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arXiv:cond-mat/0008324

Quantum Number Fractionization in Cuprates: U(1) RVB Gauge Theory and Senthil Fisher Constructions

G. Baskaran

Published Aug 22, 2000• 0 citations
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Abstract

We critically look at Senthil Fisher constructions of quantum number fractionization in cuprates. The first construction, while mathematically correct is of limited relevance to cuprates. The second one has a missing local U(1) symmetry. Once this aspect is repaired, it reveals itself as the U(1) RVB gauge theory construction of quantum number fractionization. An approximate local $Z_2$ symmetry arises in the spin sector for a reason very different from Senthil and Fisher's. The charge sector continues to have the local U(1) symmetry and the metallic spin gap phase of the 2d cuprates may, in principle, carry (irrational) fractional charge $e^* \neq e$ obeying Haldane's fractional exclusion statistics like the 1d models.

DOI
10.48550/arxiv.cond-mat/0008324

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We critically look at Senthil Fisher constructions of quantum number fractionization in cuprates. The first construction, while mathematically correct is of limited relevance to cuprates. The second one has a missing local U(1) symmetry. Once this aspect is repaired, it reveals itself as the U(1) RVB gauge theory construction of quantum number fractionization. An approximate local Z2Z_2Z2​ symmetry arises in the spin sector for a reason very different from Senthil and Fisher's. The charge sector continues to have the local U(1) symmetry and the metallic spin gap phase of the 2d cuprates may, in principle, carry (irrational) fractional charge e∗≠ee^* \neq ee∗=e obeying Haldane's fractional exclusion statistics like the 1d models.
Quantum dimer models are believed to capture the essential physics of antiferromagnetic phases dominated by short-ranged valence bond configurations. We show that these models arise as particular limits of Ising (Z2){(Z}_{2})(Z2​) gauge theories, but that in these limits the system develops a larger local U(1) invariance that has different consequences on different lattices. Conversely, we note that the standard Z2{Z}_{2}Z2​ gauge theory is a generalized quantum dimer model, in which the particular relaxation of the hardcore constraint for the dimers breaks the U(1) down to Z2.{Z}_{2}.Z2​. These mappings indicate that at least one realization of the Senthil--Fisher proposal for fractionalization is exactly the short ranged resonating valence bond (RVB) scenario of Anderson and of Kivelson, Rokhsar and Sethna. They also suggest that other realizations will require the identification of a local low energy, Ising link variable and a natural constraint. We also discuss the notion of topological order in Z2{Z}_{2}Z2​ gauge theories and its connection to earlier ideas in RVB theory. We note that this notion is not central to the experiment proposed by Senthil and Fisher to detect vortices in the conjectured Z2{Z}_{2}Z2​ gauge field.
We develop a new theoretical framework for describing and analyzing exotic phases of strongly correlated electrons which support excitations with fractional quantum numbers. Starting with a class of microscopic models believed to capture much of the essential physics of the cuprate superconductors, we derive a new gauge theory---based upon a discrete Ising or Z2{Z}_{2}Z2​ symmetry---which interpolates naturally between an antiferromagnetic Mott insulator and a conventional d-wave superconductor. We explore the intervening regime, and demonstrate the possible existence of an exotic fractionalized insulator, the nodal liquid, as well as various more conventional insulating phases exhibiting broken lattice symmetries. A crucial role is played by vortex configurations in the Z2{Z}_{2}Z2​ gauge field. Fractionalization is obtained if they are uncondensed. Within the insulating phases, the dynamics of these Z2{Z}_{2}Z2​ vortices in two dimensions is described, after a duality transformation, by an Ising model in a transverse field, the Ising spins representing the Z2{Z}_{2}Z2​ vortices. The presence of an unusual Berry's phase term in the gauge theory leads to a doping-dependent ``frustration'' in the dual Ising model, being fully frustrated at half filling. The Z2{Z}_{2}Z2​ gauge theory is readily generalized to a variety of different situations, in particular, it can also describe three-dimensional insulators with fractional quantum numbers. We point out that the mechanism of fractionalization for d>1d>1d>1 is distinct from the well-known one-dimensional spin--charge separation. Other interesting results include a description of an exotic fractionalized superconductor in two or higher dimensions.
We show that in the $SU(2)\ifmmode\times\else\texttimes\fi{}U(1)$ spin-charge gauge approach we developed earlier one can attribute consistently an exclusion statistics with parameter 1/2 to the spinless charge carriers of the $t\text{\ensuremath{-}}J$ model in two dimensions, as it occurs in one dimension. Like the one-dimensional case, the no-double occupation constraint is at the origin of this fractional exclusion statistics. With this statistics we recover a large Fermi volume of holes at high dopings, close to that of the tight binding approximation. Furthermore, the composite nature of the hole, made of charge and spin carriers only weakly bounded, can provide a natural explanation of many unusual experimental features of the hole-doped cuprates.
