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Modeling Strong Light-Matter Coupling in Correlated Systems: State-Averaged Cavity Quantum Electrodynamics Complete Active Space Self-Consistent Field Theory

The description of strongly correlated systems interacting with quantized cavity modes poses significant theoretical challenges due to the combinatorial scaling of electronic and photonic degrees of freedom. Recent advances addressing this complexity include cavity quantum electrodynamics (QED) generalizations of complete active space configuration interaction and density matrix renormalization group methods. In this work, we introduce a QED extension of state-averaged complete active space self-consistent field theory, which incorporates cavity-induced correlations through a second-order orbital optimization framework with robust convergence properties. The method is implemented using both photon number state and coherent state representations, with the latter showing robust origin invariance in the energies regardless of the completeness of the photonic Fock space. The implementation enables symmetry-free orbital relaxations to account for photon-mediated symmetry breaking in polaritonic systems. Numerical validation on lithium hydride, hydroxide anion, and magnesium hydride cation demonstrates that this method achieves significantly improved accuracy in modeling ground-state and polariton potential energy surfaces compared to QED-CASCI in a fixed orbital basis. In these studies, we reach sub-kcal/mol accuracy in potential energy surface in much smaller active spaces than are required for QED-CASCI. This advancement provides a more robust approach for studying cavity-altered chemical landscapes for ground and exited strongly coupled systems.

CASSCF↗

Upper critical magnetic field and multiband superconductivity in artificial high-𝑇 𝑐 superlattices of nano quantum wells

Artificial high-T c superlattices (AHTS) composed of quantum building blocks with tunable superconducting critical temperature have been synthesized by engineering their nanoscale geometry using the Bianconi-Perali Valletta (BPV) two-gap superconductivity theory. These quantum heterostructures consist of quantum wells made of superconducting, modulation-doped Mott insulators (S), confined by a metallic (N) potential barrier. The lattice geometry has been carefully engineered to induce the predicted Fano-Feshbach shape resonance between the gaps, near a topological Lifshitz transition. Here, we validate the BPV theory by providing compelling experimental evidence that AHTS samples, at the peak of the superconducting dome, exhibit resonant two-band, two-gap superconductivity. This is demonstrated by measuring the temperature dependence of the upper critical magnetic field, μ 0 H c2 , in samples with superlattice periods 3.3 < d < 5.28 nm and L/d ratios close to the magic value 2/3 (where L is the thickness of the superconducting La 2 CuO 4 layer and d is the superlattice period). Here, the data reveal the predicted upward concavity in H c2 (T) and a characteristic kink in the coherence length as a function of temperature, confirming the predicted two-band superconductivity with Fermi velocity ratio ≈ 0.25 and significant pair-exchange term among the two condensates.

Heterostructures↗

DBI potential, DBI inflation action and general Lagrangian relative to phantom, K-essence and quintessence

We derive a Dirac-Born-Infeld (DBI) potential and DBI inflation action by rescaling the metric. The determinant of the induced metric naturally includes the kinetic energy and the potential energy. In particular, the potential energy and kinetic energy can convert into each other in any order, which is in agreement with the limit of classical physics. This is quite different from the usual DBI action. We show that the Taylor expansion of the DBI action can be reduced into the form in the non-linear classical physics. These investigations are the support for the statement that the results of string theory are consistent with quantum mechanics and classical physics. We deduce the Phantom, K-essence, Quintessence and Generalized Klein-Gordon Equation from the DBI model.

