Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9808201 · Physical review. B, Condensed matter

Interplay between superconductivity and flux phase in the t−J model

E. Cappelluti, R. Zeyher

Published Mar 1, 1999• 104 citations
arXivHTMLPDF

Abstract

We study the phase diagram of the $t\ensuremath{-}J$ model using a mean field type approximation within the Baym-Kadanoff perturbation expansion for Hubbard X operators. The line separating the normal state from a d-wave flux or bond-order state starts near optimal doping at $T=0$ and rises quickly with decreasing doping. The transition temperature ${T}_{c}$ for d-wave superconductivity increases monotonically in the overdoped region towards optimal doping. Near optimal doping a strong competition between the two d-wave order parameters sets in leading to a strong suppression of ${T}_{c}$ in the underdoped region. Treating for simplicity the flux phase as commensurate the superconducting and flux phases coexist in the underdoped region below ${T}_{c},$ whereas a pure flux phase exists above ${T}_{c}$ with a pseudogap of d-wave symmetry in the excitation spectrum. We also find that incommensurate charge-density-wave ground states due to Coulomb interactions do not modify strongly the above phase diagram near the superconducting phase, at least, as long as the latter exists at all.

DOI
10.1103/physrevb.59.6475

Related papers

Pseudogap, competing order, and the coexistence of staggered flux and d -wave pairing in high-temperature superconductors

Sen Zhou, Ziqiang Wang

We study the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model of a doped Mott insulator in connection to high-${T}_{c}$ superconductors. The nearest neighbor Coulomb interaction $(V)$ is treated quantum mechanically on equal footing as the antiferromagnetic exchange interaction $(J)$. Motivated by the $SU(2)$ symmetry at half-filling, we construct a large-$N$ theory which allows a systematic study of the interplay between staggered flux order and superconductivity upon doping. We solve the model in the large-$N$ limit and obtain the ground state properties and the phase diagram as a function of doping. We discuss the competition and the coexistence of the staggered flux and the $d$-wave superconductivity in the underdoped regime and the disappearance of superconductivity in the overdoped regime.

arXiv:cond-mat/0401436•Physical Review B•Jul 8, 2004•16 citations

Interplay of superconductivity with structural phases in a generalized t-J model

Zeyher, R., Cappelluti, E.

The phase diagram of the t-J-V model is discussed using a 1/N expansion in terms of X operators. It is shown that a flux phase of d-wave symmetry is stabilized by the Coulomb interaction V at intermediate dopings and competes with d-wave superconductivity. Since the flux wave instability is stronger than the superconducting one optimal doping is essentially determined by the onset of the flux phase. Below optimal doping the flux phase coexists with superconductivity at low and exists as a pseudo gap phase at higher temperatures. It is also found that the flux phase boundary is much less sensitive to impurity scattering than the boundary for superconductivity in agreement with experiments in Zn doped La-214 and (Y,Ca)-123.

•Jan 1, 2000•1 citation

Interplay between d -wave superconductivity and a bond-density wave in the one-band Hubbard model

Jean Paul Latyr Faye, David Sénéchal

It is now well established that superconducting cuprates support a charge-density-wave state in the so-called underdoped region of their phase diagram. We investigate the possibility of charge order in the square-lattice Hubbard model, both alone and in coexistence with $d$-wave superconductivity. The charge order has a period of 4 in one direction, is centered on bonds, and has a $d$ form factor. We use the variational cluster approximation, an approach based on a rigorous variational principle that treats short-range correlations exactly, with two clusters of size $2\ifmmode\times\else\texttimes\fi{}6$ that together tile the infinite lattice and provide a nonbiased unit for a period-4 bond-density wave (BDW). We find that the BDW exists in a finite range of hole doping and increases in strength from $U=5$ to $U=8$. Its location and intensity depend strongly on the band dispersion. When probed simultaneously with $d$-wave superconductivity, the energy is sometimes lowered by the presence of both phases, depending on the interaction strength. Whenever they coexist, a pair-density wave (a modulation of superconducting pairing with the same period and form factor as the BDW) also exists.

arXiv:1612.02846•Physical review. B./Physical review. B•Mar 14, 2017•18 citations

Gossamer superconductivity and antiferromagnetism in the t‐J‐U model

Feng Yuan, Qingshan Yuan, C. S. Ting

The $d$-wave superconductivity (dSC) and antiferromagnetism are analytically studied in a renormalized mean-field theory for a two-dimensional $t\text{\penalty1000-\hskip0pt}J$ model plus an on-site repulsive Hubbard interaction $U$. The purpose of introducing the $U$ term is to partially impose the no-double-occupancy constraint by employing the Gutzwiller approximation. The phase diagrams as functions of doping $\ensuremath{\delta}$ and $U$ are studied. Using the standard value of $t∕J=3.0$ and in the large-$U$ limit, we show that the antiferromagnetic (AF) order emerges and coexists with the dSC in the underdoped region below the doping $\ensuremath{\delta}\ensuremath{\sim}0.1$. The dSC order parameter increases from zero as the doping increases and reaches a maximum near the optimal doping $\ensuremath{\delta}\ensuremath{\sim}0.15$. In the small-$U$ limit, only the dSC order survives while the AF order disappears. As $U$ increases up to a critical value, the AF order shows up and coexists with the dSC in the underdoped regime. At half-filling, the system is in the dSC state for small $U$ and becomes an AF insulator for large $U$. Within the present mean-field approach, we show that the ground-state energy of the coexistent state is always lower than that of the pure dSC state.

arXiv:cond-mat/0409596•Physical Review B•Mar 8, 2005•17 citations

Phase diagram of the square-lattice t − J − V model for electron-doped cuprates

Qianqian Chen, Lei Qiao, Fu‐Chun Zhang +1 more

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 $V$ 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 $V$ 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 $V$ 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 $d$-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.

