arXiv subjects
Publications and source records attributed to Alexander Rothkopf.
In this contribution to Strangeness in Quark Matter 2026, I review the open quantum systems approach, a modern theoretical framework for addressing the interaction of a quantum system with its environment, and highlight recent progress in its application to the understanding of in-medium heavy quarkonium in relativistic heavy-ion collisions.
We construct the natural generalization of stochastic quantization (in the Markovian sense) by considering jump-diffusion processes. This class of stochastic processes exhibits non-continuous paths, so-called L\'evy flights. In the presence of jumps, action landscapes with barriers can be efficiently explored, improving and even restoring ergodicity where traditional diffusion approaches become inefficient. We explore different strategies for constructing efficient jump updates, which we deploy to address the benchmark problem of topological freezing in 2d U(1) gauge theory.
We present recent results from the FASTSUM collaboration, using anisotropic lattice QCD to study spectral properties of heavy quarkonia and open heavy flavour systems at high temperature. For heavy quarkonium, our results using a number of different methods suggest a small but significant and robust negative mass shift as well as an increasing thermal width. We present the first lattice results for masses and spectral functions of B mesons at high temperature, and preliminary results for a high-precision calculation of the static quark potential.
In this contribution we present recent developments in the formulation and solution of Initial Boundary Value Problems (IBVPs). Building upon a modern variational action formulation of classical dynamics, we treat Initial Boundary Value Problems directly on the action level, bypassing governing equations. We show that by including coordinate maps as dynamical degrees of freedom together with propagating fields two key results emerge. Space-time symmetries remain protected even after discretization, leading to an exact conservation of Noether charges even for discrete IBVPs. The dynamical nature of the coordinate maps leads to an adjustment of space-time resolution, guided by Noether charge conservation, realizing a form of automatic adaptive mesh refinement. We stress that as long as SBP operators are used for the discretization, our results are independent of whether the dynamics are solved on the action or governing equation level and hold in particular also at high order. As proof-of-principle for our approach we present its application to scalar wave-propagation in 1+1 dimensions.
In this kick-off presentation for the "Recent developments in QCD" session at Baryons 2025 I will tie together the recent progress made on the extraction of parton distribution functions (PDFs) in lattice QCD and the long standing efforts in solving the inverse problem in the form of spectral function reconstruction.
A key tenet of general relativity is the dynamical nature of space-time, ideally represented as an initial value problem. Here we explore the variational formulation of classical Einstein-Hilbert gravity as initial value problem by constructing its Schwinger-Keldysh-Galley (SKG) action, including a careful treatment of boundary terms. The construction is based on a doubling of degrees of freedom and independent of a foliation. The action naturally decomposes into a bulk term furnishing Einstein's equations and a boundary term, which is related to conserved quantities, such as the Komar mass. We find that since only trivial connecting conditions must be specified on boundaries, the variational action principle for gravity as an initial value problem is rendered well-posed without the need to add additional boundary terms. The SKG approach to gravity offers a novel and complementary avenue to solve for the metric of spacetime directly from the action, bypassing the governing equations.
We demonstrate how observables with optimal sensitivity to environmental properties can be constructed explicitly from the master equation of an open-quantum system. Our approach does not rely on the explicit solution of the master equation. This makes the symmetric logarithmic derivative (SLD), the operator of optimal sensitivity and key quantity in quantum metrology, available to a large class of systems of interest, both in and out-of-equilibrium. We validate our approach by reproducing the SLD for temperature in quantum Brownian motion and demonstrate its versatility by constructing the optimal observable for the non-equilibrium relaxation rate.
Real time evolution of a scalar field theory is investigated. The severe sign problem is circumvented using the Complex Langevin equation. The naive application of the method breaks down for extended real times due to the appearance of boundary terms. We use the kernel freedom of the complex Langevin equation to push the breakdown to larger real-times. We search for the optimal kernel using machine learning methods. Thus, we extend the available range for 1+1d scalar simulations beyond the state of the art simulations.
We present a novel solution procedure for initial boundary value problems. The procedure is based on an action principle, in which coordinate maps are included as dynamical degrees of freedom. This reparametrization invariant action is formulated in an abstract parameter space and an energy density scale associated with the space-time coordinates separates the dynamics of the coordinate maps and of the propagating fields. Treating coordinates as dependent, i.e. dynamical quantities, offers the opportunity to discretize the action while retaining all space-time symmetries and also provides the basis for automatic adaptive mesh refinement (AMR). The presence of unbroken space-time symmetries after discretization also ensures that the associated continuum Noether charges remain exactly conserved. The presence of coordinate maps in addition provides new freedom in the choice of boundary conditions. An explicit numerical example for wave propagation in $1+1$ dimensions is provided, using recently developed regularized summation-by-parts finite difference operators.
We construct a discrete non-hermitean momentum operator, which implements faithfully the non self-adjoint nature of momentum for a particle in a box. Its eigenfunctions are strictly limited to the interior of the box in the continuum limit, with the quarter wave as first non-trivial eigenstate. We show how to construct the corresponding hermitean Hamiltonian for the infinite well as concrete example to realize unitary dynamics. The resulting Hilbert space can be decomposed into a physical and unphysical subspace, which are mutually orthogonal. The physical subspace in the continuum limit reproduces that of the continuum theory and we give numerical evidence that the correct probability distributions for momentum and energy are recovered.
