Superconductivity arises when electrons form Cooper pairs with phase coherence. In contrast, a lack of phase coherence in Cooper pairs can lead to an uncondensed metallic ground state known as the Bose metal state. In this paper, we investigate an attractively interacting fermionic system with nearest-neighbor hopping (t) and next-nearest-neighbor hopping (${t}^{\ensuremath{'}}$) anisotropy between two species of spins in a two-dimensional lattice. Utilizing the constrained path quantum Monte Carlo method, we demonstrate the existence of a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave Cooper pair Bose metal (CPBM) phase with ${t}^{\ensuremath{'}}/t>0.7$. The CPBM phase exhibits a domelike structure in the phase diagram of filling $n\ensuremath{\sim}0.65$, with the maximal region around an optimal ${t}^{\ensuremath{'}}/t\ensuremath{\sim}0.2$, suggesting that an appropriate value of ${t}^{\ensuremath{'}}$ facilitates the formation of the Bose metal. Furthermore, we find that a Bose metal formed by fermions with a closed Fermi surface confirms that the crucial condition for this exotic phenomenon is primarily the anisotropy of the Fermi surface rather than its topology. Our finding of the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave CPBM demonstrates the same pairing symmetry as the pseudogap behavior in cuprates, and its experimental realization in ultracold atom systems is also feasible.Superconductivity arises when electrons form Cooper pairs with phase coherence. In contrast, a lack of phase coherence in Cooper pairs can lead to an uncondensed metallic ground state known as the Bose metal state. In this paper, we investigate an attractively interacting fermionic system with nearest-neighbor hopping (t) and next-nearest-neighbor hopping (${t}^{\ensuremath{'}}$) anisotropy between two species of spins in a two-dimensional lattice. Utilizing the constrained path quantum Monte Carlo method, we demonstrate the existence of a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave Cooper pair Bose metal (CPBM) phase with ${t}^{\ensuremath{'}}/t>0.7$. The CPBM phase exhibits a domelike structure in the phase diagram of filling $n\ensuremath{\sim}0.65$, with the maximal region around an optimal ${t}^{\ensuremath{'}}/t\ensuremath{\sim}0.2$, suggesting that an appropriate value of ${t}^{\ensuremath{'}}$ facilitates the formation of the Bose metal. Furthermore, we find that a Bose metal formed by fermions with a closed Fermi surface confirms that the crucial condition for this exotic phenomenon is primarily the anisotropy of the Fermi surface rather than its topology. Our finding of the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave CPBM demonstrates the same pairing symmetry as the pseudogap behavior in cuprates, and its experimental realization in ultracold atom systems is also feasible.