In this paper we reexamine the problem of the separation of spin and charge degrees of freedom in two-dimensional strongly correlated systems. We establish a set of sufficient conditions for the occurrence of spin and charge separation. Specifically, we discuss this issue in the context of the Heisenberg model for spin-1/2 on a square lattice with nearest (J1{\mathit{J}}_{1}J1​) and next-nearest (J2{\mathit{J}}_{2}J2​) neighbor antiferromagnetic couplings. Our formulation makes explicit the existence of a local SU(2) gauge symmetry once the spin-1/2 operators are replaced by bound states of spinons. The mean-field theory for the spinons is solved numerically as a function of the ratio J2{\mathit{J}}_{2}J2​/J1{\mathit{J}}_{1}J1​ for the so-called s-RVB ansatz (s-RVB stands for short range resonating valence bond). A second-order phase transition exists in a flux state for J2{\mathit{J}}_{2}J2​/J1{\mathit{J}}_{1}J1​ larger than the critical value (J2/{\mathit{J}}_{2/}J2/​J1{\mathit{J}}_{1}J1​)cr{)}_{\mathrm{cr}})cr​ identify the range 0J2{\mathit{J}}_{2}J2​/J1{\mathit{J}}_{1}J1​(J2{\mathit{J}}_{2}J2​/J1{\mathit{J}}_{1}J1​)cr{)}_{\mathrm{cr}})cr​ as the s-RVB phase. It is characterized by the existence of a finite gap to the elementary excitations (spinons) and the breakdown of all the continuous gauge symmetries. An effective continuum theory for the spinons and the gauge degrees of freedom is constructed just below the onset of the flux phase. We argue that this effective theory is consistent with the deconfinement of the spinons carrying the fundamental charge of the gauge group. We contrast this result with the study of the one-dimensional quantum antiferromagnet within the same approach. We show that in the one-dimensional model, the spinons of the gauge picture are always confined and thus cannot be identified with the gapless spin-1/2 excitations of the quantum antiferromagnet Heisenberg model.
We describe the phases of a solvable $t\text{\ensuremath{-}}J$ model of electrons with infinite-range, and random, hopping, and exchange interactions, similar to those in the Sachdev-Ye-Kitaev models. The electron fractionalizes, as in an ``orthogonal metal,'' into a fermion fff, which carries both the electron spin and charge, and a boson $\ensuremath{\phi}$. Both fff and $\ensuremath{\phi}$ carry emergent Z2{\mathbb{Z}}_{2}Z2​ gauge charges. The model has a phase in which the $\ensuremath{\phi}$ bosons are gapped, and the fff fermions are gapless and critical, and so the electron spectral function is gapped. This phase can be considered as a toy model for the underdoped cuprates, without spatial structure. The model also has an extended, critical, ``quasi-Higgs'' phase where both $\ensuremath{\phi}$ and fff are gapless, and the electron operator $\ensuremath{\sim}f\ensuremath{\phi}$ has a Fermi liquid-like $1/\ensuremath{\tau}$ propagator in imaginary time, $\ensuremath{\tau}$. So while the electron spectral function has a Fermi liquid form, other properties are controlled by Z2{\mathbb{Z}}_{2}Z2​ fractionalization and the anomalous exponents of the fff and $\ensuremath{\phi}$ excitations. This quasi-Higgs phase is proposed as a toy model of the overdoped cuprates. We also describe the critical state separating these two phases.
Topological phases of matter are usually realized in deconfined phases of gauge theories. In this context, confined phases with strongly fluctuating gauge fields seem to be irrelevant to the physics of topological phases. For example, the low-energy theory of the two-dimensional (2D) toric code model (i.e., the deconfined phase of Z2{\mathbb{Z}}_{2}Z2​ gauge theory) is a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ Chern-Simons theory in which gauge charges (i.e., eee and mmm particles) are deconfined and the gauge fields are gapped, while the confined phase is topologically trivial. In this paper, we point out a route to constructing exotic three-dimensional (3D) gapped fermionic phases in a confining phase of a gauge theory. Starting from a parton construction with strongly fluctuating compact $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ gauge fields, we construct gapped phases of interacting fermions by condensing two linearly independent bosonic composite particles consisting of partons and $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ magnetic monopoles. This can be regarded as a 3D generalization of the 2D Bais-Slingerland condensation mechanism. Charge fractionalization results from a Debye-H\"uckel--type screening cloud formed by the condensed composite particles. Within our general framework, we explore two aspects of symmetry-enriched 3D Abelian topological phases. First, we construct a new fermionic state of matter with time-reversal symmetry and $\mathrm{\ensuremath{\Theta}}\ensuremath{\ne}\ensuremath{\pi}$, the fractional topological insulator. Second, we generalize the notion of anyonic symmetry of 2D Abelian topological phases to the charge-loop excitation symmetry (Charles\mathsf{Charles}Charles) of 3D Abelian topological phases. We show that line twist defects, which realize Charles\mathsf{Charles}Charles transformations, exhibit non-Abelian fusion properties.