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arXiv:2605.21585

Robust fluctuating intertwined charge stripes in the Emery model

Rong Zhang, Sijia Zhao, Hong‐Chen Jiang, Brian Moritz, Edwin W. Huang, Thomas Devereaux

Published May 20, 2026• 0 citations
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Abstract

The single-band Hubbard model is one of the most extensively studied models in condensed matter physics, giving rise to intertwined spin and charge stripes that coexist with, or lie in the vicinity of, superconductivity in the phase diagram. However, whether the low energy physics of the single-band Hubbard model is fully equivalent to the multi-band (multi-orbital) Emery model remains an unsettled question. While the intertwined stripes and nematicity have been studied in the single-band Hubbard model, a comprehensive picture in the Emery model is lacking. In this paper, we focus on the less investigated intertwined charge stripes using complementary density matrix renormalization group (DMRG) and determinant quantum Monte Carlo (DQMC) techniques. Our ground state DMRG confirms the presence of the oxygen-centered charge stripes at a reduced amplitude in the Emery model parameter regime widely used in the study of intertwined stripes. Close analysis of the oxygen orbital structure of the static charge correlation function from DQMC reveals the charge stripe pattern in real-space, showcasing stronger charge density modulation on $p$-orbitals pointing along ``the rivers of charge'', consistent with DMRG. For the parameter set with the largest fermion signs, we managed to reach a temperature where the system first demonstrated tendencies to form purely unidirectional spin and charge stripes, and the $B_{1g}$ component becomes dominant in the bond-charge nematic susceptibility. This observation correlates with the doping dependence of the kinetic energy anisotropy, suggesting a close relation between the nematicity and charge stripes in the Emery model.

DOI
10.48550/arxiv.2605.21585

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The single-band Hubbard model is one of the most extensively studied models in condensed matter physics, giving rise to intertwined spin and charge stripes that coexist with, or lie in the vicinity of, superconductivity in the phase diagram. However, whether the low energy physics of the single-band Hubbard model is fully equivalent to the multi-band (multi-orbital) Emery model remains an unsettled question. While the intertwined stripes and nematicity have been studied in the single-band Hubbard model, a comprehensive picture in the Emery model is lacking. In this paper, we focus on the less investigated intertwined charge stripes using complementary density matrix renormalization group (DMRG) and determinant quantum Monte Carlo (DQMC) techniques. Our ground state DMRG confirms the presence of the oxygen-centered charge stripes at a reduced amplitude in the Emery model parameter regime widely used in the study of intertwined stripes. Close analysis of the oxygen orbital structure of the static charge correlation function from DQMC reveals the charge stripe pattern in real-space, showcasing stronger charge density modulation on ppp-orbitals pointing along ``the rivers of charge'', consistent with DMRG. For the parameter set with the largest fermion signs, we managed to reach a temperature where the system first demonstrated tendencies to form purely unidirectional spin and charge stripes, and the B1gB_{1g}B1g​ component becomes dominant in the bond-charge nematic susceptibility. This observation correlates with the doping dependence of the kinetic energy anisotropy, suggesting a close relation between the nematicity and charge stripes in the Emery model.
The high-temperature superconducting cuprates host unidirectional spin- and charge-density-wave orders that can intertwine with superconductivity in nontrivial ways. While the charge components of these stripes have now been observed in nearly all cuprate families, their detailed evolution with doping varies across different materials and at high and low temperatures. We address this problem using nonperturbative determinant quantum Monte Carlo calculations for the three-band Hubbard model. Using an efficient implementation, we can resolve the model's fluctuating spin and charge modulations and map their evolution as a function of the charge transfer energy and doping. We find that the incommensurability of the charge modulations is decoupled from the spin modulations and decreases with hole doping, consistent with experimental measurements at high temperatures. These findings support the proposal that the high-temperature charge correlations are distinct from the intertwined stripe order observed at low-temperature and in the single-band Hubbard model.
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.
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.
Upon doping, Mott insulators often exhibit symmetry breaking where charge carriers and their spins organize into patterns known as stripes. For high-transition temperature cuprate superconductors, stripes are widely suspected to exist in a fluctuating form. We used numerically exact determinant quantum Monte Carlo calculations to demonstrate dynamical stripe correlations in the three-band Hubbard model, which represents the local electronic structure of the copper-oxygen plane. Our results, which are robust to varying parameters, cluster size, and boundary conditions, support the interpretation of experimental observations such as the hourglass magnetic dispersion and the Yamada plot of incommensurability versus doping in terms of the physics of fluctuating stripes. These findings provide a different perspective on the intertwined orders emerging from the cuprates' normal state.
Strongly correlated electron systems host a variety of poorly understood correlations in their high-temperature normal state. Unlike ordered phases defined by order parameters, regions of the normal state are often defined through unconventional properties such as strange metallic transport or spectroscopic pseudogaps. Characterizing the microscopic correlations in the normal state is necessary to elucidate mechanisms that lead to these properties and their connection to ground-state orders. Here we establish the presence of intertwined charge and spin stripes in the strange metal normal state of the Hubbard model using determinant quantum Monte Carlo calculations. The charge and spin density waves constituting the stripes are fluctuating and short ranged; yet they obey a mutual commensurability relation and remain microscopically interlocked, as evidenced through measurements of three-point spin-spin-hole correlation functions. Our findings demonstrate the ability of many-body numerical simulations to unravel the microscopic correlations that define quantum states of matter.
We perform extensive auxiliary-field quantum Monte Carlo (AFQMC) calculations for the three-band Hubbard (Emery) model in order to study the ground-state properties of copper-oxygen planes in the cuprates. Employing cutting-edge AFQMC techniques with a self-consistent gauge constraint in auxiliary-field space to control the sign problem, we reach supercells containing $\ensuremath{\approx}500$ atoms to capture collective modes in the charge and spin orders and characterize the behavior in the thermodynamic limit. The self-consistency scheme interfacing with generalized Hartree-Fock calculations allows high accuracy in AFQMC to resolve small energy scales, which is crucial for determining the complex candidate orders in such a system. We present detailed information on the charge order, spin order, momentum distribution, and localization properties as a function of charge-transfer energy for the the underdoped regime. In contrast with the stripe and spiral orders under hole doping, we find that the corresponding 1/81/81/8 electron-doped system exhibits purely antiferromagnetic order in the three-band model, consistent with the asymmetry between electron and hole doping in the phase diagram of cuprates.