Interacting random Dirac fermions in superconducting cuprates

We investigate a zero-temperature itinerant antiferromagnetic transition where the fermions possess a d-wave gap. This problem pertains to both the nodal liquid insulating phase and the d-wave superconducting phase of the underdoped cuprates. We find that a non-trivial quantum phase transition exists, and that the quantum critical point is dominated by a long-ranged interaction ( $|x-y|^{-2d}$ ) of the Néel order parameter, which is induced by the Dirac-like fermions near gap nodes. We formulate a Ginzburg-Landau functional and estimate the critical exponents via the large-N expansion method.We investigate a zero-temperature itinerant antiferromagnetic transition where the fermions possess a d-wave gap. This problem pertains to both the nodal liquid insulating phase and the d-wave superconducting phase of the underdoped cuprates. We find that a non-trivial quantum phase transition exists, and that the quantum critical point is dominated by a long-ranged interaction ($|x-y|^{-2d}$) of the Néel order parameter, which is induced by the Dirac-like fermions near gap nodes. We formulate a Ginzburg-Landau functional and estimate the critical exponents via the large-N expansion method.

arXiv:cond-mat/9901208Jan 21, 19991 citation

Massless Dirac Fermions, Gauge Fields, and Underdoped Cuprates

Don H. Kim, Patrick A. Lee, Xiao-Gang Wen

We study $2+1$ dimensional massless Dirac fermions and bosons coupled to a U(1) gauge field as a model for underdoped cuprates. We find that the uniform susceptibility and the specific heat coefficient are logarithmically enhanced (compared to linear-in- $T$ behavior) due to the fluctuation of a transverse gauge field which is the only massless mode at finite boson density. We analyze existing data, and find good agreement in the spin gap phase. Within our picture, the drop of the susceptibility below the superconducting ${T}_{c}$ arises from the suppression of gauge fluctuations.We study $2+1$ dimensional massless Dirac fermions and bosons coupled to a U(1) gauge field as a model for underdoped cuprates. We find that the uniform susceptibility and the specific heat coefficient are logarithmically enhanced (compared to linear-in- $T$ behavior) due to the fluctuation of a transverse gauge field which is the only massless mode at finite boson density. We analyze existing data, and find good agreement in the spin gap phase. Within our picture, the drop of the susceptibility below the superconducting ${T}_{c}$ arises from the suppression of gauge fluctuations.

arXiv:cond-mat/9702218Physical Review LettersSep 15, 199759 citations

Fermion Pairing and Superconductivity

Philippe A. Martin, FRANÇOIS ROTHEN

Springer eBooksJan 1, 20040 citations

Aspects of Highly-Correlated Electron Systems

Nicholas Rombes

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 $\\propto T \\ln T$ 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- $T_c$ 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 La ${}_{2-x}$ Sr ${}_x$ CuO ${}_4$ \\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 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 $\\propto T \\ln T$ 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-$T_c$ 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 La${}_{2-x}$Sr${}_x$CuO${}_4$ \\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.

Jun 1, 20200 citations

|x-y|^{-2d}

) of the Néel order parameter, which is induced by the Dirac-like fermions near gap nodes. We formulate a Ginzburg-Landau functional and estimate the critical exponents via the large-N expansion method.

We study

2+1

dimensional massless Dirac fermions and bosons coupled to a U(1) gauge field as a model for underdoped cuprates. We find that the uniform susceptibility and the specific heat coefficient are logarithmically enhanced (compared to linear-in-

T

behavior) due to the fluctuation of a transverse gauge field which is the only massless mode at finite boson density. We analyze existing data, and find good agreement in the spin gap phase. Within our picture, the drop of the susceptibility below the superconducting

{T}_{c}

arises from the suppression of gauge fluctuations.

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

\\propto T \\ln T

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-

T_c

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 La

{}_{2-x}

{}_x

CuO

{}_4

\\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.