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We examine the behavior of charged pions in neutron-rich matter using heavy-baryon chiral perturbation theory. This study is motivated by the prospect that pions, or pion-like, excitations, may be relevant in neutron-rich matter encountered in core-collapse supernovae and neutron star mergers. We find, as previously expected, that the $\pi^-$ mass increases with density and precludes s-wave condensation, and the mass of the $\pi^+$ mode decreases with increasing density. The energy difference between neutrons and protons in neutron-rich matter related to the nuclear symmetry energy alters the power counting. It enhances higher-order contributions to the pion self-energy. Previously unimportant but attractive diagrams are now enhanced. The net effect of this is the appearance of a new collective mode with the quantum numbers of the $\pi^+$, and a pronounced reduction of the $\pi^+$ mass., Comment: 13 pages, 4 figures, 1 appendix

The Monte Carlo evaluation of path integrals is one of a few general purpose methods to approach strongly coupled systems. It is used in all branches of Physics, from QCD/nuclear physics to the correlated electron systems. However, many systems of great importance (dense matter inside neutron stars, the repulsive Hubbard model away from half-filling, dynamical and non-equilibrium observables) are not amenable to the Monte Carlo method as it currently stands due to the so-called "sign-problem". We review a new set of ideas recently developed to tackle the sign problem based on the complexification of field space and the Picard-Lefshetz theory accompanying it. The mathematical ideas underpinning this approach, as well as the algorithms so far developed, are described together with non-trivial examples where the method has already been proved successful. Directions of future work, including the burgeoning use of machine learning techniques, are delineated., 35 pages, 21 figures

Path integral contour deformations have been shown to mitigate sign and signal-to-noise problems associated with phase fluctuations in lattice field theories. We define a family of contour deformations applicable to $SU(N)$ lattice gauge theory that can reduce sign and signal-to-noise problems associated with complex actions and complex observables. For observables, these contours can be used to define deformed observables with identical expectation value but different variance. As a proof-of-principle, we apply machine learning techniques to optimize the deformed observables associated with Wilson loops in two dimensional $SU(2)$ and $SU(3)$ gauge theory. We study loops consisting of up to 64 plaquettes and achieve variance reduction of up to 4 orders of magnitude., Comment: 25 pages, 12 figures

In this thesis a generic method for taming the sign problem is developed. The sign problem is the name given to the difficult task of numerically integrating a highly oscillatory integral, and the sign problem inhibits our ability to understand the properties of a wide range of systems of interest in theoretical physics. Particularly notably for nuclear physics, the sign problem prevents the calculation of the properties of QCD at finite baryon density, thereby precluding an under- standing of the dense nuclear matter found in the center of a neutron star. The central idea developed in this thesis is to use the multidimensional generalization of Cauchy’s Integral Theorem to deform the Feynman Path Integral of lattice fields theories into complexified field space to manifolds upon which the phase oscillations which cause the sign problem are gentle. Doing so allows calculations of theories with sign problems. Two practical manifold deformation methods, the holomorphic gradient flow and the sign-optimized manifold method, are developed. The holomorphic gradient flow, a generalization of the Lefschetz thimble method, continuously deforms the original path integration domain to a complex manifold via an evolution dictated by a complex first order differential equation. The sign-optimized manifold method is a way to generate a manifold with gentle phase oscillations by minimizing the sign problem in a parameterized family of manifolds through stochastic gradient ascent. With an eye toward QCD at finite density, the Cauchy’s Theorem approach is applied to relativistic quantum field theories of fermions at finite density throughout this thesis. Finally, these methods are general and can be applied to both bosonic and fermionic theories, as well as Minkowski path integrals describing real-time dynamics.

We present Monte Carlo calculations of the thermodynamics of the (2+1) dimensional Thirring model at finite density. We bypass the sign problem by deforming the domain of integration of the path integral into complex space in such a way as to maximize the average sign within a parameterized family of manifolds. We present results for lattice sizes up to $10^3$ and we find that at high densities and/or temperatures the chiral condensate is abruptly reduced., 5 pages, 4 figures

Monte Carlo studies involving real time dynamics are severely restricted by the sign problem that emerges from highly oscillatory phase of the path integral. In this letter, we present a new method to compute real time quantities on the lattice using the Schwinger-Keldysh formalism via Monte Carlo simulations. The key idea is to deform the path integration domain to a complex manifold where the phase oscillations are mild and the sign problem is manageable. We use the previously introduced "contraction algorithm" to create a Markov chain on this alternative manifold. We substantiate our approach by analyzing the quantum mechanical anharmonic oscillator. Our results are in agreement with the exact ones obtained by diagonalization of the Hamiltonian. The method we introduce is generic and in principle applicable to quantum field theory albeit very slow. We discuss some possible improvements that should speed up the algorithm., Comment: 5 pages, 3 figures