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arXiv:cond-mat/0505100 · Physical Review B

Nexus between quantum criticality and the chemical potential pinning in high- Tc cuprates

T. K. Kopeć

Published Aug 29, 2005• 1 citation
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Abstract

For strongly correlated electrons the relation between total number of charge carriers ${n}_{e}$ and the chemical potential $\ensuremath{\mu}$ reveals for large Coulomb energy the apparently paradoxical pinning of $\ensuremath{\mu}$ within the Mott gap, as observed in high-${T}_{c}$ cuprates. By unraveling consequences of the nontrivial topology of the charge gauge U(1) group and the associated ground state degeneracy we found a close kinship between the pinning of $\ensuremath{\mu}$ and the zero-temperature divergence of the charge compressibility $\ensuremath{\kappa}\ensuremath{\sim}\ensuremath{\partial}{n}_{e}∕\ensuremath{\partial}\ensuremath{\mu}$, which marks a novel quantum criticality governed by topological charges rather than Landau principle of the symmetry breaking.

DOI
10.1103/physrevb.72.060513

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For strongly correlated electrons the relation between total number of charge carriers ne{n}_{e}ne​ and the chemical potential \ensuremathμ\ensuremath{\mu}\ensuremathμ reveals for large Coulomb energy the apparently paradoxical pinning of \ensuremathμ\ensuremath{\mu}\ensuremathμ within the Mott gap, as observed in high-Tc{T}_{c}Tc​ cuprates. By unraveling consequences of the nontrivial topology of the charge gauge U(1) group and the associated ground state degeneracy we found a close kinship between the pinning of \ensuremathμ\ensuremath{\mu}\ensuremathμ and the zero-temperature divergence of the charge compressibility \ensuremathκ\ensuremath∼\ensuremath∂ne∕\ensuremath∂\ensuremathμ\ensuremath{\kappa}\ensuremath{\sim}\ensuremath{\partial}{n}_{e}∕\ensuremath{\partial}\ensuremath{\mu}\ensuremathκ\ensuremath∼\ensuremath∂ne​∕\ensuremath∂\ensuremathμ, which marks a novel quantum criticality governed by topological charges rather than Landau principle of the symmetry breaking.
The concept of topological excitations and the related ground state degeneracy are employed to establish an effective theory of the superconducting state evolving from the Mott insulator for high-Tc{T}_{c}Tc​ cuprates. The theory includes the effects of the relevant energy scales with the emphasis on the Coulomb interaction UUU governed by the electromagnetic U(1) compact group. The results are obtained for the layered $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}_{\ensuremath{\perp}}\ensuremath{-}U\ensuremath{-}J$ system of strongly correlated electrons relevant for cuprates. Casting the Coulomb interaction in terms of composite-fermions via the gauge flux attachment facility, we show that instanton events in the Matsubara ``imaginary time,'' labeled by topological winding numbers, are essential configurations of the phase field dual to the charge. This provides a nonperturbative concept of the topological quantization and displays the significance of discrete topological sectors in the theory governed by the global characteristics of the phase field. We show that for topologically ordered states these quantum numbers play the role of an order parameter in a way similar to the Landau order parameter for conventionally ordered states. In analogy to the usual phase transition that is characterized by a sudden change of the symmetry, the topological phase transitions are governed by a discontinuous change of the topological numbers signaled by the divergence of the zero-temperature topological susceptibility. This defines a quantum criticality ruled by topologically conserved numbers rather than the Landau principle of the symmetry breaking. We show that in the limit of strong correlations topological charge is linked to the average electronic filling number and the topological susceptibility to the electronic compressibility of the system. We exploit the impact of these nontrivial U(1) instanton phase field configurations for the cuprate phase diagram which displays the ``hidden'' quantum critical point covered by the superconducting lobe in addition to a sharp crossover between a compressible normal ``strange metal'' state and a region characterized by a vanishing compressibility, which marks the Mott insulator. Finally, we argue that the existence of robust quantum numbers explains the stability against small perturbation of the system and attributes to the topological ``quantum protectorate'' as observed in strongly correlated systems.
Quantum criticality of metal-insulator transitions in correlated electron systems is shown to belong to an unconventional universality class with violation of the Ginzburg-Landau-Wilson (GLW) scheme formulated for symmetry breaking transitions. This unconventionality arises from an emergent character of the quantum critical point, which appears at the marginal point between the Ising-type symmetry breaking at nonzero temperatures and the topological transition of the Fermi surface at zero temperature. We show that Hartree-Fock approximations of an extended Hubbard model on square lattices are capable of such metal-insulator transitions with unusual criticality under a preexisting symmetry breaking. The obtained universality is consistent with the scaling theory formulated for Mott transitions and with a number of numerical results beyond the mean-field level, implying that preexisting symmetry breaking is not necessarily required for the emergence of this unconventional universality. Examinations of fluctuation effects indicate that the obtained critical exponents remain essentially exact beyond the mean-field level. It further clarifies the whole structure of singularities by a unified treatment of the bandwidth-control and filling-control transitions. Detailed analyses of the criticality, containing diverging carrier density fluctuations around the marginal quantum critical point, are presented from microscopic calculations and reveal the nature as quantum critical ``opalescence.'' The mechanism of emerging marginal quantum critical point is ascribed to a positive feedback and interplay between the preexisting gap formation present even in metals and kinetic energy gain (loss) of the metallic carrier. Analyses of crossovers between GLW type at nonzero temperature and topological type at zero temperature show that the critical exponents observed in (V,Cr)2O3{(\mathrm{V},\mathrm{Cr})}_{2}{\mathrm{O}}_{3}(V,Cr)2​O3​ and $\ensuremath{\kappa}\text{\ensuremath{-}}\mathrm{ET}$-type organic conductors provide us with evidence for the existence of the present marginal quantum criticality.