We re-investigate the interactions between static color sources in a finite temperature gluonic medium using both high resolution isotropic and anisotropic quenched lattice QCD ensembles. The underlying ill-posed inverse problem, related to the extraction of spectral functions, is attacked with a range of different methods, including Bayesian inference, Pad\'e interpolation and model fits. Among the latter we include a tail amended Gaussian ansatz and a HTL-inspired fit ansatz. We reconfirm the presence of a dominant low-lying spectral feature that supports the existence of a potential picture for the in-medium evolution of the static charges at late real times. Using the raw unmodified lattice data, all applicable methods show clear signs of screening of the real-part of the potential. After applying a subtraction procedure featured in a previous study we find however that screening disappears from the extracted potential. Paths towards the resolution of this puzzle are discussed.
In a recently developed variational discretization scheme for second order initial value problems ( J. Comput. Phys. 498, 112652 (2024) ), it was shown that the Noether charge associated with time translation symmetry is exactly preserved in the interior of the simulated domain. The obtained solution also fulfils the naively discretized equations of motions inside the domain, except for the last two grid points. Here we provide an explanation for the deviations at the boundary as stemming from the Lagrange multipliers used to implement initial and connection conditions. We show explicitly that the Noether charge including the boundary corrections is exactly preserved at its continuum value over the whole simulation domain, including the boundary points.
We present a simulation strategy for the real-time dynamics of quantum fields, inspired by reinforcement learning. It builds on the complex Langevin approach, which it amends with system specific prior information, a necessary prerequisite to overcome this exceptionally severe sign problem. The optimization process underlying our machine learning approach is made possible by deploying inherently stable solvers of the complex Langevin stochastic process and a novel optimality criterion derived from insight into so-called boundary terms. This conceptual and technical progress allows us to both significantly extend the range of real-time simulations in 1+1d scalar field theory beyond the state-of-the-art and to avoid discretization artifacts that plagued previous real-time field theory simulations. Limitations of and promising future directions are discussed.
We study the correlator of temporal Wilson lines at non-zero temperature in 2+1 flavor lattice QCD with the aim to define the heavy quark-antiquark potential at non-zero temperature. For temperatures $153~{\rm MeV} \leq T \leq 352~{\rm MeV}$ the spectral representation of this correlator is consistent with a broadened peak in the spectral function, position or width of which then defines the real or imaginary parts of the heavy quark-antiquark potential at non-zero temperature, respectively. We find that the potential's real part is not screened contrary to the widely-held expectations. We comment on how this fact may modify the picture of quarkonium melting in the quark-gluon plasma.
Taking insight from the theory of general relativity, where space and time are treated on the same footing, we develop a novel geometric variational discretization for second order initial value problems (IVPs). By discretizing the dynamics along a world-line parameter, instead of physical time directly, we retain manifest translation symmetry and conservation of the associated continuum Noether charge. A non-equidistant time discretization emerges dynamically, realizing a form of automatic adaptive mesh refinement (AMR), guided by the system symmetries. Using appropriately regularized summation by parts finite difference operators, the continuum Noether charge, defined via the Killing vector associated with translation symmetry, is shown to be exactly preserved in the interior of the simulated time interval. The convergence properties of the approach are demonstrated with two explicit examples.
This study explores the utility of a kernel in complex Langevin simulations of quantum real-time dynamics on the Schwinger-Keldysh contour. We give several examples where we use a systematic scheme to find kernels that restore correct convergence of complex Langevin. The schemes combine prior information we know about the system and the correctness of convergence of complex Langevin to construct a kernel. This allows us to simulate up to $1.5\beta$ on the real-time Schwinger-Keldysh contour with the 0+1 dimensional anharmonic oscillator using $m=1$, $\lambda=24$, which was previously unattainable using the complex Langevin equation.
We present a novel strategy aimed at restoring correct convergence in complex Langevin simulations. The central idea is to incorporate system-specific prior knowledge into the simulations, in order to circumvent the NP-hard sign problem. In order to do so, we modify complex Langevin using kernels and propose the use of modern auto-differentiation methods to learn optimal kernel values. The optimization process is guided by functionals encoding relevant prior information, such as symmetries or Euclidean correlator data. Our approach recovers correct convergence in the non-interacting theory on the Schwinger-Keldysh contour for any real-time extent. For the strongly coupled quantum anharmonic oscillator we achieve correct convergence up to three-times the real-time extent of the previous benchmark study. An appendix sheds light on the fact that for correct convergence not only the absence of boundary terms, but in addition the correct Fokker-Plank spectrum is crucial.
We present results on the in-medium interactions of static quark anti-quark pairs using realistic 2+1 HISQ flavor lattice QCD. Focus is put on the extraction of spectral information from Wilson line correlators in Coulomb gauge using four complementary methods. Our results indicate that on HISQ lattices, the position of the dominant spectral peak associated with the real-part of the interquark potential remains unaffected by temperature. This is in contrast to prior work in quenched QCD and we present follow up comparisons to newly generated quenched ensembles.