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ThIr crystallizes in the orthorhombic Cmcm space group. The structure is three-dimensional. Th is bonded in a 5-coordinate geometry to seven equivalent Ir atoms. There are a spread of Th–Ir bond distances ranging from 2.91–3.34 Å. Ir is bonded in a 5-coordinate geometry to seven equivalent Th atoms.
The study of fermionic quantum field theories is an important problem for realizing the standard model of particle physics on a quantum computer. As a step towards this goal, we consider the massive Thirring and Gross--Neveu models with arbitrary number of fermion flavors, $N_f$, discretized on a spatial one-dimensional lattice of size $L$ in the Hamiltonian formulation. We compute the gate complexity using the higher-order product formula and using block-encoding/qubitization and quantum singular value transformations in the limit of large $N_f$ and $L$. We also prepare the ground states of both models with excellent fidelity for system sizes up to 20 qubits with $N_f = 1,2,3,4$ using the adaptive-variational quantum imaginary time algorithm. In addition, we also classify the dynamical Lie algebras of these relativistic fermionic models and show that they belong to the same isomorphism class. Our work is a concrete step towards the quantum simulation of real-time dynamics of large $N_f$ fermionic quantum field theories models relevant for chiral symmetry breaking, understanding dimensional transmutation, and exploring the conformal window of field theories on near-term and early fault-tolerant quantum computers.
Recently, it has been shown that rocky planets orbiting neutron stars can be habitable under plausible circumstances. If a distant, point-like source of visible light, such as a Sun-like main-sequence star or the gravitationally-lensed accretion disk of a supermassive black hole is present, possible temporal variations Δε{sub p}(t) of the planet’s axial tilt ε {sub p} to the ecliptic plane should be included in the overall habitability budget since the obliquity determines the insolation at a given latitude on a body’ s surface. I point out that, for rather generic initial spin–orbit initial configurations, general relativistic and classical spin variations induced by the post-Newtonian de Sitter and Lense–Thirring components of the field of the host neutron star and by its pull to the planetary oblateness J{sub 2}{sup p} may induce huge and very fast variations of ε {sub p} that would likely have an impact on the habitability of such worlds. In particular, for a planet’s distance of, say, 0.005 au from a 1.4 M {sub ⊙} neutron star corresponding to an orbital period P {sub b} = 0.109 day, obliquity shifts Δε {sub p} as large as ε{sub p}{sup max}−ε{sub p}{sup min}≃50{sup ∘}--100{sup ∘} over characteristic timescales as short as 10 days (J{sub 2}{sup p}) to 3 Myr (Lense–Thirring) may occur for arbitrary orientations of the orbital and spin angular momenta L, S {sub ns}, S {sub p} of the planet-neutron star system. In view of this feature of their spins, I dub such hypothetical planets as “nethotrons.”.
Abstract We discuss the worldsheet sigma-model whose target space is thed+1 dimensional Euclidean Schwarzschild black hole. We argue that in the limit where the Hawking temperature of the black hole,T, approaches the Hagedorn temperature,T H , it can be described in terms of a generalized version of the Horowitz-Polchinski effective theory. Ford≥ 6, where the Horowitz-Polchinski EFT [1, 2] does not have suitable solutions, the modified effective Lagrangian allows one to study the black hole CFT in an expansion in powers ofd− 6 andT H −T. AtT=T H , the sigma model is non-trivial for alld> 6. It exhibits an enhanced SU(2) symmetry, and is described by a non-abelian Thirring model with a radially dependent coupling. The resulting picture connects naturally to the results of [3–5], that relate Schwarzschild black holes in flat spacetime at largedto the two dimensional black hole. We also discuss an analogous open string system, in which the black hole is replaced by a system of two separated D-branes connected by a throat. In this system, the asymptotic separation of the branes plays the role of the inverse temperature. At the critical separation, the system is described by a Kondo-type model, which again exhibits an enhanced SU(2) symmetry. At larged, the brane system gives rise to the hairpin brane [6].
We develop a method for constructing a 1-parameter family of globally-defined Love symmetry generators in rotating black hole spacetimes of general dimension. The key ingredient is to focus on the vicinity of the (physical) outer horizon, matching only the radial derivative and the outer horizon pole pieces of the Klein-Gordon operator in the black hole spacetime to the SL(2, ℝ) Casimir operator. After revisiting the 4D Kerr and 5D Myers-Perry cases, the procedure is illustrated on generalized Lense-Thirring spacetimes which describe a wide variety of slowly rotating black hole metrics in any number of dimensions. Such spacetimes are known to admit an extended tower of Killing tensor and Killing vector symmetries and, as demonstrated in this paper, allow for separability of the massive scalar wave equation in Myers-Perry-like coordinates. Interestingly, separability also occurs in the horizon-penetrating Painlevé–Gullstrand coordinates associated with the freely infalling observer who registers flat space around her all the way to singularity.
