Density-density Correlation Function Research Articles

Fourier-Matsubara series expansion for imaginary-time correlation functions.

A Fourier-Matsubara series expansion is derived for imaginary-time correlation functions that constitutes the imaginary-time generalization of the infinite Matsubara series for equal-time correlation functions. The expansion is consistent with all known exact properties of imaginary-time correlation functions and opens up new avenues for the utilization of quantum Monte Carlo simulation data. Moreover, the expansion drastically simplifies the computation of imaginary-time density-density correlation functions with the finite temperature version of the self-consistent dielectric formalism. Its existence underscores the utility of imaginary-time as a complementary domain for many-body physics.

Critical exponents and fluctuations at BEC in a 2D harmonically trapped ideal gas

The critical properties displayed by an ideal 2D Bose gas trapped in a harmonic potential are determined and characterized in an exact numerical fashion. Beyond thermodynamics, addressed in terms of the global pressure and volume which are the appropriate variables of a fluid confined in a non-uniform harmonic potential, the density-density correlation function is also calculated and the corresponding correlation length is found. Evaluation of all these quantities as Bose–Einstein condensation (BEC) is approached manifest its critical continuous phase transition character. The divergence of the correlation length as the critical temperature is reached, unveils the expected spatial scale invariance proper of a critical transition. The logarithmic singularities of this transition are traced back to the non-analytic behavior of the thermodynamic variables at vanishing chemical potential, which is the onset of BEC. The critical exponents associated with the ideal BEC transition in the 2D inhomogeneous fluid reveals its own universality class.

Fermionic physics from abinitio path integral Monte Carlo simulations of fictitious identical particles.

The abinitio path integral Monte Carlo (PIMC) method is one of the most successful methods in statistical physics, quantum chemistry and related fields, but its application to quantum degenerate Fermi systems is severely hampered by an exponential computational bottleneck: the notorious fermion signproblem. Very recently, Xiong and Xiong [J. Chem. Phys. 157, 094112 (2022)] have suggested to partially circumvent the signproblem by carrying out simulations of fictitious systems guided by an interpolating continuous variable ξ ∈ [-1, 1], with the physical Fermi- and Bose-statistics corresponding to ξ = -1 and ξ = 1. It has been proposed that information about the fermionic limit might be obtained by calculations within the bosonic sector ξ > 0 combined with an extrapolation throughout the fermionic sector ξ < 0, essentially bypassing the signproblem. Here, we show how the inclusion of the artificial parameter ξ can be interpreted as an effective penalty on the formation of permutation cycles in the PIMC simulation. We demonstrate that the proposed extrapolation method breaks down for moderate to high quantum degeneracy. Instead, the method constitutes a valuable tool for the description of large Fermi-systems of weak quantum degeneracy. This is demonstrated for electrons in a 2D harmonic trap and for the uniform electron gas (UEG), where we find excellent agreement (∼0.5%) with exact configuration PIMC results in the high-density regime while attaining a speed-up exceeding 11 orders of magnitude. Finally, we extend the idea beyond the energy and analyze the radial density distribution (2D trap), as well as the static structure factor and imaginary-time density-density correlation function (UEG).

Analysing the dynamic structure of warm dense matter in the imaginary-time domain: theoretical models and simulations.

Rigorous diagnostics of experiments with warm dense matter are notoriously difficult. A key method is X-ray Thomson scattering (XRTS), but the interpretation of XRTS measurements is usually based on theoretical models that entail various approximations. Recently, Dornheim et al. [Nat. Commun. 13, 7911 (2022)] introduced a new framework for temperature diagnostics of XRTS experiments that is based on imaginary-time correlation functions. On the one hand, switching from the frequency to the imaginary-time domain gives one direct access to a number of physical properties, which facilitates the extraction of the temperature of arbitrarily complex materials without relying on any models or approximations. On the other hand, the bulk of theoretical work in dynamic quantum many-body theory is devoted to the frequency domain, and, to the best of our knowledge, the manifestation of physics properties within the imaginary-time density-density correlation function (ITCF) remains poorly understood. In the present work, we aim to fill this gap by introducing a simple, semi-analytical model for the imaginary-time dependence of two-body correlations within the framework of imaginary-time path integrals. As a practical example, we compare our new model to extensive ab initio path integral Monte Carlo results for the ITCF of a uniform electron gas, and find excellent agreement over a broad range of wavenumbers, densities and temperatures. This article is part of the theme issue 'Dynamic and transient processes in warm dense matter'.

