Effects of the residual interactions of the square-lattice quantum-liquid $c$ and $s1$ fermions: Superconductivity as a side effect of the short-range spin order
Evidence is found that for zero spin density $m=0$ and the range $0<x<x_c$ of finite hole concentrations of the short-range spin ordered phase where $x_c=1/\pi$ for $U/4t\to\infty$ and $x_c=0.27$ for $U/4t\approx 1.525$ the ground state of the Hubbard model on the square lattice is superconducting. Superconductivity emerges naturally from the effects of the residual interactions of the square-lattice quantum-liquid $c$ and $s1$ fermions and as a side product of the short-range spin correlations. The investigations of this paper rely on a description of that quantum problem in terms of objects which arise from a suitable electron - rotated-electron unitary transformation. Occupancy configurations of such objects generate the energy eigenstates and correspond to the representations of the $[SO(4)\times U(1)]/Z_2$ group which was found recently to refer to the model global symmetry. The studies of this paper focus on the effects of the residual interactions of the $c$ and $s1$ fermions of the square-lattice quantum liquid which emerges from the model in the subspace spanned by the ground state and the excited states generated by application onto it of one- and two-electron operators. An excellent overall agreement is reached for the magnitudes of all energy scales of four representative systems of the hole-doped cuprates: Bi$_{2}$Sr$_2$CaCu$_2$O$_{8+\delta}$, Tl$_2$Ba$_2$CuO$_{6+\delta}$, YBa$_2$Cu$_3$O$_{6+\delta}$, and La$_{2-x}$Sr$_x$CuO$_4$.
Effects of the residual interactions of the square-lattice quantum-liquid c and s1 fermions: Superconductivity as a side effect of the short-range spin order
Evidence is found that for zero spin density m=0 and the range 0<x<xc of finite hole concentrations of the short-range spin ordered phase where xc=1/π for U/4t→∞ and xc=0.27 for U/4t≈1.525 the ground state of the Hubbard model on the square lattice is superconducting. Superconductivity emerges naturally from the effects of the residual interactions of the square-lattice quantum-liquid c and s1 fermions and as a side product of the short-range spin correlations. The investigations of this paper rely on a description of that quantum problem in terms of objects which arise from a suitable electron - rotated-electron unitary transformation. Occupancy configurations of such objects generate the energy eigenstates and correspond to the representations of the [SO(4)×U(1)]/Z2 group which was found recently to refer to the model global symmetry. The studies of this paper focus on the effects of the residual interactions of the c and s1 fermions of the square-lattice quantum liquid which emerges from the model in the subspace spanned by the ground state and the excited states generated by application onto it of one- and two-electron operators. An excellent overall agreement is reached for the magnitudes of all energy scales of four representative systems of the hole-doped cuprates: Bi2Sr2CaCu2O8+δ, Tl2Ba2CuO6+δ, YBa2Cu3O6+δ, and La2−xSrxCuO4.
In this thesis, symmetry and related transformation laws under a suitable unitary transformation are used in the study of the superconducting and magnetic properties of strongly correlated electronic systems. Such a study profits from a recently found global SO(3)×SO(3)×U(1) = [SO(4)×U(1)]/Z2 symmetry in the Hubbard model on a bipartite lattice. The introduction of the square-lattice quantum liquid relies on a general description of the Hubbard model consistent with its exact global SO(3) × SO(3) × U(1) symmetry. Such a quantum liquid is expected to play the same role for many-electron problems with short-range interactions on a square lattice, as a Fermi liquid for three-dimensional isotropic metals. It refers to the Hubbard model on the square lattice in the subspace spanned by the ground state, and the excited states generated by application onto it of one- and two-electron operators. For such a square-lattice quantum liquid only the charge c fermions and spin-neutral two-spinon s1 fermions play an active role. Part of the studies of this thesis use an effective superconductivity theory according to which, for zero spin density m = 0 and the range 0 < x < x∗ of finite hole concentrations of a short-range spin ordered phase, the ground state of the square-lattice quantum liquid of c and s1 fermions with residual interactions is superconducting. (For U/4t > ui ≈ 3/4 the critical hole concentration x∗ increases continuously upon increasing the U/4t value from x∗ = 1/√2 ≈ 0.225 at U/4t = ui to x∗ = 1/ ≈ 0.318 for U/4t → ∞.) Superconductivity emerges naturally from the effects of the residual interactions of the c and s1 fermions and as a side product of the short-range spin correlations. The phase of the c fermion pairs reads = 0 + 1 where the fluctuations of the phases 0 and 1 become large for x → 0 and x → x∗, respectively. For 0 < x < x∗ and x = 0, the symmetry of the action for the phase is a global superconducting U(1) symmetry and a local compact gauge symmetry, respectively. The former U(1) symmetry is identified with the U(1) symmetry of the Hubbard-model global SO(3) × SO(3) × U(1) = [SO(4) × U(1)]/Z2 symmetry. The amplitudes g = g0 g1, g0 = |heiθ0i|, and g1 = |heiθ1i|, their zero-temperature values ˘g ≈ (x/x∗)(1−x/x∗), ˘g0 ≈ (x/x∗), and ˘g1 ≈ (1−x/x∗) and the fluctuations of the phase , play an important role in the physics of the square-lattice quantum liquid. Together with the energy scale 2 0, which gives the maximum magnitude of the s1 fermion spinon pairing energy, they fully control the order parameters of both the short-range spin and long-range superconducting correlations. Indeed, for temperatures T < Tc, the short-range spin ordered phase has both a macroscopic quantum phase-coherent pair superconducting order so that |heiθi| > 0, and a short-range spin order, whereas for Tc < T < T∗ only the latter order prevails yet there remain c fermion pairing correlations such that |heiθ1i| > 0 but heiθ0i = 0. Here the critical temperature Tc and the crossover pseudogap temperature T∗ are fully controlled by the absolute value of the order parameters of the two above orders, respectively. However, the square-lattice quantum liquid alone does not describe quantitatively the unusual properties of the hole-doped cuprates. To fulfill such a goal we profit from a suitable theory according to which the interplay of electronic correlations described by the square-lattice quantum liquid of charge c fermions and spin-neutral two-spinon s1 fermions with intrinsic disorder, is behind the unusual properties of the holedoped hight-temperature cuprate superconductors. By profiting from such an interplay, an excellent quantitative agreement with experiments for the physical quantities of several families of hole-doped cuprates is reached. That includes universal properties such as a dome-like critical-temperature hole-concentration dependence and normal-state linear-T resistivity near optimal doping, extending over a wide temperature window. The results confirm a coexisting two-gap scenario associated with an incoherent two-spinon pairing and a coherent twoc- fermion pairing, respectively. While most of our studies focus on the superconducting and magnetic properties of the square-lattice quantum liquid associated with the Hubbard model on the square lattice, the magnetic properties of a related model – the periodic Anderson model – are also addressed, on the cubic lattice.
