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

Acta Physica Polonica A

Cooper Pair with Nonzero Momentum in System with Spin Dependent Mass of Quasiparticles

Jan Kaczmarczyk, Józef Spałek

Published Apr 1, 2007• 3 citations
Publisher

Abstract

We consider a single Cooper pair with the spin dependent quasiparticle masses, with and without applied magnetic field.Such situation takes place for the strongly correlated electron systems, where a relatively strong Hubbard interaction differentiates the quasiparticle states in the majorityand the minority-spin subbands.In that situation, the two spin subbands in an applied magnetic field are not only shifted one with respect to the other, but also distorted differently, which results from the electronic correlations.Under these circumstances, the fermionic particles composing the Cooper pair, are quantum mechanically distinguishable.In result, the Cooper pair has a nonzero momentum (i.e.produces a stationary current in a similar manner as electron in the Bloch state), and there exists a critical value of the attractive interaction, below which the bound Cooper-pair state is not formed.The presence of the applied field (included via the Zeeman term) does not alter the picture qualitatively.Importance of these results for the formation of the Fulde-Ferrell-Larkin-Ovchinnikov phase in strongly correlated systems is mentioned.

DOI
10.12693/aphyspola.111.595

Related papers

Momentum space imaging of the FFLO state

Alireza Akbari, Peter Thalmeier

In a magnetic field superconductors (SC) with small orbital effect exhibit the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase above the Pauli limiting field. It is characterized by Cooper pairs with finite center of mass momentum and is stabilized by the gain in Zeeman energy of depaired electrons in the imbalanced Fermi gas. The ground state is a coherent superposition of paired and depaired states. This concept, although central to the FFLO state lacks a direct experimental confirmation. We propose that STM quasiparticle interference (QPI) can give a direct momentum space image of the depaired states in the FFLO wave function. For a proof of principle we investigate a 2D single orbital tight binding model with a SC s-wave order parameter. Using the equilibrium values of pair momentum and SC gap we calculate the spectral function of quasiparticles and associated QPI spectrum as function of magnetic field. We show that the characteristic depaired Fermi surface parts appear as a fingerprint in the QPI spectrum of the FFLO phase and we demonstrate its evolution with field strength. Its observation in STM experiments would constitute a direct proof for FFLO ground state wave function.

arXiv:1605.07909•New Journal of Physics•Jun 24, 2016•7 citations

Finite-momentum pairing and magnetic halos in a spin-imbalanced Holstein model

Chunhan Feng, G. G. Batrouni, Richard T. Scalettar

Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductivity, Cooper pairing at finite-momentum driven by spin imbalance, remains difficult to study numerically in two dimensions. The fermion sign problem in the spin-imbalanced attractive Hubbard model severely limits quantum Monte Carlo access to the low temperatures where pairing correlations are most pronounced. We show that the Holstein model, in which the effective attraction is retarded by phonon dynamics, strongly suppresses sign fluctuations under spin imbalance, enabling unbiased determinant quantum Monte Carlo simulations deep in this regime. With increasing magnetization, zero-momentum pairing evolves into finite-momentum FFLO pairing, evidenced by oscillating real-space pair correlations and a shift of the pairing susceptibility peak to finite $\vec{Q}$. The finite-momentum susceptibility substantially exceeds its zero-momentum value. At high density $n \sim 0.875$, near magnetization $m \approx 1-n$ where the majority-spin density approaches half filling, FFLO pairing coexists with checkerboard charge-density-wave order at short range, while staggered longitudinal spin correlations emerge as a magnetic halo at larger distances. These spatial textures provide experimentally accessible signatures of finite-momentum pairing and motivate cold-atom realizations of FFLO physics.

arXiv:2610.03678•Oct 2, 2026•0 citations

Novel Route to a Finite Center-of-Mass Momentum Pairing State for Superconductors: A Current-Driven Fulde-Ferrell-Larkin-Ovchinnikov State

Hyeonjin Doh, Matthew Song, Hae‐Young Kee

The previously studied Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is stabilized by a magnetic field via the Zeeman coupling in spin-singlet superconductors. Here we suggest a novel route to achieve nonzero center-of-mass momentum pairing states in superconductors with Fermi surface nesting. We investigate two-dimensional superconductors under a uniform external current, proportional to a finite pair momentum of q(e). We find that an FFLO state with a spontaneous pair momentum of q(s) is stabilized above a certain critical current that depends on the direction of the external current. A finite q(s) arises in order to make the total pair-momentum of q(t)(=q(s) + q(e)) perpendicular to the nesting vector, which is independent of spin states of Cooper pairs. We also discuss experimental signatures of the FFLO state.

