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arXiv:1310.8029 · Physical Review E

Linked-cluster expansion for the Green's function of the infinite- U Hubbard model

Ehsan Khatami, Edward Perepelitsky, Marcos Rigol, B. Sriram Shastry

Published Jun 2, 2014• 10 citations
arXivHTMLPDF

Abstract

We implement a highly efficient strong-coupling expansion for the Green's function of the Hubbard model. In the limit of extreme correlations, where the onsite interaction is infinite, the evaluation of diagrams simplifies dramatically enabling us to carry out the expansion to the eighth order in powers of the hopping amplitude. We compute the finite-temperature Green's function analytically in the momentum and Matsubara frequency space as a function of the electron density. Employing Padé approximations, we study the equation of state, Kelvin thermopower, momentum distribution function, quasiparticle fraction, and quasiparticle lifetime of the system at temperatures lower than, or of the order of, the hopping amplitude. We also discuss several different approaches for obtaining the spectral functions through analytic continuation of the imaginary frequency Green's function, and show results for the system near half filling. We benchmark our results for the equation of state against those obtained from a numerical linked-cluster expansion carried out to the eleventh order.

DOI
10.1103/physreve.89.063301

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We implement a highly efficient strong-coupling expansion for the Green's function of the Hubbard model. In the limit of extreme correlations, where the onsite interaction is infinite, the evaluation of diagrams simplifies dramatically enabling us to carry out the expansion to the eighth order in powers of the hopping amplitude. We compute the finite-temperature Green's function analytically in the momentum and Matsubara frequency space as a function of the electron density. Employing Padé approximations, we study the equation of state, Kelvin thermopower, momentum distribution function, quasiparticle fraction, and quasiparticle lifetime of the system at temperatures lower than, or of the order of, the hopping amplitude. We also discuss several different approaches for obtaining the spectral functions through analytic continuation of the imaginary frequency Green's function, and show results for the system near half filling. We benchmark our results for the equation of state against those obtained from a numerical linked-cluster expansion carried out to the eleventh order.
We employ the numerical linked-cluster expansion to study finite-temperature properties of the uniform cubic lattice Hubbard model in the thermodynamic limit for a wide range of interaction strengths and densities. We carry out the expansion to the 9th order and find that the convergence of the series extends to lower temperatures as the strength of the interaction increases, giving us access to regions of the parameter space that are difficult to reach by most other numerical methods. We study the precise trends in the specific heat, the double occupancy, and magnetic correlations at temperatures as low as 0.2 of the hopping amplitude in the strong-coupling regime. We show that in this regime, accurate estimates for transition temperatures to the N\'eel ordered phase, in agreement with the predicted asymptotic behavior, can be deduced from the low-temperature magnetic structure factor. We also find evidence for possible instability to the magnetically ordered phase away from, but close to, half filling. Our results have important implications for parametrizing fermionic systems in optical lattice experiments and for benchmarking other numerical methods.
We develop a diagrammatic approach for calculating the high temperature expansion of dynamic correlation functions, such as the electron Green's function and the time-dependent density-density and spin-spin correlation functions, for the infinite-U Hubbard Model with any number of spin species. The formalism relies on the use of restricted lattice sums, in which distinct vertices of the diagram represent distinct sites on the lattice. We derive a new formula for the restricted lattice sum of a disconnected diagram consisting of several connected components, and use it to prove the linked cluster theorem with respect to "generalized connected diagrams", formed by overlapping the original connected components on the lattice. This enables us to express all quantities as a sum over these generalized connected diagrams. We compute the Green's function to 4th order in βt for the case of m spin species on a d-dimensional hypercube by hand. We take the m\to\infty limit, enabling us to obtain expressions for the Dyson-Mori self-energy to 4th order in βt for the case of an infinite number of spin species. This may have connections to slave boson techniques used for the study of this model. Our approach is computationally more efficient than any used previously for the calculation of the high temperature expansion of dynamic correlation functions, and high order results for both the Green's function and the time-dependent density-density and spin-spin correlation functions shall be presented in a separate paper [25].
