Unphysical States Research Articles

Symmetry breaking and self-interaction correction in the chromium atom and dimer.

Density functional approximations to the exchange-correlation energy can often identify strongly correlated systems and estimate their energetics through energy-minimizing symmetry-breaking. In particular, the binding energy curve of the strongly correlated chromium dimer is described qualitatively by the local spin density approximation (LSDA) and almost quantitatively by the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA), where the symmetry breaking is antiferromagnetic for both. Here, we show that a full Perdew-Zunger self-interaction-correction (SIC) to LSDA seems to go too far by creating an unphysical symmetry-broken state, with effectively zero magnetic moment but non-zero spin density on each atom, which lies ∼4eV below the antiferromagnetic solution. A similar symmetry-breaking, observed in the atom, better corresponds to the 3d↑↑4s↑3d↓↓4s↓ configuration than to the standard 3d↑↑↑↑↑4s↑. For this new solution, the total energy of the dimer at its observed bond length is higher than that of the separated atoms. These results can be regarded as qualitative evidence that the SIC needs to be scaled down in many-electron regions.

N-representability violations in truncated equation-of-motion coupled-cluster methods.

One-electron reduced density matrices (1RDMs) from equation-of-motion (EOM) coupled-cluster with single and double excitations (CCSD) calculations are analyzed to assess their N-representability (i.e., whether they are derivable from a physical N-electron state). We identify EOM-CCSD stationary states whose 1RDMs violate either ensemble-state N-representability conditions or pure-state conditions known as generalized Pauli constraints. As such, these 1RDMs do not correspond to any physical N-electron state. Unphysical states are also encountered in the course of time-dependent EOM-CC simulations; when an external field drives transitions between a pair of stationary states with pure-state N-representable 1RDMs, the 1RDM of the time-dependent state can violate ensemble-state conditions. These observations point to potential challenges in interpreting the results of time-dependent EOM-CCSD simulations.

Taming pseudofermion functional renormalization for quantum spins: Finite temperatures and the Popov-Fedotov trick

The pseudo-fermion representation for $S=1/2$ quantum spins introduces unphysical states in the Hilbert space which can be projected out using the Popov-Fedotov trick. However, state-of-the-art implementation of the functional renormalization group method for pseudo-fermions have so far omitted the Popov-Fedotov projection. Instead, restrictions to zero temperature were made and absence of unphysical contributions to the ground-state was assumed. We question this belief by exact diagonalization of several small-system counterexamples where unphysical states do contribute to the ground state. We then introduce Popov-Fedotov projection to pseudo-fermion functional renormalization, enabling finite temperature computations with only minor technical modifications to the method. At large and intermediate temperatures, our results are perturbatively controlled and we confirm their accuracy in benchmark calculations. At lower temperatures, the accuracy degrades due to truncation errors in the hierarchy of flow equations. Interestingly, these problems cannot be alleviated by switching to the parquet approximation. We introduce the spin projection as a method-intrinsic quality check. We also show that finite temperature magnetic ordering transitions can be studied via finite-size scaling.

From data to noise to data for mixing physics across temperatures with generative artificial intelligence

Using simulations or experiments performed at some set of temperatures to learn about the physics or chemistry at some other arbitrary temperature is a problem of immense practical and theoretical relevance. Here we develop a framework based on statistical mechanics and generative artificial intelligence that allows solving this problem. Specifically, we work with denoising diffusion probabilistic models and show how these models in combination with replica exchange molecular dynamics achieve superior sampling of the biomolecular energy landscape at temperatures that were never simulated without assuming any particular slow degrees of freedom. The key idea is to treat the temperature as a fluctuating random variable and not a control parameter as is usually done. This allows us to directly sample from the joint probability distribution in configuration and temperature space. The results here are demonstrated for a chirally symmetric peptide and single-strand RNA undergoing conformational transitions in all-atom water. We demonstrate how we can discover transition states and metastable states that were previously unseen at the temperature of interest and even bypass the need to perform further simulations for a wide range of temperatures. At the same time, any unphysical states are easily identifiable through very low Boltzmann weights. The procedure while shown here for a class of molecular simulations should be more generally applicable to mixing information across simulations and experiments with varying control parameters.