This is a preface to A Passion for Theoretical Physics, a special issue collection of articles published in J. Phys. A in memory of Peter G. O. Freund. We dedicate this special issue to the memory of our esteemed colleague, inspiring teacher and cherished friend, Peter Freund. Peter George Oliver Freund was born on 7 September 1936 in Timisoara, Romania. He obtained his PhD under the supervision of Walter Thirring at the University of Vienna in 1960. He came to the University of Chicago in 1963, where he remained throughout his career. He passed away on 6 March 2018.
Black hole X-ray binaries (BHXRBs) display a wide range of variability phenomena, from long duration spectral state changes to short-term broad-band variability and quasi-periodic oscillations (QPOs). A particularly puzzling aspect is the production of QPOs, which – if properly understood – could be used as a powerful diagnostic tool of black hole accretion and evolution. In this work, we analyse a high-resolution 3D general relativistic magnetohydrodynamic simulation of a geometrically thin accretion disc, which is tilted by 65° with respect to the black hole spin axis. We find that the Lense–Thirring torque from the rapidly spinning 10 M ⊙ black hole causes several sub-discs to tear off within ~10–20 gravitational radii. Tearing occurs in cycles on time-scales of seconds. During each tearing cycle, the inner sub-disc precesses for 1–5 periods before it falls into the black hole. We find a precession frequency of ~3 Hz, consistent with observed low-frequency QPOs. In addition, we find a high frequency QPO (HFQPO) with centroid frequency of ~55 Hz in the power spectra of the mass-weighted radius of the inner disc and the radial mass flux. This signal is caused by radial epicyclic oscillations of a dense ring of gas at the tearing radius, which suggests a corresponding modulation of the X-ray light curve and may thus explain some of the observed HFQPOs.
Phenomenological amplitudes obtained in partial-wave analyses (PWA) of single-pion photoproduction are used to evaluate the contribution of this process to the Gerasimov-Drell-Hearn (GDH), Baldin and Gell-Mann-Goldberger-Thirring (GGT) sum rules, by integrating up to 2 GeV in photon energy. Here, our study confirms that the single-pion contribution to all these sum rules converges even before the highest considered photon energy, but the levels of saturation are very different in the three cases. Single-pion production almost saturates the GDH sum rule for the proton, while a large fraction is missing in the neutron case. The Baldin integrals for the proton and the neutron are both saturated to about four fifths of the predicted total strength. For the GGT sum rule, the wide variability in predictions precludes any definitive statement.
We obtain nonperturbative results on the sine-Gordon model using the lattice field technique. In particular, we employ the Fourier accelerated hybrid Monte Carlo algorithm for our studies. We find the critical temperature of the theory based on autocorrelation time, as well as the finite size scaling of the "thickness" observable used in an earlier lattice study by Hasenbusch et al. We study the entropy, which is smooth across all temperatures, supportive of an infinite order transition. This system has a well-known duality with the massive Thirring model, which can play the role of a toy models for Montonen-Olive duality in $\mathcal {N}$ = 4 super-Yang-Mills theory, since it relates solitons to elementary field excitations. Our research lays a groundwork for such study on the lattice.
An experimental test at the intersection of quantum physics and general relativity is proposed: measurement of relativistic frame dragging and geodetic precession using intrinsic spin of electrons. Furthermore, the behavior of intrinsic spin in spacetime dragged and warped by a massive rotating body is an experimentally open question, hence the results of such a measurement could have important theoretical consequences. Such a measurement is possible by using mm-scale ferromagnetic gyroscopes in orbit around the Earth. Under conditions where the rotational angular momentum of a ferromagnet is sufficiently small, a ferromagnet’s angular momentum is dominated by atomic electron spins and is predicted to exhibit macroscopic gyroscopic behavior. If such a ferromagnetic gyroscope is sufficiently isolated from the environment, rapid averaging of quantum uncertainty via the spin-lattice interaction enables readout of the ferromagnetic gyroscope dynamics with sufficient sensitivity to measure both the Lense-Thirring (frame dragging) and de Sitter (geodetic precession) effects due to the Earth.
Here, we provide a modification to the quantum phase estimation algorithm (QPEA) [Abrams and Lloyd, Phys. Rev. Lett. 83, 5162 (1999); Cleve et al., Proc. R. Soc. A 454, 339 (1998); Nielsen and Chuang, Quantum computation and quantum information, 2002.] inspired by classical windowing methods for spectral density estimation. From this modification we obtain an upper bound in the cost that implies a cubic improvement with respect to the algorithm's error rate. Numerical evaluation of the costs also demonstrates an improvement. Moreover, with similar techniques, we detail an iterative projective measurement method for ground state preparation that gives an exponential improvement over previous bounds using QPEA. Numerical tests that confirm the expected scaling behavior are also obtained. For these numerical tests we have used a lattice Thirring model as testing ground. Using well-known perturbation theory results, we also show how to more appropriately estimate the cost scaling with respect to state error instead of evolution operator error.
