Fermionic Correlation Functions Research Articles

Deciphering competing interactions of Kitaev-Heisenberg-Γ system in clusters: II. Dynamics of Majorana fermions

We perform a systematic and exact study of Majorana fermion dynamics in the Kitaev-Heisenberg-Γ model in a few finite-size clusters increasing in size up to twelve sites. We employ exact Jordan-Wigner transformations to evaluate certain measures of Majorana fermion correlation functions, which effectively capture matter and gauge Majorana fermion dynamics in different parameter regimes. An external magnetic field is shown to produce a profound effect on gauge fermion dynamics. Depending on certain non-zero choices of other non-Kitaev interactions, it can stabilise it to its non-interacting Kitaev limit. For all the parameter regimes, gauge fermions are seen to have slower dynamics, which could help build approximate decoupling schemes for appropriate mean-field theory. The probability of Majorana fermions returning to their original starting site shows that the Kitaev model in small clusters can be used as a test bed for the quantum speed limit.

A linear response framework for quantum simulation of bosonic and fermionic correlation functions

Response functions are a fundamental aspect of physics; they represent the link between experimental observations and the underlying quantum many-body state. However, this link is often under-appreciated, as the Lehmann formalism for obtaining response functions in linear response has no direct link to experiment. Within the context of quantum computing, and via a linear response framework, we restore this link by making the experiment an inextricable part of the quantum simulation. This method can be frequency- and momentum-selective, avoids limitations on operators that can be directly measured, and can be more efficient than competing methods. As prototypical examples of response functions, we demonstrate that both bosonic and fermionic Green’s functions can be obtained, and apply these ideas to the study of a charge-density-wave material on the ibm_auckland superconducting quantum computer. The linear response method provides a robust framework for using quantum computers to study systems in physics and chemistry.

Fermionic Correlation Functions from Randomized Measurements in Programmable Atomic Quantum Devices.

We provide an efficient randomized measurement protocol to estimate two- and four-point fermionic correlations in ultracold atom experiments. Our approach is based on combining random atomic beam splitter operations, which can be realized with programmable optical landscapes, with high-resolution imaging systems such as quantum gas microscopes. We illustrate our results in the context of the variational quantum eigensolver algorithm for solving quantum chemistry problems.

Generalized LKF Transformations for $$\boldsymbol{N}$$-Point Fermion Correlators in QED

Within the worldline approach to quantum electrodynamics (QED), a change of the photon's covariant gauge parameter $\xi$ is investigated to analyse the non-perturbative gauge dependence of the configuration space fermion correlation functions, deriving a generalization of the Landau-Kalatnikov-Fradkin transformations (LKFt). These transformations reveal how the non-perturbative gauge dependence of position space amplitudes can be absorbed into a multiplicative exponential factor.

Consequences of the order of the limit of infinite spacetime volume and the sum over topological sectors for CP violation in the strong interactions

We derive correlation functions for massive fermions with a complex mass in the presence of a general vacuum angle. For this purpose, we first build the Green's functions in the one-instanton background and then sum over the configurations of background instantons. The quantization of topological sectors follows for saddle points of finite Euclidean action in an infinite spacetime volume and the fluctuations about these. For the resulting correlation functions, we therefore take the infinite-volume limit before summing over topological sectors. In contrast to the opposite order of limits, the chiral phases from the mass terms and from the instanton effects then are aligned so that, in absence of additional phases, these do not give rise to observables violating charge-parity symmetry. This result is confirmed when constraining the correlations at coincident points by using the index theorem instead of instanton calculus.

Instabilities of quantum critical metals in the limit $N_f\rightarrow0$

We study a model in 2+1 dimensions composed of a Fermi surface of N_fNf flavors of fermions coupled to scalar fluctuations near quantum critical points (QCPs). The N_f\rightarrow0Nf→0 limit allows us to non-perturbatively calculate the long-range behavior of fermion correlation functions. We use this to calculate charge, spin and pair susceptibilities near different QCPs at zero and finite temperatures, with zero and finite order parameter gaps. While fluctuations smear out the fermionic quasiparticles, we find QCPs where the overall effect of fluctuations leads to enhanced pairing. We also find QCPs where the fluctuations induce spin and charge density wave instabilities for a finite interval of order parameter fluctuation gaps at T=0T=0. We restore a subset of the diagrams suppressed in the N_f\rightarrow0Nf→0 limit, all diagrams with internal fermion loops with at most 2 vertices, and find that this does not change the long-range behavior of correlators except right at the QCPs.