Qubit-efficient encoding scheme for quantum simulations of electronic structure

Simulating electronic structure on a quantum computer requires encoding of fermionic systems onto qubits. Common encoding methods transform a fermionic system of $N$ spin-orbitals into an $N$-qubit system, but many of the fermionic configurations do not respect the required conditions and symmetries of the system so the qubit Hilbert space in this case may have unphysical states and thus can not be fully utilized. We propose a generalized qubit-efficient encoding (QEE) scheme that requires the qubit number to be only logarithmic in the number of configurations that satisfy the required conditions and symmetries. For the case of considering only the particle-conserving and singlet configurations, we reduce the qubit count to an upper bound of $\mathcal O(m\log_2N)$, where $m$ is the number of particles. This QEE scheme is demonstrated on an H$_2$ molecule in the 6-31G basis set and a LiH molecule in the STO-3G basis set using fewer qubits than the common encoding methods. We calculate the ground-state energy surfaces using a variational quantum eigensolver algorithm with a hardware-efficient ansatz circuit. We choose to use a hardware-efficient ansatz since most of the Hilbert space in our scheme is spanned by desired configurations so a heuristic search for an eigenstate is sensible. The simulations are performed on IBM Quantum machines and the Qiskit simulator with a noise model implemented from a IBM Quantum machine. Using the methods of measurement error mitigation and error-free linear extrapolation, we demonstrate that most of the distributions of the extrapolated energies using our QEE scheme agree with the exact results obtained by Hamiltonian diagonalization in the given basis sets within chemical accuracy. Our proposed scheme and results show the feasibility of quantum simulations for larger molecular systems in the noisy intermediate-scale quantum (NISQ) era.

Total angular momentum representation for state-to-state quantum scattering of cold molecules in a magnetic field.

We show that the integral cross sections for state-to-state quantum scattering of cold molecules in a magnetic field can be efficiently computed using the total angular momentum representation despite the presence of unphysical Zeeman states in the eigenspectrum of the asymptotic Hamiltonian. We demonstrate that the unphysical states arise due to the incompleteness of the space-fixed total angular momentum basis caused by using a fixed cutoff value Jmax for the total angular momentum of the collision complex J. As a result, certain orbital angular momentum (l) basis states lack the full range of J values required by the angular momentum addition rules, resulting in the appearance of unphysical states. We find that by augmenting the basis with a full range of J-states for every l, it is possible to completely eliminate the unphysical states from quantum scattering calculations on molecular collisions in external magnetic fields. To illustrate the procedure, we use the augmented basis sets to calculate the state-to-state cross sections for rotational and spin relaxation in cold collisions of 40CaH(X2Σ+, v = 0, N = 1, MN = 1, MS = 1/2) molecules with 4He atoms in a magnetic field. We find excellent agreement with benchmark calculations, validating our proposed procedure. We find that N-conserving spin relaxation from the highest-energy to the lowest-energy Zeeman state of the N = 1 manifold, |1112〉→|1-1-12〉 is nearly completely suppressed due to the lack of spin-rotation coupling between the fully spin-stretched Zeeman states. Our results demonstrate the possibility of rigorous, computationally efficient, and unphysical state-free quantum calculations on cold molecular collisions and on near-threshold energy levels of strongly anisotropic atom-molecule collision complexes in an external magnetic field.

Emergent gauge symmetries: Yang-Mills theory

Gauge symmetries remove unphysical states and guarantee that field theories are free from the pathologies associated with these states. In this work we find a set of general conditions that guarantee the removal of unphysical states in field theories describing interacting vector fields. These conditions are obtained through the extension of a mechanism for the emergence of gauge symmetries proposed in a previous article [C. Barcel\'o J. High Energy Phys. 10 (2016) 084] in order to account for non-Abelian gauge symmetries, and are the following: low-energy Lorentz invariance, emergence of massless vector fields describable by an action quadratic in those fields and their derivatives, and self-coupling to a conserved current associated with specific rigid symmetries. Using a bootstrapping procedure, we prove that these conditions are equivalent to the emergence of gauge symmetries and, therefore, guarantee that any theory satisfying them must be equivalent to a Yang-Mills theory at low energies.

Chasing unphysical TD-DFT excited states in transition metal complexes with a simple diagnostic tool.

Transition Metal Complexes (TMCs) are known for the rich variety of their excited states showing different nature and degrees of locality. Describing the energies of these excited states with the same degree of accuracy is still problematic when using time-dependent density functional theory in conjunction with the most current density functional approximations. In particular, the presence of unphysically low lying excited states possessing a relevant Charge Transfer (CT) character may significantly affect the spectra computed at such a level of theory and, more relevantly, the interpretation of their photophysical behavior. In this work, we propose an improved version of the MAC index, recently proposed by the authors and collaborators, as a simple and computationally inexpensive diagnostic tool that can be used for the detection and correction of the unphysically predicted low lying excited states. The analysis, performed on five prototype TMCs, shows that spurious and ghost states can appear in a wide spectral range and that it is difficult to detect them only on the basis of their CT extent. Indeed, both delocalization of the excited state and CT extent are criteria that must be combined, as in the MAC index, to detect unphysical states.