Physical review. B./Physical review. BApr 16, 20252 citations

Variation of the Bose surface by filling a Cooper-pair Bose metal

Jiahao Su, Ji Liu, Jianyu Li +3 more

The Cooper pair Bose metal (CPBM) is a nonsuperfluid quantum phase in which uncondensed fermion pairs form a ``Bose surface'' in momentum space. We investigate the CPBM in the two-dimensional spin-anisotropic attractive Hubbard model by tuning the next-nearest-neighbor (NNN) hopping ${t}^{\ensuremath{'}}$, carrier filling $n$ , and spin anisotropy $\ensuremath{\alpha}$, using large-scale constrained-path quantum Monte Carlo simulations. A moderate NNN hopping ${t}^{\ensuremath{'}}/t=0.2$ substantially enlarges the CPBM region: the phase extends into weaker anisotropy regimes and coexists with a commensurate charge-density wave (CDW) near half-filling $(n>0.95)$ , where CDW order would otherwise dominate at ${t}^{\ensuremath{'}}=0$. Interestingly, ${t}^{\ensuremath{'}}$ suppresses the overall CDW peak amplitude and introduces a geometric correlation between the orientations of the Fermi and Bose surfaces: for weak Fermi-surface rotations, the Bose surface remains aligned with the lattice axes, while larger distortions drive both surfaces to rotate in tandem. Momentum-resolved pairing distributions reveal that the bosonic pairing channels are jointly controlled by ${t}^{\ensuremath{'}}$ and carrier filling $n$ . For small ${t}^{\ensuremath{'}}$, ${d}_{xy}$ -wave correlations dominate across the entire filling range. In contrast, for larger ${t}^{\ensuremath{'}}$, the dominant pairing symmetry varies with $n$ , reflecting a nontrivial interplay between frustration and density. These findings establish carrier filling and NNN hopping as complementary levers for manipulating CPBM stability and provide concrete criteria for identifying nonsuperfluid bosonic matter in cold-atom and correlated-electron systems.The Cooper pair Bose metal (CPBM) is a nonsuperfluid quantum phase in which uncondensed fermion pairs form a ``Bose surface'' in momentum space. We investigate the CPBM in the two-dimensional spin-anisotropic attractive Hubbard model by tuning the next-nearest-neighbor (NNN) hopping ${t}^{\ensuremath{'}}$, carrier filling $n$, and spin anisotropy $\ensuremath{\alpha}$, using large-scale constrained-path quantum Monte Carlo simulations. A moderate NNN hopping ${t}^{\ensuremath{'}}/t=0.2$ substantially enlarges the CPBM region: the phase extends into weaker anisotropy regimes and coexists with a commensurate charge-density wave (CDW) near half-filling $(n>0.95)$, where CDW order would otherwise dominate at ${t}^{\ensuremath{'}}=0$. Interestingly, ${t}^{\ensuremath{'}}$ suppresses the overall CDW peak amplitude and introduces a geometric correlation between the orientations of the Fermi and Bose surfaces: for weak Fermi-surface rotations, the Bose surface remains aligned with the lattice axes, while larger distortions drive both surfaces to rotate in tandem. Momentum-resolved pairing distributions reveal that the bosonic pairing channels are jointly controlled by ${t}^{\ensuremath{'}}$ and carrier filling $n$. For small ${t}^{\ensuremath{'}}$, ${d}_{xy}$-wave correlations dominate across the entire filling range. In contrast, for larger ${t}^{\ensuremath{'}}$, the dominant pairing symmetry varies with $n$, reflecting a nontrivial interplay between frustration and density. These findings establish carrier filling and NNN hopping as complementary levers for manipulating CPBM stability and provide concrete criteria for identifying nonsuperfluid bosonic matter in cold-atom and correlated-electron systems.

