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arXiv:2211.06322 · Physical Review Letters

Phase Diagram, d -Wave Superconductivity, and Pseudogap of the t − t ′ − J Model at Finite Temperature

Dai-Wei Qu, Qiaoyi Li, Shou-Shu Gong, Yang Qi, Wei Li, Gang Su

Published Dec 19, 2024• 17 citations
arXivHTMLPDF

Abstract

Recently, robust d-wave superconductive (SC) order has been unveiled in the ground state of the 2D t-t^{'}-J model-with both nearest-neighbor (t) and next-nearest-neighbor (t^{'}) hoppings-by density matrix renormalization group studies. However, there is currently a debate on whether the d-wave SC holds up strong on both t^{'}/t>0 and t^{'}/t<0 cases for the t-t^{'}-J model, which correspond to the electron- and hole-doped sides of the cuprate phase diagram, respectively. Here, we exploit state-of-the-art thermal tensor network approach to accurately obtain the phase diagram of the t-t^{'}-J model on cylinders with widths up to W=6 and down to low temperature as T/J≃0.06, pushing the boundaries of contemporary finite-T calculations. For t^{'}/t>0, we find a domelike SC regime with a diverging d-wave pairing susceptibility, χ_{SC}∝1/T^{α} below a characteristic temperature T_{c}^{*}. Near optimal doping, T_{c}^{*} reaches its highest value of about 0.15J. Above T_{c}^{*} yet below a higher crossover temperature T^{*}, the magnetic susceptibility becomes suppressed, which can be related to the onset of pseudogap (PG) behaviors. On the other hand, for t^{'}/t<0, we find the pairing correlations are much weaker, although there exhibits a node-antinode structure in the PG regime as observed in the hole-doped cuprates. The thermal tensor network calculations of the t-t^{'}-J model underscore both the similarities and differences in the finite-temperature phase diagram between the fundamental model and cuprates, yielding unique insights into their intricate behaviors.