79 ASTRONOMY AND ASTROPHYSICS↗

PT quantum mechanics - Recent results

Most quantum physicists believe that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under matrix transposition and complex conjugation) to be sure that the energy eigenvalues are real and that time evolution is unitary. However, the non-Dirac-hermitian Hamiltonian H p{sup 2}+ix{sup 3} has a real positive discrete spectrum and generates unitary time evolution and defines a fully consistent and physical quantum theory. Evidently, Dirac Hermiticity is too restrictive. While H = p{sup 2}+ix{sup 3} is not Dirac Hermitian, it is PT symmetric (invariant under combined space reflection P and time reversal T). Another PT-symmetric Hamiltonian whose energy levels are real, positive and discrete is H = p{sup 2}-x{sup 4}, which contains an upside-down potential. The quantum mechanics defined by a PT-symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics and quantum field theory are extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. In the past two years some of these properties have been verified in laboratory experiments. Here, we first discuss PT-symmetric Hamiltonians at a simple intuitive level and explain why the energy levels of such Hamiltonians may be real, positive, and discrete. Second, we describe a recent experiment in which the PT phase transition was observed. Third, we briefly mention that PT-symmetric theories can be useful at a fundamental level. While the double-scaling limit of an O(N)-symmetric g{phi}{sup 4} quantum field theory appears to be inconsistent because the critical value of g is negative, this limit is in fact not inconsistent because the critical theory is PT symmetric.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Holography: Quantum Mechanical Aspects of Black Holes (Final Report)

Current theories of fundamental physics are based on quantum field theory, unifying quantum mechanics and special relativity. A similar unification of quantum field theory with gravity has been elusive. String theory is a consistent model of quantum gravity and it is important to identify general lessons that can be applied to the real world. The main technique that emerged from such studies has been the gravitational path integral. During the period of this grant, the PI has explored the implication of the gravitational path integral in novel setups to uncover new features of rotating black holes in four dimensions, as well as increasing our understanding of some specific black holes in string theory.

79 ASTRONOMY AND ASTROPHYSICS↗

Modular invariance and uniqueness of $T\overline{T} $ deformed CFT

Any two dimensional quantum field theory that can be consistently defined on a torus is invariant under modular transformations. In this paper we study families of quantum field theories labeled by a dimensionful parameter t, that have the additional property that the energy of a state at finite t is a function only of t and of the energy and momentum of the corresponding state at t = 0, where the theory becomes conformal. We show that under this requirement, the partition sum of the theory at t = 0 uniquely determines the partition sum (and thus the spectrum) of the perturbed theory, to all orders in t, to be that of a $T\bar{T}$ deformed CFT. Non-perturbatively, we find that for one sign of t (for which the energies are real) the partition sum is uniquely determined, while for the other sign we find non-perturbative ambiguities. We characterize these ambiguities and comment on their possible relations to holography.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Consistent inclusion of triple substitutions within a coupled cluster based static quantum embedding theory

We have previously proposed the MPCC static embedding framework for quantum chemistry that self-consistently couples a high-level coupled cluster (CC) treatment of the fragment (active region) with a lower level, Møller–Plesset perturbation treatment of the environment. Our initial implementation was limited to single and double (SD) substitutions, with CCSD for the fragment and first-order perturbative SD amplitudes for the environment. Here, we extend the MPCC embedding treatment to triple substitutions, which is essential for achieving chemical accuracy in energy differences. To this end, we employ a CCSDT solver for the fragment subsystem. For the environment subsystem, we construct a perturbative estimate of the triples amplitudes, explicitly accounting for feedback from all fragment amplitudes. The resulting approach is denoted MPCCSDT(pt). We further introduce a more complete formulation in which feedback from the environment amplitudes to the fragment amplitudes is also included. This scheme involves an iterative treatment of the environment triples amplitudes and is denoted MPCCSDT(it). In addition, we assess the accuracy of the previously proposed low-level method by introducing a modified low-level approach that incorporates a lowest-order treatment of selected long-range effects, including spin fluctuations and charge polarization. All resulting approaches may be viewed as post-CCSD(T) methods. We therefore consider test cases for which CCSD(T) exhibits substantial deviations from CCSDT. These include (i) single- and triple-bond stretching in F 2 and N 2 , (ii) bond dissociation energies of selected molecules from the W4-11 dataset, and (iii) total atomization energies of transition metal hydrides. Our results demonstrate that inclusion of triples amplitudes at the fragment level alone is insufficient; a perturbative treatment of the environment triples amplitudes is required. For many energy-difference applications, feedback from the environment triples amplitudes to the fragment amplitudes is not essential, but it does play a role in the very challenging CoH and FeH molecules. A very interesting finding from our study is that in some challenging cases, we need an improved (second-order) perturbative method for the SD amplitudes, going beyond the first-order one used in our earlier work. In conclusion, considering both cost and accuracy, the MP2CCSDT(pt) model is the most promising for future applications among the candidates considered here.