arXiv:cond-mat/0604604•Physical Review Letters•Dec 18, 2006•18 citations

Pairing in two dimensions and possible consequences for superconductivity: extension of Cooper's approach

Soumyadipta Pal

Relation between the binding energy of a pair of electrons in Cooper's configuration and the magnitude of the attractive interaction between them is investigated theoretically for a two-dimensional electron gas when the electron pair has (a) zero and (b) finite center of mass momentum (CMM) first (i) in the background of a passive Fermi sea and (ii) with inclusion of the effect of Pauli exclusion principle in the presence of an active Fermi sea, at zero temperature. In the Cooper's approach (Cooper 1956 Phys. Rev. 104 1189) two pairing electrons are considered above the Fermi sea. Due to Pauli exclusion principle, these two electrons are forbidden to go into the states whose energies are less than the Fermi energy. Consequently, the role of the electrons which form the Fermi sea is passive in formation of electron pair (Fujita and Godoy 2001 Theory of High Temperature Superconductivity (Kluwer Academic Publishers)). However, in the case of an active Fermi sea, implying possible presence of multiple pairs of electrons coming out of the Fermi sea, the two electrons will have less available region in the momentum space for pair formation due to the Pauli blocking effect. In both the situations, threshold values of attractive interaction are found to exist to bind the electrons having finite CMM even with respect to the Fermi energy while with arbitrarily small value of attractive interaction they can be bound only in the case of zero CMM and that too in the presence of a passive Fermi sea. However, the presence of an active Fermi sea strongly opposes the pair formation having lower CMM. Furthermore, raising the range of the attractive interaction below and above the Fermi surface makes the pair formation more difficult due to enhanced Pauli blocking in this case. In the case of a passive Fermi sea however, with the increase in the range of the attractive interaction it becomes easier for the electron pairs to be in both Cooper pair state (CP) and true bound state (TB). These results are totally distinct from those obtained by Randeria et al (1989 Phys. Rev. Lett. 62 981, 1990 Phys. Rev. B 41 327) that a fermionic bound state in true vacuum is a necessary and sufficient condition for the Cooper pairing instability in the presence of a passive Fermi sea, independent of attractive interaction strength implying that for a potential which is attractive everywhere in two dimension, a two-body bound state exists for an arbitrarily weak attraction. Our calculations show, in the mean field approximation, that co-existence of TB state and CP state with various CMMs can occur only beyond a critical attractive interaction strength. This could lead to the formation of incoherent pairs even under strong coupling situation, which may hint onto the pseudo gap formation leading to some of the exotic features in the anomalous normal state seen in some of the exotic superconductors.

Journal of Physics Communications•Oct 16, 2017•3 citations

Unconventional superconducting phases in a correlated two-dimensional Fermi gas of nonstandard quasiparticles: a simple model

Jan Kaczmarczyk, Józef Spałek

We discuss a detailed phase diagram and other microscopic characteristics on the applied magnetic field-temperature (H(a)-T) plane for a simple model of a correlated fluid represented by a two-dimensional (2D) gas of heavy quasiparticles with masses dependent on the spin direction and the effective field generated by the electron correlations. The consecutive transitions between the Bardeen-Cooper-Schrieffer (BCS) and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases are either continuous or discontinuous, depending on the values of H(a) and T. In the latter case, weak metamagnetic transitions occur at the BCS-FFLO boundary. We single out two different FFLO phases, as well as a re-entrant behaviour of one of them at high fields. The results are compared with those for ordinary Landau quasiparticles in order to demonstrate the robustness of the FFLO states against the BCS state for the case with spin-dependent masses (SDM). We believe that the mechanism of FFLO stabilization by SDM is generic: other high-field low-temperature (HFLT) superconducting phases should benefit from SDM as well.

arXiv:0907.3589•Journal of Physics Condensed Matter•Aug 11, 2010•13 citations

Novel dynamics and Cooper-pair momentum measurement in Fulde-Ferrell-Larkin-Ovchinnikov superfluids on optical lattices