Highly efficient and accurate sum-over-poles expansion of Fermi and Bose functions at near zero temperatures: Fano spectrum decomposition scheme.

The Fano spectrum decomposition (FSD) scheme is proposed as an efficient and accurate sum-over-poles expansion of Fermi and Bose functions at cryogenic temperatures. The new method practically overcomes the discontinuity of Fermi and Bose functions near zero temperature, which causes slow convergence in conventional schemes such as the state-of-the-art Padé spectrum decomposition (PSD). The FSD scheme fragments Fermi or Bose function into a high-temperature reference and a low-temperature correction. While the former is efficiently decomposed via the standard PSD, the latter can be accurately described by several modified Fano functions. The resulting FSD scheme is found to converge overwhelmingly faster than the standard PSD method. Remarkably, the low-temperature correction supports further a recursive and scalable extension to access the near-zero temperature regime. Thus, the proposed FSD scheme, which obeys rather simple recursive relations, has a great value in efficient numerical evaluations of Fermi or Bose function-involved integrals for various low-temperature condensed physics formulations and problems. For numerical demonstrations, we exemplify FSD for the efficient unraveling of fermionic reservoir correlation functions and the exact hierarchical equations of motion simulations of spin-boson dynamics, both at extremely low temperatures.

Decay of Complex-Time Determinantal and Pfaffian Correlation Functionals in Lattices

We supplement the determinantal and Pfaffian bounds of Sims and Warzel (Commun Math Phys 347:903--931, 2016) for many-body localization of quasi-free fermions, by considering the high dimensional case and complex-time correlations. Our proof uses the analyticity of correlation functions via the Hadamard three-line theorem. We show that the dynamical localization for the one-particle system yields the dynamical localization for the many-point fermionic correlation functions, with respect to the Hausdorff distance in the determinantal case. In Sims and Warzel (2016), a stronger notion of decay for many-particle configurations was used but only at dimension one and for real times. Considering determinantal and Pfaffian correlation functionals for complex times is important in the study of weakly interacting fermions.

A note on Sachdev–Ye–Kitaev like model without random coupling

We study a description of the large N limit of the Sachdev–Ye–Kitaev (SYK) model in terms of quantum mechanics without quenched disorder. Instead of random couplings, we introduce massive scalar fields coupled to fermions, and study a small mass limit of the theory. We show that, under a certain condition, the correlation functions of fermions reproduce those of the SYK model with a temperature dependent coupling constant in the large N limit. We also discuss a supersymmetric generalization of our quantum mechanical model. As a byproduct, we develop an efficient way of estimating the large N behavior of correlators in the SYK model.

\u201c1kF \u201d singularities and finite density ABJM theory at strong coupling

We study non-analytic behavior in the static charge susceptibility in finite density states of the ABJM theory using its holographic dual. Emphasis is placed on a particular state characterized by vanishing entropy density at zero temperature, and Fermi surface-like singularities in various fermionic correlation functions. The susceptibility exhibits branch points in the complex momentum plane, with a real part quantitatively very similar to the location of the Fermi surface singularities.

Continuous-time quantum Monte Carlo for fermion-boson lattice models: Improved bosonic estimators and application to the Holstein model

We extend the continuous-time interaction-expansion quantum Monte Carlo method with respect to measuring observables for fermion-boson lattice models. Using generating functionals, we express expectation values involving boson operators, which are not directly accessible because simulations are done in terms of a purely fermionic action, as integrals over fermionic correlation functions. We also demonstrate that certain observables can be inferred directly from the vertex distribution, and present efficient estimators for the total energy and the phonon propagator of the Holstein model. Furthermore, we generalize the covariance estimator of the fidelity susceptibility, an unbiased diagnostic for phase transitions, to the case of retarded interactions. The new estimators are applied to half-filled spinless and spinful Holstein models in one dimension. The observed renormalization of the phonon mode across the Peierls transition in the spinless model suggests a soft-mode transition in the adiabatic regime. The critical point is associated with a minimum in the phonon kinetic energy and a maximum in the fidelity susceptibility.

