We explore the universal properties of interacting fermionic lattice systems, mostly focusing on the development of pairing correlations from attractive interactions. Using renormalization group, we identify a large number of fixed points and show that they correspond to resonant scattering in multiple channels. Pairing resonances in finite-density band insulators occur between quasiparticles and quasiholes living at different symmetry-related wave vectors in the Brillouin zone. This allows a BCS-BEC crossover interpretation of both Cooper and particle-hole pairing. We show that, in two dimensions, the runaway flows of relevant attractive interactions lead to charged-boson-dominated low-energy dynamics in the insulating states and superfluid transitions in the bosonic mean-field or $\mathit{XY}$ universality classes. Analogous phenomena in higher dimensions are restricted to the strong-coupling limit, while, at weak couplings, the transition is in the pair-breaking BCS class. The models discussed here can be realized with ultracold gases of alkali atoms tuned to a broad Feshbach resonance in an optical lattice, enabling experimental studies of pairing correlations in insulators, especially in their universal regimes. In turn, these simple and tractable models capture the emergence of fluctuation-driven superconducting transitions in fermionic systems, which is of interest in the context of high-temperature superconductors.We explore the universal properties of interacting fermionic lattice systems, mostly focusing on the development of pairing correlations from attractive interactions. Using renormalization group, we identify a large number of fixed points and show that they correspond to resonant scattering in multiple channels. Pairing resonances in finite-density band insulators occur between quasiparticles and quasiholes living at different symmetry-related wave vectors in the Brillouin zone. This allows a BCS-BEC crossover interpretation of both Cooper and particle-hole pairing. We show that, in two dimensions, the runaway flows of relevant attractive interactions lead to charged-boson-dominated low-energy dynamics in the insulating states and superfluid transitions in the bosonic mean-field or $\mathit{XY}$ universality classes. Analogous phenomena in higher dimensions are restricted to the strong-coupling limit, while, at weak couplings, the transition is in the pair-breaking BCS class. The models discussed here can be realized with ultracold gases of alkali atoms tuned to a broad Feshbach resonance in an optical lattice, enabling experimental studies of pairing correlations in insulators, especially in their universal regimes. In turn, these simple and tractable models capture the emergence of fluctuation-driven superconducting transitions in fermionic systems, which is of interest in the context of high-temperature superconductors.

arXiv:1006.2378Physical Review BFeb 28, 201110 citations

Pair density wave instability and Cooper pair insulators in gapped fermion systems

Predrag Nikolić, A. A. Burkov, Arun Paramekanti

By analyzing simple models of fermions in lattice potentials we argue that the zero-temperature pairing instability of any ideal band-insulator occurs at a finite momentum. The resulting supersolid state is known as "pair density wave". The pairing momentum at the onset of instability is generally incommensurate as a result of phase-space restrictions and relative strengths of interband and intraband pairing. However, commensurate pairing occurs in the strong-coupling limit and becomes a Cooper-channel analogue of the Halperin-Rice exciton condensation instability in indirect bandgap semiconductors. The exceptional sensitivity of incommensurate pairing to quantum fluctuations can lead to a strongly-correlated insulating regime and a non-BCS transition, even in the case of weak coupling as shown by an exact renormalization group analysis.By analyzing simple models of fermions in lattice potentials we argue that the zero-temperature pairing instability of any ideal band-insulator occurs at a finite momentum. The resulting supersolid state is known as "pair density wave". The pairing momentum at the onset of instability is generally incommensurate as a result of phase-space restrictions and relative strengths of interband and intraband pairing. However, commensurate pairing occurs in the strong-coupling limit and becomes a Cooper-channel analogue of the Halperin-Rice exciton condensation instability in indirect bandgap semiconductors. The exceptional sensitivity of incommensurate pairing to quantum fluctuations can lead to a strongly-correlated insulating regime and a non-BCS transition, even in the case of weak coupling as shown by an exact renormalization group analysis.