Accurate localization of Kosterlitz-Thouless-type quantum phase transitions for one-dimensional spinless fermions

We investigate the charge-density wave (CDW) transition for one-dimensional spinless fermions at half band-filling with nearest-neighbor electron transfer amplitude $t$ and interaction $V$. The model is equivalent to the anisotropic XXZ Heisenberg model for which the Bethe Ansatz provides an exact solution. For $V> V_{\rm c}= 2t$, the CDW order parameter and the single-particle gap are finite but exponentially small, as is characteristic for a Kosterlitz-Thouless transition. It is notoriously difficult to locate such infinite-order phase transitions in the phase diagram using approximate analytical and numerical approaches. Second-order Hartree-Fock theory is qualitatively applicable for all interaction strengths, and predicts the CDW transition to occur at $V_{\rm c,2}^{(2)}\approx 1.5t$. Second-order Hartree Fock theory is almost variational because the density of quasi-particle excitations is small. We apply the density-matrix renormalization group (DMRG) for periodic boundary conditions for system sizes up to 514 sites which permits a reliable extrapolation of all physical quantities to the thermodynamic limit, apart from the critical region. We investigate the ground-state energy, the gap, the order parameter, the momentum distribution, the quasi-particle density, and the density-density correlation function to locate $V_{\rm c}$ from the DMRG data. Tracing the breakdown of the Luttinger liquid and the peak in the quasi-particle density at the band edge permits us to reproduce $V_{\rm c}$ with an accuracy of one percent.

Periodically driven Rydberg chains with staggered detuning

We study the stroboscopic dynamics of a periodically driven finite Rydberg chain with staggered ($\Delta$) and time-dependent uniform ($\lambda(t)$) detuning terms using exact diagonalization (ED). We show that at intermediate drive frequencies ($\omega_D$), the presence of a finite $\Delta$ results in violation of the eigenstate thermalization hypothesis (ETH) via clustering of Floquet eigenstates. Such clustering is lost at special commensurate drive frequencies for which $\hbar \omega_d=n \Delta$ ($n \in Z$) leading to restoration of ergodicity. The violation of ETH in these driven finite-sized chains is also evident from the dynamical freezing displayed by the density-density correlation function at specific $\omega_D$. Such a correlator exhibits stable oscillations with perfect revivals when driven close to the freezing frequencies for initial all spin-down ($|0\rangle$) or Neel ($|{\mathbb Z}_2\rangle$, with up-spins on even sites) states. The amplitudes of these oscillations vanish at the freezing frequencies and reduces upon increasing $\Delta$; their frequencies, however, remains pinned to $\Delta/\hbar$ in the large $\Delta$ limit. In contrast, for the $|{\bar {\mathbb Z}_2}\rangle$ (time-reversed partner of $|{\mathbb Z}_2\rangle$) initial state, we find complete absence of such oscillations leading to freezing for a range of $\omega_D$; this range increases with $\Delta$. We also study the properties of quantum many-body scars in the Floquet spectrum of the model as a function of $\Delta$ and show the existence of novel mid-spectrum scars at large $\Delta$. We supplement our numerical results with those from an analytic Floquet Hamiltonian computed using Floquet perturbation theory (FPT) and also provide a semi-analytic computation of the quantum scar states within a forward scattering approximation (FSA).

One- and two-particle properties of the weakly interacting two-dimensional Hubbard model in proximity to the van Hove singularity

We study the weak-coupling limit of the $t-t^\prime-U$ Hubbard model on a two-dimensional square lattice using a direct perturbative approach. Aided by symbolic computational tools, we compute the longitudinal density-density correlation functions in the $\chi_{\uparrow \uparrow}$ and $\chi_{\uparrow \downarrow}$ basis from which we can obtain the dynamical spin and charge susceptibilities at arbitrary doping and temperature. We find that for non-zero $t^\prime$, the zero frequency commensurate $\mathbf{q} = (\pi, \pi)$ spin and charge excitations are each strongest at different densities and we observe a clear behavioral change that appears tied to the van Hove singularity of the non-interacting dispersion upon which the perturbative expansion is built. We find a strongly reduced compressibility in the vicinity of the van Hove singularity as well as a behavioral change in the double occupancy. For finite $t^\prime$, the observed van Hove singularity occurs away from half-filling leading us to conclude that that this reduction in compressibility is distinct from Mott insulating physics that one expects in the strong-coupling regime. We compute the full dynamical spin and charge excitations and observe distinct structure for electron and hole doped scenarios in agreement with experiments on cuprate materials. Finally, we observe a peculiar splitting in spin and charge excitations in the vicinity of the van Hove singularity, the origin of which is traced to a splitting near the bottom of the band.

Short-range and long-range correlations in driven dense colloidal mixtures in narrow pores.

