspective and present the applications of non-perturbative numerical methods ranging from exact diagonalization to quantum Monte Carlo techniques. Emphasis is given to the effect of interactions. A major goal of this work is to compare theoretical predictions with available experimental results. We begin by introducing the effective interactions in the many-body alkali-gas system. As simplifications of the real interaction between two alkali atoms, the resonance properties of various short range models, the zero-range model and the two-channel model are investigated. The fundamental result is that under appropriate conditions, the true interaction potential V(r) of two atoms may be replaced by a regularized delta-function. In collaboration with experimentalists in our department, we developed a unique method to construct localized single-particle wave functions using imaginary time projection and thereby determine lattice Hamiltonian parameters. Our method enables an efficient coarse-grained mapping from a continuum system to a lattice model. We apply the method to a specific disordered potential generated by an optical lattice experiment and calculate for each instance of disorder the equivalent lattice model parameters. Detailed statistical analysis is performed on the resulting probability distributions of the Hubbard parameters. In the final part, we study the pairing and ferromagnetic instabilities in systems of spin-1/2 fermions. To interpret a recent experimental study of the possibility of itinerant ferromagnetism in cold atom systems, we investigate the energy spectrum of a system of four spin-1/2 fermions with short range attractive interactions both exactly and within the scattering length approximation. The formation of molecular bound states and the ferromagnetic transition of the excited scattering state are examined systematically as a function of the two-body scattering length. We show that an adiabatic ferromagnetic transition occurs, but at a critical transition point k_{F}a much higher than predictions from the scattering length approximations. The exact critical interaction strength calculated in the four-particle system is consistent with that reported by experiment. Finally, by constructing a many-body wavefunction satisfying the Bethe-Peierls boundary conditions, we attempted the first variational Monte Carlo calculations of the zero-range model of unitary Fermion gases, eliminating the need for short-range approximations employed by existing QMC calculations.
Lattice field theory methods, usually associated with non-perturbative studies of quantum chromodynamics, are becoming increasingly common in the calculation of ground-state and thermal properties of strongly interacting non-relativistic few- and many-body systems, blurring the interfaces between condensed matter, atomic and low-energy nuclear physics. While some of these techniques have been in use in the area of condensed matter physics for a long time, others, such as hybrid Monte Carlo and improved effective actions, have only recently found their way across areas. With this topical review, we aim to provide a modest overview and a status update on a few notable recent developments. For the sake of brevity we focus on zero-temperature, non-relativistic problems. After a short introduction, we lay out some general considerations and proceed to discuss sampling algorithms, observables, and systematic effects. We show selected results on ground- and excited-state properties of fermions in the limit of unitarity. The appendix contains details on group theory on the lattice.
We analyze low-energy scattering for arbitrary short-range interactions plus an attractive 1/r61/{r}^{6}1/r6 tail. We derive the constraints of causality and unitarity and find that the van der Waals length scale dominates over parameters characterizing the short-distance physics of the interaction. This separation of scales suggests a separate universality class for physics characterizing interactions with an attractive 1/r61/{r}^{6}1/r6 tail. We argue that a similar universality class exists for any attractive potential $1/{r}^{\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\geqslant}2$. We also discuss the extension to multichannel systems near a magnetic Feshbach resonance. We discuss the implications for effective field theory with attractive singular power-law tails.
At long distances, interactions between neutral ground-state atoms can be described by the van der Waals potential. In the ultracold regime, atom-atom scattering is dominated by sss-waves phase shifts given by an effective range expansion in terms of the scattering length ${\ensuremath{\alpha}}_{0}$ and the effective range r0{r}_{0}r0​. We show that while the scattering length cannot be predicted for these potentials, the effective range is given by the universal low-energy theorem ${r}_{0}=A+B/{\ensuremath{\alpha}}_{0}+C/{\ensuremath{\alpha}}_{0}^{2}$, where AAA, BBB, and CCC depend on the dispersion coefficients Cn{C}_{n}Cn​ and the reduced diatom mass. We confront this formula to about 100 determinations of r0{r}_{0}r0​ and ${\ensuremath{\alpha}}_{0}$ and show why the result is dominated by the leading dispersion coefficient C6{C}_{6}C6​. Universality and scaling extend much beyond naive dimensional analysis estimates.