Motivated by conjectures about near-horizon dynamics in quantum gravity, we search for lines of perturbatively accessible fixed points emanating from models of N free fermions. Through two loops we find a new class of models, apart from the well-known Abelian Thirring models. Further study is needed to see whether these can lead to true conformal manifolds, or perhaps a new class of large- N fixed points. Published by the American Physical Society 2024
Li9V6(P8O29)2 crystallizes in the triclinic P1 space group. The structure is three-dimensional. there are nine inequivalent Li sites. In the first Li site, Li is bonded to four O atoms to form distorted LiO4 tetrahedra that share corners with four PO4 tetrahedra and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 1.89–2.24 Å. In the second Li site, Li is bonded to four O atoms to form distorted LiO4 tetrahedra that share a cornercorner with one LiO4 tetrahedra, corners with four PO4 tetrahedra, and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 1.88–2.32 Å. In the third Li site, Li is bonded to four O atoms to form LiO4 tetrahedra that share corners with three VO6 octahedra and corners with four PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 64–68°. There are a spread of Li–O bond distances ranging from 1.86–2.27 Å. In the fourth Li site, Li is bonded to four O atoms to form distorted LiO4 tetrahedra that share a cornercorner with one LiO4 tetrahedra, corners with four PO4 tetrahedra, and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 1.92–2.22 Å. In the fifth Li site, Li is bonded to four O atoms to form distorted LiO4 tetrahedra that share a cornercorner with one LiO4 tetrahedra, corners with four PO4 tetrahedra, and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 1.89–2.21 Å. In the sixth Li site, Li is bonded to four O atoms to form distorted LiO4 tetrahedra that share a cornercorner with one LiO4 tetrahedra, corners with four PO4 tetrahedra, and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 1.89–2.21 Å. In the seventh Li site, Li is bonded to four O atoms to form distorted LiO4 tetrahedra that share a cornercorner with one LiO4 tetrahedra, corners with four PO4 tetrahedra, and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 1.89–2.27 Å. In the eighth Li site, Li is bonded to four O atoms to form distorted LiO4 tetrahedra that share a cornercorner with one LiO4 tetrahedra, corners with four PO4 tetrahedra, and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 1.94–2.22 Å. In the ninth Li site, Li is bonded to four O atoms to form distorted LiO4 tetrahedra that share corners with two LiO4 tetrahedra, corners with four PO4 tetrahedra, and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 1.94–2.30 Å. There are six inequivalent V sites. In the first V site, V is bonded to six O atoms to form VO6 octahedra that share corners with six PO4 tetrahedra and edges with two LiO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.87–1.99 Å. In the second V site, V is bonded to six O atoms to form VO6 octahedra that share a cornercorner with one LiO4 tetrahedra, corners with six PO4 tetrahedra, and an edgeedge with one LiO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.91–1.99 Å. In the third V site, V is bonded to six O atoms to form VO6 octahedra that share a cornercorner with one LiO4 tetrahedra and corners with six PO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.86–1.99 Å. In the fourth V site, V is bonded to six O atoms to form VO6 octahedra that share corners with six PO4 tetrahedra and an edgeedge with one LiO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.83–2.02 Å. In the fifth V site, V is bonded to six O atoms to form VO6 octahedra that share corners with six PO4 tetrahedra and edges with two LiO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.86–2.01 Å. In the sixth V site, V is bonded to six O atoms to form VO6 octahedra that share a cornercorner with one LiO4 tetrahedra, corners with six PO4 tetrahedra, and edges with two LiO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.82–1.96 Å. There are sixteen inequivalent P sites. In the first P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 41–44°. There are a spread of P–O bond distances ranging from 1.47–1.61 Å. In the second P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 39–45°. There are a spread of P–O bond distances ranging from 1.48–1.61 Å. In the third P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 38–44°. There are a spread of P–O bond distances ranging from 1.49–1.62 Å. In the fourth P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with three VO6 octahedra and corners with four LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 36–37°. There are a spread of P–O bond distances ranging from 1.51–1.58 Å. In the fifth P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with three VO6 octahedra and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 28–33°. There are a spread of P–O bond distances ranging from 1.49–1.61 Å. In the sixth P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with three LiO4 tetrahedra. The corner-sharing octahedral tilt angles are 45°. There are a spread of P–O bond distances ranging from 1.48–1.62 Å. In the seventh P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 39–42°. There are a spread of P–O bond distances ranging from 1.50–1.62 Å. In the eighth P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with three LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 43–44°. There are a spread of P–O bond distances ranging from 1.50–1.61 Å. In the ninth P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 39–44°. There are a spread of P–O bond distances ranging from 1.48–1.62 Å. In the tenth P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 38–41°. There are a spread of P–O bond distances ranging from 1.50–1.61 Å. In the eleventh P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 40–45°. There are a spread of P–O bond distances ranging from 1.48–1.62 Å. In the twelfth P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with three VO6 octahedra and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 30–36°. There are a spread of P–O bond distances ranging from 1.49–1.62 Å. In the thirteenth P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with three VO6 octahedra and a cornercorner with one LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 25–33°. There are a spread of P–O bond distances ranging from 1.48–1.62 Å. In the fourteenth P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 36–45°. There are a spread of P–O bond distances ranging from 1.48–1.62 Å. In the fifteenth P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 36–45°. There are a spread of P–O bond distances ranging from 1.47–1.62 Å. In the sixteenth P site, P is bonded to four O atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with three LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 41–48°. There are a spread of P–O bond distances ranging from 1.51–1.62 Å. There are fifty-eight inequivalent O sites. In the first O site, O is bonded in a bent 120 degrees geometry to one Li and one P atom. In the second O site, O is bonded in a distorted trigonal planar geometry to one Li, one V, and one P atom. In the third O site, O is bonded in a bent 120 degrees geometry to one Li and one P atom. In the fourth O site, O is bonded in a bent 150 degrees geometry to one V and one P atom. In the fifth O site, O is bonded in a distorted trigonal planar geometry to one Li, one V, and one P atom. In the sixth O site, O is bonded in a 3-coordinate geometry to one Li, one V, and one P atom. In the seventh O site, O is bonded in a bent 150 degrees geometry to one V and one P atom. In the eighth O site, O is bonded in a trigonal planar geometry to two Li and one P atom. In the ninth O site, O is bonded in a distorted trigonal planar geometry to one Li, one V, and one P atom. In the tenth O site, O is bonded in a bent 150 degrees geometry to one V and one P atom. In the eleventh O site, O is bonded in a bent 150 degrees geometry to one V and one P atom. In the twelfth O site, O is bonded in a 3-coordinate geometry to one Li, one V, and one P atom. In the thirteenth O site, O is bonded in a linear geometry to one Li and one P atom. In the fourteenth O site, O is bonded in a single-bond geometry to one P atom. In the fifteenth O site, O is bonded in a 3-coordinate geometry to one Li, one V, and one P atom. In the sixteenth O site, O is bonded in a distorted trigonal planar geometry to one Li, one V, and one P atom. In the seventeenth O site, O is bonded in a distorted trigonal non-coplanar geometry to one Li, one V, and one P atom. In the eighteenth O site, O is bonded in a distorted trigonal planar geometry to one Li, one V, and one P atom. In the nineteenth O site, O is bonded in a bent 120 degrees geometry to two P atoms. In the twentieth O site, O is bonded in a trigonal planar geometry to two Li and one P atom. In the twenty-first O site, O is bonded in a bent 120 degrees geometry to one Li and one P atom. In the twenty-second O site, O is bonded in a trigonal planar geometry to one Li, one V, and one P atom. In the twenty-third O site, O is bonded in a 3-coordinate geometry to one Li, one V, and one P atom. In the twenty-fourth O site, O is bonded in a distorted trigonal planar geometry to one Li, one V, and one P atom. In the twenty-fifth O site, O is bonded in a bent 150 degrees geometry to one V and one P atom. In the twenty-sixth O site, O is bonded in a bent 150 degrees geometry to one V and one P atom. In the twenty-seventh O site, O is bonded in a distorted bent 120 degrees geometry to two P atoms. In the twenty-eighth O site, O is bonded in a bent 120 degrees geometry to one Li and one P atom. In the twenty-ninth O site, O is bonded in a bent 120 degrees geometry to one Li and one P atom. In the thir
Li4V3P8O29 crystallizes in the triclinic P1 space group. The structure is three-dimensional. there are eight inequivalent Li1+ sites. In the first Li1+ site, Li1+ is bonded in a 4-coordinate geometry to four O2- atoms. There are a spread of Li–O bond distances ranging from 1.91–2.33 Å. In the second Li1+ site, Li1+ is bonded in a 4-coordinate geometry to four O2- atoms. There are a spread of Li–O bond distances ranging from 1.89–2.35 Å. In the third Li1+ site, Li1+ is bonded to four O2- atoms to form distorted LiO4 tetrahedra that share a cornercorner with one LiO4 tetrahedra, corners with four PO4 tetrahedra, and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 1.99–2.19 Å. In the fourth Li1+ site, Li1+ is bonded in a 4-coordinate geometry to four O2- atoms. There are a spread of Li–O bond distances ranging from 1.88–2.32 Å. In the fifth Li1+ site, Li1+ is bonded to four O2- atoms to form distorted LiO4 tetrahedra that share a cornercorner with one