arXiv:2608.28750

Floquet engineering of competing antiferromagnetism and $d$-wave superconductivity on the square lattice

Zhaoyu Han, Subir Sachdev A

Published Aug 28, 2026• 0 citations

Abstract

We propose a driven Lieb--Hubbard quantum simulator whose prethermal dynamics realize a square-lattice spin-1/2 fermion model with independently tunable repulsive on-site and attractive bond interactions. Periodically modulating the charge-transfer offset between the site ($d$) and bond ($p$) orbitals of the Lieb lattice brings a $p$-orbital doublon energetically close to a pair in the $d$ manifold while keeping all $p$-orbital singlons off resonance, thereby creating a synthetic ``negative-$U$'' center on a Lieb-lattice bond site. Two controlled eliminations then generate a compact bond attraction in the reduced $d$-only model on the square lattice, despite the microscopic repulsion in the $p$ orbital. The resulting interaction $J$ is tunable independently of the Hubbard repulsion $U_d$ on the $d$ orbitals, while interference between photon-assisted paths provides access to an intermediate-coupling regime in which $J$ and $U_d$ are both comparable to the effective hopping. At half filling, a mean-field calculation in this regime finds adjacent antiferromagnetic and $d$-wave superconducting phases, as well as narrow coexistence regions, suggesting close competition between these orders. We discuss the branch-preparation, prethermal, and higher-band conditions required to translate the formal construction into an optical-lattice protocol. More broadly, our work identifies a structural similarity between Floquet systems and electron-phonon problems that may guide the design of novel quantum-simulation protocols.

DOI: 10.48550/arxiv.2608.28750

Related papers

Unconventional superconducting phases for the two-dimensional extended Hubbard model on a square lattice

Wen-Min Huang, Chen-Yen Lai, Chuntai Shi +1 more

arXiv:1303.7468Physical Review BAug 12, 201323 citations

Floquet engineering magnetism and superconductivity in the square-lattice Hubbard model

Jan-Niklas Herre, Takuya Okugawa, Ammon Fischer +2 more

arXiv:2511.11798Physical review. B./Physical review. BFeb 17, 20261 citation

Periodic drive induced unconventional superconductivity in a half-filled system

Suryashekhar Kusari, Arti Garg

arXiv:2602.19920Feb 23, 20260 citations

Repulsive fermions in optical lattices: Phase separation versus coexistence of antiferromagnetism and d -wave superfluidity

S. Y. Chang, Sandeep Pathak, Nandini Trivedi

arXiv:1103.5725Physical Review AJan 18, 20121 citation

Tuning competition between charge order and superconductivity in the square-lattice t − t ′ − J model

Xin Lu, Huaiming Guo, Wei-Qiang Chen +2 more

arXiv:2409.15270Physical review. B./Physical review. BJan 21, 20253 citations

Probing Topological Floquet-Bands and Quantum Magnetism with Ultracold Fermions

Gregor Jotzu

Jan 1, 20150 citations

arXiv:2608.28750

Floquet engineering of competing antiferromagnetism and $d$ -wave superconductivity on the square latticeFloquet engineering of competing antiferromagnetism and $d$-wave superconductivity on the square lattice

Zhaoyu Han, Subir Sachdev A

Published Aug 28, 2026• 0 citations

arXiv HTML PDF

Abstract

d

) and bond (

p

) orbitals of the Lieb lattice brings a

p

-orbital doublon energetically close to a pair in the

d

manifold while keeping all

p

-orbital singlons off resonance, thereby creating a synthetic ``negative-

U

'' center on a Lieb-lattice bond site. Two controlled eliminations then generate a compact bond attraction in the reduced

d

-only model on the square lattice, despite the microscopic repulsion in the

p

orbital. The resulting interaction

J

is tunable independently of the Hubbard repulsion

U_d

on the

d

orbitals, while interference between photon-assisted paths provides access to an intermediate-coupling regime in which

J

and

U_d

are both comparable to the effective hopping. At half filling, a mean-field calculation in this regime finds adjacent antiferromagnetic and

d

-wave superconducting phases, as well as narrow coexistence regions, suggesting close competition between these orders. We discuss the branch-preparation, prethermal, and higher-band conditions required to translate the formal construction into an optical-lattice protocol. More broadly, our work identifies a structural similarity between Floquet systems and electron-phonon problems that may guide the design of novel quantum-simulation protocols.

