Unphysical Branch Research Articles

SemiH: DFT Hamiltonian neural network training with semi-supervised learning

Abstract Over the past decades, density functional theory (DFT) calculations have been utilized in various fields such as materials science and semiconductor devices. However, due to the inherent nature of DFT calculations, which rigorously consider interactions between atoms, they require significant computational cost. To address this, extensive research has recently focused on training neural networks to replace DFT calculations. However, previous methods for training neural networks necessitated an extensive number of DFT simulations to acquire the ground truth (Hamiltonians). Conversely, when dealing with a limited amount of training data, deep learning models often display increased errors in predicting Hamiltonians and band structures for testing data. This phenomenon poses the potential risk of generating inaccurate physical interpretations, including the emergence of unphysical branches within band structures. To tackle this challenge, we propose a novel deep learning-based method for calculating DFT Hamiltonians, specifically tailored to produce accurate results with limited training data. Our framework not only employs supervised learning with the calculated Hamiltonian but also generates pseudo Hamiltonians (targets for unlabeled data) and trains the neural networks on unlabeled data. Particularly, our approach, which leverages unlabeled data, is noteworthy as it marks the first attempt in the field of neural network Hamiltonians. Our framework showcases the superior performance of our framework compared to the state-of-the-art approach across various datasets, such as MoS2, Bi2Te3, HfO2, and InGaAs. Moreover, our framework demonstrates enhanced generalization performance by effectively utilizing unlabeled data, achieving noteworthy results when evaluated on data more complex than the training set, such as configurations with more atoms and temperature ranges outside the training data.

Fundamental competition of smooth and non-smooth bifurcations and their ghosts in vibro-impact pairs

A combined analysis of smooth and non-smooth bifurcations captures the interplay of different qualitative transitions in a canonical model of an impact pair, a forced capsule in which a ball moves freely between impacts on either end of the capsule. The analysis, generic for the impact pair context, is also relevant for applications. It is applied to a model of an inclined vibro-impact energy harvester device, where the energy is generated via impacts of the ball with a dielectric polymer on the capsule ends. While sequences of bifurcations have been studied extensively in single- degree-of-freedom impacting models, there are limited results for two-degree-of-freedom impacting systems such as the impact pair. Using an analytical characterization of impacting solutions and their stability based on the maps between impacts, we obtain sequences of period doubling and fold bifurcations together with grazing bifurcations, a particular focus here. Grazing occurs when a sequence of impacts on either end of the capsule are augmented by a zero-velocity impact, a transition that is fundamentally different from the smooth bifurcations that are instead characterized by eigenvalues of the local behavior. The combined analyses allow identification of bifurcations also on unstable or unphysical solutions branches, which we term ghost bifurcations. While these ghost bifurcations are not observed experimentally or via simple numerical integration of the model, nevertheless they can influence the birth or death of complex behaviors and additional grazing transitions, as confirmed by comparisons with the numerical results. The competition between the different bifurcations and their ghosts influences the parameter ranges for favorable energy output; thus, the analyses of bifurcation sequences yield important design information.

Anomalous bootstrap on the half-line

We study carefully the problem of the bootstrap on the half-line. We show why one needs the full set of constraints derived from the Stieltjes theorem on the moment problem by reexamining previous results on the hydrogen atom. We also study the hydrogen atom at continuous angular momentum. We show that the constraints on the moment problem alone do not fix the boundary conditions in all cases and at least one of the positive matrices needs to be slightly enlarged to remove unphysical branches. We explain how to solve the more general problem of the bootstrap for Robin boundary conditions. The recursion relations that are usually used receive additional anomalous contributions. These corrections are necessary to compute the moments of the measure. We apply these to the linear potential and we show how the bootstrap matches the analytical results, based on the Airy function, for this example.

Pentagon functions for one-mass planar scattering amplitudes

We present analytic results for all planar two-loop Feynman integrals contributing to five-particle scattering amplitudes with one external massive leg. We express the integrals in terms of a basis of algebraically-independent transcendental functions, which we call one-mass pentagon functions. We construct them by using the properties of iterated integrals with logarithmic kernels. The pentagon functions are manifestly free of unphysical branch cuts, do not require analytic continuation, and can be readily evaluated over the whole physical phase space of the massive-particle production channel. We develop an efficient algorithm for their numerical evaluation and present a public implementation suitable for direct phenomenological applications.