Shee, Avijit [University of California, Berkeley, ↗

An infinite swampland of U(1) charge spectra in 6D supergravity theories

We analyze the anomaly constraints on 6D supergravity theories with a single abelian U(1) gauge factor. For theories with charges restricted to q = ±1, ±2 and no tensor multiplets, anomaly-free models match those models that can be realized from F-theory compactifications almost perfectly. For theories with tensor multiplets or with larger charges, the F-theory constraints are less well understood. We show, however, that there is an infinite class of distinct massless charge spectra in the “swampland” of theories that satisfy all known quantum consistency conditions but do not admit a realization through F-theory or any other known approach to string compactification. We also compare the spectra of charged matter in abelian theories with those that can be realized from breaking nonabelian SU(2) and higher rank gauge symmetries.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

The QTP family of consistent functionals and potentials in Kohn-Sham density functional theory

This manuscript presents the second, consistent density functional in the QTP (Quantum Theory Project) family, that is, the CAM-QTP(01). It is a new range-separated exchange-correlation functional in which the non-local exchange contribution is 100% at large separation. It follows the same basic principles of this family that the Kohn-Sham eigenvalues of the occupied orbitals approximately equal the vertical ionization energies, which is not fulfilled by most of the traditional density functional methods. This new CAM-QTP(01) functional significantly improves the accuracy of the vertical excitation energies especially for the Rydberg states in the test set. It also reproduces many other properties such as geometries, reaction barrier heights, and atomization energies.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Quantum histories without contrary inferences

In the consistent histories formulation of quantum theory it was shown that it is possible to retrodict contrary properties. We show that this problem do not appear in our formalism of generalized contexts for quantum histories. - Highlights: • We prove ordinary quantum mechanics has no contrary properties. • Contrary properties in consistent histories are reviewed. • We prove generalized contexts for quantum histories have no contrary properties.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

On exotic consistent anomalies in (1+1)$d$: A ghost story

We revisit ’t Hooft anomalies in (1+1) d d non-spin quantum field theory, starting from the consistency and locality conditions, and find that consistent U(1) and gravitational anomalies cannot always be canceled by properly quantized (2+1) d d classical Chern-Simons actions. On the one hand, we prove that certain exotic anomalies can only be realized by non-reflection-positive or non-compact theories; on the other hand, without insisting on reflection-positivity, the exotic anomalies present a caveat to the inflow paradigm. For the mixed U(1) gravitational anomaly, we propose an inflow mechanism involving a mixed U(1) \times × SO(2) classical Chern-Simons action with a boundary condition that matches the SO(2) gauge field with the (1+1) d d spin connection. Furthermore, we show that this mixed anomaly gives rise to an isotopy anomaly of U(1) topological defect lines. The isotopy anomaly can be canceled by an extrinsic curvature improvement term, but at the cost of creating a periodicity anomaly. We survey the holomorphic bc b c ghost system which realizes all the exotic consistent anomalies, and end with comments on a subtlety regarding the anomalies of finite subgroups of U(1).

