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Magnetic incommensurability and fluctuating charge density waves in the repulsive Hubbard model

A. Sherman, Michael Schreiber

Published Aug 3, 2013• 0 citations
OpenAlex

Abstract

Abstract. Magnetic and charge susceptibilities of the two-dimensional repulsive Hubbard model are investigated applying a strong coupling diagram technique in which the expansion in powers of the hopping constants is used. For small lattices and high temperatures results are in agreement with Monte Carlo simulations. With the departure from half-filling x the low-frequency magnetic susceptibility becomes incommensurate and the incommensurability parameter grows with x. The incommensurability, its dependence on frequency and on x resemble experimental results in lanthanum cuprates. Also for finite x sharp maxima appear in the static charge susceptibility. The maxima are finite which points to the absence of the long-range charge ordering (static stripes). However, for x ≈ 0.12 the maxima are located near the momenta (0, ±π/2), (±π/2, 0). In this case an interaction of carriers with tetragonal distortions can stabilize stripes with the wavelength of four lattice spacings, as observed in the low-temperature tetragonal phase of cuprates. As follows from the obtained results, the magnetic incommensurability is not a consequence of the stripes. The Hubbard model is thought to be appropriate to describe the main features of electron correlations in narrow energy bands, leading to collective effects such as magnetism and

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The charge susceptibility of the two-dimensional repulsive Hubbard model is investigated by using the diagram technique developed for the case of strong correlations. In this technique, a power series in the hopping constant is used. It is shown that once the Fermi level crosses one of the Hubbard subbands, a sharp peak appears in the momentum dependence of the static susceptibility. With further departure from half-filling, the peak transforms to a ridge around the $\ensuremath{\Gamma}$ point. Within the considered range $0\ensuremath{\le}|1\ensuremath{-}\overline{n}|\ensuremath{\lesssim}0.2$ of the electron filling $\overline{n}$, the static susceptibility is finite, which points to the absence of long-range charge ordering. However, for $|1\ensuremath{-}\overline{n}|\ensuremath{\approx}0.12$, the susceptibility maxima are located halfway between the center and the boundaries of the Brillouin zone. In this case, an interaction of the carriers with the tetragonal distortions can stabilize the charge-density wave with the wavelength of four lattice spacings, as experimentally observed in the low-temperature tetragonal phase of lanthanum cuprates. Within the range of parameters inherent in cuprate perovskites, the character of the susceptibility evolution with $\overline{n}$ depends only weakly on the ratio of the nearest-neighbor hopping constant to the Hubbard repulsion and on details of the initial band structure. The location of the susceptibility maxima in the Brillouin zone is mainly determined by the value of $\overline{n}$.

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The magnetic susceptibility of the two-dimensional repulsive Hubbard model with nearest-neighbor hopping is investigated using the diagram technique developed for the case of strong correlations. In this technique, a power series in the hopping constant is used. At half-filling, the calculated zero-frequency susceptibility and the square of the site spin reproduce adequately the results of Monte Carlo simulations. Also, in agreement with numerical simulations, no evidence of ferromagnetic correlations was found in the considered range of electron concentrations $0.8\ensuremath{\lesssim}\overline{n}\ensuremath{\lesssim}1.2$ for the repulsion parameters $8\ensuremath{\mid}t\ensuremath{\mid}\ensuremath{\leqslant}U\ensuremath{\leqslant}16\ensuremath{\mid}t\ensuremath{\mid}$. However, for larger $U∕\ensuremath{\mid}t\ensuremath{\mid}$ and $\ensuremath{\mid}1\ensuremath{-}\overline{n}\ensuremath{\mid}\ensuremath{\approx}0.2$, the nearest-neighbor correlations become ferromagnetic. For $\overline{n}\ensuremath{\lesssim}0.94$ and $\overline{n}\ensuremath{\gtrsim}1.06$, the imaginary part of the real-frequency susceptibility becomes incommensurate for small frequencies. The incommensurability parameter grows with departure from half-filling and decreases with increasing frequency. This behavior of the susceptibility can explain the observed low-frequency incommensurate response observed in normal-state lanthanum cuprates.