Physical review. B./Physical review. BNov 21, 20250 citations

Variation of Bose surface by Filling in Cooper pair Bose metal

Jiahao Su, Liu Ji, J. Y. Li +3 more

The Cooper pair Bose metal (CPBM) is a non-superfluid quantum phase in which uncondensed fermion pairs form a "Bose surface" in momentum space. We investigate the CPBM in the two-dimensional spin-anisotropic attractive Hubbard model by tuning the next-nearest-neighbor (NNN) hopping t', carrier filling n, and spin anisotropy alpha, using large-scale constrained-path quantum Monte Carlo simulations. A moderate NNN hopping (t'/t = 0.2) substantially enlarges the CPBM region: the phase extends into weaker anisotropy regimes and coexists with a commensurate charge-density wave (CDW) near half-filling (n > 0.95), where CDW order would otherwise dominate at t' = 0. Interestingly, t' suppresses the overall CDW peak amplitude and introduces a geometric correlation between the orientations of the Fermi and Bose surfaces: for weak Fermi-surface rotations, the Bose surface remains aligned with the lattice axes, while larger distortions drive both surfaces to rotate in tandem. Momentum-resolved pairing distributions reveal that the bosonic pairing channels are jointly controlled by t' and carrier filling n. For small t', d_xy-wave correlations dominate across the entire filling range. In contrast, for larger t', the dominant pairing symmetry varies with n, reflecting a nontrivial interplay between frustration and density. These findings establish carrier filling and NNN hopping as complementary levers for manipulating CPBM stability and provide concrete criteria for identifying non-superfluid bosonic matter in cold-atom and correlated-electron systems.The Cooper pair Bose metal (CPBM) is a non-superfluid quantum phase in which uncondensed fermion pairs form a "Bose surface" in momentum space. We investigate the CPBM in the two-dimensional spin-anisotropic attractive Hubbard model by tuning the next-nearest-neighbor (NNN) hopping t', carrier filling n, and spin anisotropy alpha, using large-scale constrained-path quantum Monte Carlo simulations. A moderate NNN hopping (t'/t = 0.2) substantially enlarges the CPBM region: the phase extends into weaker anisotropy regimes and coexists with a commensurate charge-density wave (CDW) near half-filling (n > 0.95), where CDW order would otherwise dominate at t' = 0. Interestingly, t' suppresses the overall CDW peak amplitude and introduces a geometric correlation between the orientations of the Fermi and Bose surfaces: for weak Fermi-surface rotations, the Bose surface remains aligned with the lattice axes, while larger distortions drive both surfaces to rotate in tandem. Momentum-resolved pairing distributions reveal that the bosonic pairing channels are jointly controlled by t' and carrier filling n. For small t', d_xy-wave correlations dominate across the entire filling range. In contrast, for larger t', the dominant pairing symmetry varies with n, reflecting a nontrivial interplay between frustration and density. These findings establish carrier filling and NNN hopping as complementary levers for manipulating CPBM stability and provide concrete criteria for identifying non-superfluid bosonic matter in cold-atom and correlated-electron systems.

arXiv:2505.24405May 30, 20250 citations

Exotic d-wave Cooper Pair Bose Metal in two dimensions

Zhangkai Cao, Jiahao Su, Jianyu Li +5 more

The study of non-Fermi liquids sheds light on unconventional phenomena in condensed matter systems that lie beyond the scope of Landau Fermi liquid theory. One intriguing example is the Bose metal, characterized by an uncondensed bosonic ground state. However, constructing a Bose metal phase in two dimensions (2D) remains a significant challenge.} Utilizing constraint path quantum Monte Carlo and functional renormalization group methods on a fermionic system with spin anisotropy in a 2D lattice, we reveal the emergence of a Cooper pair Bose metal (CPBM) phase in a highly anisotropic regime ( $α< 0.30$ ) with wide range of filling, as proposed in [A. E. Feiguin and M. P. A. Fisher, Phys. Rev. Lett. 103, 025303 (2009)]. Our findings exhibit a visible nonzero momentum Bose surface in the Cooper-pair distribution function, accompanied by a distinct signal of $d_{xy}$ correlation between pairs. Our results highlight that spin-dependent anisotropy in the Fermi surface leads to versatile pairing forms. Platforms such as ultracold atoms in optical lattices and recently proposed altermagnets hold promise for realizing this intriguing phase.The study of non-Fermi liquids sheds light on unconventional phenomena in condensed matter systems that lie beyond the scope of Landau Fermi liquid theory. One intriguing example is the Bose metal, characterized by an uncondensed bosonic ground state. However, constructing a Bose metal phase in two dimensions (2D) remains a significant challenge.} Utilizing constraint path quantum Monte Carlo and functional renormalization group methods on a fermionic system with spin anisotropy in a 2D lattice, we reveal the emergence of a Cooper pair Bose metal (CPBM) phase in a highly anisotropic regime ($α< 0.30$) with wide range of filling, as proposed in [A. E. Feiguin and M. P. A. Fisher, Phys. Rev. Lett. 103, 025303 (2009)]. Our findings exhibit a visible nonzero momentum Bose surface in the Cooper-pair distribution function, accompanied by a distinct signal of $d_{xy}$ correlation between pairs. Our results highlight that spin-dependent anisotropy in the Fermi surface leads to versatile pairing forms. Platforms such as ultracold atoms in optical lattices and recently proposed altermagnets hold promise for realizing this intriguing phase.