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Topics in Theoretical Physics

This award supported a broadly based research effort in theoretical particle physics, including research aimed at uncovering the laws of nature at short (subatomic) and long (cosmological) distances. These theoretical developments apply to experiments in laboratories such as CERN, the facility that operates the Large Hadron Collider outside Geneva, as well as to cosmological investigations done using telescopes and satellites. The results reported here apply to physics beyond the so-called Standard Model of particle physics; physics of high energy collisions such as those observed at the Large Hadron Collider; theoretical and mathematical tools and frameworks for describing the laws of nature at short distances; cosmology and astrophysics; and analytic and computational methods to solve theories of short distance physics. Some specific research accomplishments include + Theories of the electroweak interactions, the forces that give rise to many forms of radioactive decay; + Physics of the recently discovered Higgs boson. + Models and phenomenology of dark matter, the mysterious component of the universe, that has so far been detected only by its gravitational effects. + High energy particles in astrophysics and cosmology. + Algorithmic research and Computational methods for physics of and beyond the Standard Model. + Theory and applications of relativity and its possible limitations. + Topological effects in field theory and cosmology. + Conformally invariant systems and AdS/CFT. This award also supported significant training of students and postdoctoral fellows to lead the research effort in particle theory for the coming decades. These students and fellows worked closely with other members of the group as well as theoretical and experimental colleagues throughout the physics community. Many of the research projects funded by this grant arose in response to recently obtained experimental results in the areas of particle physics and cosmology. We describe a few of these below. Relativity is founded on a symmetry property of nature called "Lorentz Invariance". Like all symmetry properties, it is essential to determine precisely how symmetric nature actually is; that is, do the laws of nature fully respect the symmetry or is there room for tiny symmetry violating effects? An important consequence of Lorentz invariance is the existence of a universal limiting velocity for all physical particles. Light travels at this limiting velocity so it is frequently referred to as simply "the speed of light", but relativity requires that ALL particles travel more slowly than this speed. Once the Higgs particle was discovered in 2012 a natural question was whether or not this particle's speed was consistent with relativity. Although the speed of the Higgs particle is not measurable directly, Cohen has shown that, if the maximal speed of the Higgs particle was not precisely the same as the speed of light, then the Higgs would have some unusual properties. In some cases the Higgs would be unstable to some unusual decay modes; in other cases the interactions of the Higgs with other particles would change the properties of these other particles in ways that could be observed in so-called cosmic rays, very energetic particles (such as photons, protons and other atomic nuclei) coming from space. Once these particles hit the upper atmosphere they produce a "shower" of particles that can be seen by ground-based instruments. If the Higgs has a maximal speed that differs even a tiny bit from the speed of light these showers would look quite different from what is observed. In this way Cohen was able to establish that the Higgs travels with a maximal speed that cannot differ from the speed of light by more than one part in a thousand-trillion. This is by far the most precisely determined property of the Higgs particle. Cohen and Schmaltz reviewed evidence from the Large Hadron Collider (LHC), a particle physics experiment operating at the CERN laboratory near Geneva, for a new particle sometimes called a W'. This evidence included certain unexpected by-products in collisions of protons at very high energy. While the evidence was not significant enough to claim a discovery, it was sufficiently intriguing that many particle theorists worked to construct explanations for this signal. Cohen and Schmaltz were able to determine that such explanations are highly constrained by previous experiments involving collisions of very energetic particles. Nevertheless they were able to construct a theory that adequately explains the LHC data and remain consistent with prior experiments. Their explanation predicts the existence of yet another new particle, called a Z', with a mass slightly greater than that of the W'. This additional particle, if it exists, should be seen as more data is collected from the LHC. Amusingly, there is one collision by-product that has already been seen by the CMS experiment at the LHC that supports the existence of this new particle; however, it is not unlikely that this single event is a so-called "background" event, that is a somewhat atypical by-product of a conventional Standard Model process. This theory for the anomalous LHC data will either be confirmed or excluded with further data-taking at the LHC. The ratio of the number of electrons produced in bottom quark decays over the number of muons produced has been measured at the LHC. This ratio is interesting because it can be predicted very precisely from a basic property of the Standard Model: lepton universality. If lepton universality is correct, the ratio of electrons to muons is predicted to be equal to 1. The first measurements of this ratio find a value different from 1 with a statistical significance of about 3 standard deviations. Schmaltz and collaborator proposed a new extension of the Standard Model