We find evidence that for zero spin density m=0, intermediate U/4t values, and a range x∈(xc,x∗) of finite hole concentrations the ground state of the virtual-electron pair quantum liquid obtained from perturbing the square-lattice quantum liquid of Ref. \cite{companion2} by weak three-dimensional (3D) uniaxial anisotropy and intrinsic disorder has long-range d-wave superconducting order. Here t is the effective nearest-neighbor transfer integral and U the effective on-site repulsion. The long-range d-wave superconducting order emerges below a critical temperature Tc for a hole concentration range centered at xop=(xc+x∗)/2. It results from the effects of the residual interactions of the charge c fermions and spin-neutral two-spinon s1 fermions of Ref. \cite{companion2}, as a by-product of the short-range spin correlations. Rather than the U(1) symmetry contained in the η-spin SU(2) symmetry of the SO(4)=[SU(2)×SU(2)]/Z2 symmetry, the U(1) symmetry broken at Tc is the c fermion U(1) symmetry of Ref. \cite{bipartite}. It is contained in the extended global SO(3)×SO(3)×U(1)=[SO(4)×U(1)]/Z2 symmetry of the Hubbard model on the square lattice. Our preliminary results seem to indicate that combining the electronic correlations described by the square-lattice quantum liquid perturbed by 3D uniaxial anisotropy with the very weak effects of intrinsic disorder or superfluid-density anisotropy leads for the hole-concentration range x∈(xc,x∗) to a successful description of the universal properties of the hole-doped cuprate superconductors.
A description of the electronic correlations contained in the Hubbard model on the square-lattice perturbed by very weak three-dimensional uniaxial anisotropy in terms of the residual interactions of charge c fermions and spin-neutral composite two-spinon s1 fermions is used to access further information on the origin of quantum critical behavior in the hole-doped cuprate superconductors. Excellent quantitative agreement with their anisotropic linear-ω one-electron scattering rate and normal-state linear-T resistivity is achieved. Our results provide strong evidence that the normal-state linear-T resistivity is indeed a manifestation of low-temperature scale-invariant physics.
A description of the Hubbard model on a square lattice consistent with its global SO(3)×SO(3)×U(1) symmetry
In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral t, on-site repulsion U, and Na2≫1 sites consistent with its exact global SO(3)×SO(3)×U(1) symmetry is constructed. Our studies profit from the interplay of that recently found global symmetry of the model on any bipartite lattice with the transformation laws under a suitable electron - rotated-electron unitary transformation of a well-defined set of operators and quantum objects. For U/4t>0 the occupancy configurations of these objects generate the energy eigenstates that span the one- and two-electron subspace. Such a subspace as defined in this paper contains nearly the whole spectral weight of the excitations generated by application onto the zero-spin-density ground state of one- and two-electron operators. Our description involves three basic objects: charge c fermions, spin-1/2 spinons, and η-spin-1/2 η-spinons. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the η-spinon (and spinons) that are not invariant under that transformation have η spin 1/2 (and spin 1/2) but are part of η-spin-neutral (and spin-neutral) 2ν-η-spinon (and 2ν-spinon) composite ην fermions (and sν fermions) where ν=1,2,... is the number of η-spinon (and spinon) pairs.The description introduced here is consistent with a Mott-Hubbard insulating ground state with antiferromagnetic long-range order for half filling at x=0 hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations x>0.
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.
In this paper the spin configurations of the ground state and one- and two-electron excited states of the Hubbard model on the square lattice are studied. We profit from a general rotated-electron description, which is consistent with the model global SO(3)×SO(3)×U(1) symmetry. For rotated electrons, doubly and single occupancy are good quantum numbers for on-site repulsion U>0. The above states are within that description generated by occupancy configurations of charge c fermions and spin-singlet two-spinon s1 bond particles. Those describe the charge and spin degrees of freedom, respectively, of the rotated electrons that singly occupy sites. While the c fermions have no internal structure, that of the spin-neutral s1 bond-particle occupancy configurations is here described in terms of spinon occupancies of a well-defined effective spin lattice. In reference \cite{companion} it is confirmed that our results contribute to the further understanding of the role of electronic correlations in the spin spectrum of the parent compound La2CuO4. They are also of interest for studies of ultra-cold fermionic atoms on an optical lattice.