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arXiv:cond-mat/9811236 · Physical review. B, Condensed matter

Dual order parameter for the nodal liquid

Leon Balents, Matthew P. A. Fisher, Chetan Nayak

Published Jul 15, 1999• 162 citations
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Abstract

The guiding conception of vortex-condensation-driven Mott insulating behavior is central to the theory of the nodal liquid. We amplify our earlier description of this idea and show how vortex condensation in two-dimensional (2D) electronic systems is a natural extension of 1D Mott insulating and 2D bosonic Mott insulating behavior. For vortices in an underlying superconducting pair field, there is an important distinction between the condensation of flux $hc/2e$ and flux $hc/e$ vortices. The former case leads to spin-charge confinement, exemplified by the band insulator and the charge-density wave. In the latter case, spin and charge are liberated, leading directly to a 2D Mott insulator exhibiting spin-charge separation. Possible upshots include not only the nodal liquid, but also an undoped antiferromagnetic insulating phase with gapped excitations exhibiting spin-charge separation.

DOI
10.1103/physrevb.60.1654

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The guiding conception of vortex-condensation-driven Mott insulating behavior is central to the theory of the nodal liquid. We amplify our earlier description of this idea and show how vortex condensation in two-dimensional (2D) electronic systems is a natural extension of 1D Mott insulating and 2D bosonic Mott insulating behavior. For vortices in an underlying superconducting pair field, there is an important distinction between the condensation of flux hc/2ehc/2ehc/2e and flux hc/ehc/ehc/e vortices. The former case leads to spin-charge confinement, exemplified by the band insulator and the charge-density wave. In the latter case, spin and charge are liberated, leading directly to a 2D Mott insulator exhibiting spin-charge separation. Possible upshots include not only the nodal liquid, but also an undoped antiferromagnetic insulating phase with gapped excitations exhibiting spin-charge separation.
Motivated by the recent paper of Mross and Senthil [Phys. Rev. B 84, 165126 (2011)], which provides a dual description for a Mott transition from a Fermi liquid to a quantum-spin liquid in two space dimensions, we extend their approach to higher-dimensional cases, and we provide explicit formalism in three space dimensions. Instead of the vortices driving a conventional Fermi liquid into quantum-spin-liquid states in two dimensions (2D), it is the vortex lines that lead to the instability of a Fermi liquid in three dimensions (3D). The extended formalism can result in rich consequences when the vortex lines condense in different degrees of freedom. For example, when the vortex lines condense in charge-phase degrees of freedom, the resulting effective fermionic action is found to be equivalent to that obtained by well-studied slave-particle approaches for Hubbard and/or Anderson lattice models, which confirm the validity of the extended dual formalism in 3D. When the vortex lines condense in spin-phase degrees of freedom, a doublon metal with a spin gap and an instability to the unconventional superconducting pairing can be obtained. In addition, when the vortex lines condense in both phase degrees, an exotic doubled U(1)U(1)U(1) gauge theory occurs, which describes a separation of spin-opposite fermionic excitations. It is noted that the first two features have been discussed in a similar way in 2D and that the last one has not been reported in previous papers. The present paper is expected to be useful in understanding the Mott transition happening beyond two space dimensions.
The pseudogap phase is considered to be a new state of matter in the phase string model of the doped Mott insulator, which is composed of two distinct regimes known as the upper and lower pseudogap phases, respectively. The former corresponds to the formation of spin-singlet pairing, the magnetic characterizations of which have been recently studied [Phys. Rev. B 72, 104520 (2005)]. The latter, as a low-temperature regime of the pseudogap phase, is systematically explored in this work, which is characterized by the formation of the Cooper pair amplitude and described by a generalized Ginzburg-Landau theory. Elementary excitation in this phase is a charge-neutral object carrying spin-1∕21∕21∕2 and locking with a supercurrent vortex, known as a spinon-vortex composite. Such a lower pseudogap phase can be regarded as a vortex liquid state due to the presence of free spinon vortices. Here thermally excited spinon vortices destroy the phase coherence and are responsible for the nontrivial Nernst effect and diamagnetism. The transport entropy and core energy associated with a spinon vortex are determined by the spin degrees of freedom. Such a spontaneous vortex liquid phase can be also considered as a spin liquid with a finite correlation length and gaped S=1∕2S=1∕2S=1∕2 excitations, where a resonancelike nonpropagating spin mode emerges at the antiferromagnetic wavevector $(\ensuremath{\pi},\ensuremath{\pi})$ with a doping-dependent characteristic energy. The superconducting phase is closely related to the lower pseudogap phase by a topological transition with spinon vortices and antivortices forming bound pairs and the emergence of fermionic quasiparticles as holon-spinon-vortex bound objects. A quantitative phase diagram in the parameter space of doping, temperature, and magnetic field is determined. Comparisons with experiments are also made.