which can explain the new data. In addition, Schmaltz and collaborators proposed several new measurements of ratios of decay rates which can confirm or rule out the surprising results from the earlier LHC data. The most recent and precise measurements of the cosmic microwave background from the Planck satellite, from a combination of measurements of the dark matter distribution in the universe, and from a measurement of the expansion rate of the universe today show some disagreement when interpreted in terms of the so-called LambdaCDM model. Schmaltz and collaborators proposed an alternative model to LambdaCDM in which the usual cold dark matter is replaced by a new ``dark sector". This sector consists of a cold dark matter particle which interacts with a newly postulated dark radiation component of the universe. The dark radiation can help explain the discrepancy in measurements of the expansion rate, and the dark matter interactions subtly modify the clumping of dark matter at large scales, thus potentially explaining both kinds of tensions in the data. In two publications Schmaltz described the new model and then performed a precision comparison of the predictions of the model with all currently available cosmological data. The results favor the new model at the level of three standard deviations with current data. Quantum Field Theory (QFT) is the language we use to describe quantum systems which are consistent with Einstein’s theory of Special Relativity. In particular, the requirement of Einstein’s theory that signals not travel faster than the speed of light constrains the types of interactions which particles can engage in. One consequence of relativity is that these interactions cannot preserve particle number. The stronger the interactions, the more severe the particle number violation in a given Relativistic QFT. When particle number violation is strong, it becomes very difficult to adequately parameterize the quantum wave function (which characterizes the state of a quantum system). For example, though we can formulate the QFT which describes the strong force as a set of interactions between quark and gluon particles, we have no clear idea how to express the proton state in terms of these quarks and gluons. This is because the proton, though a bound state of quarks and gluons, is not a state of a fixed number of particles due to strong interactions. Yet, understanding the proton state is very important in order to theoretically predict the reaction rates observed at the LHC in Geneva, which is a proton-proton collider. Katz has formulated a new approach to QFT, which among other things offers a way to adequately approximate the quantum wave function of a bound state at strong coupling. The approximation scheme is related to the fact that any sensible QFT (including that of the strong interactions) is at short distances approximately self-similar upon rescaling of space and time. It turns out that keeping track of the response upon this rescaling is important in efficiently parameterizing the state. Katz and collaborators have used this observation to approximate the state of the proton in toy versions of the strong force. In the late 60s Sheldon Glashow, Abdus Salam and Steven Weinberg (1979 Nobel Prize awardees) proposed a theory unifying weak and electromagnetic interaction which assumed the existence of new particles, the W and Z bosons. The W and Z bosons were eventually detected in high-energy collision in a particle accelerator at CERN, and the recent discovery of the Higgs meson at the Large Hadron Collider (LHC), always at CERN, completed the picture. However, deep theoretical considerations indicate that the theory by Glashow, Weinberg and Salam, often referred to as "the standard model" cannot be the whole story: the existence of new particles and new interactions at yet higher energies is widely anticipated. The experiments at the LHC are looking for these, while theorists, like Brower, Rebbi and collaborators, are investigating models for these new interactions. Working in a large national collaboration with access to the most powerful DOE computers Brower, Rebbi and colleagues have been using calculational techniques, similar to those successfully employed for many years to investigate the interactions among quarks in nucleons, to study theories that can describe the expected "beyond the standard model" (BSM) interactions. Their results, which include also a model for dark matter, have been published in several refereed papers in prestigious journals. Various ideas in topologically interesting field theories predict hypothetical objects such as fractional charges and Majorana excitations. However, such fascinating objects have not been seen in particle physics. Nevertheless, these objects demonstrate possible phenomena that quantum field theory can support. Pi used condensed matter physics as a laboratory to study possible realizations and observable effects of these objects predicted by quantum field theory. In recent times there has developed considerable interest among condensed matter field theorists in precisely the same geometrical and topological structures, which were first discovered in particle physics field theories. From particle physicists' point of view, this is an interesting development, since condensed matter provides an arena in which one can concretely realize particle physics ideas. Moreover, particle physicists can learn new ideas from condensed matter physics. Higgs phenomenon is precisely an important particle physics realization of condensed matter ideas. In contrast to the small distance characterizing condensed matter systems, field theory also describes large distance physics characterizing cosmology. Pi worked on various geometrical effects in the standard theory of cosmology, viz general relativity.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Is quantum mechanics exact?