DOI: 10.48550/arxiv.2608.28750

Related papers

Unconventional superconducting phases for the two-dimensional extended Hubbard model on a square lattice

Wen-Min Huang, Chen-Yen Lai, Chuntai Shi +1 more

We study the phase diagram of the extended Hubbard model on a two-dimensional square lattice, including on-site ( $U$ ) and nearest-neighbor ( $V$ ) interactions, at weak couplings. We show that the charge-density wave phase that is known to occur at half filling when $4V>U$ gives way to a ${d}_{xy}$ -wave superconducting instability away from half filling, when the Fermi surface is not perfectly nested, and for sufficiently large repulsive $V$ and a range of on-site repulsive interaction $U$ . In addition, when nesting is further suppressed and in the presence of a nearest-neighbor attraction, a triplet time-reversal breaking $({p}_{x}+i{p}_{y})$ -wave pairing instability emerges, competing with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing state that is known to dominate at fillings just slightly away from half. At even smaller fillings, where the Fermi surface no longer presents any nesting, the $({p}_{x}+i{p}_{y})$ -wave superconducting phase dominates in the whole regime of on-site repulsions and nearest-neighbor attractions, while ${d}_{xy}$ pairing occurs in the presence of on-site attraction. Our results suggest that zero-energy Majorana fermions can be realized on a square lattice in the presence of a magnetic field. For a system of cold fermionic atoms on a two-dimensional square optical lattice, both an on-site repulsion and a nearest-neighbor attraction would be required, in addition to rotation of the system to create vortices. We discuss possible ways of experimentally engineering the required interaction terms in a cold atom system.We study the phase diagram of the extended Hubbard model on a two-dimensional square lattice, including on-site ($U$) and nearest-neighbor ($V$) interactions, at weak couplings. We show that the charge-density wave phase that is known to occur at half filling when $4V>U$ gives way to a ${d}_{xy}$-wave superconducting instability away from half filling, when the Fermi surface is not perfectly nested, and for sufficiently large repulsive $V$ and a range of on-site repulsive interaction $U$. In addition, when nesting is further suppressed and in the presence of a nearest-neighbor attraction, a triplet time-reversal breaking $({p}_{x}+i{p}_{y})$-wave pairing instability emerges, competing with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing state that is known to dominate at fillings just slightly away from half. At even smaller fillings, where the Fermi surface no longer presents any nesting, the $({p}_{x}+i{p}_{y})$-wave superconducting phase dominates in the whole regime of on-site repulsions and nearest-neighbor attractions, while ${d}_{xy}$ pairing occurs in the presence of on-site attraction. Our results suggest that zero-energy Majorana fermions can be realized on a square lattice in the presence of a magnetic field. For a system of cold fermionic atoms on a two-dimensional square optical lattice, both an on-site repulsion and a nearest-neighbor attraction would be required, in addition to rotation of the system to create vortices. We discuss possible ways of experimentally engineering the required interaction terms in a cold atom system.

arXiv:1303.7468Physical Review BAug 12, 201323 citations

Floquet engineering magnetism and superconductivity in the square-lattice Hubbard model

