We solve the problem of fermionic pairing mediated by a massless boson in the limit of large coupling constant. At weak coupling, the transition temperature is exponentially small and superconductivity is robust against phase fluctuation. In the strong coupling limit, the pair formation occurs at a temperature of the order of the Fermi energy, however, the actual transition temperature is much smaller due to phase and amplitude fluctuations of the pairing gap. Our model calculations describe superconductivity due to color magnetic interactions in quark matter and in systems close to a ferromagnetic quantum critical point with Ising symmetry. Our strong-coupling results are, however, more general and can be applied to other systems as well, including the antiferromagnetic exchange in 2D used for description of the cuprates.
We solve the problem of fermionic pairing mediated by a massless boson in the limit of large coupling constant. At weak coupling, the transition temperature is exponentially small and superconductivity is robust against phase fluctuation. In the strong coupling limit, the pair formation occurs at a temperature of the order of the Fermi energy, however, the actual transition temperature is much smaller due to phase and amplitude fluctuations of the pairing gap. Our model calculations describe superconductivity due to color magnetic interactions in quark matter and in systems close to a ferromagnetic quantum critical point with Ising symmetry. Our strong-coupling results are, however, more general and can be applied to other systems as well, including the antiferromagnetic exchange in 2D used for description of the cuprates.
We solve the problem of fermionic pairing mediated by a massless boson in the limit of large coupling constant. At weak coupling, the transition temperature is exponentially small and superconductivity is robust against phase fluctuation. In the strong coupling limit, the pair formation occurs at a temperature of the order of the Fermi energy, however, the actual transition temperature is much smaller due to phase and amplitude fluctuations of the pairing gap. Our model calculations describe superconductivity due to color magnetic interactions in quark matter and in systems close to a ferromagnetic quantum critical point with Ising symmetry. Our strong coupling results are, however, more general and can be applied to other systems as well, including the antiferromagnetic exchange in 2D used for description of the cuprates.
We study the interplay between superconductivity and non-Fermi liquid behavior of a Fermi surface coupled to a massless SU(N) matrix boson near the quantum critical point. The presence of thermal infrared singularities in both the fermionic self-energy and the gap equation invalidates the Eliashberg approximation, and makes the quantum-critical pairing problem qualitatively different from that at zero temperature. Taking the large N limit, we solve the gap equation beyond the Eliashberg approximation, and obtain the superconducting temperature Tc as a function of N. Our results show an anomalous scaling between the zero-temperature gap and Tc. For N greater than a critical value, we find that Tc vanishes with a Berezinskii-Kosterlitz-Thouless scaling behavior, and the system retains non-Fermi liquid behavior down to zero temperature. This confirms and extends previous renormalization-group analyses done at T=0, and provides a controlled example of a naked quantum critical point. We discuss the crucial role of thermal fluctuations in relating our results with earlier work where superconductivity always develops due to the special role of the first Matsubara frequency.
