Fermion Green Research Articles

Denoising of imaginary time response functions with Hankel projections

Imaginary-time response functions of finite-temperature quantum systems are often obtained with methods that exhibit stochastic or systematic errors. Reducing these errors comes at a large computational cost—in quantum Monte Carlo simulations, the reduction of noise by a factor of two incurs a simulation cost of a factor of four. In this paper, we relate certain imaginary-time response functions to an inner product on the space of linear operators on Fock space. We then show that data with noise typically does not respect the positive definiteness of its associated Gramian. The Gramian has the structure of a Hankel matrix. As a method for denoising noisy data, we introduce an alternating projection algorithm that finds the closest positive definite Hankel matrix consistent with noisy data. We test our methodology at the example of fermion Green's functions for continuous-time quantum Monte Carlo data and show remarkable improvements of the error, reducing noise by a factor of up to 20 in practical examples. We argue that Hankel projections should be used whenever finite-temperature imaginary-time data of response functions with errors is analyzed, be it in the context of quantum Monte Carlo, quantum computing, or in approximate semianalytic methodologies. Published by the American Physical Society 2024

ZN lattice gauge theories with matter fields

Motivated by the exotic phenomenology of certain quantum materials and recent advances in programmable quantum emulators, we here study fermions and bosons in ZN lattice gauge theories. We introduce a family of exactly soluble models, and characterize their orthogonal (semi)metallic ground states, the excitation spectrum, and the correlation functions. We further study integrability-breaking perturbations using an appropriately derived set of Feynman diagrammatic rules and borrowing physics associated to Anderson's orthogonality catastrophe. In the context of the ground states, we revisit Luttinger's theorem following Oshikawa's flux insertion argument and furthermore demonstrate the existence of a Luttinger surface of zeros in the fermionic Green's function. Upon inclusion of perturbations, we address the transition from the orthogonal metal to the normal state by condensation of certain excitations in the gauge sectors, so-called “e particles.” We furthermore discuss the effect of dynamics in the dual “m-particle” excitations, which ultimately leads to the formation of charge-neutral hadronic N-particle bound states. We present analytical arguments for the most important phases and estimates for phase boundaries of the model. Specifically, and in sharp distinction to quasi-one-dimensional ZN lattice gauge theories, renormalization group arguments imply that the phase diagram does not include an emergent deconfining U(1) phase in the limit of large number of fermion flavors. Therefore, in regards to lattice QED problems, ZN quantum emulators with N<∞ can at best be used for approximate solutions at intermediate length scales. Published by the American Physical Society 2024

Engineering quasi-steady-state correlations in uncorrelated thermal states using stochastic driving

Nonequilibrium quantum dynamics can give rise to the emergence of novel steady states. We propose a scheme for driving an initially uncorrelated thermal state to generate customized correlation functions by determining and reverse engineering the steady-state two-point functions for a class of Markov processes. We also extend the formalism to the calculation of four-point functions. We then apply our method to generating power-law correlated fermionic Green's functions. Furthermore, we find that the power-law patterns emerge at much shorter times than the convergence to the steady state, at which point the disorder in the two-point correlations disappears. On the other hand, the density-density correlations exhibit steady-state disorder while following a power-law trend line. These ideal steady states appear as intermediate-time quasi-steady states in the presence of perturbations.

Ana_cont: Python package for analytic continuation

We present the Python package ana_cont for the analytic continuation of fermionic and bosonic many-body Green's functions by means of either the Padé approximants or the maximum entropy method. The determination of hyperparameters and the implementation are described in detail. The code is publicly available on GitHub, where also documentation and learning resources are provided. Program summaryProgram Title: ana_contCPC Library link to program files:https://doi.org/10.17632/vphzykvvf2.1Developer's repository link:https://github.com/josefkaufmann/ana_contLicensing provisions: MITProgramming language:PythonExternal dependencies:Python(≥3.6), numpy, scipy, matplotlib, h5py, PyQt5, CythonSupplementary material: Test case files, tutorials, and instructionsNature of problem: Analytic continuation of correlation functions from Matsubara frequencies/imaginary time to real frequencies.Solution method: Padé interpolation, maximum entropy methodAdditional comments including restrictions and unusual features: The most important features can be accessed through the graphical user interface. For more flexibility, it is recommended to use the code as a library and write problem specific scripts.