Jan-Niklas Herre, Takuya Okugawa, Ammon Fischer +2 more

We study the interplay of magnetic order and superconductivity in the square-lattice Hubbard model under periodic driving with circularly polarized light. Formulating diagrammatic techniques based on the random-phase approximation in terms of Floquet Green's functions, allows us to analyze fluctuation-driven unconventional pairing for weak to moderate interactions. The interplay of repulsive interactions and photoassisted hopping of electrons gives rise to a rich magnetic phase diagram featuring an antiferromagnetic-to-ferromagnetic transition prior to a Floquet Lifshitz transition. Close to the antiferromagnetic transition, topological d + i d -wave superconductivity prevails in the phase diagram for a wide range of drive parameters. At intermediate- to high-frequency driving near the Floquet Lifshitz transition, superconducting orders are tuned from spin-singlet d -wave to spin-triplet p -wave character, providing an effective protocol for Floquet engineering topological superconductivity.We study the interplay of magnetic order and superconductivity in the square-lattice Hubbard model under periodic driving with circularly polarized light. Formulating diagrammatic techniques based on the random-phase approximation in terms of Floquet Green's functions, allows us to analyze fluctuation-driven unconventional pairing for weak to moderate interactions. The interplay of repulsive interactions and photoassisted hopping of electrons gives rise to a rich magnetic phase diagram featuring an antiferromagnetic-to-ferromagnetic transition prior to a Floquet Lifshitz transition. Close to the antiferromagnetic transition, topological d + i d -wave superconductivity prevails in the phase diagram for a wide range of drive parameters. At intermediate- to high-frequency driving near the Floquet Lifshitz transition, superconducting orders are tuned from spin-singlet d -wave to spin-triplet p -wave character, providing an effective protocol for Floquet engineering topological superconductivity.

arXiv:2511.11798Physical review. B./Physical review. BFeb 17, 20261 citation

Periodic drive induced unconventional superconductivity in a half-filled system

Suryashekhar Kusari, Arti Garg

The non-equilibrium control of electronic properties has emerged as a transformative paradigm for engineering novel quantum phases. The most intriguing example of such a phase is light-induced superconductivity (SC) in non-superconducting materials. However, realizing unconventional SC at commensurate half-filling remains a formidable challenge even in non-equilibrium, as the regime is typically dominated by the robust stability of the antiferromagnetic (AFM) Mott insulating (MI) state. Here, we provide a novel non-equilibrium route to realize unconventional d-wave SC in a half-filled system through Floquet engineering. We analyze the periodically driven Fermi-Hubbard model on a bipartite lattice and demonstrate that a high-frequency drive can transform a weakly interacting insulator into a regime of strong correlations by the drive-induced renormalization of nearest-neighbor hopping. Furthermore, the drive induces staggered higher range hoppings that can frustrate the AFM order while simultaneously generate staggered potential that lifts the kinetic constraints inherent to the half-filled system, fostering the charge dynamics required to stabilize d-wave pairing against the competing AFM state. The resulting SC phase is protected by high-frequency prethermalization, maintaining stability over timescales exponentially large in the drive frequency. This protocol circumvents the need for chemical doping, offering a 'disorder-free' alternative for realizing unconventional pairing with direct applications in optimizing the performance of superconducting quantum computers, qubit arrays and other upcoming quantum technologies.The non-equilibrium control of electronic properties has emerged as a transformative paradigm for engineering novel quantum phases. The most intriguing example of such a phase is light-induced superconductivity (SC) in non-superconducting materials. However, realizing unconventional SC at commensurate half-filling remains a formidable challenge even in non-equilibrium, as the regime is typically dominated by the robust stability of the antiferromagnetic (AFM) Mott insulating (MI) state. Here, we provide a novel non-equilibrium route to realize unconventional d-wave SC in a half-filled system through Floquet engineering. We analyze the periodically driven Fermi-Hubbard model on a bipartite lattice and demonstrate that a high-frequency drive can transform a weakly interacting insulator into a regime of strong correlations by the drive-induced renormalization of nearest-neighbor hopping. Furthermore, the drive induces staggered higher range hoppings that can frustrate the AFM order while simultaneously generate staggered potential that lifts the kinetic constraints inherent to the half-filled system, fostering the charge dynamics required to stabilize d-wave pairing against the competing AFM state. The resulting SC phase is protected by high-frequency prethermalization, maintaining stability over timescales exponentially large in the drive frequency. This protocol circumvents the need for chemical doping, offering a 'disorder-free' alternative for realizing unconventional pairing with direct applications in optimizing the performance of superconducting quantum computers, qubit arrays and other upcoming quantum technologies.