arXiv:2405.13405May 22, 20240 citations

Exotic d -wave Cooper pair Bose metal in two dimensions

Zhangkai Cao, Jiahao Su, Jianyu Li +5 more

The study of non-Fermi liquids sheds light on unconventional phenomena in condensed matter systems that lie beyond the scope of Landau Fermi liquid theory. One intriguing example is the Bose metal, characterized by an uncondensed bosonic ground state. However, constructing a Bose metal phase in two dimensions (2D) remains a significant challenge. Utilizing constraint path quantum Monte Carlo and functional renormalization group methods on a fermionic system with spin anisotropy in a 2D lattice, we reveal the emergence of a Cooper pair Bose metal (CPBM) phase in a highly anisotropic regime ($\ensuremath{\alpha}<0.30$) with wide range of filling, as proposed in [A. E. Feiguin et al., Phys. Rev. Lett. 103, 025303 (2009)]. Our findings exhibit a visible nonzero momentum Bose surface in the Cooper-pair distribution function, accompanied by a distinct signal of ${d}_{xy}$ correlation between pairs. Our results highlight that spin-dependent anisotropy in the Fermi surface leads to versatile pairing forms. Platforms such as ultracold atoms in optical lattices and recently proposed altermagnets hold promise for realizing this intriguing phase.The study of non-Fermi liquids sheds light on unconventional phenomena in condensed matter systems that lie beyond the scope of Landau Fermi liquid theory. One intriguing example is the Bose metal, characterized by an uncondensed bosonic ground state. However, constructing a Bose metal phase in two dimensions (2D) remains a significant challenge. Utilizing constraint path quantum Monte Carlo and functional renormalization group methods on a fermionic system with spin anisotropy in a 2D lattice, we reveal the emergence of a Cooper pair Bose metal (CPBM) phase in a highly anisotropic regime ($\ensuremath{\alpha}<0.30$) with wide range of filling, as proposed in [A. E. Feiguin et al., Phys. Rev. Lett. 103, 025303 (2009)]. Our findings exhibit a visible nonzero momentum Bose surface in the Cooper-pair distribution function, accompanied by a distinct signal of ${d}_{xy}$ correlation between pairs. Our results highlight that spin-dependent anisotropy in the Fermi surface leads to versatile pairing forms. Platforms such as ultracold atoms in optical lattices and recently proposed altermagnets hold promise for realizing this intriguing phase.

Physical review. B./Physical review. BDec 26, 20243 citations

Oriented Triplet $p$ -Wave Pairing from Fermi surface Anisotropy and Nonlocal AttractionOriented Triplet $p$-Wave Pairing from Fermi surface Anisotropy and Nonlocal Attraction

Shuning Tan, Ji Liu, Minghuan Zeng +3 more

Using constrained-path quantum Monte Carlo, we map the ground-state phase diagram versus the nearest-neighbor (NN) attraction $V$ and spin-dependent hopping anisotropy $α$ for the two-dimensional attractive $t$ -- $U$ -- $V$ Hubbard model at filling $n\simeq0.85$ . We identify an onsite $s$ -wave superfluid, a Cooper pair Bose metal with an uncondensed Bose surface, and an oriented equal-spin triplet $p$ -wave pairing phase. The NN attraction activates the odd-parity channel, while hopping anisotropy suppresses the competing $s$ -wave coherence and selects a $p_x/p_y$ polar axis, and thus lowers the critical $|V_c|$ for the onset of triplet-dominant $p$ -wave pairing. A channel-resolved Landau analysis provides a criterion for the Landau $p$ -wave scale $V_c^{\mathrm L}(α)$ , consistent with the observed anisotropy dependence of $|V_c|$ . Our results establish how NN interaction and Fermi surface anisotropy cooperate to generate the oriented triplet $p$ -wave pairing, and suggest that cold-atom and altermagnetic platforms could potentially realize this mechanism.Using constrained-path quantum Monte Carlo, we map the ground-state phase diagram versus the nearest-neighbor (NN) attraction $V$ and spin-dependent hopping anisotropy $α$ for the two-dimensional attractive $t$--$U$--$V$ Hubbard model at filling $n\simeq0.85$. We identify an onsite $s$-wave superfluid, a Cooper pair Bose metal with an uncondensed Bose surface, and an oriented equal-spin triplet $p$-wave pairing phase. The NN attraction activates the odd-parity channel, while hopping anisotropy suppresses the competing $s$-wave coherence and selects a $p_x/p_y$ polar axis, and thus lowers the critical $|V_c|$ for the onset of triplet-dominant $p$-wave pairing. A channel-resolved Landau analysis provides a criterion for the Landau $p$-wave scale $V_c^{\mathrm L}(α)$, consistent with the observed anisotropy dependence of $|V_c|$. Our results establish how NN interaction and Fermi surface anisotropy cooperate to generate the oriented triplet $p$-wave pairing, and suggest that cold-atom and altermagnetic platforms could potentially realize this mechanism.