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Stripe correlations in the two-dimensional Hubbard-Holstein model

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Abstract Several state-of-the-art numerical methods have observed static or fluctuating spin and charge stripes in doped two-dimensional Hubbard models, suggesting that these orders play a significant role in shaping the cuprate phase diagram. Many experiments, however, also indicate that the cuprates have strong electron-phonon (e-ph) coupling, and it is unclear how this interaction influences stripe correlations. We study static and fluctuating stripe orders in the doped single-band Hubbard-Holstein model using zero temperature variational Monte Carlo and finite temperature determinant quantum Monte Carlo. We find that the lattice couples more strongly with the charge component of the stripes, leading to an enhancement or suppression of stripe correlations, depending on model parameters like the next-nearest-neighbor hopping $${t}^{{\prime} }$$ t′ or phonon energy Ω. Our results help elucidate how thee-ph interaction can tip the delicate balance between stripe and superconducting correlations in the Hubbard-Holstein model with implications for our understanding of the high-Tccuprates.

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Abstract. Magnetic and charge susceptibilities of the two-dimensional repulsive Hubbard model are investigated applying a strong coupling diagram technique in which the expansion in powers of the hopping constants is used. For small lattices and high temperatures results are in agreement with Monte Carlo simulations. With the departure from half-filling x the low-frequency magnetic susceptibility becomes incommensurate and the incommensurability parameter grows with x. The incommensurability, its dependence on frequency and on x resemble experimental results in lanthanum cuprates. Also for finite x sharp maxima appear in the static charge susceptibility. The maxima are finite which points to the absence of the long-range charge ordering (static stripes). However, for x ≈ 0.12 the maxima are located near the momenta (0, ±π/2), (±π/2, 0). In this case an interaction of carriers with tetragonal distortions can stabilize stripes with the wavelength of four lattice spacings, as observed in the low-temperature tetragonal phase of cuprates. As follows from the obtained results, the magnetic incommensurability is not a consequence of the stripes. The Hubbard model is thought to be appropriate to describe the main features of electron correlations in narrow energy bands, leading to collective effects such as magnetism and
The charge susceptibility of the two-dimensional repulsive Hubbard model is investigated by using the diagram technique developed for the case of strong correlations. In this technique, a power series in the hopping constant is used. It is shown that once the Fermi level crosses one of the Hubbard subbands, a sharp peak appears in the momentum dependence of the static susceptibility. With further departure from half-filling, the peak transforms to a ridge around the $\ensuremath{\Gamma}$ point. Within the considered range $0\ensuremath{\le}|1\ensuremath{-}\overline{n}|\ensuremath{\lesssim}0.2$ of the electron filling n‾\overline{n}n, the static susceptibility is finite, which points to the absence of long-range charge ordering. However, for $|1\ensuremath{-}\overline{n}|\ensuremath{\approx}0.12$, the susceptibility maxima are located halfway between the center and the boundaries of the Brillouin zone. In this case, an interaction of the carriers with the tetragonal distortions can stabilize the charge-density wave with the wavelength of four lattice spacings, as experimentally observed in the low-temperature tetragonal phase of lanthanum cuprates. Within the range of parameters inherent in cuprate perovskites, the character of the susceptibility evolution with n‾\overline{n}n depends only weakly on the ratio of the nearest-neighbor hopping constant to the Hubbard repulsion and on details of the initial band structure. The location of the susceptibility maxima in the Brillouin zone is mainly determined by the value of n‾\overline{n}n.
The magnetic susceptibility of the two-dimensional repulsive Hubbard model with nearest-neighbor hopping is investigated using the diagram technique developed for the case of strong correlations. In this technique, a power series in the hopping constant is used. At half-filling, the calculated zero-frequency susceptibility and the square of the site spin reproduce adequately the results of Monte Carlo simulations. Also, in agreement with numerical simulations, no evidence of ferromagnetic correlations was found in the considered range of electron concentrations $0.8\ensuremath{\lesssim}\overline{n}\ensuremath{\lesssim}1.2$ for the repulsion parameters $8\ensuremath{\mid}t\ensuremath{\mid}\ensuremath{\leqslant}U\ensuremath{\leqslant}16\ensuremath{\mid}t\ensuremath{\mid}$. However, for larger $U∕\ensuremath{\mid}t\ensuremath{\mid}$ and $\ensuremath{\mid}1\ensuremath{-}\overline{n}\ensuremath{\mid}\ensuremath{\approx}0.2$, the nearest-neighbor correlations become ferromagnetic. For $\overline{n}\ensuremath{\lesssim}0.94$ and $\overline{n}\ensuremath{\gtrsim}1.06$, the imaginary part of the real-frequency susceptibility becomes incommensurate for small frequencies. The incommensurability parameter grows with departure from half-filling and decreases with increasing frequency. This behavior of the susceptibility can explain the observed low-frequency incommensurate response observed in normal-state lanthanum cuprates.