arXiv:2602.19920Feb 23, 20260 citations

Repulsive fermions in optical lattices: Phase separation versus coexistence of antiferromagnetism and d -wave superfluidity

S. Y. Chang, Sandeep Pathak, Nandini Trivedi

We investigate a system of fermions on a two-dimensional optical square lattice in the strongly repulsive coupling regime. In this case, the interactions can be controlled by laser intensity as well as by Feshbach resonance. We compare the energetics of states with resonating valence bond $d$ -wave superfluidity, antiferromagnetic long-range order, and a homogeneous state with coexistence of superfluidity and antiferromagnetism. Using a variational formalism, we show that the energy density of a hole ${e}_{\mathrm{hole}}(x)$ has a minimum at doping $x={x}_{c}$ that signals phase separation between the antiferromagnetic and $d$ -wave paired superfluid phases. The energy of the phase-separated ground state is, however, found to be very close to that of a homogeneous state with coexisting antiferromagnetic and superfluid orders. We explore the dependence of the energy on the interaction strength and on the three-site hopping terms and compare with the nearest-neighbor hopping t-J model.We investigate a system of fermions on a two-dimensional optical square lattice in the strongly repulsive coupling regime. In this case, the interactions can be controlled by laser intensity as well as by Feshbach resonance. We compare the energetics of states with resonating valence bond $d$-wave superfluidity, antiferromagnetic long-range order, and a homogeneous state with coexistence of superfluidity and antiferromagnetism. Using a variational formalism, we show that the energy density of a hole ${e}_{\mathrm{hole}}(x)$ has a minimum at doping $x={x}_{c}$ that signals phase separation between the antiferromagnetic and $d$-wave paired superfluid phases. The energy of the phase-separated ground state is, however, found to be very close to that of a homogeneous state with coexisting antiferromagnetic and superfluid orders. We explore the dependence of the energy on the interaction strength and on the three-site hopping terms and compare with the nearest-neighbor hopping t-J model.

arXiv:1103.5725Physical Review AJan 18, 20121 citation

Tuning competition between charge order and superconductivity in the square-lattice t − t ′ − J model