TRIQS/SOM 2.0: Implementation of the stochastic optimization with consistent constraints for analytic continuation

TRIQS/SOM 2.0: Implementation of the stochastic optimization with consistent constraints for analytic continuation

Correlations at PT-Symmetric Quantum Critical Point.

We consider a PT-symmetric Fermi gas with an exceptional point, representing the critical point between PT-symmetric and symmetry broken phases. The low energy spectrum remains linear in momentum and is identical to that of a Hermitian Fermi gas. The fermionic Green's function decays in a power law fashion for large distances, as expected from gapless excitations, although the exponent is reduced from -1 due to the quantum Zeno effect. In spite of the gapless nature of the excitations, the ground state entanglement entropy saturates to a finite value, independent of the subsystem size due to the non-Hermitian correlation length intrinsic to the system. Attractive or repulsive interaction drives the system into the PT-symmetry broken regime or opens up a gap and protects PT symmetry, respectively. Our results challenge the concept of universality in non-Hermitian systems, where quantum criticality can be masked due to non-Hermiticity.

Photon polarization tensor in a magnetized plasma: Absorptive part

We calculate the absorptive part of the photon polarization tensor in a hot magnetized relativistic plasma. In the derivation, we utilize a Landau-level representation for the fermion Green's function in a mixed coordinate-momentum space and obtain a closed-form expression for the one-loop polarization tensor. At the leading order in the coupling, its absorptive part is determined by particle and antiparticle splitting processes ($e^{-} \leftrightarrow e^{-}+\gamma$ and $e^{+} \leftrightarrow e^{+}+\gamma $, respectively), as well as by particle-antiparticle annihilation processes ($e^{-} + e^{+}\leftrightarrow \gamma$). The interpretation in terms of quantum transitions between Landau levels is also given. By making use of the photon polarization tensor, we study the differential photon emission rate in the quantum limit of magnetized relativistic plasma. At low energies, the photon emission has a prolate profile with the symmetry axis along the line of the magnetic field. At high energies, on the other hand, the photon emission has an oblate profile. The underlying reasons for such emission profiles are given in both regimes. The general result for the photon polarization tensor is also used to calculate the longitudinal and transverse components of magneto-optical conductivity.

Chiral kinetic theory with small mass corrections and quantum coherent states

We study the effect of a small fermion mass in the formulation of the on-shell effective field theory (OSEFT). This is our starting point to derive small mass corrections to the chiral kinetic theory. In the massless case, only four Wigner functions are needed to describe positive and negative energy fermions of left and right chirality, corresponding to the vectorial components of a fermionic two-point Green's function. As soon as mass corrections are introduced, tensorial components are also needed, while the scalar components strictly vanish in the OSEFT. The tensorial components are conveniently parametrized in the so-called spin coherent functions, which describe quantum coherent mixtures of left-right and right-left chiral fermions, of either positive or negative energy. We show that, up to second order in the energy expansion, vectorial and tensorial components are decoupled, and obey the same dispersion law and transport equation, depending on their respective chirality. We study the mass modifications of the reparametrization invariance of the OSEFT, and check that vector and tensorial components are related by the associated symmetry transformations. We study how the macroscopic properties of the system are described in terms of the whole set of Wigner functions, and check that our framework allows to account for the mass modifications to the chiral anomaly equation.

Dynamics of entanglement after exceptional quantum quench

We investigate a quantum quench from a critical to an exceptional point. The initial state, prepared in the ground state of a critical hermitian system, is time evolved with a non-hermitian SSH model, tuned to its exceptional point. The single particle density matrix exhibits supersonic modes and multiple light cones, characteristic to non-hermitian time evolution. These propagate with integer multiples of the original Fermi velocity. In the long time limit, the fermionic Green's function decays spatially as $1/x^2$, in sharp contrast to the usual $1/x$ decay of non-interacting fermions. The entanglement entropy is understood as if all these supersonic modes arise from independent quasiparticles (though they do not), traveling with the corresponding supersonic light cone velocity. The entropy production rate decreases with time and develops plateaus during the time evolution, signaling the distinct velocities in the propagation of non-local quantum correlations. At late times, the entanglement entropy saturates to a finite value, satisfying a volume law.

