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arXiv:1004.0923

d-wave superconductivity in the virtual-electron pair quantum liquid

J. M. P. Carmelo

Published Apr 6, 2010• 0 citations
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Abstract

We find evidence that for zero spin density $m=0$, intermediate $U/4t$ values, and a range $x\in (x_c,x_*)$ 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 $T_c$ for a hole concentration range centered at $x_{op}= (x_c+x_*)/2$. 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 $SO(4) = [SU(2)\times SU(2)]/Z_2$ symmetry, the U(1) symmetry broken at $T_c$ is the $c$ fermion U(1) symmetry of Ref. \cite{bipartite}. It is contained in the extended global $SO(3)\times SO(3)\times U(1)=[SO(4)\times U(1)]/Z_2$ 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 $x\in (x_c,x_*)$ to a successful description of the universal properties of the hole-doped cuprate superconductors.

DOI
10.48550/arxiv.1004.0923

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The Hubbard model on a square lattice is one of the most studied condensed-matter quantum problems.Here we find evidence that for intermediate U/4tU/4tU/4t values and a hole-concentration range x∈(xc,x∗)x\in (x_c,x_*)x∈(xc​,x∗​) the ground state of the Hubbard model on the square lattice perturbed by weak three-dimensional (3D) uniaxial anisotropy has long-range d-wave superconducting order. Here ttt is the effective nearest-neighbor transfer integral and UUU the effective on-site repulsion. The lower critical concentration xcx_cxc​ involves the Ginzburg number Gi and is approximately given by xc≈Gi+x0≈0.05x_c\approx {\rm Gi}+x_0\approx 0.05xc​≈Gi+x0​≈0.05. Here $x_0
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.
Effects of the residual interactions of the square-lattice quantum-liquid ccc and s1s1s1 fermions: Superconductivity as a side effect of the short-range spin order
Evidence is found that for zero spin density m=0m=0m=0 and the range 0<x<xc0<x<x_c0<x<xc​ of finite hole concentrations of the short-range spin ordered phase where xc=1/πx_c=1/\pixc​=1/π for U/4t→∞U/4t\to\inftyU/4t→∞ and xc=0.27x_c=0.27xc​=0.27 for U/4t≈1.525U/4t\approx 1.525U/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 ccc and s1s1s1 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[SO(4)\times U(1)]/Z_2[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 ccc and s1s1s1 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: Bi2_{2}2​Sr2_22​CaCu2_22​O8+δ_{8+\delta}8+δ​, Tl2_22​Ba2_22​CuO6+δ_{6+\delta}6+δ​, YBa2_22​Cu3_33​O6+δ_{6+\delta}6+δ​, and La2−x_{2-x}2−x​Srx_xx​CuO4_44​.
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 ccc fermions and spin-neutral composite two-spinon s1s1s1 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-ωωω one-electron scattering rate and normal-state linear-TTT resistivity is achieved. Our results provide strong evidence that the normal-state linear-TTT resistivity is indeed a manifestation of low-temperature scale-invariant physics.
We describe the confining instabilities of a proposed quantum spin liquid underlying the pseudogap metal state of the hole-doped cuprates. The spin liquid can be described by a SU(2) gauge theory of N f = 2 massless Dirac fermions carrying fundamental gauge charges—this is the low-energy theory of a mean-field state of fermionic spinons moving on the square lattice with π -flux per plaquette in the ℤ 2 center of SU(2). This theory has an emergent SO(5) f global symmetry and is presumed to confine at low energies to the Néel state. At nonzero doping (or smaller Hubbard repulsion U at half-filling), we argue that confinement occurs via the Higgs condensation of bosonic chargons carrying fundamental SU(2) gauge charges also moving in π ℤ 2 -flux. At half-filling, the low-energy theory of the Higgs sector has N b = 2 relativistic bosons with a possible emergent SO(5) b global symmetry describing rotations between a d -wave superconductor, period-2 charge stripes, and the time-reversal breaking “ d -density wave” state. We propose a conformal SU(2) gauge theory with N f = 2 fundamental fermions, N b = 2 fundamental bosons, and a SO(5) f ×SO(5) b global symmetry, which describes a deconfined quantum critical point between a confining state which breaks SO(5) f and a confining state which breaks SO(5) b . The pattern of symmetry breaking within both SO(5)s is determined by terms likely irrelevant at the critical point, which can be chosen to obtain a transition between Néel order and d -wave superconductivity. A similar theory applies at nonzero doping and large U , with longer-range couplings of the chargons leading to charge order with longer periods.
Unravelling competing orders emergent in doped Mott insulators and their interplay with unconventional superconductivity is one of the major challenges in condensed matter physics. To explore the possible superconducting state in a doped Mott insulator, we study the square-lattice t-J model with both the nearest-neighbor and next-nearest-neighbor electron hoppings and spin interactions. By using the state-of-the-art density matrix renormalization group calculation with imposing charge U(1) and spin SU(2) symmetries on the six-leg cylinders, we establish a quantum phase diagram including three phases: a stripe charge density wave phase, a superconducting phase without static charge order, and a superconducting phase coexistent with a weak charge stripe order. Crucially, we demonstrate that the superconducting phase has a power-law pairing correlation that decays much slower than the charge density and spin correlations, which is a quasi-1D descendant of the uniform d-wave superconductor in two dimensions. These findings reveal that enhanced charge and spin fluctuations with optimal doping is able to produce robust d-wave superconductivity in doped Mott insulators, providing a foundation for connecting theories of superconductivity to models of strongly correlated systems.