Dilatonic states near holographic phase transitions

The spectrum of bound states of special strongly coupled confining field theories might include a parametrically light dilaton, associated with the formation of enhanced condensates that break (approximate) scale invariance spontaneously. It has been suggested in the literature that such a state may arise in connection with the theory being close to the unitarity bound in holographic models. We extend these ideas to cases where the background geometry is non-AdS, and the gravity description of the dual confining field theory has a top-down origin in supergravity. We exemplify this programme by studying the circle compactification of Romans six-dimensional half-maximal supergravity. We uncover a rich space of solutions, many of which were previously unknown in the literature. We compute the bosonic spectrum of excitations, and identify a tachyonic instability in a region of parameter space for a class of regular background solutions. A tachyon only exists along an energetically disfavoured (unphysical) branch of solutions of the gravity theory; we find evidence of a first-order phase transition that separates this region of parameter space from the physical one. Along the physical branch of regular solutions, one of the lightest scalar particles is approximately a dilaton, and it is associated with a condensate in the underlying theory. Yet, because of the location of the phase transition, its mass is not parametrically small, and it is, coincidentally, the next-to-lightest scalar bound state, rather than the lightest one.

Pentagon functions for scattering of five massless particles

We complete the analytic calculation of the full set of two-loop Feynman integrals required for computation of massless five-particle scattering amplitudes. We employ the method of canonical differential equations to construct a minimal basis set of transcendental functions, pentagon functions, which is sufficient to express all planar and nonplanar massless five-point two-loop Feynman integrals in the whole physical phase space. We find analytic expressions for pentagon functions which are manifestly free of unphysical branch cuts. We present a public library for numerical evaluation of pentagon functions suitable for immediate phenomenological applications.

Simulation‐based equations of state for the Lennard‐Jones fluid: Apparent success and hidden failure

AbstractIn this work, we utilize concepts from bifurcation theory to pinpoint hidden defects in accurate multiparameter simulation‐based equations of state for the Lennard‐Jones (LJ) fluid. The proposed bifurcation diagrams track the evolution of volume roots as temperatures vary at constant pressure. We critically evaluate four distinct types of LJ‐based equations of state: modified Benedict‐Webb‐Rubin equation (with three different parameter sets), Kolafa and Nezbeda, Mecke et al, and Thol et al. For each model, we mainly construct two bifurcation diagrams at subcritical and supercritical isobars. The unphysical behaviors associated with the studied equations involve spurious two‐phase separation regions, distorted volume‐temperature behavior, unphysical branches, unphysical turning points, and multiplicity in volume roots. Our proposed bifurcation diagram provides a reliable and simple technique to pinpoint hidden defects in equations of state‐based merely on temperature, volume, and pressure without the need of their partial derivatives or thermodynamic potentials.

Dessin d’enfant and a description of mutually nonlocal 7-branes without using branch cuts

We consider the special roles of the zero loci of the Weierstrass invariants $g_2(\tau(z))$, $g_3(\tau(z))$ in F-theory on an elliptic fibration over $P^1$ or a further fibration thereof. They are defined as the zero loci of the coefficient functions $f(z)$ and $g(z)$ of a Weierstrass equation. They are thought of as complex co-dimension one objects and correspond to the two kinds of critical points of a dessin d'enfant of Grothendieck. The $P^1$ base is divided into several cell regions bounded by some domain walls extending from these planes and D-branes, on which the imaginary part of the $J$-function vanishes. This amounts to drawing a dessin with a canonical triangulation. We show that the dessin provides a new way of keeping track of mutual non-localness among 7-branes without employing unphysical branch cuts or their base point. With the dessin we can see that weak- and strong-coupling regions coexist and are located across an $S$-wall from each other. We also present a simple method for computing a monodromy matrix for an arbitrary path by tracing the walls it goes through.

Nonphysical Behavior in Several Statistical Mechanically Based Equations of State