Signatures and conditions for phase band crossings in periodically driven integrable systems

We present generic conditions for phase band crossings for a class of periodically driven integrable systems represented by free fermionic models subjected to arbitrary periodic drive protocols characterized by a frequency $\omega_D$. These models provide a representation for the Ising and $XY$ models in $d=1$, the Kitaev model in $d=2$, several kinds of superconductors, and Dirac fermions in graphene and atop topological insulator surfaces. Our results demonstrate that the presence of a critical point/region in the system Hamiltonian (which is traversed at a finite rate during the dynamics) may change the conditions for phase band crossings that occur at the critical modes. We also show that for $d>1$, phase band crossings leave their imprint on the equal-time off-diagonal fermionic correlation functions of these models; the Fourier transforms of such correlation functions, $F_{\vec k_0}( \omega_0)$, have maxima and minima at specific frequencies which can be directly related to $\omega_D$ and the time at which the phase bands cross at $\vec k = \vec k_0$. We discuss the significance of our results in the contexts of generic Hamiltonians with $N>2$ phase bands and the underlying symmetry of the driven Hamiltonian.

High-order correlation of chaotic bosons and fermions

We theoretically study the high-order correlation functions of chaotic bosons and fermions. Based on the different parity of the Stirling number, the products of the first-order correlation functions are well classified and employed to represent the high-order correlation function. The correlation of bosons conduces a bunching effect, which will be enhanced as order $N$ increases. Different from bosons, the anticommutation relation of fermions leads to the parity of the Stirling number, which thereby results in a mixture of bunching and antibunching behaviors in high-order correlation. By further investigating third-order ghost diffraction and ghost imaging, the differences between the high-order correlations of bosons and fermions are discussed in detail. A larger $N$ will dramatically improve the ghost image quality for bosons, but a good strategy should be carefully chosen for the fermionic ghost imaging process due to its complex correlation components.

Nature of the many-body excitations in a quantum wire: Theory and experiment

The natural excitations of an interacting one-dimensional system at low energy are hydrodynamic modes of Luttinger liquid, protected by the Lorentz invariance of the linear dispersion. We show that beyond low energies, where quadratic dispersion reduces the symmetry to Galilean, the main character of the many-body excitations changes into a hierarchy: calculations of dynamic correlation functions for fermions (without spin) show that the spectral weights of the excitations are proportional to powers of $\mathcal{R}^{2}/L^{2}$, where $\mathcal{R}$ is a length-scale related to interactions and $L$ is the system length. Thus only small numbers of excitations carry the principal spectral power in representative regions on the energy-momentum planes. We have analysed the spectral function in detail and have shown that the first-level (strongest) excitations form a mode with parabolic dispersion, like that of a renormalised single particle. The second-level excitations produce a singular power-law line shape to the first-level mode and multiple power-laws at the spectral edge. We have illustrated crossover to Luttinger liquid at low energy by calculating the local density of state through all energy scales: from linear to non-linear, and to above the chemical potential energies. In order to test this model, we have carried out experiments to measure momentum-resolved tunnelling of electrons (fermions with spin) from/to a wire formed within a GaAs heterostructure. We observe well-resolved spin-charge separation at low energy with appreciable interaction strength and only a parabolic dispersion of the first-level mode at higher energies. We find structure resembling the second-level excitations, which dies away rapidly at high momentum in line with the theoretical predictions here.

