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arXiv:1804.04130 · Physical review. B./Physical review. B

Z2 fractionalized phases of a solvable disordered t−J model

Wenbo Fu, Yingfei Gu, Subir Sachdev A, Grigory Tarnopolsky

Published Aug 28, 2018• 14 citations
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Abstract

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 $f$, which carries both the electron spin and charge, and a boson $\ensuremath{\phi}$. Both $f$ and $\ensuremath{\phi}$ carry emergent ${\mathbb{Z}}_{2}$ gauge charges. The model has a phase in which the $\ensuremath{\phi}$ bosons are gapped, and the $f$ 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 $f$ 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 ${\mathbb{Z}}_{2}$ fractionalization and the anomalous exponents of the $f$ 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.

DOI
10.1103/physrevb.98.075150

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We describe the phases of a solvable t\ensuremath-Jt\text{\ensuremath{-}}Jt\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 $f$, which carries both the electron spin and charge, and a boson $\ensuremath{\phi}$. Both $f$ and $\ensuremath{\phi}$ carry emergent ${\mathbb{Z}}_{2}$ gauge charges. The model has a phase in which the $\ensuremath{\phi}$ bosons are gapped, and the $f$ 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 \ensuremathϕ\ensuremath{\phi}\ensuremathϕ and fff are gapless, and the electron operator \ensuremath∼f\ensuremathϕ\ensuremath{\sim}f\ensuremath{\phi}\ensuremath∼f\ensuremathϕ has a Fermi liquid-like 1/\ensuremathτ1/\ensuremath{\tau}1/\ensuremathτ propagator in imaginary time, \ensuremathτ\ensuremath{\tau}\ensuremathτ. 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 \ensuremathϕ\ensuremath{\phi}\ensuremathϕ 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.
We investigate a model of electrons with random and all-to-all hopping and spin exchange interactions, with a constraint of no double occupancy. The model is studied in a Sachdev–Ye–Kitaev-like large- M limit with SU( M ) spin symmetry. The saddle-point equations of this model are similar to approximate dynamic mean-field equations of realistic, nonrandom, t - J models. We use numerical studies on both real and imaginary frequency axes, along with asymptotic analyses, to establish the existence of a critical non–Fermi-liquid metallic ground state at large doping, with the spin correlation exponent varying with doping. This critical solution possesses a time-reparameterization symmetry, akin to Sachdev–Ye–Kitaev (SYK) models, which contributes a linear-in-temperature resistivity over the full range of doping where the solution is present. It is therefore an attractive mean-field description of the overdoped region of cuprates, where experiments have observed a linear- T resistivity in a broad region. The critical metal also displays a strong particle–hole asymmetry, which is relevant to Seebeck coefficient measurements. We show that the critical metal has an instability to a low-doping spin-glass phase and compute a critical doping value. We also describe the properties of this metallic spin-glass phase.
This thesis is devoted to the study of Sachdev-Ye-Kitaev (SYK) models, which describe matter without quasiparticles. The SYK model is an exactly solvable model that is hoped to bring insights into the understanding of strongly correlated materials.\nThe main part discusses different aspects of SYK models. Chapter 2 presents numerical studies of the SYK models, shows its non-Fermi liquid nature and verifies the analytical predictions on two-point function and thermal entropies. Chapter 3 considers the thermodynamic and transport properties of the complex SYK models in higher dimensional generalizations, we show some universal relations obtained from SYK models which are consistent with holographic theories. Chapter 4 discusses a supersymmetric generalization of the SYK model. We show a different a scaling from the normal SYK model both analytically and numerically. We also discuss a different origin of the extensive zero temperature entropy in mathcalN=2\\mathcal{N}=2mathcalN=2 theory. Chapter 5 considers a continuous set of boson-fermion SYK models with a tunable low-energy scaling dimension. From the explicit computation of out of time ordered correlators, we find the theory is less chaotic in the free fermion limit. Chapter 6 describes the phases of a solvable ttt-JJJ model of electrons with an SYK-like form. We find a mathbbZ2\\mathbb{Z}_2mathbbZ2​ fractionalization picture where the deconfined phase is similar to the pseudo gap regime of underdoped cuprates, while the `quasi-Higgs' phase is similar to the metallic phase of overdoped cuprates.
Building upon techniques employed in the construction of the Sachdev-Ye-Kitaev model, which is a solvable (0 + 1)-dimensional model of a non-Fermi liquid, we develop a solvable infinite-ranged random-hopping model of fermions coupled to fluctuating U(1)U(1)U(1) gauge fields. In a specific large-NNN limit, our model realizes a gapless non-Fermi-liquid phase, which combines the effects of hopping and interaction terms. We derive the thermodynamic properties of the non-Fermi-liquid phase realized by this model and the charge transport properties of an infinite-dimensional version with spatial structure.