In this work, we utilize bifurcation diagrams to study the role of mathematical artifacts in deteriorating the physical behavior in statistical mechanically based equations of state of pure fluids. We study the impact of common empirical approximations usually employed to overcome some of the mathematical and physical challenges such as the parametrization of mean field models or pair correlations functions at contact. The proposed diagrams elucidate how the reduced molar volume bifurcates with the variation of temperature at constant pressure. We generate bifurcation diagrams for the modified van der Waals equation of state (EOS) of Poole et al, SAFT-VR Mie, Soft-SAFT, CK-SAFT, and the original SAFT EOSs for spherical and nonspherical molecules. We find that the bifurcation diagram can serve as a useful schematic tool to reveal the unphysical PVT behavior, demonstrate the existence of physical and spurious two-phase separation regions, and illustrate how the number of molar volume roots vary with temperatures. Our method shows that the presence of unphysical branches can cause spurious two-phase separation regions and create erroneous behavior in the stability limit of vapor–liquid equilibrium. We demonstrate that the existence of customary and spurious phase envelopes is accompanied by S-shaped behavior in the volume-temperature bifurcation diagrams. The study reveals that none of the SAFT models is free from producing unphysical behavior. While the SAFT-VR Mie EOS exhibits solid–liquid-like behavior for nonspherical molecules, the CK-SAFT EOS shows liquid–liquid demixing behavior for spherical and nonspherical compounds. For the soft-SAFT EOS, three different two-phase separation regions are observed in addition to the common vapor–liquid phase separation region.

Practical consequences of the Luttinger-Ward functional multivaluedness for cluster DMFT methods

The Luttinger-Ward functional (LWF) has been a starting point for conserving approximations in many-body physics for 50 years. The recent discoveries of its multivaluedness and the associated divergence of the two-particle irreducible vertex function $\mathrm{\ensuremath{\Gamma}}$ have revealed an inherent limitation of this approach. Here we demonstrate how these undesirable properties of the LWF can lead to a failure of computational methods based on an approximation of the LWF. We apply the nested cluster scheme (NCS) to the Hubbard model and observe the existence of an additional stationary point of the self-consistent equations, associated with an unphysical branch of the LWF. In the strongly correlated regime, starting with the first divergence of $\mathrm{\ensuremath{\Gamma}}$, this unphysical stationary point becomes attractive in the standard iterative technique used to solve DMFT. This leads to an incorrect solution, even in the large cluster size limit, for which we discuss diagnostics.

O(N)model within theΦ-derivable expansion to orderλ2: On the existence and UV/IR sensitivity of the solutions to self-consistent equations

We discuss various aspects of the O(N)-model in the vacuum and at finite temperature within the Phi-derivable expansion scheme to order lambda^2. In continuation to an earlier work, we look for a physical parametrization in the N=4 case that allows to accommodate the lightest mesons. Using zero-momentum curvature masses to approximate the physical masses, we find that, in the parameter range where a relatively large sigma mass is obtained, the scale of the Landau pole is lower compared to that obtained in the two-loop truncation. This jeopardizes the insensitivity of the observables to the ultraviolet regulator and could hinder the predictivity of the model. Both in the N=1 and N=4 cases, we also find that, when approaching the chiral limit, the (iterative) solution to the Phi-derivable equations is lost in an interval around the would-be transition temperature. In particular, it is not possible to conclude at this order of truncation on the order of the transition in the chiral limit. Because the same issue could be present in other approaches, we investigate it thoroughly by considering a localized version of the Phi-derivable equations, whose solution displays the same qualitative features, but allows for a more analytical understanding of the problem. In particular, our analysis reveals the existence of unphysical branches of solutions which can coalesce with the physical one at some temperatures, with the effect of opening up a gap in the admissible values for the condensate. Depending on its rate of growth with the temperature, this gap can eventually engulf the physical solution.

Nonexistence of the Luttinger-Ward functional and misleading convergence of skeleton diagrammatic series for hubbard-like models.

The Luttinger-Ward functional Φ[G], which expresses the thermodynamic grand potential in terms of the interacting single-particle Green's function G, is found to be ill defined for fermionic models with the Hubbard on-site interaction. In particular, we show that the self-energy Σ[G]∝δΦ[G]/δG is not a single-valued functional of G: in addition to the physical solution for Σ[G], there exists at least one qualitatively distinct unphysical branch. This result is demonstrated for several models: the Hubbard atom, the Anderson impurity model, and the full two-dimensional Hubbard model. Despite this pathology, the skeleton Feynman diagrammatic series for Σ in terms of G is found to converge at least for moderately low temperatures. However, at strong interactions, its convergence is to the unphysical branch. This reveals a new scenario of breaking down of diagrammatic expansions. In contrast, the bare series in terms of the noninteracting Green's function G0 converges to the correct physical branch of Σ in all cases currently accessible by diagrammatic Monte Carlo calculations. In addition to their conceptual importance, these observations have important implications for techniques based on the explicit summation of the diagrammatic series.