arXiv:2601.11267Jan 16, 20260 citations

n

, and spin anisotropy $\ensuremath{\alpha}$, using large-scale constrained-path quantum Monte Carlo simulations. A moderate NNN hopping ${t}^{\ensuremath{'}}/t=0.2$ substantially enlarges the CPBM region: the phase extends into weaker anisotropy regimes and coexists with a commensurate charge-density wave (CDW) near half-filling

(n>0.95)

, where CDW order would otherwise dominate at ${t}^{\ensuremath{'}}=0$. Interestingly, ${t}^{\ensuremath{'}}$ suppresses the overall CDW peak amplitude and introduces a geometric correlation between the orientations of the Fermi and Bose surfaces: for weak Fermi-surface rotations, the Bose surface remains aligned with the lattice axes, while larger distortions drive both surfaces to rotate in tandem. Momentum-resolved pairing distributions reveal that the bosonic pairing channels are jointly controlled by ${t}^{\ensuremath{'}}$ and carrier filling

n

. For small ${t}^{\ensuremath{'}}$,

{d}_{xy}

-wave correlations dominate across the entire filling range. In contrast, for larger ${t}^{\ensuremath{'}}$, the dominant pairing symmetry varies with

n

, reflecting a nontrivial interplay between frustration and density. These findings establish carrier filling and NNN hopping as complementary levers for manipulating CPBM stability and provide concrete criteria for identifying nonsuperfluid bosonic matter in cold-atom and correlated-electron systems.

α< 0.30

) with wide range of filling, as proposed in [A. E. Feiguin and M. P. A. Fisher, Phys. Rev. Lett. 103, 025303 (2009)]. Our findings exhibit a visible nonzero momentum Bose surface in the Cooper-pair distribution function, accompanied by a distinct signal of

d_{xy}

correlation between pairs. Our results highlight that spin-dependent anisotropy in the Fermi surface leads to versatile pairing forms. Platforms such as ultracold atoms in optical lattices and recently proposed altermagnets hold promise for realizing this intriguing phase.

{d}_{xy}

Oriented Triplet

p

-Wave Pairing from Fermi surface Anisotropy and Nonlocal Attraction

Using constrained-path quantum Monte Carlo, we map the ground-state phase diagram versus the nearest-neighbor (NN) attraction

V

and spin-dependent hopping anisotropy

α

for the two-dimensional attractive

t

U

V

Hubbard model at filling

n\simeq0.85

. We identify an onsite

s

-wave superfluid, a Cooper pair Bose metal with an uncondensed Bose surface, and an oriented equal-spin triplet

p

-wave pairing phase. The NN attraction activates the odd-parity channel, while hopping anisotropy suppresses the competing

s

-wave coherence and selects a

p_x/p_y

polar axis, and thus lowers the critical

|V_c|

for the onset of triplet-dominant

p

-wave pairing. A channel-resolved Landau analysis provides a criterion for the Landau

p

-wave scale

V_c^{\mathrm L}(α)

, consistent with the observed anisotropy dependence of

|V_c|

. Our results establish how NN interaction and Fermi surface anisotropy cooperate to generate the oriented triplet

p

-wave pairing, and suggest that cold-atom and altermagnetic platforms could potentially realize this mechanism.