Multi-replica biased sampling for photoswitchable π-conjugated polymers.

In recent years, π-conjugated polymers are attracting considerable interest in view of their light-dependent torsional reorganization around the π-conjugated backbone, which determines peculiar light-emitting properties. Motivated by the interest in designing conjugated polymers with tunable photoswitchable pathways, we devised a computational framework to enhance the sampling of the torsional conformational space and, at the same time, estimate ground- to excited-state free-energy differences. This scheme is based on a combination of Hamiltonian Replica Exchange Method (REM), parallel bias metadynamics, and free-energy perturbation theory. In our scheme, each REM samples an intermediate unphysical state between the ground and the first two excited states, which are characterized by time-dependent density functional theory simulations at the B3LYP/6-31G* level of theory. We applied the method to a 5-mer of 9,9-dioctylfluorene and found that upon irradiation, this system can undergo a dihedral inversion from -155° to 155°, crossing a barrier that decreases from 0.1 eV in the ground state (S0) to 0.05 eV and 0.04 eV in the first (S1) and second (S2) excited states. Furthermore, S1 and even more S2 were predicted to stabilize coplanar dihedrals, with a local free-energy minimum located at ±44°. The presence of a free-energy barrier of 0.08 eV for the S1 state and 0.12 eV for the S2 state can trap this conformation in a basin far from the global free-energy minimum located at 155°. The simulation results were compared with the experimental emission spectrum, showing a quantitative agreement with the predictions provided by our framework.

Frustrated quantum spins at finite temperature: Pseudo-Majorana functional renormalization group approach

The pseudofermion functional renormalization group (PFFRG) method has proven to be a powerful numerical approach to treat frustrated quantum spin systems. In its usual implementation, however, the complex fermionic representation of spin operators introduces unphysical Hilbert-space sectors which render an application at finite temperatures inaccurate. In this work we formulate a general functional renormalization group approach based on Majorana fermions to overcome these difficulties. We, particularly, implement spin operators via an $\text{SO}(3)$ symmetric Majorana representation which does not introduce any unphysical states and, hence, remains applicable to quantum spin models at finite temperatures. We apply this scheme, dubbed pseudo-Majorana functional renormalization group (PMFRG) method, to frustrated Heisenberg models on small spin clusters as well as square and triangular lattices. Computing the finite-temperature behavior of spin correlations and thermodynamic quantities such as free energy and heat capacity, we find good agreement with exact diagonalization and the high-temperature series expansion down to moderate temperatures. We observe a significantly enhanced accuracy of the PMFRG compared to the PFFRG at finite temperatures. More generally, we conclude that the development of functional renormalization group approaches with Majorana fermions considerably extends the scope of applicability of such methods.

Robust recovery of primitive variables in relativistic ideal magnetohydrodynamics

Modern simulation codes for general relativistic ideal magnetohydrodynamics are all facing a long standing technical problem given by the need to recover fundamental variables from those variables that are evolved in time. In the relativistic case, this requires the numerical solution of a system of nonlinear equations. Although several approaches are available, none has proven completely reliable. A recent study comparing different methods showed that all can fail, in particular for the important case of strong magnetization and moderate Lorentz factors. Here, we propose a new robust, efficient, and accurate solution scheme, along with a proof for the existence and uniqueness of a solution, and analytic bounds for the accuracy. Further, the scheme allows us to reliably detect evolution errors leading to unphysical states and automatically applies corrections for typical harmless cases. A reference implementation of the method is made publicly available as a software library. The aim of this library is to improve the reliability of binary neutron star merger simulations, in particular in the investigation of jet formation and magnetically driven winds.

A Local Reduction of the Dirac Equation Applied to the Study of Quark Interactions

A general procedure of local reduction for the Dirac equation is introduced to study one- and n-body interacting systems. In the one-body case we show that the reduction allows for an approximate solution of the Dirac equation, correlating the upper and the lower components of the wave function. The two-body case is studied in more detail. We show that the method prevents from introducing spurious, unphysical states. The reduction is also applied to another relativistic equation. Finally, the method is used to construct a specific model in order to study the Charmonium spectrum.