arXiv:1007.2433Journal of Physics Conference SeriesJan 1, 20110 citations

A study of Fermionic Cooper pairs

Ramon Carbó‐Dorca, Debraj Nath

Journal of Mathematical ChemistryJul 21, 20230 citations

Condensed Cooper Pairs and Macroscopic Quantum Phenomena

Alfred Rieckers

NATO ASI series. Series B : PhysicsJan 1, 19915 citations

Origin of topological order in a Cooper-pair insulator

Siddhartha Patra, Siddhartha Lal

We present a microscopic derivation of the Cooper-pair insulator (CPI), a topologically ordered counterpart of the $s$ -wave superconductor. For this, we study a generalized model of electrons with attractive interactions with the unitary renormalization group method. The effective Hamiltonian for the CPI corresponds to a gapped, insulating state of quantum matter arising solely from interparticle interactions (and without any need for disorder or coupling to an external magnetic field). The CPI ground-state manifold displays signatures of topological order, including a fourfold degeneracy. The CPI effective Hamiltonian can be written in terms of Wilson loops and contains a topological $\ensuremath{\theta}$-term known to be equivalent to the Chern-Simons term in two spatial dimensions. We show that the long-ranged many-particle entanglement content of the CPI ground state is driven by interhelicity two-particle scattering processes. The state with $\ensuremath{\theta}=0$ possesses the largest bipartite entanglement entropy (EE) and scales logarithmically with subsystem size ( $L$ ). Passage from the CPI to the $s$ -wave BCS superconducting ground state at $T=0$ under RG is demonstrated through the replacement of the long-ranged entanglement of the former by the short-ranged entanglement of the latter. A study of the renormalization of the entanglement in $k$ space shows that the CPI state possesses a hierarchy of scales of entanglement, and that this hierarchy collapses in the BCS state. Our work offers clear evidence for the microscopic origins of topological order in a prototypical system and lays the foundation for similar investigations in other systems of correlated electrons.We present a microscopic derivation of the Cooper-pair insulator (CPI), a topologically ordered counterpart of the $s$-wave superconductor. For this, we study a generalized model of electrons with attractive interactions with the unitary renormalization group method. The effective Hamiltonian for the CPI corresponds to a gapped, insulating state of quantum matter arising solely from interparticle interactions (and without any need for disorder or coupling to an external magnetic field). The CPI ground-state manifold displays signatures of topological order, including a fourfold degeneracy. The CPI effective Hamiltonian can be written in terms of Wilson loops and contains a topological $\ensuremath{\theta}$-term known to be equivalent to the Chern-Simons term in two spatial dimensions. We show that the long-ranged many-particle entanglement content of the CPI ground state is driven by interhelicity two-particle scattering processes. The state with $\ensuremath{\theta}=0$ possesses the largest bipartite entanglement entropy (EE) and scales logarithmically with subsystem size ($L$). Passage from the CPI to the $s$-wave BCS superconducting ground state at $T=0$ under RG is demonstrated through the replacement of the long-ranged entanglement of the former by the short-ranged entanglement of the latter. A study of the renormalization of the entanglement in $k$ space shows that the CPI state possesses a hierarchy of scales of entanglement, and that this hierarchy collapses in the BCS state. Our work offers clear evidence for the microscopic origins of topological order in a prototypical system and lays the foundation for similar investigations in other systems of correlated electrons.

arXiv:2102.12193Physical review. B./Physical review. BOct 29, 20217 citations

\mathit{XY}

universality classes. Analogous phenomena in higher dimensions are restricted to the strong-coupling limit, while, at weak couplings, the transition is in the pair-breaking BCS class. The models discussed here can be realized with ultracold gases of alkali atoms tuned to a broad Feshbach resonance in an optical lattice, enabling experimental studies of pairing correlations in insulators, especially in their universal regimes. In turn, these simple and tractable models capture the emergence of fluctuation-driven superconducting transitions in fermionic systems, which is of interest in the context of high-temperature superconductors.

We present a microscopic derivation of the Cooper-pair insulator (CPI), a topologically ordered counterpart of the

s

-wave superconductor. For this, we study a generalized model of electrons with attractive interactions with the unitary renormalization group method. The effective Hamiltonian for the CPI corresponds to a gapped, insulating state of quantum matter arising solely from interparticle interactions (and without any need for disorder or coupling to an external magnetic field). The CPI ground-state manifold displays signatures of topological order, including a fourfold degeneracy. The CPI effective Hamiltonian can be written in terms of Wilson loops and contains a topological $\ensuremath{\theta}$-term known to be equivalent to the Chern-Simons term in two spatial dimensions. We show that the long-ranged many-particle entanglement content of the CPI ground state is driven by interhelicity two-particle scattering processes. The state with $\ensuremath{\theta}=0$ possesses the largest bipartite entanglement entropy (EE) and scales logarithmically with subsystem size (

L

). Passage from the CPI to the

s

-wave BCS superconducting ground state at

T=0

under RG is demonstrated through the replacement of the long-ranged entanglement of the former by the short-ranged entanglement of the latter. A study of the renormalization of the entanglement in

k

space shows that the CPI state possesses a hierarchy of scales of entanglement, and that this hierarchy collapses in the BCS state. Our work offers clear evidence for the microscopic origins of topological order in a prototypical system and lays the foundation for similar investigations in other systems of correlated electrons.