The system of a driven dense colloid mixture in a tube with diameter comparable to particle size is modeled by a generalization of the asymmetric simple exclusion process (ASEP) model. The generalization goes in two directions: relaxing the exclusion constraint by allowing several (but few) particles on a site and by considering two species of particles, which differ in size and transport coefficients. We calculate the nearest-neighbor correlations using a variant of the Kirkwood approximation and show by comparison with numerical simulations that the approximation provides quite accurate results. However, for long-range correlations, we show that the Kirkwood approximation is useless, as it predicts exponential decay of the density-density correlation function with distance, while simulation data indicate that the decay is algebraic. For the one-component system, we show that the decay is governed by a power law with universal exponent close to 2. In the two-component system, the correlation function behaves in a more complicated manner: Its sign oscillates but the envelope decays again very slowly and the decay is compatible with a power law with an exponent somewhat lower than 2. Therefore, our generalization of the ASEP belongs to a different universality class from the ensemble of generalized ASEP models which are mappable to zero-range processes.

Decoding polymer self-dynamics using a two-step approach.

The self-correlation function and corresponding self-intermediate scattering function in Fourier space are important quantities for describing the molecular motions of liquids. This work draws attention to a largely overlooked issue concerning the analysis of these space-time density-density correlation functions of polymers. We show that the interpretation of non-Gaussian behavior of polymers is generally complicated by intrachain averaging of distinct self-dynamics of different segments. By the very nature of the mathematics involved, the averaging process not only conceals critical dynamical information, but also contributes to the observed non-Gaussian dynamics. To fully expose this issue and provide a thorough benchmark of polymer self-dynamics, we perform analyses of coarse-grained molecular dynamics simulations of linear and ring polymer melts as well as several theoretical models using a "two-step" approach, where interchain and intrachain averagings of segmental self-dynamics are separated. While past investigations primarily focused on the average behavior, our results indicate that a more nuanced approach to polymer self-dynamics is clearly required.

Monitored open fermion dynamics: Exploring the interplay of measurement, decoherence, and free Hamiltonian evolution

The interplay of unitary evolution and local measurements in many-body systems gives rise to a stochastic state evolution and to measurement-induced phase transitions in the pure state entanglement. In realistic settings, however, this dynamics may be spoiled by decoherence, e.g., dephasing, due to coupling to an environment or measurement imperfections. We investigate the impact of dephasing and the inevitable evolution into a non-Gaussian, mixed state, on the dynamics of monitored fermions. We approach it from three complementary perspectives: (i) the exact solution of the conditional master equation for small systems, (ii) quantum trajectory simulations of Gaussian states for large systems, and (iii) a renormalization group analysis of a bosonic replica field theory. For weak dephasing, constant monitoring preserves a weakly mixed state, which displays a robust measurement-induced phase transition between a critical and a pinned phase, as in the decoherence-free case. At strong dephasing, we observe the emergence of a new scale describing an effective temperature, which is accompanied with an increased mixedness of the fermion density matrix. Remarkably, observables such as density-density correlation functions or the subsystem parity still display scale invariant behavior even in this strongly mixed phase. We interpret this as a signature of gapless, classical diffusion, which is stabilized by the balanced interplay of Hamiltonian dynamics, measurements, and decoherence.

Measuring postquench entanglement entropy through density correlations

Following a sudden change of interactions in an integrable system of one-dimensional fermions, we analyze the dependence of the static structure factor on the observation time after the quantum quench. At small waiting times after the quench, we map the system to noninteracting bosons such that we are able to extract their occupation numbers from the Fourier transform of the density-density correlation function, and use these to compute a bosonic entropy from a diagonal ensemble. By comparing this bosonic entropy with the asymptotic steady-state entanglement entropy per fermion computed with exact diagonalization, we find excellent agreement.Received 8 December 2021Revised 17 March 2022Accepted 22 March 2022DOI:https://doi.org/10.1103/PhysRevResearch.4.L022023Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBosonizationCharge density wavesEntanglement entropyQuantum quenchTechniquesExact diagonalizationLuttinger liquid modelCondensed Matter, Materials & Applied PhysicsQuantum InformationStatistical Physics

Density correlation functions and the spatial structure of the two-dimensional BEC-BCS crossover

The spatial structure of a two-dimensional homogeneous mixture of fermionic atoms in two hyperfine states is analyzed throughout the BEC-BCS crossover. Within the BCS-Leggett mean-field model we consider three functions: the pair wave function and the density-density correlation functions between atoms of the same and of different hyperfine states. For the correlation functions we derive analytical expressions which allow to unveil the rich spatial structure of the superfluid. Mainly, we are able to study the large-distance behavior of the three functions, which exhibits an exponential decay and a well-defined oscillatory behavior. We report closed-form expressions for the correlation lengths and mean pair radius. Differences and similarities emerge when comparing with the three dimensional case. Particularly, we find an expression for the large-distance correlation length, in terms of the chemical potential and the gap, valid in two and three dimensions, but whose dependence on the corresponding scattering lengths differ significantly.