Xin Lu, Huaiming Guo, Wei-Qiang Chen +2 more

Recently, a flurry of works have found strong competition between charge density wave (CDW) and superconductivity (SC) in the doped Hubbard and $t\text{\ensuremath{-}}J$ models on the square lattice. Interestingly, some recent results suggest that the electron-phonon coupling may suppress CDW order and enhance SC. In this work, we consider the square-lattice Hubbard model with the Holstein or Su-Schrieffer-Heeger electron-phonon coupling at the large- $U$ and antiadiabatic (infinite phonon frequency) limits, which gives an effective $t\text{\ensuremath{-}}J$ model with either a density attractive interaction $V$ or a ${J}_{P}$ term that contributes a larger spin exchange and a density repulsive interaction. To explore how these effective couplings may tune the competition between CDW and SC, we implement the density matrix renormalization group simulations on the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model with $V$ or ${J}_{P}$ coupling. We focus on the six-leg cylinder system with the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, which hosts partially filled stripe and $d$ -wave SC in phase diagram. By tuning ${t}^{\ensuremath{'}}/t>0$ and $V$ or ${J}_{P}$ , we establish two quantum phase diagrams at $\frac{1}{12}$ doping ratio. In the SC phases, the increased $V$ or ${J}_{P}$ coupling can enhance the quasi-long-range SC order, consistent with some previous findings. Nonetheless, no SC emerges when the partially filled stripe phase disappears with increased $V$ or ${J}_{P}$ . Instead, the system has a transition to either a phase-separation-like regime or a filled stripe phase. On the other hand, with increased ${t}^{\ensuremath{'}}/t$, not only the partially filled stripe but the phase separation and filled stripe can also be tuned to SC phase. Our results suggest that although $V$ and ${J}_{P}$ couplings may strengthen hole binding, the hole dynamics controlled by ${t}^{\ensuremath{'}}/t$ appears to play a more crucial role for obtaining a SC in the $t\text{\ensuremath{-}}J$ model.Recently, a flurry of works have found strong competition between charge density wave (CDW) and superconductivity (SC) in the doped Hubbard and $t\text{\ensuremath{-}}J$ models on the square lattice. Interestingly, some recent results suggest that the electron-phonon coupling may suppress CDW order and enhance SC. In this work, we consider the square-lattice Hubbard model with the Holstein or Su-Schrieffer-Heeger electron-phonon coupling at the large-$U$ and antiadiabatic (infinite phonon frequency) limits, which gives an effective $t\text{\ensuremath{-}}J$ model with either a density attractive interaction $V$ or a ${J}_{P}$ term that contributes a larger spin exchange and a density repulsive interaction. To explore how these effective couplings may tune the competition between CDW and SC, we implement the density matrix renormalization group simulations on the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model with $V$ or ${J}_{P}$ coupling. We focus on the six-leg cylinder system with the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, which hosts partially filled stripe and $d$-wave SC in phase diagram. By tuning ${t}^{\ensuremath{'}}/t>0$ and $V$ or ${J}_{P}$, we establish two quantum phase diagrams at $\frac{1}{12}$ doping ratio. In the SC phases, the increased $V$ or ${J}_{P}$ coupling can enhance the quasi-long-range SC order, consistent with some previous findings. Nonetheless, no SC emerges when the partially filled stripe phase disappears with increased $V$ or ${J}_{P}$. Instead, the system has a transition to either a phase-separation-like regime or a filled stripe phase. On the other hand, with increased ${t}^{\ensuremath{'}}/t$, not only the partially filled stripe but the phase separation and filled stripe can also be tuned to SC phase. Our results suggest that although $V$ and ${J}_{P}$ couplings may strengthen hole binding, the hole dynamics controlled by ${t}^{\ensuremath{'}}/t$ appears to play a more crucial role for obtaining a SC in the $t\text{\ensuremath{-}}J$ model.