Shuning Tan, Jiayi Shi, Peng Zou +2 more

Determining the center-of-mass (COM) momentum of Cooper pairs in unconventional superconductors or superfluids is a topic of great interest in condensed matter physics and ultracold atomic gases. Theoretically, we investigate the dynamical excitations of a two-dimensional spin-polarized attractive Hubbard model on a square optical lattice under an effective Zeeman field by computing the density and spin dynamical structure factors, focusing on phase transition from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid. In the FFLO superfluid, besides the phonon mode in the density channel, a low-energy bogolon mode emerges in the spin channel, which is associated with Bogoliubov quasiparticles on a Bogoliubov Fermi surface. Moreover, the dynamical excitations exhibit pronounced anisotropy in momentum space due to the finite COM momentum. At half filling, the roton mode around [ π , π ] evolves from a point-like minimum into a ring structure shifted by the COM momentum across the BCS-FFLO transition, providing a roton-based protocol to extract the COM momentum. These predictions provide key insights for confirming the existence of FFLO superfluids and understanding their dynamical excitation spectra.

arXiv:2602.01280•iScience•May 26, 2026•0 citations
We consider a single Cooper pair with the spin dependent quasiparticle masses, with and without applied magnetic field.Such situation takes place for the strongly correlated electron systems, where a relatively strong Hubbard interaction differentiates the quasiparticle states in the majorityand the minority-spin subbands.In that situation, the two spin subbands in an applied magnetic field are not only shifted one with respect to the other, but also distorted differently, which results from the electronic correlations.Under these circumstances, the fermionic particles composing the Cooper pair, are quantum mechanically distinguishable.In result, the Cooper pair has a nonzero momentum (i.e.produces a stationary current in a similar manner as electron in the Bloch state), and there exists a critical value of the attractive interaction, below which the bound Cooper-pair state is not formed.The presence of the applied field (included via the Zeeman term) does not alter the picture qualitatively.Importance of these results for the formation of the Fulde-Ferrell-Larkin-Ovchinnikov phase in strongly correlated systems is mentioned.
In a magnetic field superconductors (SC) with small orbital effect exhibit the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase above the Pauli limiting field. It is characterized by Cooper pairs with finite center of mass momentum and is stabilized by the gain in Zeeman energy of depaired electrons in the imbalanced Fermi gas. The ground state is a coherent superposition of paired and depaired states. This concept, although central to the FFLO state lacks a direct experimental confirmation. We propose that STM quasiparticle interference (QPI) can give a direct momentum space image of the depaired states in the FFLO wave function. For a proof of principle we investigate a 2D single orbital tight binding model with a SC s-wave order parameter. Using the equilibrium values of pair momentum and SC gap we calculate the spectral function of quasiparticles and associated QPI spectrum as function of magnetic field. We show that the characteristic depaired Fermi surface parts appear as a fingerprint in the QPI spectrum of the FFLO phase and we demonstrate its evolution with field strength. Its observation in STM experiments would constitute a direct proof for FFLO ground state wave function.
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductivity, Cooper pairing at finite-momentum driven by spin imbalance, remains difficult to study numerically in two dimensions. The fermion sign problem in the spin-imbalanced attractive Hubbard model severely limits quantum Monte Carlo access to the low temperatures where pairing correlations are most pronounced. We show that the Holstein model, in which the effective attraction is retarded by phonon dynamics, strongly suppresses sign fluctuations under spin imbalance, enabling unbiased determinant quantum Monte Carlo simulations deep in this regime. With increasing magnetization, zero-momentum pairing evolves into finite-momentum FFLO pairing, evidenced by oscillating real-space pair correlations and a shift of the pairing susceptibility peak to finite Q⃗\vec{Q}Q​. The finite-momentum susceptibility substantially exceeds its zero-momentum value. At high density n∼0.875n \sim 0.875n∼0.875, near magnetization m≈1−nm \approx 1-nm≈1−n where the majority-spin density approaches half filling, FFLO pairing coexists with checkerboard charge-density-wave order at short range, while staggered longitudinal spin correlations emerge as a magnetic halo at larger distances. These spatial textures provide experimentally accessible signatures of finite-momentum pairing and motivate cold-atom realizations of FFLO physics.
The previously studied Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is stabilized by a magnetic field via the Zeeman coupling in spin-singlet superconductors. Here we suggest a novel route to achieve nonzero center-of-mass momentum pairing states in superconductors with Fermi surface nesting. We investigate two-dimensional superconductors under a uniform external current, proportional to a finite pair momentum of q(e). We find that an FFLO state with a spontaneous pair momentum of q(s) is stabilized above a certain critical current that depends on the direction of the external current. A finite q(s) arises in order to make the total pair-momentum of q(t)(=q(s) + q(e)) perpendicular to the nesting vector, which is independent of spin states of Cooper pairs. We also discuss experimental signatures of the FFLO state.