Cutting rules and perturbative unitarity of noncommutative electric-type field theories from string theory

We discuss the breakdown of the perturbative unitarity of noncommutative quantum field theories in an electric-type background in the light of string theory. We consider the analytic structure of string loop two-point functions using a suitable off-shell continuation, and then study the zero slope limit of Seiberg and Witten. In this way we pick up how the unphysical tachyonic branch cut appears in noncommutative field theory. We briefly discuss discontinuities and cutting rules for the full string theory amplitude and relate them to the noncommutative field theoretical results, and also discuss the insight one gains into the magnetic case too.

Selection of the Scaling Solution in a Cluster Coalescence Model

The scaling properties of the cluster size distribution of a system of diffusing clusters is studied in terms of a simple kinetic mean field model. It is shown that a one parameter family of mathematically valid scaling solutions exists. Despite this, the kinetics reaches a unique scaling solution independent of initial conditions. This selected scaling solution is marginally physical, i.e., it is the borderline solution between the unphysical and physical branches of the family of solutions.

Improving lattice quark actions

We explore the first stage of the Symanzik improvement program for lattice Dirac fermions, namely the construction of doubler-free, highly improved classical actions on isotropic as well as anisotropic lattices (where the temporal lattice spacing, a t , is smaller than the spatial one). Using field transformations to eliminate doublers, we derive the previously presented isotropic D234 action with O(a 3) errors, as well as anisotropic D234 actions with O(a 4) or O(a 3,a 4) errors. Besides allowing the simulation of heavy quarks within a relativistic framework, anisotropic lattices alleviate potential problems due to unphysical branches of the quark dispersion relation (which are generic to improved actions), facilitate studies of lattice thermodynamics, and allow accurate mass determinations for particles with bad signal/noise properties, like glueballs and P-state mesons. We also show how field transformations can be used to completely eliminate unphysical branches of the dispersion relation. Finally, we briefly discuss future steps in the improvement program.

Towards highly improved quark actions

We describe two ideas useful in the construction of highly improved quark actions for simulations on coarse lattices: (i) Field transformations solve the doubler problem without destroying tree-level improvement for on- or off-shell quantities. The simplest example is the Sheikholeslami-Wohlert (clover) action. Going to the next order of improvement yields the class of D234 actions. (ii) Anisotropic lattices with a t < a s useful because they push up the energy of unphysical branches of the dispersion relation (which are generic to highly improved actions), allow accurate mass determinations for particles with bad signal/noise properties (glueballs, P-state mesons), and enable one to simulate heavy quarks within a relativistic framework. We present first simulation results for the quenched light hadron and charmonium spectra obtained with a D234 action on anisotropic lattices.

Analytic structure of the ac quasienergy in the complex-field plane.

Motivated by the possibility of developing new computational techniques for studying multiphoton ionization of atoms by monochromatic radiation, we have analyzed, in a nonrigorous fashion, the behavior of the ac quasienergy as a function of the complex field strength F. We elaborate upon earlier work of Manakov and Fainshtein [Theor. Math. Phys. 48, 815 (1981)] and conjecture that the ac quasienergy is a multivalued analytic function ${\mathit{E}}_{\mathrm{ac}}$(F) whose branches originate from different unperturbed (real or ``shadow'') atomic levels for real values of F. We further conjecture that the branch points of ${\mathit{E}}_{\mathrm{ac}}$(F) are of the square-root type. These branch points occur at complex values of F where two branches coalesce, and, as F sweeps along the real axis, the passage past a branch point coincides with the passage past either an intermediate multiphoton resonance (R) or a multiphoton ionization threshold (T). Branch points of type-R group into quadruplets, while branch points of type-T group into pairs. The two branches that intersect at a type-R branch point are both physically accessible, and originate from real levels, while only one of the two branches that intersect at a type-T branch point is physically accessible\char22{}the unphysical type-T branch is a ``shadow'' eigenvalue, which corresponds to a state with unphysical boundary conditions. We discuss the probability for the atom to make a transition from one branch to another when F is a slowly varying function of time. Normally, a type-R branch point enhances the ionization signal, while a type-T branch point diminishes the signal. In partial support of some of our conjectures, we present results of an accurate numerical study of the quasienergy (and its perturbation expansion) for the ground state of both the hydrogen atom and a model atom (an electron bound to a zero-range potential).

