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arXiv:2605.15060

Dynamical scaling near the pseudogap quantum critical point of the two-dimensional Hubbard model

Mathias Pelz, Gabriel Kotliar, Jan von Delft, Andreas Gleis

Published May 14, 2026• 0 citations
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Abstract

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 $x=ω/T$ scaling of the form $χ''(ω,T)\sim \tanh(x/2)$, and the cluster contribution to the optical conductivity obeys $Tσ'_{\mathrm{cl}}(ω,T) \sim \tanh(x/2)/x$, implying a $1/T$ 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 $1/T$ 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.

DOI
10.48550/arxiv.2605.15060

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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.
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
Finding microscopic models for metallic states that exhibit quantum critical properties is a major theoretical challenge. We calculate the dynamical local spin susceptibility $\ensuremath{\chi}(T,\ensuremath{\omega})$ for a Hubbard model at half-filling using dynamical mean-field theory, which is exact in infinite dimensions. Qualitatively distinct behavior is found in the different regions of the phase diagram: Mott insulator, Fermi liquid metal, bad metal, and a quantum critical region above the finite-temperature critical point. The signature of the latter is $\ensuremath{\omega}/T$ scaling, where $\ensuremath{\omega}$ is the frequency and TTT is the temperature. Our results are consistent with previous results showing scaling of the dc electrical conductivity, and they are relevant to experiments on organic charge-transfer salts.
A quantum critical point is found in the phase diagram of the two-dimensional Hubbard model [Vidhyadhiraja et al., Phys. Rev. Lett. 102, 206407 (2009)]. It is due to the vanishing of the critical temperature associated with a phase-separation transition, and it separates the non-Fermi-liquid region from the Fermi liquid. Near the quantum critical point, the pairing is enhanced since the real part of the bare ddd-wave pairing susceptibility exhibits an algebraic divergence with decreasing temperature, replacing the logarithmic divergence found in a Fermi liquid [Yang et al., Phys. Rev. Lett. 106, 047004 (2011)]. In this paper, we explore the single-particle and transport properties near the quantum critical point using high-quality estimates of the self-energy obtained by direct analytic continuation of the self-energy from the continuous-time quantum Monte Carlo method. We focus mainly on a van Hove singularity coming from the relatively flat dispersion that crosses the Fermi level near the quantum critical filling. The flat part of the dispersion orthogonal to the antinodal direction remains pinned near the Fermi level for a range of doping that increases when we include a negative next-near-neighbor hopping ${t}^{\ensuremath{'}}$ in the model. For comparison, we calculate the bare ddd-wave pairing susceptibility for noninteracting models with the usual two-dimensional tight-binding dispersion and a hypothetical quartic dispersion. We find that neither model yields a van Hove singularity that completely describes the critical algebraic behavior of the bare ddd-wave pairing susceptibility found in the numerical data. The resistivity, thermal conductivity, thermopower, and the Wiedemann-Franz law are examined in the Fermi liquid, marginal Fermi liquid, and pseudogap doping regions. A negative next-near-neighbor hopping ${t}^{\ensuremath{'}}$ increases the doping region with marginal Fermi liquid character. Both TTT and negative ${t}^{\ensuremath{'}}$ are relevant variables for the quantum critical point, and both the transport and the displacement of the van Hove singularity with filling suggest that they are qualitatively similar in their effect.
We present a numerical study on the non-Fermi liquid behavior of a three-dimensional strongly correlated system. The Hubbard model in a simple cubic lattice is simulated by the dynamical cluster approximation; in particular, the quasi-particle weight is calculated at finite dopings for a range of temperatures. By fitting the quasi-particle weight to the marginal Fermi liquid form at finite doping near the putative quantum critical point, we find evidence of a separatrix between Fermi liquid and non-Fermi liquid regions. Our results suggest that a marginal Fermi liquid and possibly a quantum critical point exist in the non-symmetry broken solution of the three-dimensional interacting electron systems. We also calculate the spectral function, close to the half-filling, and we obtain evidence of pseudogap.
Cluster dynamical mean-field and maximum entropy analytical continuation methods are used to obtain theoretical estimates for the many-body density of states, electron self-energy, in-plane and ccc-axis optical conductivity and the B1g{B}_{1g}B1g​ and B2g{B}_{2g}B2g​ Raman-scattering spectra of the two-dimensional square lattice Hubbard model at intermediate interaction strengths and carrier concentrations near half filling. The calculations are based on an eight-site cluster approximation which gives access to both the zone-diagonal and zone-face portions of the Fermi surface. At low dopings, the zone-face regions exhibit a ``pseudogap,'' a suppression of the many-body density of states for energies near the Fermi surface. The pseudogap magnitude is largest near half filling and decreases smoothly with doping but as temperature is increased, the gap fills in rather than closes. The calculated response functions bear a strong qualitative resemblance to data taken in the pseudogap regime of high-Tc{T}_{c}Tc​ cuprates, strongly suggesting that the intermediate coupling Hubbard model accounts for much of the exotic behavior observed in high-Tc{T}_{c}Tc​ materials.
We study charge transport across the metal-insulator crossover in the half-filled two-dimensional Hubbard model, with particular emphasis on precision control. The dynamic current-current correlation function is obtained directly in the thermodynamic limit, and the optical conductivity is extracted using numerical analytic continuation. To achieve this, we develop a multiscale approach: the non-perturbative low-frequency behavior is computed using the unbiased diagrammatic Monte Carlo technique, while the high-frequency physics is captured via a self-consistent (semi-)analytic diagrammatic theory. We found that across a broad temperature range where the DC resistivity displays anomalous scaling, ∼Tα\sim T^α∼Tα with 0<α≲10<α\lesssim 10<α≲1, the Nernst-Einstein relation implies the diffusion constant with the characteristic ∼1/T\sim 1/\sqrt{T}∼1/T​ "strange metal" behavior. It was also revealed that the insulating regime is entered through a peculiar non-Fermi liquid state-which we call a Pseudogap Metal-characterized by insulating charge compressibility coexisting with metallic transport. Diagrammatically, the high-temperature incoherent transport is captured by the dressed polarization bubble, whereas near the metal-insulator crossover, the effective interaction vertex between opposite-spin particles is responsible for transferring the Drude weight to a high-frequency continuum.