Parametrically coupled fermionic oscillators: Correlation functions and phase-space description

A fermionic analog of a parametric amplifier is used to describe the joint quantum state of the two interacting fermionic modes. Based on a two-mode generalization of the time-dependent density operator, time evolution of the fermionic density operator is determined in terms of its two-mode Wigner and $P$ function. It is shown that the equation of motion of the Wigner function corresponds to a fermionic analog of Liouville's equation. The equilibrium density operator for fermionic fields developed by Cahill and Glauber is thus extended to a dynamical context to show that the mathematical structures of both the correlation functions and the weight factors closely resemble their bosonic counterpart. It has been shown that the fermionic correlation functions are marked by a characteristic upper bound due to Fermi statistics, which can be verified in the matter wave counterpart of photon down-conversion experiments.

Note on Rome-Southampton renormalization with smeared gauge fields

We have calculated continuum-limit step scaling functions of bilinear and four-fermion operators renormalized in a Rome-Southampton scheme using various smearing prescriptions for the gauge field. Also, for the first time, we have calculated nonperturbative anomalous dimensions of operators renormalized in a Rome-Southampton scheme. The effect of such smearing first enters connected fermionic correlation functions via radiative corrections. We use off-shell renormalization as a probe, and observe that the upper edge of the Rome-Southampton window is reduced by link smearing. This can be interpreted as arising due to the fermions decoupling from the high-momentum gluons and we observe that the running of operators with the scale at large lattice momenta shows enhanced lattice artifacts. We find that the effect is greater for HEX smearing than for Stout smearing, but that in both cases additional care must be taken when using off-shell renormalization with smeared gauge fields compared to thin-link simulations.

Holographic models for undoped Weyl semimetals

Abstract We continue our recently proposed holographic description of single-particle correlation functions for four-dimensional chiral fermions with Lifshitz scaling at zero chemical potential, paying particular attention to the dynamical exponent z = 2. We present new results for the spectral densities and dispersion relations at non-zero momenta and temperature. In contrast to the relativistic case with z = 1, we find the existence of a quantum phase transition from a non-Fermi liquid into a Fermi liquid in which two Fermi surfaces spontaneously form, even at zero chemical potential. Our findings show that the boundary system behaves like an undoped Weyl semimetal.

Spin-foam fermions: PCT symmetry, Dirac determinant and correlation functions

We discuss fermion coupling in the framework of spin-foam quantum gravity. We analyze the gravity-fermion spin-foam model and its fermion correlation functions. We show that there is a spin-foam analogue of PCT symmetry for the fermion fields on a spin-foam model, which is proved for spin-foam fermion correlation functions. We compute the determinant of the Dirac operator for the fermions, where two presentations of the Dirac determinant are given in terms of diagram expansions. We compute the fermion correlation functions and show that they can be given by Feynman diagrams on the spin-foams, where the Feynman propagators can be represented by a discretized path integral of a world-line action along the edges of the underlying 2-complex.

Holography and ARPES sum-rules

Abstract We study correlation functions of elementary fermions in strongly interacting field theories using the AdS/CFT correspondence. This correspondence generically associates bulk fields to composite operators in field theory. We modify the holographic prescription in order to obtain correlators that correspond to fermonic single-particle excitations by introducing a dynamical fermionic source localized on a UV brane in a holographic background. We work out the conditions when these correlators obey the zeroth frequency sum-rule satisfied by angle-resolved photo-emission spectroscopy (ARPES) and are thus directly relevant to the AdS/CMT correspondence. To illustrate our techniques, we study field theories at zero chemical potential with an arbitrary dynamical exponent z, i.e., the Lifshitz invariant conformal field theories, including the usual relativistic case z = 1.

Many-particle correlations in a non-equilibrium Luttinger liquid

We develop an operator-based approach to the problem of a Luttinger liquid conductor in anon-equilibrium stationary state. We show that the coherent-state many-bodyfermionic density matrix and all fermionic correlation functions out of equilibrium aregiven by one-dimensional functional determinants of the Fredholm type. Thus,the model constitutes a remarkable example of a many-body problem where allthe correlation functions can be evaluated exactly. On the basis of the generalformalism, we investigate four-point correlation functions of the fermions comingout of the Luttinger liquid wire. The correlations obtained among the fermionicdistribution functions represent the combined effect of interaction and non-equilibriumconditions.