Near a quantum critical point in a metal, a strong fermion-fermion interaction, mediated by a soft boson, acts in two different directions: it destroys fermionic coherence and it gives rise to an attraction in one or more pairing channels. The two tendencies compete with each other. We analyze a class of quantum critical models, in which momentum integration and the selection of a particular pairing symmetry can be done explicitly, and the competition between non-Fermi liquid and pairing can be analyzed within an effective model with dynamical electron-electron interaction $V({\mathrm{\ensuremath{\Omega}}}_{m})\ensuremath{\propto}1/{|{\mathrm{\ensuremath{\Omega}}}_{m}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). In this paper, the first in the series, we consider the case T=0 and $0<\ensuremath{\gamma}<1$. We argue that tendency to pairing is stronger, and the ground state is a superconductor. We argue, however, that a superconducting state is highly nontrivial as there exists a discrete set of topologically distinct solutions for the pairing gap ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ ($n=0,1,2,...,\ensuremath{\infty}$). All solutions have the same spatial pairing symmetry, but differ in the time domain: ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ changes sign n times as a function of Matsubara frequency ${\ensuremath{\omega}}_{m}$. The n=0 solution ${\mathrm{\ensuremath{\Delta}}}_{0}({\ensuremath{\omega}}_{m})$ is sign preserving and tends to a finite value at ${\ensuremath{\omega}}_{m}=0$, like in BCS theory. The $n=\ensuremath{\infty}$ solution corresponds to an infinitesimally small $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})$, which oscillates down to the lowest frequencies as $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})\ensuremath{\propto}|{\ensuremath{\omega}}_{m}{|}^{\ensuremath{\gamma}/2}cos[2\ensuremath{\beta}\phantom{\rule{0.16em}{0ex}}log(|{\ensuremath{\omega}}_{m}|/{\ensuremath{\omega}}_{0})]$, where $\ensuremath{\beta}=O(1)$ and ${\ensuremath{\omega}}_{0}$ is of order of fermion-boson coupling. As a proof, we obtain the exact solution of the linearized gap equation at T=0 on the entire frequency axis for all $0<\ensuremath{\gamma}<1$, and an approximate solution of the nonlinear gap equation. We argue that the presence of an infinite set of solutions opens up a new channel of gap fluctuations. We extend the analysis to the case where the pairing component of the interaction has additional factor 1/N and show that there exists a critical Ncr>1, above which superconductivity disappears, and the ground state becomes a non-Fermi liquid. We show that all solutions develop simultaneously once N gets smaller than Ncr.
We begin by scrutinizing a recent proposal that presents an alternate description of the half-filled Landau level in terms of massless Dirac fermions. In Chapter \\ref{chap:HFLL}, we examine the possibility of pairing of these Dirac fermions by numerically solving the coupled Eliashberg equations unlike a related previous calculation (Wang and Chakravarty, 2016). In addition, vertex corrections are calculated to be zero from the Ward identity. We find that pairing is possible in non-zero angular momentum channels; only differences are minor numerical shifts. As before, the pairing leads to the gapped Pfaffian and anti-Pfaffian states. However, in our approximation scheme, pairing is not possible in the putative particle–hole symmetric state for l=0 angular momentum. The specific heat at low temperatures of a system of massless Dirac fermions interacting with a transverse gauge field, expected to be relevant for the half-filled Landau level, is calculated. Using the Luttinger formula, it is found to be proptoTlnT in the leading low temperature limit, due to the exchange of transverse gauge bosons. The result agrees with the corresponding one in the nonrelativistic composite fermion theory of Halperin, Lee and Read of the half-filled Landau level.The rest of the thesis concerns the cuprate high-Tc superconductors (``cuprates''). Conventional wisdom says that beyond the superconducting ``dome'' in cuprates, the material behaves as a Fermi liquid. However, this picture does not help explain the disappearance of the superconducting order parameter, and there are some anomalous measurements that cannot be explained by a Fermi liquid phase. It was proposed by Kopp, Ghosal, and Chakravarty that there is a \\emph{ferromagnetic} phase at zero temperature beyond the superconducting dome, and that fluctuations of the ferromagnetic order parameter compete with the superconducting order parameter and work to suppress the superconducting transition temperature \\cite{Kopp:2007fn}. In Chapter \\ref{chap:ferro} we summarize the experimental evidence for ferromagnetic fluctuations in the overdoped cuprates, and present several calculations supporting the existence of a ferromagnetic ground state in the 2D single-band Hubbard model, which model is thought to provide an adequate description of the cuprate superconductors.Another region of interest in the cuprate phase diagram is the pseudogap phase. It is unclear which of a host of competing order parameters is responsible for the behavior in this phase, such as the recent observation of an anomalous thermal Hall conductance in the cuprate La2−xSrxCuO4 \\cite{Grissonnanche:2019gx}. One promising candidate is the d-density wave state. In Chapter \\ref{chap:DMI}, we investigate the effect that density wave states have on the localized spins of a square lattice. We derive the effective Dzyaloshinskii-Moriya (DM) interaction from first principles and study its effects on both ferromagnetic and antiferromagnetic backgrounds. We find that topologically nontrivial density wave states can induce stable DM interactions among the localized spins of the lattice when an external magnetic field is present. Furthermore, these density wave-induced DM vectors point along the external magnetic field's direction--implying that they break time-reversal and spin rotation symmetries in the same manner. Due to these symmetry considerations alone we find that the underlying magnon excitations cannot induce any thermal Hall effect. Utilizing a Holstein-Primakoff substitution about a mean-field ground state expansion we calculate the topological density wave corrections to magnetic ground state energy, spin canting angles, and the dispersion of the magnons for both the ferromagnetic and antiferromagnetic cases.
