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arXiv:0806.0815 · Physical Review B

Bond excitations in the pseudogap phase of the Hubbard model

Alexandru Macridin, Mark Jarrell

Published Dec 2, 2008• 12 citations
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Abstract

Using the dynamical cluster approximation, we calculate the correlation functions associated with the nearest-neighbor bond operator which measure the $z$ component of the spin exchange in the two-dimensional Hubbard model with $U$ equal to the bandwidth. We find that in the pseudogap region, the local bond susceptibility diverges at $T=0$. This shows the existence of degenerate bond spin excitation and implies quantum criticality and bond order formation when long-range correlations are considered. The strong correlation between excitations on parallel neighboring bonds suggests bond singlet dimerization. The suppression of divergence for $n<\ensuremath{\approx}0.78$ implies that for these model parameters, this is a quantum critical point which separates the unconventional pseudogap region characterized by bond order from a conventional Fermi liquid.

DOI
10.1103/physrevb.78.241101

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Using the dynamical cluster approximation, we calculate the correlation functions associated with the nearest-neighbor bond operator which measure the zzz component of the spin exchange in the two-dimensional Hubbard model with UUU equal to the bandwidth. We find that in the pseudogap region, the local bond susceptibility diverges at T=0T=0T=0. This shows the existence of degenerate bond spin excitation and implies quantum criticality and bond order formation when long-range correlations are considered. The strong correlation between excitations on parallel neighboring bonds suggests bond singlet dimerization. The suppression of divergence for n<\ensuremath≈0.78n<\ensuremath{\approx}0.78n<\ensuremath≈0.78 implies that for these model parameters, this is a quantum critical point which separates the unconventional pseudogap region characterized by bond order from a conventional Fermi liquid.
We investigate the two-dimensional Hubbard model with next-nearest-neighbor hopping, ${t}^{\ensuremath{'}}$, using the dynamical cluster approximation. We confirm the existence of a first-order phase-separation transition terminating at a second-order critical point at filling ${n}_{c}({t}^{\ensuremath{'}})$ and temperature ${T}_{ps}({t}^{\ensuremath{'}})$. We find that as ${t}^{\ensuremath{'}}$ approaches zero, ${T}_{ps}({t}^{\ensuremath{'}})$ vanishes and ${n}_{c}({t}^{\ensuremath{'}})$ approaches the filling associated with the quantum critical point separating the Fermi liquid from the pseudogap phase. We propose that the quantum critical point under the superconducting dome is the zero-temperature limit of the line of second-order critical points.
We study dynamical scaling in the quantum-critical fan of the pseudogap-metal to Fermi-liquid transition of the two-dimensional Hubbard model. Using a four-patch dynamical cluster approximation with the numerical renormalization group as a cluster impurity solver, we access real-frequency dynamics over several decades at arbitrary temperatures. Close to the critical doping, the local spin and cluster-current susceptibility spectra exhibit x=ω/Tx=ω/Tx=ω/T scaling of the form χ′′(ω,T)∼tanh⁡(x/2)χ''(ω,T)\sim \tanh(x/2)χ′′(ω,T)∼tanh(x/2), and the cluster contribution to the optical conductivity obeys Tσcl′(ω,T)∼tanh⁡(x/2)/xTσ'_{\mathrm{cl}}(ω,T) \sim \tanh(x/2)/xTσcl′​(ω,T)∼tanh(x/2)/x, implying a 1/T1/T1/T cluster dc conductivity. In the scaling regime, the vertex contribution to the cluster optical response is much larger than the bubble contribution. We further find evidence for a marginal-Fermi-liquid nodal self-energy. This, together with the 1/T1/T1/T vertex contribution to the conductivity, implies strange-metal optical transport in the quantum critical region. Our results describe several qualitative aspects of several experimental observations.
Despite decades of intense theoretical and experimental investigation, the two-dimensional Fermi-Hubbard model still resists a complete microscopic understanding. Conventional approaches typically probe global observables and locally resolved correlation functions. Here, we develop a complementary perspective based on the measurement of entanglement. Using both an ultracold-atom quantum simulator and numerical simulations based on the dynamical vertex approximation, we find that entanglement is closely tied to the onset of the enigmatic pseudogap regime: spin-singlet entanglement emerges only as the pseudogap sets in and, in contrast to classical correlations, remains confined to nearest-neighbour sites in this regime. Our results, therefore, disfavour purely classical-fluctuation theories of the pseudogap and constrain microscopic models to those that develop nearest-neighbour spin-singlet entanglement at the pseudogap onset.
Using the dynamical cluster approximation and quantum Monte Carlo simulations we calculate the single-particle spectra of the Hubbard model with next-nearest neighbor hopping . In the underdoped region, we find that the pseudogap along the zone diagonal in the electron doped systems is due to long-range antiferromagnetic correlations. The physics in the proximity of (0, pi) is dramatically influenced by t' and determined by the short range correlations. The effect t' of on the low-energy angle-resolved photoemission spectroscopy spectra is weak except close to the zone edge. The short range correlations are sufficient to yield a pseudogap signal in the magnetic susceptibility and produce a concomitant gap in the single-particle spectra near (pi, pi/2), but not necessarily at a location in the proximity of the Fermi surface.
We study the normal-state, doping-driven phase diagram of the square-lattice Hubbard model using the dynamical cluster approximation combined with the numerical renormalization group as a cluster solver, which gives direct access to real-frequency dynamics at essentially zero temperature. In a parameter regime relevant for cuprates, U=7tU=7tU=7t and t′=−0.3tt'=-0.3tt′=−0.3t, we find a critical doping p∗p^{\ast}p∗ that marks a continuous quantum phase transition between a pseudogap metal and a normal Fermi liquid. The transition is identified by a continuous collapse, from both sides, of the Fermi-liquid scale extracted from charge, spin, and dx2−y2d_{x^2-y^2}dx2−y2​-wave pairing susceptibilities. This collapse produces a non-Fermi-liquid regime at intermediate energy scales, which appears to extend to arbitrarily low scales at p∗p^{\ast}p∗. As p∗p^{\ast}p∗ is crossed from the normal Fermi liquid at p>p∗p>p^{\ast}p>p∗ into the pseudogap metal at $p
Using large-scale dynamical cluster quantum Monte Carlo simulations, we study the Lifshitz transition of the two-dimensional Hubbard model with next-nearest-neighbor hopping (${t}^{\ensuremath{'}}$), chemical potential, and temperature as control parameters. At ${t}^{\ensuremath{'}}\ensuremath{\le}0$, we identify a line of Lifshitz transition points associated with a change in the Fermi surface topology at zero temperature. In the overdoped region, the Fermi surface is complete and electron-like; across the Lifshitz transition, the Fermi surface becomes hole-like and develops a pseudogap. At (or very close to) the Lifshitz transition points, a van Hove singularity in the density of states crosses the Fermi level. The van Hove singularity occurs at finite doping due to correlation effects and becomes more singular when ${t}^{\ensuremath{'}}$ becomes more negative. The resulting temperature dependence on the bare ddd-wave pairing susceptibility close to the Lifshitz points is significantly different from that found in the traditional van Hove scenarios. Such unambiguous numerical observation of the Lifshitz transition at ${t}^{\ensuremath{'}}\ensuremath{\le}0$ extends our understanding of the quantum critical region in the phase diagram and shines lights on future investigations of the nature of the quantum critical point in the two-dimensional Hubbard model.