arXiv:2312.05893•Physical review. B./Physical review. B•Jul 17, 2024•7 citations

Competition between superconductivity and the pseudogap phase in the t–J model

R. Zeyher, A. Greco

Abstract The t–J model in the large N limit (N denotes the number of spin components) yields a pseudogap phase in the underdoped region which is related to a d‐wave charge density wave (d‐CDW). We present results for the doping dependence of the superconducting and d‐CDW order parameters as well as for collective excitations in the presence of these two order parameters. We argue that the electronic Raman spectrum with B1g symmetry probes the amplitude fluctuations of the d‐CDW at zero momentum.

arXiv:cond-mat/0210110•physica status solidi (b)•Mar 1, 2003•5 citations
We study the phase diagram of the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model using a mean field type approximation within the Baym-Kadanoff perturbation expansion for Hubbard X operators. The line separating the normal state from a d-wave flux or bond-order state starts near optimal doping at T=0T=0T=0 and rises quickly with decreasing doping. The transition temperature Tc{T}_{c}Tc​ for d-wave superconductivity increases monotonically in the overdoped region towards optimal doping. Near optimal doping a strong competition between the two d-wave order parameters sets in leading to a strong suppression of Tc{T}_{c}Tc​ in the underdoped region. Treating for simplicity the flux phase as commensurate the superconducting and flux phases coexist in the underdoped region below Tc,{T}_{c},Tc​, whereas a pure flux phase exists above Tc{T}_{c}Tc​ with a pseudogap of d-wave symmetry in the excitation spectrum. We also find that incommensurate charge-density-wave ground states due to Coulomb interactions do not modify strongly the above phase diagram near the superconducting phase, at least, as long as the latter exists at all.
We study the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model of a doped Mott insulator in connection to high-Tc{T}_{c}Tc​ superconductors. The nearest neighbor Coulomb interaction (V)(V)(V) is treated quantum mechanically on equal footing as the antiferromagnetic exchange interaction (J)(J)(J). Motivated by the SU(2)SU(2)SU(2) symmetry at half-filling, we construct a large-NNN theory which allows a systematic study of the interplay between staggered flux order and superconductivity upon doping. We solve the model in the large-NNN limit and obtain the ground state properties and the phase diagram as a function of doping. We discuss the competition and the coexistence of the staggered flux and the ddd-wave superconductivity in the underdoped regime and the disappearance of superconductivity in the overdoped regime.
The phase diagram of the t-J-V model is discussed using a 1/N expansion in terms of X operators. It is shown that a flux phase of d-wave symmetry is stabilized by the Coulomb interaction V at intermediate dopings and competes with d-wave superconductivity. Since the flux wave instability is stronger than the superconducting one optimal doping is essentially determined by the onset of the flux phase. Below optimal doping the flux phase coexists with superconductivity at low and exists as a pseudo gap phase at higher temperatures. It is also found that the flux phase boundary is much less sensitive to impurity scattering than the boundary for superconductivity in agreement with experiments in Zn doped La-214 and (Y,Ca)-123.
It is now well established that superconducting cuprates support a charge-density-wave state in the so-called underdoped region of their phase diagram. We investigate the possibility of charge order in the square-lattice Hubbard model, both alone and in coexistence with ddd-wave superconductivity. The charge order has a period of 4 in one direction, is centered on bonds, and has a ddd form factor. We use the variational cluster approximation, an approach based on a rigorous variational principle that treats short-range correlations exactly, with two clusters of size $2\ifmmode\times\else\texttimes\fi{}6$ that together tile the infinite lattice and provide a nonbiased unit for a period-4 bond-density wave (BDW). We find that the BDW exists in a finite range of hole doping and increases in strength from U=5U=5U=5 to U=8U=8U=8. Its location and intensity depend strongly on the band dispersion. When probed simultaneously with ddd-wave superconductivity, the energy is sometimes lowered by the presence of both phases, depending on the interaction strength. Whenever they coexist, a pair-density wave (a modulation of superconducting pairing with the same period and form factor as the BDW) also exists.
The ddd-wave superconductivity (dSC) and antiferromagnetism are analytically studied in a renormalized mean-field theory for a two-dimensional $t\text{\penalty1000-\hskip0pt}J$ model plus an on-site repulsive Hubbard interaction UUU. The purpose of introducing the UUU term is to partially impose the no-double-occupancy constraint by employing the Gutzwiller approximation. The phase diagrams as functions of doping $\ensuremath{\delta}$ and UUU are studied. Using the standard value of t∕J=3.0t∕J=3.0t∕J=3.0 and in the large-UUU limit, we show that the antiferromagnetic (AF) order emerges and coexists with the dSC in the underdoped region below the doping $\ensuremath{\delta}\ensuremath{\sim}0.1$. The dSC order parameter increases from zero as the doping increases and reaches a maximum near the optimal doping $\ensuremath{\delta}\ensuremath{\sim}0.15$. In the small-UUU limit, only the dSC order survives while the AF order disappears. As UUU increases up to a critical value, the AF order shows up and coexists with the dSC in the underdoped regime. At half-filling, the system is in the dSC state for small UUU and becomes an AF insulator for large UUU. Within the present mean-field approach, we show that the ground-state energy of the coexistent state is always lower than that of the pure dSC state.
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.
Abstract The t–J model in the large N limit (N denotes the number of spin components) yields a pseudogap phase in the underdoped region which is related to a d‐wave charge density wave (d‐CDW). We present results for the doping dependence of the superconducting and d‐CDW order parameters as well as for collective excitations in the presence of these two order parameters. We argue that the electronic Raman spectrum with B1g symmetry probes the amplitude fluctuations of the d‐CDW at zero momentum.