Relation between the binding energy of a pair of electrons in Cooper's configuration and the magnitude of the attractive interaction between them is investigated theoretically for a two-dimensional electron gas when the electron pair has (a) zero and (b) finite center of mass momentum (CMM) first (i) in the background of a passive Fermi sea and (ii) with inclusion of the effect of Pauli exclusion principle in the presence of an active Fermi sea, at zero temperature. In the Cooper's approach (Cooper 1956 Phys. Rev. 104 1189) two pairing electrons are considered above the Fermi sea. Due to Pauli exclusion principle, these two electrons are forbidden to go into the states whose energies are less than the Fermi energy. Consequently, the role of the electrons which form the Fermi sea is passive in formation of electron pair (Fujita and Godoy 2001 Theory of High Temperature Superconductivity (Kluwer Academic Publishers)). However, in the case of an active Fermi sea, implying possible presence of multiple pairs of electrons coming out of the Fermi sea, the two electrons will have less available region in the momentum space for pair formation due to the Pauli blocking effect. In both the situations, threshold values of attractive interaction are found to exist to bind the electrons having finite CMM even with respect to the Fermi energy while with arbitrarily small value of attractive interaction they can be bound only in the case of zero CMM and that too in the presence of a passive Fermi sea. However, the presence of an active Fermi sea strongly opposes the pair formation having lower CMM. Furthermore, raising the range of the attractive interaction below and above the Fermi surface makes the pair formation more difficult due to enhanced Pauli blocking in this case. In the case of a passive Fermi sea however, with the increase in the range of the attractive interaction it becomes easier for the electron pairs to be in both Cooper pair state (CP) and true bound state (TB). These results are totally distinct from those obtained by Randeria et al (1989 Phys. Rev. Lett. 62 981, 1990 Phys. Rev. B 41 327) that a fermionic bound state in true vacuum is a necessary and sufficient condition for the Cooper pairing instability in the presence of a passive Fermi sea, independent of attractive interaction strength implying that for a potential which is attractive everywhere in two dimension, a two-body bound state exists for an arbitrarily weak attraction. Our calculations show, in the mean field approximation, that co-existence of TB state and CP state with various CMMs can occur only beyond a critical attractive interaction strength. This could lead to the formation of incoherent pairs even under strong coupling situation, which may hint onto the pseudo gap formation leading to some of the exotic features in the anomalous normal state seen in some of the exotic superconductors.
We discuss a detailed phase diagram and other microscopic characteristics on the applied magnetic field-temperature (H(a)-T) plane for a simple model of a correlated fluid represented by a two-dimensional (2D) gas of heavy quasiparticles with masses dependent on the spin direction and the effective field generated by the electron correlations. The consecutive transitions between the Bardeen-Cooper-Schrieffer (BCS) and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases are either continuous or discontinuous, depending on the values of H(a) and T. In the latter case, weak metamagnetic transitions occur at the BCS-FFLO boundary. We single out two different FFLO phases, as well as a re-entrant behaviour of one of them at high fields. The results are compared with those for ordinary Landau quasiparticles in order to demonstrate the robustness of the FFLO states against the BCS state for the case with spin-dependent masses (SDM). We believe that the mechanism of FFLO stabilization by SDM is generic: other high-field low-temperature (HFLT) superconducting phases should benefit from SDM as well.
Determining the center-of-mass (COM) momentum of Cooper pairs in unconventional superconductors or superfluids is a topic of great interest in condensed matter physics and ultracold atomic gases. Theoretically, we investigate the dynamical excitations of a two-dimensional spin-polarized attractive Hubbard model on a square optical lattice under an effective Zeeman field by computing the density and spin dynamical structure factors, focusing on phase transition from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid. In the FFLO superfluid, besides the phonon mode in the density channel, a low-energy bogolon mode emerges in the spin channel, which is associated with Bogoliubov quasiparticles on a Bogoliubov Fermi surface. Moreover, the dynamical excitations exhibit pronounced anisotropy in momentum space due to the finite COM momentum. At half filling, the roton mode around [ π , π ] evolves from a point-like minimum into a ring structure shifted by the COM momentum across the BCS-FFLO transition, providing a roton-based protocol to extract the COM momentum. These predictions provide key insights for confirming the existence of FFLO superfluids and understanding their dynamical excitation spectra.