Shock wave structure in gas mixtures with large mass disparity

The structure of normal shock waves is considered when the ratio of molecular masses mp/m of a binary mixture of inert monatomic gases is large and the density ratio ρp/ρ is of order unity or below. Generalized hydrodynamic equations, valid for arbitrary intermolecular potentials, are obtained from a hypersonic closure of the kinetic equation for the heavy gas and a near-equilibrium closure for the light component. Because the Prandtl number of the light gas and the Schmidt number of the mixture are nearly constant, the only independent transport coefficient arising in the model is the viscosity μ of the light gas, which is absorbed into a new independent position variable s. Knowledge of μ as a function of temperature thus determines the shock structure independently from the details of the intermolecular potential, allowing comparison with experiments in the complete absence of free parameters. In terms of the ratio M (frozen Mach number) between the speed of propagation and the sound speed of the light gas in the unperturbed medium, one finds that: (i) When M &gt; 1, the behaviour is similar to that of a ‘dusty gas’, with a broad relaxation layer (outer solution) following a sharp boundary layer through which the speed of the heavy gas is almost constant (a shock within a shock). (ii) When (1 + ρp/ρ)s−½ &lt; M &lt; 1, the boundary layer disappears, yielding a so-called ‘fully dispersed wave’. (iii) Because the internal energy of the heavy gas is negligible, the present problem differs from previous shock studies in that, for the first time, the structure of the relaxation region is obtained algebraically in phase space, thus permitting an exhaustive study of the behaviour. From it, the overshooting solution found by Sherman (1960) is related to the unphysical degenerate branch of the outer solution arising when M &gt; 1, showing a failure of the Chapman–Enskog theory, even for weak shocks, when the heavy gas is dilute. Also, an algebraic explanation arises for the ‘double hump structure’ observed in He–Xe shocks. (iv) When M is nearly unity, the initial boundary layer spreads out, and the structure must be obtained by integration of a numerically unstable system of three differential equations. However, the reduction of order brought about by the weak variation of the light-gas entropy at the head of the shock, results in a stable system of equations that we integrate numerically. Excellent phase-space agreement with recent shock-tube experiments of Tarczynski, Herczynski &amp; Walenta (1986) is found for both weak and strong shocks.

Exact near-onset analysis of the spin-density-wave instability in ferromagnetic superconductors: The linearly polarized state

Using an approach similar to Abvikosov's theory of the vortex state near ${H}_{c2}$, we have performed an exact, near-onset analysis of a spin-density-wave instability leading to the "linearly polarized state" of Greenside et al. in ferromagnetic superconductors. The approach is based on a generalized Ginzburg-Landau theory for such materials, as formulated by Blount and Varma. Two models have been considered. In the ($\ensuremath{\alpha}$,$\ensuremath{\beta}$) model, where the bulk magnetic energy is taken to be $\frac{1}{2}{\ensuremath{\alpha}}_{m}{M}^{2}+\frac{1}{4}{\ensuremath{\beta}}_{m}{M}^{4}$, we find the transition to be second order, and obtain explicit formulas for various physical quantities to leading order in the deviation from onset. We have also rigorously analyzed the most favored spatial structure just below onset, among all possibilities allowed by the instability, and have concluded that a plane-wave-like structure is favored in a physical limit considered. In the ($\ensuremath{\alpha}$,$\ensuremath{\gamma}$) model, where the bulk magnetic energy is taken to be $\frac{1}{2}{\ensuremath{\alpha}}_{m}{M}^{2}+\frac{1}{6}{\ensuremath{\gamma}}_{m}{M}^{6}$, as is supported by recent experiments for Er${\mathrm{Rh}}_{4}$${\mathrm{B}}_{4}$, we find the transition to be first order. This approach is then confined to an unphysical branch, which does not permit us to calculate various physical quantities on the physical branch.

Partial wave dispersion relations for photon-nucleon scattering and nucleon-antinucleon production by photons

Partial wave dispersion relations for the processes γ Open image in new window →→γ+ Open image in new window and γ+γ→ Open image in new window + Open image in new window to the ordere2 are derived with the help of the Mandelstam representation. The angular momentum decomposition for both processes is carried out and the analytic properties of the partial wave amplitudes are determined. The absorptive parts of these amplitudes along part of the physical branch cuts are calculated by means of the unitarity condition. For γ+ Open image in new window →γ+ Open image in new window the resulting expressions involve amplitudes for single pion photoproduction; for γ+γ→ Open image in new window + Open image in new window they involve theT=0s-wave pion-pion phase shift and the total cross-section for the process γ+π→π+π. The discontinuities along the unphysical branch cuts are calculated by means of crossing symmetry and of analytic continuation through Legendre polynomials. The limits of convergence of the Legendre expansions are discussed and reasons for the importance of the two-pion exchange contribution in photon-nucleon scattering are presented.