We study the electron spectral function of various zero-temperature spin-charge separated phases in two dimensions. In these phases, the electron is not a fundamental excitation of the system, but rather ``decays'' into a spin-1/2 chargeless fermion (the spinon) and a spinless charge e boson (the chargon). Using low-energy effective theories for the spinons (d-wave pairing plus possible N\'eel order) and the chargons (condensed or quantum-disordered bosons), we explore three phases of possible relevance to the cuprate superconductors: (1) AF∗,{\mathrm{AF}}^{*},AF∗, a fractionalized antiferromagnet where the spinons are paired into a state with long-ranged N\'eel order and the chargons are 1/2-filled and (Mott) insulating; (2) the nodal liquid, a fractionalized insulator where the spinons are d-wave paired and the chargons are uncondensed; and (3) the d-wave superconductor, where the chargons are condensed and the spinons retain a d-wave gap. Working within the Z2{Z}_{2}Z2​ gauge theory of such fractionalized phases, our results should be valid at scales below the energy gap of the vison---the basic vortex excitation in the theory. However, on a phenomenological level, our results should apply to any spin-charge separated system where the excitations have these low-energy effective forms. Comparison with angle-resolved photoemission spectroscopy data in the undoped, pseudogapped, and superconducting regions is made.
The Kitaev model on a honeycomb lattice with bond-dependent Ising interactions offers an exactly solvable model of a quantum spin liquid (QSL) with fractionalized excitations: gapped Z2{Z}_{2}Z2​ fluxes and gapless linearly dispersing Majorana fermions in the isotropic limit (Kx=Ky=Kz{K}_{x}={K}_{y}={K}_{z}Kx​=Ky​=Kz​). We explore the phase diagram along two axes, an external magnetic field hhh, applied out of plane of the honeycomb, and anisotropic interactions Kz{K}_{z}Kz​ larger than the other two (${K}_{x}={K}_{y}\ensuremath{\equiv}K$). For ${K}_{z}/K\ensuremath{\gg}2$ and h=0h=0h=0, the matter Majorana fermions have the largest gap, and the system is described by a gapped Z2{Z}_{2}Z2​ toric code. One of the central questions we address is whether the fractionalized excitations in the different phases have sharp signatures that can be detected in experiments. We show that while the response to single-spin excitations is broad, the spectral function corresponding to two-spin excitations across a bond has sharp signatures that can be attributed to specific anyons. In the toric code regime, the $\ensuremath{\epsilon}=e\ifmmode\times\else\texttimes\fi{}m$ fermion, formed from the bosonic Ising electric (eee) and magnetic (mmm) charges, disperses along a specific one-dimensional direction that provides a fingerprint of fractionalization. At lower Kz{K}_{z}Kz​ in the center of the Abelian phase, in a regime we dub the primordial fractionalized regime, the field generates a hybridization between the $\ensuremath{\epsilon}$ fermion and the Majorana matter fermion, resulting in a $\ensuremath{\psi}$ fermion which too has a distinct quasi-one-dimensional dispersion. All the other phases in the field-anisotropy plane are naturally obtained from this primordial soup. These highly constrained fractonlike dispersions can be observable by inelastic light and neutron scattering, thereby providing ``smoking gun'' signatures of fractionalization in the QSL phase. Our analysis is based on calculations of susceptibilities, topological entanglement entropy, and excitation dynamics, obtained using exact diagonalization and density matrix renormalization group, and supported by perturbation theory.
A Z2{\mathbb{Z}}_{2}Z2​ fractionalized Fermi liquid (FL*) is a novel state of strongly correlated quantum matter. Although it is metallic, it violates Luttinger's theorem on the volume enclosed by the Fermi surface obeyed by conventional metals; this is possible due to the presence of emergent gauge excitations. The authors study here the superconducting states that can arise out of instabilities of such a topological metal. In particular, they focus on a FL* topological metal that has favorable energetics on the square lattice with nematic order, relevant to the cuprate superconductors. They find that a Higgs transition out of this FL* results in confinement of the anyonic degrees of freedom, and the resulting state is a superconductor with broken translation symmetry or time-reversal invariance. In the process, they also establish a complete mapping between bosonic and fermionic descriptions of time-reversal invariant gapped insulating Z2{\mathbb{Z}}_{2}Z2​ spin liquid states on the rectangular lattice, which, on doping with charge carriers, can give rise to FL* phases. They also note a possible connection to the recent observation of pair density waves in the superconducting state of the underdoped cuprates.