Tensor-network study of correlation-spreading dynamics in the two-dimensional Bose-Hubbard model

Recent developments in analog quantum simulators based on cold atoms and trapped ions call for cross-validating the accuracy of quantum-simulation experiments with use of quantitative numerical methods; however, it is particularly challenging for dynamics of systems with more than one spatial dimension. Here we demonstrate that a tensor-network method running on classical computers is useful for this purpose. We specifically analyze real-time dynamics of the two-dimensional Bose-Hubbard model after a sudden quench starting from the Mott insulator by means of the tensor-network method based on infinite projected entangled pair states. Calculated single-particle correlation functions are found to be in good agreement with a recent experiment. By estimating the phase and group velocities from the single-particle and density-density correlation functions, we predict how these velocities vary in the moderate interaction region, which serves as a quantitative benchmark for future experiments and numerical simulations.

Measurement-Induced Dark State Phase Transitions in Long-Ranged Fermion Systems.

We identify an unconventional algebraic scaling phase in the quantum dynamics of long-range hopping, free fermions, which are exposed to continuous local measurements. The algebraic phase occurs for hopping decay exponents 1<p≲3/2, and features an algebraic entanglement entropy growth, and a slow algebraic decay of the density-density correlation function, both with a fractional exponent. It is separated from a critical phase with logarithmic entanglement growth at small, and an area law phase with constant entanglement entropy at large monitoring rates. A perturbative renormalization group analysis predicts that the transitions to the long-range phase correspond to an unconventional, modified sine-Gordon theory. Exact numerical simulations of the monitored wave functions are in excellent agreement with an analytical replica field theory approach, which confirms the view of the measurement-induced phase transition as a quantum phase transition in the dark state of an effective, non-Hermitian Hamiltonian.

Liquid-Crystal Ordering and Microphase Separation in the Lamellar Phase of Rod-Coil-Rod Triblock Copolymers. Molecular Theory and Computer Simulations.

A molecular model of the orientationally ordered lamellar phase exhibited by asymmetric rod-coil-rod triblock copolymers has been developed using the density-functional approach and generalizing the molecular-statistical theory of rod-coil diblock copolymers. An approximate expression for the free energy of the lamellar phase has been obtained in terms of the direct correlation functions of the system, the Flory-Huggins parameter and the Maier-Saupe orientational interaction potential between rods. A detailed derivation of several rod-rod and rod-coil density-density correlation functions required to evaluate the free energy is presented. The orientational and translational order parameters of rod and coil segments depending on the temperature and triblock asymmetry have been calculated numerically by direct minimization of the free energy. Different structure and ordering of the lamellar phase at high and low values of the triblock asymmetry is revealed and analyzed in detail. Asymmetric rod-coil-rod triblock copolymers have been simulated using the method of dissipative particle dynamics in the broad range of the Flory-Huggins parameter and for several values of the triblock asymmetry. It has been found that the lamellar phase appears to be the most stable one at strong segregation. The density distribution of the coil segments and the segments of the two different rods have been determined for different values of the segregation strength. The simulations confirm the existence of a weakly ordered lamellar phase predicted by the density-functional theory, in which the short rods separate from the long ones and are characterized by weak positional ordering.

Nonlinear sigma model with particle-hole asymmetry for the disordered two-dimensional electron gas

The non-linear sigma model is a well-established theoretical tool for studies of transport and thermodynamics in disordered electronic systems. The conventional sigma model approach for interacting systems does not account for particle-hole asymmetry. It is therefore not suited for studying quantities that are sensitive to this effect such as the thermoelectric transport coefficient. Here, we derive a minimal extension of the Keldysh non-linear sigma model tailored for two-dimensional interacting systems. We argue that this model can be used to systematically study the combined effect of interactions and disorder on thermoelectric transport. As a first step in this direction, we use the model to analyze the structure of the heat density-density correlation function and calculate interaction corrections to its static part. The calculation of interaction corrections to the dynamical part of the correlation function and the thermodynamic transport coefficient is left for future work.

