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arXiv:2411.15158 · Nature Communications

Forestalled phase separation as the precursor to stripe order

Aritra Sinha, Alexander Wietek

Published Nov 28, 2025• 6 citations
arXivHTMLPDF

Abstract

Stripe order is a prominent feature in the phase diagram of the cuprate superconductors and has been established as the lowest-energy state of the two-dimensional Fermi-Hubbard model in relevant regimes. As temperature rises, stripes and superconductivity give way to the interesting strange-metal and pseudogap regimes. Here we investigate a crucial aspect of these regimes using numerical simulations of the square lattice Hubbard model. Using infinite projected entangled pair states with purification in the thermodynamic limit, we show that the thermodynamic charge susceptibility develops a broad maximum above the stripe regime at filling near n = 0.90, that strengthens upon cooling. Minimally entangled typical thermal states on finite cylinders attribute this enhancement to the emergence of large, fluctuating charge clusters, while the charge susceptibility remains finite. Upon further cooling, the cluster sizes stop broadening and lock to the stripe wavelength, indicating that interaction-driven clustering organizes the intermediate-temperature landscape but is ultimately forestalled by stripe order at low temperature.

DOI
10.1038/s41467-025-66563-5

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Stripe order is a prominent feature in the phase diagram of the cuprate superconductors and has been established as the lowest-energy state of the two-dimensional Fermi-Hubbard model in relevant regimes. As temperature rises, stripes and superconductivity give way to the interesting strange-metal and pseudogap regimes. Here we investigate a crucial aspect of these regimes using numerical simulations of the square lattice Hubbard model. Using infinite projected entangled pair states with purification in the thermodynamic limit, we show that the thermodynamic charge susceptibility develops a broad maximum above the stripe regime at filling near n = 0.90, that strengthens upon cooling. Minimally entangled typical thermal states on finite cylinders attribute this enhancement to the emergence of large, fluctuating charge clusters, while the charge susceptibility remains finite. Upon further cooling, the cluster sizes stop broadening and lock to the stripe wavelength, indicating that interaction-driven clustering organizes the intermediate-temperature landscape but is ultimately forestalled by stripe order at low temperature.
The high-temperature superconducting cuprates are governed by intertwined spin, charge, and superconducting orders. While various state-of-the-art numerical methods have demonstrated that these phases also manifest themselves in doped Hubbard models, they differ on which is the actual ground state. Finite cluster methods typically indicate that stripe order dominates while embedded quantum cluster methods, which access the thermodynamic limit by treating long-range correlations with a dynamical mean field, conclude that superconductivity does. Here, we report the observation of fluctuating spin and charge stripes in the doped single-band Hubbard model using a quantum Monte Carlo dynamical cluster approximation (DCA) method. By resolving both the fluctuating spin and charge orders using DCA, we demonstrate for the first time that they survive in the doped Hubbard model in the thermodynamic limit. This discovery also provides a new opportunity to study the influence of fluctuating stripe correlations on the model's pairing correlations within a unified numerical framework.
Evolution of superconductivity from charge clusters to stripes in the ttt-t′t't′-JJJ model
Competition and coexistence of charge orders and superconductivity are hallmarks in many strongly correlated electron systems. Here, we unravel the precise role of charge fluctuations on the superconducting state in the ttt-t′t't′-JJJ model of the high-temperature cuprate superconductors. Using finite-temperature tensor network simulations, we investigate thermal snapshots in the underdoped regime where the ground state features a superconducting stripe phase. At intermediate temperatures, where stripes have melted and hole clustering is observed, we find that pairing correlations are tightly localized on the hole clusters. Upon entering the stripe regime at lower temperatures, pairing increasingly delocalizes across different hole clusters to ultimately become coherent across the full system in the ground state. This pair-charge locking gives rise to an intuitive picture of the parent state of the superconducting stripe phase: pairing is localized on hole clusters formed via hole attraction due to the onset of magnetic correlations at intermediate temperature. We discuss how this microscopic picture is consistent with a broad range of experimental observations in cuprate superconductors, including scanning tunneling microscopy (STM) evidence for local pairing above TcT_cTc​ and nuclear magnetic resonance (NMR) signatures of charge clustering in the underdoped regime.
The high-temperature superconducting cuprates are governed by intertwined spin, charge, and superconducting orders. While various state-of-the-art numerical methods have demonstrated that these phases also manifest themselves in doped Hubbard models, they differ on which is the actual ground state. Finite-cluster methods typically indicate that stripe order dominates, while embedded quantum-cluster methods, which access the thermodynamic limit by treating long-range correlations with a dynamical mean field, conclude that superconductivity does. Here, we report the observation of fluctuating spin and charge stripes in the doped single-band Hubbard model using a quantum Monte Carlo dynamical cluster approximation (DCA) method. By resolving both the fluctuating spin and charge orders using DCA, we demonstrate that they survive in the doped Hubbard model in the thermodynamic limit. This discovery also provides an opportunity to study the influence of fluctuating stripe correlations on the model's pairing correlations within a unified numerical framework. Using this approach, we also find evidence for pair-density-wave correlations whose strength is correlated with that of the stripes.
Electronic Green's function across the pseudogap to stripe transition in the ttt-t′t't′-JJJ model
Superconducting domes in strongly correlated electronic systems are often accompanied by charge density waves and peculiar features in the electronic structure. The appearance of a pseudogap, in particular, and its relation to charge density waves remains insufficiently understood. Here, we investigate the electronic Green's function of the underdoped ttt-t′t't′-JJJ model of the cuprate superconductors using tensor network algorithms for finite temperature dynamics on cylinders of width 444. We find the prominent momentum differentiation, the hallmark of the pseudogap, to be strongly dependent on t′t't′, which develops into a momentum-dependent opening of a gap upon decreasing temperature, consistent with the formation of a Fermi arc at intermediate temperatures. The nodal gap around k=(π/2,π/2){\bf k}=(π/2,π/2)k=(π/2,π/2) closes and fills with increasing temperature as coherent stripe order melts into a regime of fluctuating charge clusters.
Physics The Hubbard model describes the physics of interacting particles on a lattice and is thought to contain elements essential to the superconductivity of the cuprates. Despite its apparent simplicity, solving the Hubbard model remains extremely challenging, especially at finite concentrations of hole dopants and intermediate temperatures. Wietek et al. used the minimally entangled typical thermal states (METTS) method to calculate the structural factors and thermodynamic properties for the two-dimensional Hubbard model with strong correlations at a finite doping and for a range of temperatures. The researchers found that a stripe order formed at low temperatures, whereas at higher temperatures, a phase resembling the experimentally observed pseudogap state took over. Phys. Rev. X. 11 , 0031007 (2021).
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.