LiO4 tetrahedra, corners with four PO4 tetrahedra, and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 1.89–2.27 Å. In the sixth Li1+ site, Li1+ is bonded to four O2- atoms to form distorted LiO4 tetrahedra that share a cornercorner with one LiO4 tetrahedra, corners with four PO4 tetrahedra, and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 1.89–2.24 Å. In the seventh Li1+ site, Li1+ is bonded to four O2- atoms to form distorted LiO4 tetrahedra that share a cornercorner with one LiO4 tetrahedra, corners with four PO4 tetrahedra, and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 1.91–2.21 Å. In the eighth Li1+ site, Li1+ is bonded in a 4-coordinate geometry to four O2- atoms. There are a spread of Li–O bond distances ranging from 1.97–2.29 Å. There are six inequivalent V+4.67+ sites. In the first V+4.67+ site, V+4.67+ is bonded to six O2- atoms to form VO6 octahedra that share corners with six PO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.86–1.94 Å. In the second V+4.67+ site, V+4.67+ is bonded to six O2- atoms to form VO6 octahedra that share corners with six PO4 tetrahedra and edges with two LiO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.88–1.99 Å. In the third V+4.67+ site, V+4.67+ is bonded to six O2- atoms to form VO6 octahedra that share corners with six PO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.88–1.90 Å. In the fourth V+4.67+ site, V+4.67+ is bonded to six O2- atoms to form VO6 octahedra that share corners with six PO4 tetrahedra and an edgeedge with one LiO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.84–1.99 Å. In the fifth V+4.67+ site, V+4.67+ is bonded to six O2- atoms to form VO6 octahedra that share corners with six PO4 tetrahedra and an edgeedge with one LiO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.89–2.02 Å. In the sixth V+4.67+ site, V+4.67+ is bonded to six O2- atoms to form VO6 octahedra that share corners with six PO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.84–1.98 Å. There are sixteen inequivalent P5+ sites. In the first P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one LiO4 tetrahedra, and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 39–41°. There are a spread of P–O bond distances ranging from 1.48–1.62 Å. In the second P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one LiO4 tetrahedra, and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 40–46°. There are a spread of P–O bond distances ranging from 1.48–1.61 Å. In the third P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 38–43°. There are a spread of P–O bond distances ranging from 1.48–1.62 Å. In the fourth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with three VO6 octahedra and a cornercorner with one LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 30–37°. There are a spread of P–O bond distances ranging from 1.49–1.63 Å. In the fifth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with three VO6 octahedra and a cornercorner with one LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 29–35°. There are a spread of P–O bond distances ranging from 1.49–1.61 Å. In the sixth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 40–43°. There are a spread of P–O bond distances ranging from 1.48–1.62 Å. In the seventh P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 41–42°. There are a spread of P–O bond distances ranging from 1.50–1.62 Å. In the eighth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 39–46°. There are a spread of P–O bond distances ranging from 1.50–1.61 Å. In the ninth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 38–40°. There are a spread of P–O bond distances ranging from 1.47–1.62 Å. In the tenth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 38–40°. There are a spread of P–O bond distances ranging from 1.50–1.61 Å. In the eleventh P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 38–41°. There are a spread of P–O bond distances ranging from 1.50–1.62 Å. In the twelfth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with three VO6 octahedra and a cornercorner with one LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 30–35°. There are a spread of P–O bond distances ranging from 1.49–1.60 Å. In the thirteenth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with three VO6 octahedra and a cornercorner with one LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 27–36°. There are a spread of P–O bond distances ranging from 1.49–1.61 Å. In the fourteenth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 39–43°. There are a spread of P–O bond distances ranging from 1.48–1.60 Å. In the fifteenth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one LiO4 tetrahedra, and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 38–44°. There are a spread of P–O bond distances ranging from 1.48–1.63 Å. In the sixteenth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one LiO4 tetrahedra, and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 42–47°. There are a spread of P–O bond distances ranging from 1.51–1.62 Å. There are fifty-eight inequivalent O2- sites. In the first O2- site, O2- is bonded in a bent 120 degrees geometry to one Li1+ and one P5+ atom. In the second O2- site, O2- is bonded in a distorted trigonal planar geometry to one Li1+, one V+4.67+, and one P5+ atom. In the third O2- site, O2- is bonded in a single-bond geometry to one P5+ atom. In the fourth O2- site, O2- is