A strong-coupling expansion for the Green's functions, self-energies, and correlation functions of the Bose-Hubbard model is developed. We illustrate the general formalism, which includes all possible (normal-phase) inhomogeneous effects in the formalism, such as disorder or a trap potential, as well as effects of thermal excitations. The expansion is then employed to calculate the momentum distribution of the bosons in the Mott phase for an infinite homogeneous periodic system at zero temperature through third order in the hopping. By using scaling theory for the critical behavior at zero momentum and at the critical value of the hopping for the Mott insulator--to--superfluid transition along with a generalization of the random-phase-approximation-like form for the momentum distribution, we are able to extrapolate the series to infinite order and produce very accurate quantitative results for the momentum distribution in a simple functional form for one, two, and three dimensions. The accuracy is better in higher dimensions and is on the order of a few percent relative error everywhere except close to the critical value of the hopping divided by the on-site repulsion. In addition, we find simple phenomenological expressions for the Mott-phase lobes in two and three dimensions which are much more accurate than the truncated strong-coupling expansions and any other analytic approximation we are aware of. The strong-coupling expansions and scaling-theory results are benchmarked against numerically exact quantum Monte Carlo simulations in two and three dimensions and against density-matrix renormalization-group calculations in one dimension. These analytic expressions will be useful for quick comparison of experimental results to theory and in many cases can bypass the need for expensive numerical simulations.
The equation for the electron Green's function of the fermionic Hubbard model, derived using the strong coupling diagram technique, is solved self-consistently for the near-neighbor form of the kinetic energy and for half-filling. In this case the Mott transition occurs at the Hubbard repulsion Uc ≈ 6.96t, where t is the hopping constant. The calculated spectral functions, density of states (DOS) and momentum distribution are compared with results of Monte Carlo simulations. A satisfactory agreement was found for U > Uc and for temperatures, at which magnetic ordering and spin correlations are suppressed. For U < Uc and lower temperatures the theory describes qualitatively correctly the positions and widths of spectral continua, variations of spectral shapes and occupation numbers with changing wave vector and repulsion. The locations of spectral maxima turn out to be close to the positions of δ-function peaks in the Hubbard-I approximation.
A strong-coupling series expansion for the Green's function and the extremely correlated Fermi liquid (ECFL) theory are used to calculate the moments of the electronic spectral functions of the infinite-UUU Hubbard model. Results from these two complementary methods agree very well at both low densities, where the ECFL solution is the most accurate, and at high to intermediate temperatures, where the series converge. We find that a modified first moment, which underestimates the contributions from the occupied states and is accessible in the series through the time-dependent Green's function, best describes the peak location of the spectral function in the strongly correlated regime. This is examined by the ECFL results at low temperatures, where it is shown that the spectral function is largely skewed towards the occupied states.
We introduce a method, numerically exact in principle, for computing the momentum- and frequency-resolved single-particle Green's function of a polaron at finite temperature. The method, which we refer to as the finite-temperature Green's function cluster expansion, combines two ingredients: the generalized Green's function cluster expansion, a numerically exact extension of the momentum average family of methods that solves the polaron problem at zero temperature through a hierarchy of equations of motion for restricted phonon cloud configurations; and the thermofield double formalism, which maps the thermal trace onto a pure-state expectation value over a doubled Hilbert space. The resulting equations of motion have the same algebraic structure as those of the multi-boson zero-temperature theory, with the temperature entering through a Bogoliubov-type mixing angle that controls the coupling to a set of fictitious bath bosons. We implement the method in our open-source software package and benchmark it on the one-dimensional Holstein polaron, computing spectral functions, dispersions, lifetimes, and effective masses across coupling regimes and temperatures up to T/Ω∼1T/Ω\sim 1T/Ω∼1. Where finite-temperature density matrix renormalization group results are available, we find quantitative agreement at affordable computational cost. The method recovers momentum-resolved spectra directly in frequency space, with no time evolution or analytic continuation. We also discuss the practical costs of the approach. In particular, since the doubled phonon Hilbert space has a non-trivial configuration structure in which real and fictitious clouds compete, convergence in the corresponding cloud parameters requires care.