We investigate the effects of pairing fluctuations in fermionic superfluids/superconductors where pairing occurs among three species (colors) of fermions. Such color superfluids/superconductors can be realized in three-component atomic Fermi gases and in dense quark matter. The superfluidity/superconductivity is characterized by a three-component order parameter which denotes the pairing among the three colors of fermions. Because of the SU(3) symmetry of the Hamiltonian, one color does not participate in pairing. This branch of fermionic excitation is gapless in the naive BCS mean-field description. In this paper, we adopt a pairing fluctuation theory to investigate the pairing fluctuation effects on the unpaired color in strongly coupled atomic color superfluids and quark color superconductors. At low temperature, a large pairing gap of the paired colors suppresses the pairing fluctuation effects for the unpaired color, and the spectral density of the unpaired color shows a single Fermi-liquid peak, which indicates the naive mean-field picture remains valid. As the temperature is increased, the spectral density of the unpaired color generally exhibits a three-peak structure: The Fermi-liquid peak remains but gets suppressed, and two pseudogaplike peaks appears. At and above the superfluid transition temperature, the Fermi-liquid peak disappears completely and all three colors exhibit pseudogaplike spectral density. The coexistence of Fermi liquid and pseudogap behavior is generic for both atomic color superfluids and quark color superconductors.
Bosonization of Cooper pairs and novel Bose-liquid superconductivity and superfluidity in high-Tc cuprates and other systems
The criteria for bosonization of Cooper pairs and novel Bose-liquid superconductivity and superfluidity in high-Tc cuprates and other pseudogap matters are formulated by using the uncertainty principle and the boson mean field theory. We argue that the BCS-type s- or d-wave superconductivity occurring in the fermionic limit of Cooper pairs (which exist in ordinary metals and heavily overdoped cuprates with large Fermi energies) is not characteristic of underdoped to overdoped cuprates with low Fermi energies. The superconducting order parameter in high-Tc cuprates and other pseudogap matters cannot be determined as the BCS-like (s- or d-wave) gap. We show that the unconventional superconductivity/superfluidity occurring in the bosonic limit of Cooper pairs exists in low Fermi energy systems where the bosonic Cooper pairs are formed at a pseudogap temperature T* above the uperconducting/superfluid transition temperature Tc and then such bosons condenses into a Bose superfluid at Tc. The pair condensation of attracting bosons occurs at Tc and then their single particle condensation sets in at Tc∗ lower than Tc (in three dimensions (3D)) or at T=0 (in two dimensions (2D)). The coherent single particle and pair condensates of bosons exist as two distinct superfluid phases below Tc. By solving the mean field equations for high-Tc cuprates, the novel superconducting states (i.e., a vortex-like state existing below the temperature Tv=Tc2D lower than T* but higher than Tc=Tc3D and two superconducting phases below Tc) and their properties characterized by the boson superfluid stiffness are determined and compared with the key experimental findings. The novel Bose-liquid superconductivity and superfluidity exists also in other exotic superconductors/superfluids, including quantum liquids, atomic Fermi gases and low-density nuclear matter.