Gauss’s law, duality, and the Hamiltonian formulation of U(1) lattice gauge theory

Quantum computers have the potential to explore the vast Hilbert space of entangled states that play an important role in the behavior of strongly interacting matter. This opportunity motivates reconsidering the Hamiltonian formulation of gauge theories, with a suitable truncation scheme to render the Hilbert space finite-dimensional. Conventional formulations lead to a Hilbert space largely spanned by unphysical states; given the current inability to perform large scale quantum computations, we examine here how one might restrict wave function evolution entirely or mostly to the physical subspace. We consider such constructions for the simplest of these theories containing dynamical gauge bosons -- U(1) lattice gauge theory without matter in $d=2,3$ spatial dimensions -- and find that electric-magnetic duality naturally plays an important role. We conclude that this approach is likely to significantly reduce computational overhead in $d=2$ by a reduction of variables and by allowing one to regulate magnetic fluctuations instead of electric. The former advantage does not exist in $d=3$, but the latter might be important for asymptotically-free gauge theories.

Size segregation in compressible granular shear flows of binary particle systems

This paper deals with the modelling and simulation of segregation in granular materials. The basis is a hydrodynamic model for granular material flows, which is extended to capture the dynamic process of segregation in shear flows of systems with small and large particles. The granular flow equations consist of a set of compressible Navier–Stokes-like equations as well as an equation for the granular temperature. With the help of the granular temperature equation, the granular flow equations are able to cover a wide range of regimes, starting from dilute to arresting flows. However, this paper focuses on dry granular shear flows. It extends this hydrodynamic system in a dense shear flow regime by a segregation equation using the framework of mixture theory. Special focus is lain on the segregation direction. A procedure from mechanics is adapted to obtain the segregation direction from the granular flow system independent of the choice of the coordinate system. In particular, this is done in three-dimensional space. Due to the compressibility of the granular flow system and the structure of the derived segregation equation, solving the segregation equation requires special numerical treatment. Therefore, a suitable numerical scheme is presented which prevents the system from reaching unphysical states.

A pressure-based solution framework for sub- and supersonic flows considering real-gas effects and phase separation under engine-relevant conditions

The present study describes a pressure-based solution framework suitable for the numerical investigation of sub- and supersonic injection processes under high-pressure conditions. The focal point is put on the introduction of a fully consistent and rigorous thermodynamic closure based on the cubic equation of state. Hereby, both real-gas effects as well as phase separation processes can be considered resulting in a more thorough investigation of physical phenomena and an increased solver stability as the inadvertently penetration of unphysical states, i.e., the crossing of the spinodal, is prevented. To demonstrate the advantage of the numerical framework, recent experimental investigations of an underexpanded argon jet showing both real-gas and phase separation effects are used. The agreement between the experiment and the Large-Eddy Simulation is good. In particular, the phase separation process shows excellent agreement and is studied thoroughly based on the numerical results.

Benchmarking the simplest slave-particle theory with Hubbard dimer**Project supported by the National Natural Science Foundation of China (Grant Nos. 11674139, 11704166, and 11834005), the Fundamental Research Funds for the Central Universities, China, and Program for Changjiang Scholars and Innovative Research Team in University, China (Grant No. IRT-16R35).

Slave-particle method is a powerful tool to tackle the correlation effect in quantum many-body physics. Although it has been successfully used to comprehend various intriguing problems, such as Mott metal–insulator transition and Kondo effect, there is still no convincing theory so far on the availability and limitation of this method. The abuse of slave-particle method may lead to wrong physics. As the simplest slave-particle method, slave spin, which is widely applied to many strongly correlated problems, is highly accessible and researchable. In this work, we will uncover the nature of the slave-spin method by studying a two-site Hubbard model. After exploring aspects of properties of this toy model, we make a comparative analysis of the results obtained by three methods: (i) slave-spin method on mean-field level, (ii) slave-spin method with gauge constraint, and (iii) the exact solution as a benchmark. We find that, protected by the particle–hole symmetry, the slave-spin mean-field method can recover the static properties of ground state exactly at half filling. Furthermore, in the parameter space where both U and T are small enough, the slave-spin mean-field method is also reliable in calculating the dynamic and thermal dynamic properties. However, when U or T is considerably large, the mean-field approximation gives ill-defined behaviors, which result from the unphysical states in the enlarged Hilbert space. These findings lead to our conclusion that the accuracy of slave particle can be guaranteed if we can exclude all unphysical states by enforcing gauge constraints. Our work demonstrates the promising prospect of slave-particle method in studying complex strongly correlated models with specific symmetry or in certain parameter space.