We formulate physically motivated axioms for a physical theory which for systems with a finite number of degrees of freedom uniquely lead to quantum mechanics as the only nontrivial consistent theory. Complex numbers and the existence of the Planck constant common to all systems arise naturally in this approach. The axioms are divided into two groups covering kinematics and basic measurement theory, respectively. We show that even if the second group of axioms is dropped, there are no deformations of quantum mechanics which preserve the kinematic axioms. Thus, any theory going beyond quantum mechanics must represent a radical departure from the usual a priori assumptions about the laws of nature.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Time-reversal symmetry, anomalies, and dualities in (2+1)$d$

We study continuum quantum field theories in 2+1 dimensions with time-reversal symmetry \cal T . The standard relation {\cal T}^2=(-1)^F is satisfied on all the “perturbative operators” i.e. polynomials in the fundamental fields and their derivatives. However, we find that it is often the case that acting on more complicated operators {\cal T}^2=(-1)^F {\cal M} with \cal M a non-trivial global symmetry. For example, acting on monopole operators, \cal M could be \pm1 ± 1 depending on the magnetic charge. We study in detail U(1) U ( 1 ) gauge theories with fermions of various charges. Such a modification of the time-reversal algebra happens when the number of odd charge fermions is 2 ~{\rm mod }~4 , e.g. in QED with two fermions. Our work also clarifies the dynamics of QED with fermions of higher charges. In particular, we argue that the long-distance behavior of QED with a single fermion of charge 2 2 is a free theory consisting of a Dirac fermion and a decoupled topological quantum field theory. The extension to an arbitrary even charge is straightforward. The generalization of these abelian theories to SO(N) S O ( N ) gauge theories with fermions in the vector or in two-index tensor representations leads to new results and new consistency conditions on previously suggested scenarios for the dynamics of these theories. Among these new results is a surprising non-abelian symmetry involving time-reversal.

Cordova, Clay↗

Control simulations of many-body quantum systems by a synergism of discrete real-time learning and optimal control theory

We present a self-consistent algorithm for optimal control simulations of many-body quantum systems. The algorithm features a two-step synergism that combines discrete real-time machine learning (DRTL) with Quantum Optimal Control Theory (QOCT) using the time-dependent Schrödinger equation. Specifically, in step (1), DRTL is employed to identify a compact working space (i.e., the important portion of the Hilbert space) for the time evolution of the many-body quantum system in the presence of a control field (i.e., the initial or previously updated field), and in step (2), QOCT utilizes the DRTL-determined working space to find a newly updated control field for a chosen objective. Steps 1 and 2 are iterated until a self-consistent control objective value is reached such that the resulting optimal control field yields the same targeted objective value when the corresponding working space is systematically enlarged. Furthermore, to demonstrate this two-step self-consistent DRTL-QOCT synergistic algorithm, we perform optimal control simulations of strongly interacting 1D as well as 2D Heisenberg spin systems. In both scenarios, only a single spin (at the left end site for 1D and the upper left corner site for 2D) is driven by the time-dependent control fields to create an excitation at the opposite site as the target. It is found that, starting from all spin-down zero excitation states, the synergistic method is able to identify working spaces and convergence of the desired controlled dynamics with just a few iterations of the overall algorithm. In the cases studied, the dimensionality of the working space scales only quasi-linearly with the number of spins.