DOI
10.1103/physrevlett.133.256003

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Recently, robust d-wave superconductive (SC) order has been unveiled in the ground state of the 2D t-t^{'}-J model-with both nearest-neighbor (t) and next-nearest-neighbor (t^{'}) hoppings-by density matrix renormalization group studies. However, there is currently a debate on whether the d-wave SC holds up strong on both t^{'}/t>0 and t^{'}/t<0 cases for the t-t^{'}-J model, which correspond to the electron- and hole-doped sides of the cuprate phase diagram, respectively. Here, we exploit state-of-the-art thermal tensor network approach to accurately obtain the phase diagram of the t-t^{'}-J model on cylinders with widths up to W=6 and down to low temperature as T/J≃0.06, pushing the boundaries of contemporary finite-T calculations. For t^{'}/t>0, we find a domelike SC regime with a diverging d-wave pairing susceptibility, χ_{SC}∝1/T^{α} below a characteristic temperature T_{c}^{}. Near optimal doping, T_{c}^{} reaches its highest value of about 0.15J. Above T_{c}^{} yet below a higher crossover temperature T^{}, the magnetic susceptibility becomes suppressed, which can be related to the onset of pseudogap (PG) behaviors. On the other hand, for t^{'}/t<0, we find the pairing correlations are much weaker, although there exhibits a node-antinode structure in the PG regime as observed in the hole-doped cuprates. The thermal tensor network calculations of the t-t^{'}-J model underscore both the similarities and differences in the finite-temperature phase diagram between the fundamental model and cuprates, yielding unique insights into their intricate behaviors.
The Hubbard and closely related t-J models are exciting platforms for unconventional superconductivity (SC). Through state-of-the-art density matrix renormalization group calculations using the grand canonical ensemble, we address open issues regarding the ground-state phase diagram of the extended t-J model on a square lattice in the parameter regime relevant to cuprate superconductors. On large 8-leg cylinders, we demonstrate that the pure t-J model with only nearest-neighbor hoppings and superexchange interactions, for a wide range of doping ( δ = 0.1 to 0.2 ), hosts robust d-wave superconductivity possibly coexisting with weak unidirectional pair density wave. Furthermore, a small next nearest neighbor hopping t 2 suppresses pair and charge density waves, resulting in a uniform d-wave SC phase in both electron- and hole-doped cuprate model systems. Our work validates the t-J model as a proper minimum model for the emergence of superconductivity in cuprate superconductors.
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.
The square-lattice Hubbard and closely related t-J models are considered as basic paradigms for understanding strong correlation effects and unconventional superconductivity (SC). Recent large-scale density matrix renormalization group simulations on the extended t-J model have identified d-wave SC on the electron-doped side (with the next-nearest-neighbor hopping t_{2}>0) but a dominant charge density wave (CDW) order on the hole-doped side (t_{2}<0), which is inconsistent with the SC of hole-doped cuprate compounds. We re-examine the ground-state phase diagram of the extended t-J model by employing the state-of-the-art density matrix renormalization group calculations with much enhanced bond dimensions, allowing more accurate determination of the ground state. On six-leg cylinders, while different CDW phases are identified on the hole-doped side for the doping range δ=1/16-1/8, a SC phase emerges at a lower doping regime, with algebraically decaying pairing correlations and d-wave symmetry. On the wider eight-leg systems, the d-wave SC also emerges on the hole-doped side at the optimal 1/8 doping, demonstrating the winning of SC over CDW by increasing the system width. Our results not only suggest a new path to SC in general t-J model through weakening the competing charge orders, but also provide a unified understanding on the SC of both hole- and electron-doped cuprate superconductors.
We report results of large-scale ground-state density matrix renormalization group (DMRG) calculations on t-[Formula: see text]-J cylinders with circumferences 6 and 8. We determine a rough phase diagram that appears to approximate the two-dimensional (2D) system. While for many properties, positive and negative [Formula: see text] values ([Formula: see text]) appear to correspond to electron- and hole-doped cuprate systems, respectively, the behavior of superconductivity itself shows an inconsistency between the model and the materials. The [Formula: see text] (hole-doped) region shows antiferromagnetism limited to very low doping, stripes more generally, and the familiar Fermi surface of the hole-doped cuprates. However, we find [Formula: see text] strongly suppresses superconductivity. The [Formula: see text] (electron-doped) region shows the expected circular Fermi pocket of holes around the [Formula: see text] point and a broad low-doped region of coexisting antiferromagnetism and d-wave pairing with a triplet p component at wavevector [Formula: see text] induced by the antiferromagnetism and d-wave pairing. The pairing for the electron low-doped system with [Formula: see text] is strong and unambiguous in the DMRG simulations. At larger doping another broad region with stripes in addition to weaker d-wave pairing and striped p-wave pairing appears. In a small doping region near [Formula: see text] for [Formula: see text], we find an unconventional type of stripe involving unpaired holes located predominantly on chains spaced three lattice spacings apart. The undoped two-leg ladder regions in between mimic the short-ranged spin correlations seen in two-leg Heisenberg ladders.
In this paper, the effect of the next-earest (t ′ ) and third-earest (t ′ ′ ) hopping terms on superconductivity (SC) of the two-dimensional (2D) extended t-J model is studied by the variational Monte-Carlo (VMC) method. For hole-doped systems, the d-wave pairing correlation is suppressed by t ′ and t ′ ′ for underdoped cases, while it can be greatly enhanced by proper values of t ′ and t ′ ′ for the overdoped ones. This behavior results from the deformation of the Fermi surfaces. PACS numbers: 74.20.-z, 71.18.+y The 2D t-J model is one of the most popular models for studying the properties of the high-temperature superconductivity (HTS) [1, 2]. Yet the problem whether this model can give such high Tc for the cuprates or not is still under debate [4–7]. It has been well accepted that the next-nearest-neighbor (t ′ ) and the third-nearest-neighbor (t ′ ′) hopping terms are necessary for models to explain several experiments like angle-resolved photoemission (ARPES) of HTS [3]. However, the effect of t ′ and t ′ ′ on pairing correlation is also controversial. Previous DMRG studies of the two-dimensional (2D) extended t-J model show that the SC is suppressed (enhanced) for negative (positive) next-nearestneighbor hopping t ′ /t [8]. But results from first-principle calculation show that t ′ /t = −0.3 maximizes SC at optimal doping [9]. The Hamiltonian of the extended t-J model is H = Ht + HJ = − ∑ ij tij(˜c † i,σ ˜cj,σ + H.C.) + J ∑
Motivated by significant discrepancies between experimental observations of electron-doped cuprates and numerical results of the Hubbard and $t\text{\ensuremath{-}}J$ models, we investigate the role of intersite interactions VVV by studying the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model on square lattices. Based on large-scale density matrix renormalization group simulations, we identify the ground-state phase diagram across varying intersite interactions VVV and doping concentration $\ensuremath{\delta}$. We find that the phase diagram with finite intersite interactions $2\ensuremath{\lesssim}V/J\ensuremath{\lesssim}3$ offers a more accurate description of electron-doped cuprates than the conventional Hubbard and $t\text{\ensuremath{-}}J$ models. Moreover, we reveal the role of intersite interactions VVV at varying doping levels: at light doping, intersite interactions favor N\'eel antiferromagnetic order, and suppress both superconductivity and charge density wave; around optimal doping, these interactions support a pseudogap-like phase while suppressing superconductivity, and we further perform the slave-boson mean-field analysis to understand the numerical results microscopically; at higher doping, the effects of intersite interactions become insignificant, with our numerical predictions suggesting the emergence of an incommensurate spin density wave phase. Our specific focus around optimal doping with various intersite interactions identifies successive phases including phase separation, uniform ddd-wave SC, and a pseudogap-like phase, and reveals a relative insensitivity of the charge density wave to superconductivity. Our study suggests the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model as the minimal model to capture the essential physics of electron-doped cuprates.