bonded in a bent 150 degrees geometry to one V+4.67+ and one P5+ atom. In the fifth O2- site, O2- is bonded in a bent 150 degrees geometry to one V+4.67+ and one P5+ atom. In the sixth O2- site, O2- is bonded in a bent 150 degrees geometry to one V+4.67+ and one P5+ atom. In the seventh O2- site, O2- is bonded in a bent 150 degrees geometry to one V+4.67+ and one P5+ atom. In the eighth O2- site, O2- is bonded in a trigonal planar geometry to two Li1+ and one P5+ atom. In the ninth O2- site, O2- is bonded in a distorted trigonal planar geometry to one Li1+, one V+4.67+, and one P5+ atom. In the tenth O2- site, O2- is bonded in a bent 150 degrees geometry to one V+4.67+ and one P5+ atom. In the eleventh O2- site, O2- is bonded in a bent 150 degrees geometry to one V+4.67+ and one P5+ atom. In the twelfth O2- site, O2- is bonded in a 3-coordinate geometry to one Li1+, one V+4.67+, and one P5+ atom. In the thirteenth O2- site, O2- is bonded in a single-bond geometry to one P5+ atom. In the fourteenth O2- site, O2- is bonded in a single-bond geometry to one P5+ atom. In the fifteenth O2- site, O2- is bonded in a 3-coordinate geometry to one Li1+, one V+4.67+, and one P5+ atom. In the sixteenth O2- site, O2- is bonded in a bent 150 degrees geometry to one V+4.67+ and one P5+ atom. In the seventeenth O2- site, O2- is bonded in a distorted trigonal non-coplanar geometry to one Li1+, one V+4.67+, and one P5+ atom. In the eighteenth O2- site, O2- is bonded in a distorted trigonal planar geometry to one Li1+, one V+4.67+, and one P5+ atom. In the nineteenth O2- site, O2- is bonded in a distorted bent 120 degrees geometry to two P5+ atoms. In the twentieth O2- site, O2- is bonded in a trigonal planar geometry to two Li1+ and one P5+ atom. In the twenty-first O2- site, O2- is bonded in a bent 120 degrees geometry to one Li1+ and one P5+ atom. In the twenty-second O2- site, O2- is bonded in a bent 150 degrees geometry to one V+4.67+ and one P5+ atom. In the twenty-third O2- site, O2- is bonded in a 3-coordinate geometry to one Li1+, one V+4.67+, and one P5+ atom. In the twenty-fourth O2- site, O2- is bonded in a distorted trigonal planar geometry to one Li1+, one V+4.67+, and one P5+ atom. In the twenty-fifth O2- site, O2- is bonded in a bent 150 degrees geometry to one V+4.67+ and one P5+ atom. In the twenty-sixth O2- site, O2- is bonded in a bent 150 degrees geometry to one V+4.67+ and one P5+ atom. In the twenty-seventh O2- site, O2- is bonded in a distorted bent 150 degrees geometry to two P5+ atoms. In the twenty-eighth O2- site, O2- is bonded in a bent 120 degrees geometry to one Li1+ and one P5+ atom. In the twenty-ninth O2- site, O2- is bonded in a bent 120 degrees geometry to one Li1+ and one P5+ atom. In the thirtieth O2- site, O2- is bonded in a bent 150 degrees geometry to one V+4.67+ and one P5+ atom. In the thirty-first O2- site, O2- is bonded in a bent 150 degrees geometry to one V+4.67+ and one P5+ atom. In the thirty-second O2- site, O2- is bonded in a distorted trigonal planar geometry to one Li1+, one V+4.67+, and one P5+ atom. In the thirty-third O2- site, O2- is bonded in a 3-coordinate geometry to one Li1+, one V+4.67+, and one P5+ atom. In the thirty-fourth O2- site, O2- is bonded in a bent 150 degrees geometry to one V+4.67+ and one P5+ atom. In the thir
Li6V3P8O29 crystallizes in the triclinic P1 space group. The structure is three-dimensional. there are twelve inequivalent Li1+ sites. In the first Li1+ site, Li1+ is bonded in a 4-coordinate geometry to four O2- atoms. There are a spread of Li–O bond distances ranging from 1.95–2.24 Å. In the second Li1+ site, Li1+ is bonded in a distorted see-saw-like geometry to four O2- atoms. There are a spread of Li–O bond distances ranging from 1.94–2.09 Å. In the third Li1+ site, Li1+ is bonded in a 4-coordinate geometry to four O2- atoms. There are a spread of Li–O bond distances ranging from 2.00–2.19 Å. In the fourth Li1+ site, Li1+ is bonded to four O2- atoms to form distorted LiO4 tetrahedra that share corners with four PO4 tetrahedra and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 1.97–2.25 Å. In the fifth Li1+ site, Li1+ is bonded in a 4-coordinate geometry to four O2- atoms. There are a spread of Li–O bond distances ranging from 1.91–2.28 Å. In the sixth Li1+ site, Li1+ is bonded to four O2- atoms to form distorted LiO4 tetrahedra that share corners with four PO4 tetrahedra and an edgeedge with one VO6 octahedra. There are a spread of Li–O bond distances ranging from 2.01–2.21 Å. In the seventh Li1+ site, Li1+ is bonded in a distorted see-saw-like geometry to four O2- atoms. There are a spread of Li–O bond distances ranging from 2.00–2.09 Å. In the eighth Li1+ site, Li1+ is bonded to four O2- atoms to form LiO4 tetrahedra that share corners with three VO6 octahedra and corners with four PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 67–69°. There are a spread of Li–O bond distances ranging from 1.86–2.31 Å. In the ninth Li1+ site, Li1+ is bonded in a 4-coordinate geometry to four O2- atoms. There are a spread of Li–O bond distances ranging from 2.00–2.24 Å. In the tenth Li1+ site, Li1+ is bonded in a 4-coordinate geometry to four O2- atoms. There are a spread of Li–O bond distances ranging from 1.99–2.20 Å. In the eleventh Li1+ site, Li1+ is bonded in a 5-coordinate geometry to five O2- atoms. There are a spread of Li–O bond distances ranging from 1.98–2.59 Å. In the