The influence of spin and charge fluctuations on spectra of the two-dimensional fermionic Hubbard model is considered using the strong coupling diagram technique. Infinite sequences of diagrams containing ladder inserts, which describe the interaction of electrons with these fluctuations, are summed, and obtained equations are self-consistently solved for the ranges of Hubbard repulsions [Formula: see text], temperatures [Formula: see text] and electron concentrations [Formula: see text] with t the intersite hopping constant. For all considered U the system exhibits a transition to the long-range antiferromagnetic order at [Formula: see text]. At the same time no indication of charge ordering is observed. Obtained solutions agree satisfactorily with results of other approaches and obey moments sum rules. In the considered region of the U-T plane, the curve separating metallic solutions passes from [Formula: see text] at the highest temperatures to U = 2t at [Formula: see text] for half-filling. If only short-range fluctuations are allowed for the remaining part of this region is occupied by insulating solutions. Taking into account long-range fluctuations leads to strengthening of maxima tails, which transform a part of insulating solutions into bad-metal states. For low T, obtained results allow us to trace the gradual transition from the regime of strong correlations with the pronounced four-band structure and well-defined Mott gap for [Formula: see text] to the Slater regime of weak correlations with the spectral intensity having a dip along the boundary of the magnetic Brillouin zone due to an antiferromagnetic ordering for [Formula: see text]. For [Formula: see text] and [Formula: see text] doping leads to the occurrence of a pseudogap near the Fermi level, which is a consequence of the splitting out of a narrow band from a Hubbard subband. Obtained spectra feature waterfalls and Fermi arcs, which are similar to those observed in hole-doped cuprates.
Motivated by the stripe developments in cuprates, we review analytical results for the spin-charge solitonic superstructures derived in the framework of the Hubbard model in our studies of weakly coupled (quasi) one-dimensional repulsive electron systems on a lattice. These results demonstrate that close to half filling, in the high temperature regime above the mean field transition temperature, short range repulsions favor charge density fluctuations with wavevectors bearing special relations with those of the spin density fluctuations. In the low temperature regime, besides the wavevectors, mutual phases of the charge and spin densities also become coupled due to a quantum interference phenomenon, leading to the stripe phase instability. It is shown that away from half filling, periodic lattice potential causes cooperative condensation of the spin and charge superlattices. "Switching off" this potential leads to the vanishing of the stripe order. The leading spin-charge coupling term in the effective Landau functional is derived microscopically. Results of the 1D renormalization group ("parquet") analysis away from half filling are also presented. They reveal transient-scale correlations resembling the mean-field pattern. Possible correspondence of our theory with the experimental data on stripe phase in high Tc cuprates is discussed.
Abstract We consider the extended Hubbard model on a two-dimensional square lattice at half-filling. The model is investigated using the strong coupling diagram technique. We sum infinite series of ladder diagrams allowing for full-scale charge and spin fluctuations and the actual short-range antiferromagnetic order for nonzero temperatures. In agreement with earlier results, we find the first-order phase transition in the charge subsystem occurring at v = v c ≳ U/4 with v and U the intersite and on-site Coulomb repulsion constants. The transition reveals itself in an abrupt sign change of a sharp maximum in the zero-frequency charge susceptibility at the corner of the Brillouin. States arising at the transition have alternating deviations of electron occupations from the mean value on neighboring sites. Due to fluctuations, these alternating occupation deviations have short-range order. For the considered parameters, such behavior is found for U ≲ 5t with t the hopping constant. For the insulating case U ≳ 6t, in which the transition is not observed, we find a continuous growth of the Mott gap with v. The evolution of the electron density of states with increasing v is also considered.
Abstract Several state-of-the-art numerical methods have observed static or fluctuating spin and charge stripes in doped two-dimensional Hubbard models, suggesting that these orders play a significant role in shaping the cuprate phase diagram. Many experiments, however, also indicate that the cuprates have strong electron-phonon (e-ph) coupling, and it is unclear how this interaction influences stripe correlations. We study static and fluctuating stripe orders in the doped single-band Hubbard-Holstein model using zero temperature variational Monte Carlo and finite temperature determinant quantum Monte Carlo. We find that the lattice couples more strongly with the charge component of the stripes, leading to an enhancement or suppression of stripe correlations, depending on model parameters like the next-nearest-neighbor hopping t′{t}^{{\prime} }t′ t′ or phonon energy Ω. Our results help elucidate how thee-ph interaction can tip the delicate balance between stripe and superconducting correlations in the Hubbard-Holstein model with implications for our understanding of the high-Tccuprates.