Holographic spectral functions with momentum relaxation

We study (fermionic) spectral functions in two holographic models, the Gubser-Rocha-linear axion model and the linear axion model, where translational symmetry is broken by axion fields linear to the boundary coordinates (${\ensuremath{\psi}}_{I}=\ensuremath{\beta}{\ensuremath{\delta}}_{Ii}{x}^{i}$). Here, $\ensuremath{\beta}$ corresponds to the strength of momentum relaxation. The spectral function is computed by the fermionic Green's function of the bulk Dirac equation, where a fermion mass, $m$, and a dipole coupling, $p$, are introduced as input parameters. By classifying the shape of spectral functions, we construct complete phase diagrams in ($m$, $p$, $\ensuremath{\beta}$) space for both models. We find that two phase diagrams are similar even though their background geometries are different. This similarity might be due to temperature effect, since our analysis has been done at small but finite temperature ($T/\ensuremath{\mu}=0.1$). We also find that the effect of momentum relaxation on the (spectral function) phases of two models are similar even though the effect of momentum relaxation on the dc conductivities of two models are very different. We suspect that this is because holographic fermion does not backreact to geometry in our framework.

Radiative effects on false vacuum decay in Higgs-Yukawa theory

We derive fermionic Green's functions in the background of the Euclidean solitons describing false vacuum decay in a prototypal Higgs-Yukawa theory. In combination with appropriate counterterms for the masses, couplings and wave-function normalization, these can be used to calculate radiative corrections to the soliton solutions and transition rates that fully account for the inhomogeneous background provided by the nucleated bubble. We apply this approach to the archetypal example of transitions between the quasi-degenerate vacua of a massive scalar field with a quartic self-interaction. The effect of fermion loops is compared with those from additional scalar fields, and the loop effects accounting for the spacetime inhomogeneity of the tunneling configuration are compared with those where gradients are neglected. We find that scalar loops lead to an enhancement of the decay rate, whereas fermion loops lead to a suppression. These effects get relatively amplified by a perturbatively small factor when gradients are accounted for. In addition, we observe that the radiative corrections to the solitonic field profiles are smoother when the gradients are included. The method presented here for computing fermionic radiative corrections should be applicable beyond the archetypal example of vacuum decay. In particular, we work out methods that are suitable for calculations in the thin-wall limit, as well as others that take account of the full spherical symmetry of the solution. For the latter case, we construct the Green's functions based on spin hyperspherical harmonics, which are eigenfunctions of the appropriate angular momentum operators that commute with the Dirac operator in the solitonic background.

Symmetric Fermion Mass Generation as Deconfined Quantum Criticality

Massless 2+1D Dirac fermions arise in a variety of systems from graphene to the surfaces of topological insulators, where generating a mass is typically associated with breaking a symmetry. However, with strong interactions, a symmetric gapped phase can arise for multiples of eight Dirac fermions. A continuous quantum phase transition from the massless Dirac phase to this massive phase, which we term Symmetric Mass Generation (SMG), is necessarily beyond the Landau paradigm and is hard to describe even at the conceptual level. Nevertheless, such transition has been consistently observed in several numerical studies recently. Here, we propose a theory for the SMG transition which is reminiscent of deconfined criticality and involves emergent non-Abelian gauge fields coupled both to Dirac fermions and to critical Higgs bosons. We motivate the theory using an explicit parton construction and discuss predictions for numerics. Additionally, we show that the fermion Green's function is expected to undergo a zero to pole transition across the critical point.

Effective locality and gauge invariance

A few years ago, based on standard functional manipulations, an unexpected property was proven to be satisfied by the fermionic Green's functions of QCD, and dubbed effective locality. This feature of QCD is non-perturbative as it results from a full integration of the fermionic and gluonic degrees of freedom. Previous derivations of effective locality are reviewed, corrected, and enlarged. Focussing on the way nonabelian gauge invariance is realized in the non-perturbative regime of QCD, the meaning of effective locality is proposed.

Quantum quench of the Sachdev-Ye-Kitaev model

We describe the non-equilibrium dynamics of the Sachdev-Ye-Kitaev models of fermions with all-to-all interactions. These provide tractable models of the dynamics of quantum systems without quasiparticle excitations. The Kadanoff-Baym equations show that the final state is thermal, and their numerical analysis appears consistent with a thermalization rate proportional to the absolute temperature of the final state. We also obtain an exact analytic solution of the non-equilibrium dynamics in the large $q$ limit of a model with $q$ fermion interactions: in this limit, the thermalization of the fermion Green's function is instantaneous.