Local Simulations of MRI turbulence with Meshless Methods

Abstract The magneto-rotational instability (MRI) is one of the most important processes in sufficiently ionized astrophysical disks. Grid-based simulations, especially those using the local shearing box approximation, provide a powerful tool to study the nonlinear turbulence the MRI produces. On the other hand, meshless methods have been widely used in cosmology, galactic dynamics, and planet formation, but have not been fully deployed on the MRI problem. We present local unstratified and vertically stratified MRI simulations with two meshless MHD schemes: a recent implementation of smoothed-particle magnetohydrodynamics (SPH MHD), and a meshless finite-mass (MFM) MHD scheme with constrained gradient divergence cleaning, as implemented in the GIZMO code. Concerning variants of the SPH hydro force formulation, we consider both the “vanilla” SPH and the PSPH variant included in GIZMO. We find, as expected, that the numerical noise inherent in these schemes significantly affects turbulence. Furthermore, a high-order kernel, free of the pairing instability, is necessary. Both schemes adequately simulate MRI turbulence in unstratified shearing boxes with net vertical flux. The turbulence, however, dies out in zero-net-flux unstratified boxes, probably due to excessive numerical dissipation. In zero-net-flux vertically stratified simulations, MFM can reproduce the MRI dynamo and its characteristic butterfly diagram for several tens of orbits before ultimately decaying. In contrast, extremely strong toroidal fields, as opposed to sustained turbulence, develop in equivalent simulations using SPH MHD. The latter unphysical state is likely caused by a combination of excessive artificial viscosity, numerical resistivity, and the relatively large residual errors in the divergence of the magnetic field.

Development and Testing of the OPLS-AA/M Force Field for RNA.

Significant improvements have been made to the OPLS-AA force field for modeling RNA. New torsional potentials were optimized based on density functional theory (DFT) scans at the ωB97X-D/6-311++G(d,p) level for potential energy surfaces of the backbone α and γ dihedral angles. In combination with previously reported improvements for the sugar puckering and glycosidic torsion terms, the new force field was validated through diverse molecular dynamics simulations for RNAs in aqueous solution. Results for dinucleotides and tetranucleotides revealed both accurate reproduction of 3 J couplings from NMR and the avoidance of several unphysical states observed with other force fields. Simulations of larger systems with noncanonical motifs showed significant structural improvements over the previous OPLS-AA parameters. The new force field, OPLS-AA/M, is expected to perform competitively with other recent RNA force fields and to be compatible with OPLS-AA models for proteins and small molecules.

Unphysical properties of the static quark-antiquark four-point correlator in Landau gauge

We consider the four point connected correlator representing a static quark-antiquark pair separated by a spatial distance $R$, propagating for a Euclidean time $T$. This function is computed by lattice Monte Carlo in SU(2) pure gauge theory at lattice couplings $\ensuremath{\beta}=2.2$ and $\ensuremath{\beta}=2.5$ in both Coulomb and Landau gauges. The Coulomb gauge correlator is well behaved, and is dominated at large $T$ by a state whose energy grows linearly as $\ensuremath{\sigma}R$, with $\ensuremath{\sigma}$ the known asymptotic string tension. The connected correlator in Landau gauge behaves differently. At intermediate $R$ there is clear evidence of a linear potential, but the corresponding string tension extrapolates to zero at large $T$. At large $R$ the connected correlator becomes negative; moreover there are strong finite size effects. These numerical results suggest that unphysical states dominate the large Euclidean time behavior of this Landau gauge correlator.

Exploring half-metallic Co-based full Heusler alloys using a DFT+U method combined with linear response approach.

A density functional theory (DFT)+U method based on linear response (LR) theory was applied to investigate the electronic structures of a Co-based ternary full Heusler alloy Co2YSi to explore half-metallic (HM) ferromagnets with a wide HM gap. The LR-based DFT+U calculations tend to obtain a reasonable correlation parameter for the Y site, while the correlation of the Co site misdirects to the unphysical ground state due to the overestimated parameter value that arises from the delocalized electronic structure of Co. Furthermore, we found that the HM gap of Co2MnSi originates from the Co orbital in the conduction state and the Co–Mn hybridizing t2g orbital in the valence state around the Fermi energy. This means that the HM gap is a tunable property by selecting the Y element and/or mixing several elements into the Y site through t2g atomic-orbital coupling. Our LR-based DFT+U method was extended to other ternary Co2YSi and quaternary Co2(Y,Mn)Si. We found that Co2(Ti0.25,Mn0.75)Si and Co2(Fe0.25,Mn0.75)Si show HM nature, with the Fermi energy being at almost the center of the minority band gap, which leads to high thermal stability.