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arXiv:2603.20368

Evolution of superconductivity from charge clusters to stripes in the $t$-$t'$-$J$ model

Aritra Sinha, Hannes Karlsson, Martin Ulaga, Alexander Wietek

Published Mar 20, 2026• 0 citations
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Abstract

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 $t$-$t'$-$J$ 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 $T_c$ and nuclear magnetic resonance (NMR) signatures of charge clustering in the underdoped regime.

DOI
10.48550/arxiv.2603.20368

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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.
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Fluctuating Pair Density Wave in Finite-temperature Phase Diagram of the ttt-t′t^\primet′ Hubbard Model
The Hubbard model and its extensions are canonical theoretical frameworks for understanding correlated electronic states, including those in high-TcT_cTc​ cuprates. Here, we use state-of-the-art thermal tensor network method to map out the temperature-doping phase diagram of the ttt-t′t^\primet′ Hubbard model. On the electron-doped side, we find a ddd-wave superconducting (dSC) regime, supporting the scenario of high-TcT_cTc​ superconductivity. In contrast, on the hole-doped side, no robust dSC phase is detected. Instead, a finite-temperature regime dominated by strong pair-density-wave (PDW) fluctuations emerges, which may eventually give way to charge density wave order upon further cooling. The PDW state exhibits inter-arc pairing with net momentum near (0,π)(0, π)(0,π), distinct from the zero-momentum pairing in conventional dSC. Furthermore, these fluctuating PDW states occupy the lower portion of the pseudogap regime on the hole-doped side. We provide a comprehensive finite-temperature perspective consistent with previous ground-state studies, shedding new light on pairing instabilities and exotic electronic states in high-TcT_cTc​ superconductors.
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.
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.