Area term of the entanglement entropy of a supersymmetricO(N)vector model in three dimensions

We studied the leading area term of the entanglement entropy of $\mathcal{N}=1$ supersymmetric $O(N)$ vector model in $2+1$ dimensions close to the line of second order phase transition in the large $N$ limit. We found that the area term is independent of the varying interaction coupling along the critical line, unlike what is expected in a perturbative theory. Along the way, we studied non-commuting limits $n-1\to 0$ verses UV cutoff $r\to 0$ when evaluating the gap equation and found a match only when appropriate counter term is introduced and whose coupling is chosen to take its fixed point value. As a bonus, we also studied Fermionic Green's functions in the conical background. We made the observation of a map between the problem and the relativistic hydrogen atom.

Spectral properties of four-time fermionic Green's functions

The spectral relations for the four-time fermionic Green's functions are derived in the most general case. The terms which correspond to the zero-frequency anomalies, known before only for the bosonic Green's functions, are separated and their connection with the second cumulants of the Boltzmann distribution function is elucidated. The high-frequency expansions of the four-time fermionic Green's functions are provided for different directions in the frequency space.

Numerical study of fermion and boson models with infinite-range random interactions

We present numerical studies of fermion and boson models with random all-to-all interactions (the SYK models). The high temperature expansion and exact diagonalization of the $N$-site fermion model are used to compute the entropy density: our results are consistent with the numerical solution of $N=\infty$ saddle point equations, and the presence of a non-zero entropy density in the limit of vanishing temperature. The exact diagonalization results for the fermion Green's function also appear to converge well to the $N=\infty$ solution. For the hard-core boson model, the exact diagonalization study indicates spin glass order. Some results on the entanglement entropy and the out-of-time-order correlators are also presented.

Strange metals from quantum geometric fluctuations of interfaces

Our current understanding of strongly correlated electron systems is based on a homogeneous framework. Here we take a step going beyond this paradigm by incorporating inhomogeneity from the beginning. Specifying to systems near the Mott metal-insulator transition, we propose a real space picture of itinerant electrons functioning in the fluctuating geometries bounded by interfaces between metallic and insulating regions. In 2+1-dimensions, the interfaces are closed bosonic strings, and we have a system of strings coupled to itinerant electrons. When the interface tension vanishes, the geometric fluctuations become critical, which gives rise to non-Fermi liquid behavior for the itinerant electrons. In particular, the poles of the fermion Green's function can be converted to zeros, indicating the absence of propagating quasiparticles. Furthermore, the quantum geometric fluctuations mediate Cooper pairing among the itinerant electrons, indicating the intrinsic instability of electronic systems near the Mott transition.

Fermi surface behavior in the ABJM M2-brane theory

We calculate fermionic Green's functions for states of the three-dimensional ABJM M2-brane theory at large N using the gauge-gravity correspondence. We embed extremal black brane solutions in four-dimensional maximally supersymmetric gauged supergravity, obtain the linearized Dirac equations for each spin-1/2 mode that cannot mix with a gravitino, and solve these equations with infalling boundary conditions to calculate retarded Green's functions. For generic values of the chemical potentials, we find Fermi surfaces with universally non-Fermi liquid behavior, matching the situation for four-dimensional N=4 Super-Yang-Mills. Fermi surface singularities appear and disappear discontinuously at the point with all chemical potentials equal, reminiscent of a quantum critical point. One limit of parameter space has zero entropy at zero temperature, and fermionic fluctuations are perfectly stable inside an energy region around the Fermi surface. An ambiguity in the quantization of the fermions is resolved by supersymmetry.

Topological number and fermion Green's function for strongly interacting topological superconductors

It has been understood that short range interactions can reduce the classification of topological superconductors in all dimensions. In this Rapid Communication we demonstrate by explicit calculations that when the topological phase transition between two distinct phases in the noninteracting limit is gapped out by interaction, the bulk fermion Green's function $G(i\ensuremath{\omega})$ at the ``transition'' approaches zero as $G(i\ensuremath{\omega})\ensuremath{\sim}\ensuremath{\omega}$ at certain momentum $\stackrel{P\vec}{k}$ in the Brillouin zone.