We study the nature of the zero-temperature phase transition between a d-wave superconductor and a Mott insulator in two dimensions. In this ``quantum confinement transition,'' spin and charge are confined to form the electron in the Mott insulator. Within a dual formulation, direct transitions from d-wave superconductors at half-filling to insulators with spin-Peierls (as well as other) order emerge naturally. The possibility of striped superconductors is also discussed within the dual formulation. The transition is described by nodal fermions and bosonic vortices, interacting via a long-ranged statistical interaction modeled by two coupled Chern-Simons gauge fields, and the critical properties of this model are discussed.
The pseudogap state of cuprate high-temperature superconductors has been often viewed as either a yet unknown competing order or a precursor state to superconductivity. While awaiting the resolution of the pseudogap problem in cuprates, we demonstrate that local pairing fluctuations, vortex liquid dynamics, and other precursor phenomena can emerge quite generally whenever fermionic excitations remain gapped across the superconducting transition, regardless of the gap origin. Our choice of a tractable model is a lattice band insulator with short-range attractive interactions between fermions in the sss-wave channel. An effective crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) regimes can be identified in any band insulator above two dimensions, while in two dimensions only the BEC regime exists. The superconducting transition is ``unconventional'' (non-pair-breaking) in the BEC regime, identified by either the bosonic mean field or XY\mathit{XY}XY universality class. The insulator adjacent to the superconductor in the BEC regime is a bosonic Mott insulator of Cooper pairs, which may be susceptible to charge density wave ordering. We construct a function of the many-body excitation spectrum whose nonanalytic changes define a sharp distinction between band and Mott insulators. The corresponding ``second-order transition'' can be observed out of equilibrium by driving a Cooper pair laser in the Mott insulator. We explicitly show that the gap for charged bosonic excitations lies below the threshold for Cooper pair breakup in any BEC regime, despite quantum fluctuations. Our discussion ends with a view of possible consequences for cuprates, where antinodal pair dynamics has certain features in common with our simple sss-wave picture.
We present a theoretical approach to describing the Mott transition of electrons on a two-dimensional lattice that begins with the low-energy effective theory of the Fermi liquid. The approach to the Mott transition must be characterized by the suppression of density and current fluctuations that correspond to specific shape deformations of the Fermi surface. We explore the nature of the Mott insulator and the corresponding Mott transition when these shape fluctuations of the Fermi surface are suppressed without making any a priori assumptions about other Fermi surface shape fluctuations. Building on insights from the theory of the Mott transition of bosons, we implement this suppression by identifying and condensing vortex degrees of freedom in the phase of the low-energy electron operator. We show that the resulting Mott insulator is a quantum spin liquid with a spinon Fermi surface coupled to a U(1)U(1)U(1) gauge field, which is usually described within a slave particle formulation. Our approach thus provides a coarse-grained treatment of the Mott transition and the proximate spin liquid that is nevertheless equivalent to the standard slave particle construction. This alternate point of view provides further insight into the novel physics of the Mott transition and the spin liquid state that is potentially useful. We describe a generalization that suppresses spin antisymmetric fluctuations of the Fermi surface that leads to a spin-gapped charge metal previously also discussed in terms of slave particle constructions.
We investigate a microscopic model for strongly correlated electrons with both on-site and nearest-neighbor Coulomb repulsion on a two-dimensional (2D) square lattice. This exhibits a state in which electrons undergo a ``somersault'' in their internal spin space (spin flux) as they traverse a closed loop in external coordinate space. When this spin-12\frac{1}{2}21​ antiferromagnetic (AFM) insulator is doped, the ground state is a liquid of charged, bosonic meron vortices, which for topological reasons are created in vortex-antivortex pairs. The magnetic exchange energy of the distorted AFM background leads to a logarithmic vortex-antivortex attraction which overcomes the direct Coulomb repulsion between holes localized on the vortex cores. This leads to the appearance of preformed charged pairs. We use the configuration interaction (CI) method to study the quantum translational and rotational motion of various charged magnetic solitons and soliton pairs. The CI method systematically describes fluctuation and quantum tunneling corrections to the Hartree-Fock approximation. We find that the lowest-energy charged meron-antimeron pairs exhibit d-wave rotational symmetry, consistent with the symmetry of the cuprate superconducting order parameter. For a single hole in the 2D AFM plane, we find a precursor to spin-charge separation in which a conventional charged spin polaron dissociates into a singly charged meron-antimeron bound pair. This model provides a unified microscopic basis for (i) non-Fermi-liquid transport properties, (ii) d-wave preformed charged carrier pairs, (iii) midinfrared optical absorption, (iv) destruction of AFM long-range order with doping and other magnetic properties, and (v) certain aspects of angle-resolved photoemission spectroscopy.