twelfth Li1+ site, Li1+ is bonded in a 6-coordinate geometry to six O2- atoms. There are a spread of Li–O bond distances ranging from 1.93–2.83 Å. There are six inequivalent V4+ sites. In the first V4+ site, V4+ is bonded to six O2- atoms to form VO6 octahedra that share a cornercorner with one LiO4 tetrahedra, corners with six PO4 tetrahedra, and edges with two LiO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.87–2.01 Å. In the second V4+ site, V4+ is bonded to six O2- atoms to form VO6 octahedra that share corners with six PO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.89–2.01 Å. In the third V4+ site, V4+ is bonded to six O2- atoms to form VO6 octahedra that share corners with six PO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.86–2.01 Å. In the fourth V4+ site, V4+ is bonded to six O2- atoms to form VO6 octahedra that share a cornercorner with one LiO4 tetrahedra and corners with six PO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.89–2.02 Å. In the fifth V4+ site, V4+ is bonded to six O2- atoms to form VO6 octahedra that share a cornercorner with one LiO4 tetrahedra and corners with six PO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.85–2.04 Å. In the sixth V4+ site, V4+ is bonded to six O2- atoms to form VO6 octahedra that share corners with six PO4 tetrahedra. There are a spread of V–O bond distances ranging from 1.90–2.06 Å. There are sixteen inequivalent P5+ sites. In the first P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 35–41°. There are a spread of P–O bond distances ranging from 1.50–1.60 Å. In the second P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one LiO4 tetrahedra, and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedral tilt angles are 39°. There are a spread of P–O bond distances ranging from 1.52–1.61 Å. In the third P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one LiO4 tetrahedra, and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 41–46°. There are a spread of P–O bond distances ranging from 1.51–1.62 Å. In the fourth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with three VO6 octahedra and a cornercorner with one LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 32–35°. There are a spread of P–O bond distances ranging from 1.49–1.59 Å. In the fifth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with three VO6 octahedra. The corner-sharing octahedra tilt angles range from 36–38°. There are a spread of P–O bond distances ranging from 1.50–1.60 Å. In the sixth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one LiO4 tetrahedra, and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 38–43°. There are a spread of P–O bond distances ranging from 1.51–1.63 Å. In the seventh P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one LiO4 tetrahedra, and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 38–47°. There are a spread of P–O bond distances ranging from 1.52–1.61 Å. In the eighth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 40–41°. There are a spread of P–O bond distances ranging from 1.52–1.62 Å. In the ninth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 46–48°. There are a spread of P–O bond distances ranging from 1.49–1.61 Å. In the tenth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 37–40°. There are a spread of P–O bond distances ranging from 1.50–1.60 Å. In the eleventh P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one LiO4 tetrahedra, and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 45–49°. There are a spread of P–O bond distances ranging from 1.49–1.62 Å. In the twelfth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with three VO6 octahedra and a cornercorner with one LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 30–36°. There are a spread of P–O bond distances ranging from 1.48–1.59 Å. In the thirteenth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with three VO6 octahedra and a cornercorner with one LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 31–37°. There are a spread of P–O bond distances ranging from 1.50–1.60 Å. In the fourteenth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 41–48°. There are a spread of P–O bond distances ranging from 1.50–1.62 Å. In the fifteenth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra and a cornercorner with one PO4 tetrahedra. The corner-sharing octahedra tilt angles range from 37–46°. There are a spread of P–O bond distances ranging from 1.52–1.61 Å. In the sixteenth P5+ site, P5+ is bonded to four O2- atoms to form PO4 tetrahedra that share corners with two VO6 octahedra, a cornercorner with one PO4 tetrahedra, and corners with two LiO4 tetrahedra. The corner-sharing octahedra tilt angles range from 46–48°. There are a spread of P–O bond distances ranging from 1.51–1.62 Å. There are fifty-eight inequivalent O2- sites. In the first O2- site, O2- is bonded in a distorted trigonal planar geometry to two Li1+ and one P5+ atom. In the second O2- site, O2- is bonded in a bent 150 degrees geometry to one V4+ and one P5+ atom. In the third O2- site, O2- is bonded in a 3-coordinate geometry to two Li1+ and one P5+ atom. In the fourth O2- site, O2- is bonded in a bent 150 degrees geometry to one V4+ and one P5+ atom. In the fifth O2- site, O2- is bonded in a bent 150 degrees geometry to one V4+ and one P5+ atom. In the sixth O2- site, O2- is bonded in a 3-coordinate geometry to one Li1+, one V4+, and one P5+ atom. In the seventh O2- site, O2- is bonded in a bent 150 degrees geometry to one V4+ and one P5+ atom. In the eighth O2- site, O2- is bonded in a distorted trigonal planar geometry to two Li1+ and one P5+ atom. In the ninth O2- site, O2- is bonded in a distorted trigonal planar geometry to one Li1+, one V4+, and one P5+ atom. In the tenth O2- site, O2- is bonded in a distorted trigonal non-coplanar geometry to one Li1+, one V4+, and one P5+ atom. In the eleventh O2- site, O2- is bonded in a bent 150 degrees geometry to one V4+ and one P5+ atom. In the twelfth O2- site, O2- is bonded in a distorted trigonal non-coplanar geometry to one Li1+, one V4+, and one P5+ atom. In the thirteenth O2- site, O2- is bonded in a single-bond geometry to one P5+ atom. In the fourteenth O2- site, O2- is bonded in a single-bond geometry to one P5+ atom. In the fifteenth O2- site, O2- is bonded in a 3-coordinate geometry to one Li1+, one V4+, and one P5+ atom. In the sixteenth O2- site, O2- is bonded in a bent 150 degrees geometry to one V4+ and one P5+ atom. In the seventeenth O2- site, O2- is bonded in a 3-coordinate geometry to one Li1+, one V4+, and one P5+ atom. In the eighteenth O2- site, O2- is bonded in a distorted trigonal planar geometry to one Li1+, one V4+, and one P5+ atom. In the nineteenth O2- site, O2- is bonded in a distorted bent 120 degrees geometry to two P5+ atoms. In the twentieth O2- site, O2- is bonded in a 3-coordinate geometry to three Li1+ and one P5+ atom. In the twenty-first O2- site, O2- is bonded in a 3-coordinate geometry to two Li1+ and one P5+ atom. In the twenty-second O2- site, O2- is bonded in a bent 150 degrees geometry to one V4+ and one P5+ atom. In the twenty-third O2- site, O2- is bonded in a distorted trigonal non-coplanar geometry to one Li1+, one V4+, and one P5+ atom. In the twenty-fourth O2- site, O2- is bonded in a distorted trigonal planar geometry to one Li1+, one V4+, and one P5+ atom. In the twenty-fifth O2- site, O2- is bonded in a bent 150 degrees geometry to one V4+ and one P5+ atom. In the twenty-sixth O2- site, O2- is bonded in a distorted trigonal planar geometry to one Li1+, one V4+, and one P5+ atom. In the twenty-seventh O2- site, O2- is bonded in a distorted bent 150 degrees geometry to two P5+ atoms. In the twenty-eighth O2- site, O2- is bonded in a 4-coordinate geometry to three Li1+ and one P5+ atom. In the twenty-ninth O2- site, O2- is bonded in a 3-coordinate geometry to three Li1+ and one P5+ atom. In the thirtieth O2- site, O2- is bonded in a distorted trigonal planar geometry to one Li1+, one V4+, and one P5+ atom. In the thirty-first O2- site, O2- is bonded in a bent 150 degrees geometry to one V4+ and one P5+ atom. In the thir
The angular momentum of gas feeding a black hole (BH) may be misaligned with respect to the BH spin, resulting in a tilted accretion disk. Rotation of the BH drags the surrounding spacetime, manifesting as Lense–Thirring torques that lead to disk precession and warping. We study these processes by simulating a thin (H/r = 0.02), highly tilted ($\tau$ = 65°) accretion disk around a rapidly rotating (a = 0.9375) BH at extremely high resolutions, which we performed using the general-relativistic magnetohydrodynamic code H-AMR. The disk becomes significantly warped and continuously tears into two individually precessing subdisks. We find that mass accretion rates far exceed the standard α-viscosity expectations. We identify two novel dissipation mechanisms specific to warped disks that are the main drivers of accretion, distinct from the local turbulent stresses that are usually thought to drive accretion. In particular, we identify extreme scale height oscillations that occur twice an orbit throughout our disk. When the scale height compresses, "nozzle" shocks form, dissipating orbital energy and driving accretion. Separate from this phenomenon, there is also extreme dissipation at the location of the tear. This leads to the formation of low-angular momentum "streamers" that rain down onto the inner subdisk, shocking it. The addition of low-angular momentum gas to the inner subdisk causes it to rapidly accrete, even when it is transiently aligned with the BH spin and thus unwarped. These mechanisms, if general, significantly modify the standard accretion paradigm. Additionally, they may drive structural changes on much shorter timescales than expected in α-disks, potentially explaining some of the extreme variability observed in active galactic nuclei.
We present an extension of the Evolving density matrices on Qubits (E$ρ$OQ) framework that enables efficient fault-tolerant preparation of fermionic quantum states. The original method circumvents state preparation by stochastic sampling, but faces a sign problem in fermionic systems leading to a large number of circuits necessary. We resolve this by combining classical stochastic sampling with a linear combination of unitaries method that avoids the exponential circuit scaling that plagued naïve implementations. The resulting algorithm requires $\mathcal{O}(M^2)$$R_Z$ rotations for circuit preparation, where $M$ is the number of retained basis states. We validate the method for ground and excited states in the Thirring model, including by computing two-point correlation functions relevant to scattering. In this model for fixed accuracy $\varepsilon$, $M$ is found to scale empirically as $M \propto \frac{1}{mg}\log(1/g)\log(1/m)$.