Artificial neural networks↗

De Sitter diagrammar and the resummation of time

Light scalars in inflationary spacetimes suffer from logarithmic infrared divergences at every order in perturbation theory. This corresponds to the scalar field values in different Hubble patches undergoing a random walk of quantum fluctuations, leading to a simple toy “landscape” on superhorizon scales, in which we can explore questions relevant to eternal inflation. However, for a sufficiently long period of inflation, the infrared divergences appear to spoil computability. Some form of renormalization group approach is thus motivated to resum the log divergences of conformal time. Such a resummation may provide insight into De Sitter holography. We present here a novel diagrammatic analysis of these infrared divergences and their resummation. Basic graph theory observations and momen- tum power counting for the in-in propagators allow a simple and insightful determination of the leading-log contributions. One thus sees diagrammatically how the superhorizon sector consists of a semiclassical theory with quantum noise evolved by a first-order, interacting classical equation of motion. This rigorously leads to the “Stochastic Inflation” ansatz developed by Starobinsky to cure the scalar infrared pathology nonperturbatively. Our approach is a controlled approximation of the underlying quantum field theory and is systematically improvable.

79 ASTRONOMY AND ASTROPHYSICS↗

Evidence for Dirac flat band superconductivity enabled by quantum geometry

In a flat band superconductor, the charge carriers’ group velocity v F is extremely slow. Superconductivity therein is particularly intriguing, being related to the long-standing mysteries of high-temperature superconductors and heavy-fermion systems. Yet the emergence of superconductivity in flat bands would appear paradoxical, as a small v F in the conventional Bardeen–Cooper–Schrieffer theory implies vanishing coherence length, superfluid stiffness and critical current. Here, using twisted bilayer graphene, we explore the profound effect of vanishingly small velocity in a superconducting Dirac flat band system. Using Schwinger-limited non-linear transport studies we demonstrate an extremely slow normal state drift velocity v n ≈ 1,000 m s –1 for filling fraction ν between -1/2 and -3/4 of the moiré superlattice. In the superconducting state, the same velocity limit constitutes a new limiting mechanism for the critical current, analogous to a relativistic superfluid. Importantly, our measurement of superfluid stiffness, which controls the superconductor’s electrodynamic response, shows that it is not dominated by the kinetic energy but instead by the interaction-driven superconducting gap, consistent with recent theories on a quantum geometric contribution. Here we find evidence for small Cooper pairs, characteristic of the Bardeen–Cooper–Schrieffer to Bose–Einstein condensation crossover, with an unprecedented ratio of the superconducting transition temperature to the Fermi temperature exceeding unity and discuss how this arises for ultra-strong coupling superconductivity in ultra-flat Dirac bands.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Operator product expansion algebra

We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean φ{sup 4}-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of hep-th/1105.3375, that the 3-point OPE, =Σ{sub C}C{sub A{sub 1A{sub 2A{sub 3}{sup C}}}} , usually interpreted only as an asymptotic short distance expansion, actually converges at finite, and even large, distances. We further show that the factorization identity C{sub A{sub 1A{sub 2A{sub 3}{sup B}}}}=Σ{sub C}C{sub A{sub 1A{sub 2}{sup C}}}C{sub CA{sub 3}{sup B}} is satisfied for suitable configurations of the spacetime arguments. Again, the infinite sum is shown to be convergent. Our proofs rely on explicit bounds on the remainders of these expansions, obtained using refined versions, mostly due to Kopper et al., of the renormalization group flow equation method. These bounds also establish that each OPE coefficient is a real analytic function in the spacetime arguments for non-coinciding points. Our results hold for arbitrary but finite loop orders. They lend support to proposals for a general axiomatic framework of quantum field theory, based on such “consistency conditions” and akin to vertex operator algebras, wherein the OPE is promoted to the defining structure of the theory.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