arXiv:2409.15270Physical review. B./Physical review. BJan 21, 20253 citations

Probing Topological Floquet-Bands and Quantum Magnetism with Ultracold Fermions

Gregor Jotzu

Complex quantum many-body systems are ubiquitous in nature, yet their behaviour often remains very challenging to predict with analytical or numerical calculations.In this thesis a different approach is presented, using ultracold fermionic 40 K atoms in a lattice with a flexible geometry, formed by interfering laser beams.This system constitutes a precisely tunable, yet very accessible complex system, which can be used to probe -or "quantum simulate" -phenomena predicted for condensed-matter systems.We demonstrate, theoretically and experimentally, how a periodically modulated system can be described by an effective Hamiltonian on longer time-scales.This approach based on Floquet theory applies even when driving the system far from equilibrium.Using oscillating magnetic field gradients, we create an effectively spin-dependent lattice, where the tunnelling rate for different internal atomic states can be freely tuned.The flexible geometry of the lattice is used to create a honeycomb lattice and study the linear band crossings -known as Dirac cones -characteristic of its band structure.We distort the lattice and use momentum-resolved interband transitions probe the deformation and movement of the Dirac cones and their eventual disappearance when moving towards a square or dimerised geometry.We show that circular modulation of a honeycomb lattice induces imaginary next-nearest neighbour tunnelling, which breaks time-reversal symmetry.In addition, a tunable site-offset is used to break inversion symmetry.We thereby realize a model proposed by F.D.M. Haldane in 1988 and study the predicted topological transition from a trivial to a Chern-insulating regime, characterised by a closing band gap.Furthermore, effects of the Berry curvature are observed in both regimes, using transverse drifts of a driven atomic cloud.Using a mixture of two internal states with tunable scattering between them, we study the Hubbard model on a variety of geometries and with a range of observables.The crossover between metal, Mott-insulator and band insulator in a two-and threedimensional Honeycomb lattice is probed by observing the double occupancy, compressibility and excitation spectrum.The addition of a site-offset leads to the appearance of a competing type of density order -an explicit charge-density wave -which is directly probed and leads to new types of excitations.At lower temperatures, magnetic correlations appear.By freezing out the dynamics of the system and adiabatically merging neighbouring sites,the probability to find singlet or triplet states on different patterns of bonds is directly probed.Using this method we study the dependence of magnetic correlations on geometry in seven different lattice structures, and crossovers between them.We find that lower connectivity quantum effects enhance short-range correlations and compare our results to sophisticated numerical approximations.Ramping the lattice from one geometry to another allows for probing the quantum dynamics of re-distributing magnetic correlations, from the sudden to the adiabatic regime.vComplex quantum many-body systems are ubiquitous in nature, yet their behaviour often remains very challenging to predict with analytical or numerical calculations.In this thesis a different approach is presented, using ultracold fermionic 40 K atoms in a lattice with a flexible geometry, formed by interfering laser beams.This system constitutes a precisely tunable, yet very accessible complex system, which can be used to probe -or "quantum simulate" -phenomena predicted for condensed-matter systems.We demonstrate, theoretically and experimentally, how a periodically modulated system can be described by an effective Hamiltonian on longer time-scales.This approach based on Floquet theory applies even when driving the system far from equilibrium.Using oscillating magnetic field gradients, we create an effectively spin-dependent lattice, where the tunnelling rate for different internal atomic states can be freely tuned.The flexible geometry of the lattice is used to create a honeycomb lattice and study the linear band crossings -known as Dirac cones -characteristic of its band structure.We distort the lattice and use momentum-resolved interband transitions probe the deformation and movement of the Dirac cones and their eventual disappearance when moving towards a square or dimerised geometry.We show that circular modulation of a honeycomb lattice induces imaginary next-nearest neighbour tunnelling, which breaks time-reversal symmetry.In addition, a tunable site-offset is used to break inversion symmetry.We thereby realize a model proposed by F.D.M. Haldane in 1988 and study the predicted topological transition from a trivial to a Chern-insulating regime, characterised by a closing band gap.Furthermore, effects of the Berry curvature are observed in both regimes, using transverse drifts of a driven atomic cloud.Using a mixture of two internal states with tunable scattering between them, we study the Hubbard model on a variety of geometries and with a range of observables.The crossover between metal, Mott-insulator and band insulator in a two-and threedimensional Honeycomb lattice is probed by observing the double occupancy, compressibility and excitation spectrum.The addition of a site-offset leads to the appearance of a competing type of density order -an explicit charge-density wave -which is directly probed and leads to new types of excitations.At lower temperatures, magnetic correlations appear.By freezing out the dynamics of the system and adiabatically merging neighbouring sites,the probability to find singlet or triplet states on different patterns of bonds is directly probed.Using this method we study the dependence of magnetic correlations on geometry in seven different lattice structures, and crossovers between them.We find that lower connectivity quantum effects enhance short-range correlations and compare our results to sophisticated numerical approximations.Ramping the lattice from one geometry to another allows for probing the quantum dynamics of re-distributing magnetic correlations, from the sudden to the adiabatic regime.v