The electronic $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U\text{\ensuremath{-}}J$ model for the CuO2\mathrm{Cu}{\mathrm{O}}_{2}CuO2​ plane in high-Tc{T}_{c}Tc​ superconducting cuprates is elaborated. It incorporates both charge transfer UUU and exchange correlation JJJ with direct ttt and diagonal ${t}^{\ensuremath{'}}$ hopping. Making use of the Yang concept of the off diagonal long range order (ODLRO) in the reduced density matrix of the electronic system we study the loss of quantum wave coherence (and thereby superconductivity) due to the Coulomb interaction. To this aim the second-quantized Hamiltonian of the model is translated to the phase representation with the help of the topologically constrained path integral formalism. As a result the electrons emerge as composite particles consisting of of spin-carrying neutral fermions and charged topological bosons in a form of ``flux tubes'' with the quantum phase variable dual to the local electron density. The charge carrying bose field forms a collective variable representing the correlated ``sliding'' motion of the entire electronic system. We found that the spin gap $\ensuremath{\Delta}$ which appears in one-particle spectrum is the result of valence bond pairing of neutral fermions without ODLRO and deduce the solution for $\ensuremath{\Delta}$ with ``ddd-wave'' symmetry. In the bosonic sector we derive the effective quantum rotor model along with microscopic phase stiffnesses setting the energy scale for superconductivity and point out the special role played by the diagonal hopping ${t}^{\ensuremath{'}}$. We explore the relationship of the quantum rotor phase model and the Bose-Hubbard system emphasizing the ring topology of quantum phase variable. Furthermore, we obtain the ground state phase diagrams which facilitate the Mott insulator-superconductor transition scenario. We elucidate the role of the chemical potential (doping) and the huge degeneracy of the effective bosonic system in achieving ODLRO with long-range phase coherence in the presence of large Coulomb repulsion. Finally, we discuss a number of relevant issues pertaining to the the present work which include: theoretical aspects of the hierarchy of the energy scales in cuprates, ``spin-charge separation'' concepts, ``local-pair'' and ``phase-fluctuation'' scenarios as well as experimental findings related to the role of the ``ddd-wave'' symmetry of the spin gap and diagonal hopping ${t}^{\ensuremath{'}}$ in cuprates.
Opportunities for topological phase transitions in strongly correlated Fermi systems near a quantum critical point are explored as an alternative to collective scenarios for experimentally observed departures from standard Fermi-liquid behavior. Attention is focused on a quantum critical point at which the effective mass is divergent due to vanishing of the quasiparticle group velocity at the Fermi surface. Working within the original Landau quasiparticle theory, it is demonstrated that the quasiparticle picture can remain meaningful beyond the quantum critical point through rearrangements of the unstable normal Fermi surface and quasiparticle spectrum. Two possibilities emerge at zero temperature, depending on whether the quasiparticle interaction is regular or singular at zero momentum transfer. In the regular case, one type of topological phase transformation leads to a state with a multiconnected Fermi surface. In the singular case, another type of topological phase transition leads to an exceptional state containing a fermion condensate – the Fermi surface swells into a volume in momentum space, within which partial occupation prevails and quasiparticle energies are pinned to the chemical potential. As the temperature increases from zero to a characteristic value Tm, a crossover can occur from the state with multiple Fermi surfaces to that containing a fermion condensate.
We show that the presence of two topological quantum critical points (QCP's) in 2D electronic system on a square lattice imposes strong constraints on density-wave (DW) correlations in the high-T(c) cuprates. Electronic properties of the corresponding strongly correlated system are highly reminiscent of the experimental trends in the high-T(c) cuprates. The most interesting results are the existence at low doping of two single-particle gaps different by order of magnitude both increasing towards low doping and of the specific insulating state characterized by a small chemical potential jump.
Transport measurements on the cuprates suggest the presence of a quantum critical point (QCP) hiding underneath the superconducting dome near optimal hole doping. We provide numerical evidence in support of this scenario via a dynamical cluster quantum Monte Carlo study of the extended two-dimensional Hubbard model. Single-particle quantities, such as the spectral function, the quasi-particle weight and the entropy, display a crossover between two distinct ground states: a Fermi liquid at low filling and a nonFermi liquid with a pseudo-gap at high filling. Both states are found to cross over to a marginal Fermi-liquid state at higher temperatures. For finite next-nearest-neighbour hopping t � , we find a classical critical point at temperature Tc. This classical critical point is found to be associated with a phase-separation transition between a compressible Mott gas and an incompressible Mott liquid corresponding to the Fermi liquid and the pseudo-gap state, respectively. Since the critical temperature Tc extrapolates to zero