Correlations between a Hawking particle and its partner in a 1+1D Bose-Einstein condensate analog black hole

The Fourier transform of the density-density correlation function in a Bose-Einstein condensate (BEC) analog black hole is a useful tool to investigate correlations between the Hawking particles and their partners. It can be expressed in terms of $\langle ^{\text{out}}\hat{a}_{\text{up}}^{\text{ext}}\ ^{\text{out}}\hat{a}_{\text{up}}^{\text{int}} \rangle$, where $^{\text{out}}\hat{a}_{\text{up}}^{\text{ext}}$ is the annihilation operator for the Hawking particle and $^{\text{out}}\hat{a}_{\text{up}}^{\text{int}} $ is the corresponding one for the partner. This basic quantity is calculated for three different models for the BEC flow. It is shown that in each model the inclusion of the effective potential in the mode equations makes a significant difference. Furthermore, particle production induced by this effective potential in the interior of the black hole is studied for each model and shown to be nonthermal. An interesting peak that is related to the particle production and is present in some models is discussed.

Light extinction and scattering from aggregates composed of submicron particles

The influence of the internal structure of inhomogeneous particles on their radiative properties is an open issue repeatedly questioned in many fields of science and technology. The importance of a refined description of the particle composition and structure, going beyond mean-field approximations, is generally recognized. Here, we focus on describing internal inhomogeneities from a statistical point of view. We introduce an analytical description based on the two-point density-density correlation function, or the corresponding static structure factor, to calculate the extinction cross sections. The model agrees with numerical predictions and is validated experimentally with colloidal aggregates in the 0.3–6 μm size range, which serve as an inhomogeneous model system that can be characterized enough to work without any free parameters. The model can be tightly compared to measurements with single particle extinction and scattering and spectrophotometry and suggests a simple behavior for 90° scattering from fractal aggregates as a function of extinction, which is also confirmed experimentally and numerically. We also discuss the case of absorbing particles and report the experimental results for water suspensions of black carbon for both the forward and 90° scattering properties. In this case, the total scattering and the extinction cross sections determine the single scattering albedo, which agrees with numerical simulations. The three parameters necessary to feed radiative transfer models, namely, extinction, asymmetry parameter, and single scattering albedo, can all be set by the analytical model, with explicit dependence on a few parameters. Results are applicable to radiative transfer problems in climate, paleoclimate, star and planetary formation, and nanoparticle optical characterization for science and industry, including the intercomparison of different optical methods such as those adopted by ISO standards.

Acoustic analog of Hawking radiation in quantized circular superflows of Bose-Einstein condensates

We propose emulation of Hawking radiation (HR) by means of acoustic excitations propagating on top of persistent current in an atomic Bose-Einstein condensate (BEC) loaded in an annular confining potential. The setting is initially created as a spatially uniform one, and then switches into a nonuniform configuration, while maintaining uniform BEC density. The eventual setting admits the realization of sonic black and white event horizons with different slopes of the local sound speed. A smooth slope near the white-hole horizon suppresses instabilities in the supersonic region. It is found that tongue-shaped patterns of the density-density correlation function, which represent the acoustic analog of HR, are strongly affected by the radius of the ring-shaped configuration and number of discrete acoustic modes admitted by it. There is a minimum radius that enables the emulation of HR. We also briefly discuss a possible similarity of properties of the matter-wave sonic black holes to the known puzzle of the stability of Planck-scale primordial black holes in quantum gravity.

Density-density correlation function of strongly inhomogeneous Luttinger liquids

In this work, we show in pedagogical detail that the most singular contributions to the slow part of the asymptotic density-density correlation function of Luttinger liquids with fermions interacting mutually with only short-range forward scattering and also with localised scalar static impurities (where backward scattering takes place) has a compact analytical expression in terms of simple functions that have second order poles and involve only the scale-independent bare transmission and reflection coefficients. This proof uses conventional fermionic perturbation theory resummed to all orders, together with the idea that for such systems, the (connected) moments of the density operator all vanish beyond the second order—the odd ones vanish identically and the higher order even moments are less singular than the second order moment which is the only one included. This important result is the crucial input to the recently introduced ‘Non-Chiral Bosonization Technique’ (NCBT) to study such systems. The results of NCBT cannot be easily compared with the results obtained using conventional bosonization as the former only extracts the most singular parts of the correlation functions albeit for arbitrary impurity strengths and mutual interactions. The latter, ambitiously attempts to study all the parts of the asymptotic correlation functions and is thereby unable to find simple analytical expressions and is forced to operate in the vicinity of the homogeneous system or the half line (the opposite extreme). For a fully homogeneous system or its antithesis viz. the half-line, all the higher order connected moments of the density vanish identically which means the results of chiral bosonization and NCBT ought to be the same and indeed they are.