Jan 1, 20150 citations

Floquet engineering of competing antiferromagnetism and

d

-wave superconductivity on the square lattice

d

) and bond (

p

) orbitals of the Lieb lattice brings a

p

-orbital doublon energetically close to a pair in the

d

manifold while keeping all

p

-orbital singlons off resonance, thereby creating a synthetic ``negative-

U

'' center on a Lieb-lattice bond site. Two controlled eliminations then generate a compact bond attraction in the reduced

d

-only model on the square lattice, despite the microscopic repulsion in the

p

orbital. The resulting interaction

J

is tunable independently of the Hubbard repulsion

U_d

on the

d

orbitals, while interference between photon-assisted paths provides access to an intermediate-coupling regime in which

J

and

U_d

are both comparable to the effective hopping. At half filling, a mean-field calculation in this regime finds adjacent antiferromagnetic and

d

We study the phase diagram of the extended Hubbard model on a two-dimensional square lattice, including on-site (

U

) and nearest-neighbor (

V

) interactions, at weak couplings. We show that the charge-density wave phase that is known to occur at half filling when

4V>U

gives way to a

{d}_{xy}

-wave superconducting instability away from half filling, when the Fermi surface is not perfectly nested, and for sufficiently large repulsive

V

and a range of on-site repulsive interaction

U

. In addition, when nesting is further suppressed and in the presence of a nearest-neighbor attraction, a triplet time-reversal breaking

({p}_{x}+i{p}_{y})

-wave pairing instability emerges, competing with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing state that is known to dominate at fillings just slightly away from half. At even smaller fillings, where the Fermi surface no longer presents any nesting, the

({p}_{x}+i{p}_{y})

-wave superconducting phase dominates in the whole regime of on-site repulsions and nearest-neighbor attractions, while

{d}_{xy}

pairing occurs in the presence of on-site attraction. Our results suggest that zero-energy Majorana fermions can be realized on a square lattice in the presence of a magnetic field. For a system of cold fermionic atoms on a two-dimensional square optical lattice, both an on-site repulsion and a nearest-neighbor attraction would be required, in addition to rotation of the system to create vortices. We discuss possible ways of experimentally engineering the required interaction terms in a cold atom system.

d

-wave superfluidity, antiferromagnetic long-range order, and a homogeneous state with coexistence of superfluidity and antiferromagnetism. Using a variational formalism, we show that the energy density of a hole

{e}_{\mathrm{hole}}(x)

has a minimum at doping

x={x}_{c}

that signals phase separation between the antiferromagnetic and

d

-wave paired superfluid phases. The energy of the phase-separated ground state is, however, found to be very close to that of a homogeneous state with coexisting antiferromagnetic and superfluid orders. We explore the dependence of the energy on the interaction strength and on the three-site hopping terms and compare with the nearest-neighbor hopping t-J model.

U

and antiadiabatic (infinite phonon frequency) limits, which gives an effective $t\text{\ensuremath{-}}J$ model with either a density attractive interaction

V

or a

{J}_{P}

term that contributes a larger spin exchange and a density repulsive interaction. To explore how these effective couplings may tune the competition between CDW and SC, we implement the density matrix renormalization group simulations on the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model with

V

{J}_{P}

coupling. We focus on the six-leg cylinder system with the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, which hosts partially filled stripe and

d

-wave SC in phase diagram. By tuning ${t}^{\ensuremath{'}}/t>0$ and

V

{J}_{P}

, we establish two quantum phase diagrams at

\frac{1}{12}

doping ratio. In the SC phases, the increased

V

{J}_{P}

coupling can enhance the quasi-long-range SC order, consistent with some previous findings. Nonetheless, no SC emerges when the partially filled stripe phase disappears with increased

V

{J}_{P}

. Instead, the system has a transition to either a phase-separation-like regime or a filled stripe phase. On the other hand, with increased ${t}^{\ensuremath{'}}/t$, not only the partially filled stripe but the phase separation and filled stripe can also be tuned to SC phase. Our results suggest that although

V

and

{J}_{P}

couplings may strengthen hole binding, the hole dynamics controlled by ${t}^{\ensuremath{'}}/t$ appears to play a more crucial role for obtaining a SC in the $t\text{\ensuremath{-}}J$ model.