Fractionalization, entanglement, and separation: understanding the collective excitations in a spin-orbital chain
Cheng-Chien Chen, Michel van Veenendaal, Thomas P. Devereaux, Krzysztof Wohlfeld
备注：
Original: 9 pages, 9 figures (including supplementary material); a slightly revised version w.r.t. v2 (conclusions unchanged). Revised: the manuscript (v4) is accepted for publication in Physical Review B on March 20, 2015
Using a combined analytical and numerical approach, we study the collective spin and orbital excitations in a spin-orbital chain under a crystal field. Irrespective of the crystal field strength, these excitations can be universally described by fractionalized fermions. The fractionalization phenomenon persists and contrasts strikingly with the case of a spin chain, where fractionalized spinons cannot be individually observed but confined to form magnons in a strong magnetic field. In the spin-orbital chain, each of the fractional quasiparticles carries both spin and orbital quantum numbers, and the two variables are always entangled in the collective excitations. Our result further shows that the recently reported separation phenomenon occurs when crystal fields fully polarize the orbital degrees of freedom. In this case, however, the spinon and orbiton dynamics are decoupled solely because of a redefinition of the spin and orbital quantum numbers.
Helical Luttinger liquids, appearing at the edge of two-dimensional topological insulators, represent a new paradigm of one-dimensional systems, where peculiar quantum phenomena can be investigated. Motivated by recent experiments on charge fractionalization, we propose a setup based on helical Luttinger liquids that allows to time-resolve, in addition to charge fractionalization, also spin-charge separation and pure spin fractionalization. This is due to the combined presence of spin-momentum locking and interactions. We show that electric time-resolved measurements can reveal both charge and…
In quantum Hall edge states and in other one-dimensional interacting systems, charge fractionalization can occur due to the fact that an injected charge pulse decomposes into eigenmodes propagating at different velocities. If the original charge pulse has some spatial width due to injection with a given source-drain voltage, a finite time is needed until the separation between the fractionalized pulses is larger than their width. In the formalism of non-equilibrium bosonization, the above physics is reflected in the separation of initially overlapping square pulses in the effective scattering …
We describe two dimensional models with a metallic Fermi surface which display quantum phase transitions controlled by strongly interacting critical field theories below their upper critical dimension. The primary examples involve transitions with a topological order parameter associated with dislocations in collinear spin density wave ("stripe") correlations: the gapping of the order parameter fluctuations leads to a fractionalization of spin and charge collective modes, and this transition has been proposed as a candidate for the cuprates near optimal doping. The coupling between the order p…
Fractionalization of an electronic quasiparticle into spin, charge and orbital parts is a fundamental and characteristic property of interacting electrons in one dimension. However, real materials are never strictly one-dimensional and the fractionalization phenomena are hard to observe. Here we studied the spin and orbital excitations of the anisotropic ladder material CaCu2O3, whose electronic structure is not one-dimensional. Combining high-resolution resonant inelastic x-ray scattering experiments with theoretical model calculations we show that: (i) spin-orbital fractionalization occurs i…
We discuss possible patterns of electron fractionalization in strongly interacting electron systems. A popular possibility is one in which the charge of the electron has been liberated from its Fermi statistics. Such a fractionalized phase contains in it the seed of superconductivity. Another possibility occurs when the spin of the electron, rather than its charge, is liberated from its Fermi statistics. Such a phase contains in it the seed of magnetism, rather than superconductivity. We consider models in which both of these phases occur and study possible phase transitions between them. We d…
It is well established that at low energies one-dimensional (1D) fermionic systems are described by the Luttinger liquid (LL) theory, that predicts phenomena like spin-charge separation, and charge fractionalization into chiral modes. Here we show through the time evolution of an electron injected into a 1D t-J model, obtained with time-dependent density matrix renormalization group, that a further fractionalization of both charge and spin takes place beyond the hydrodynamic limit. Its dynamics can be understood at the supersymmetric point (J=2t) in terms of the excitations of the Bethe-Ansatz…
The fractionalization of quantum numbers in interacting quantum-many body systems is a central motif in condensed matter physics with prominent examples including the fractionalization of the electron in quantum Hall liquids or the emergence of magnetic monopoles in spin-ice materials. Here we discuss the fractionalization of magnetic moments in three-dimensional Kitaev models into Majorana fermions (and a $\mathbb Z_2$ gauge field) and their emergent collective behavior. We analytically demonstrate that the Majorana fermions form a Weyl superconductor for the Kitaev model on the recently synt…
We construct models of interacting itinerant non-Abelian anyons moving along one-dimensional chains, focusing in particular on itinerant Ising anyon chains, and derive effective anyonic t-J models for the low energy sectors. Solving these models by exact diagonalization, we find a fractionalization of the anyons into charge and (non-Abelian) anyonic degrees of freedom -- a generalization of spin-charge separation of electrons which occurs in Luttinger liquids. A detailed description of the excitation spectrum by combining spectra for charge and anyonic sectors requires a subtle coupling betwee…
We study density response N(k,omega) and one-particle spectra A(k,omega) for a Wigner lattice model at quarter filling using exact diagonalization. We investigate these observables for models with short and long-range electron-electron interaction and show that truncation of the electron repulsion can lead to very different results. The spectra show clear signatures of charge fractionalization into pairs of domain walls, whose interaction can be attractive or repulsive and is controlled by the formal fractional charges. In striking contrast to a bound exciton in N(k,omega), we find an antiboun…
We construct explicit examples of microscopic models that stabilize a variety of fractionalized phases of strongly correlated systems in spatial dimension bigger than one, and in zero external magnetic field. These include models of charge fractionalization in boson-only systems, and various kinds of spin-charge separation in electronic systems. We determine the excitation spectrum and show the consistency with that expected from field theoretic descriptions of fractionalization. Our results are further substantiated by direct numerical calculation of the phase diagram of one of the models.
Electrons have three quantized properties -- charge, spin, and Fermi statistics -- that are directly responsible for a vast array of phenomena. Here we show how these properties can be coherently and dynamically stripped from the electron as it enters certain exotic states of matter known as a quantum spin liquid (QSL). In a QSL, electron spins collectively form a highly entangled quantum state that gives rise to emergent gauge forces and fractionalization of spin, charge, and statistics. We show that certain QSLs host distinct, topologically robust boundary types, some of which allow the elec…
Recently performed resonant inelastic x-ray scattering experiment (RIXS) at the copper L3 edge in the quasi-1D Mott insulator Sr2CuO3 has revealed a significant dispersion of a single orbital excitation (orbiton). This large and unexpected orbiton dispersion has been explained using the concept of spin-orbital fractionalization in which orbiton, which is intrinsically coupled to the spinon in this material, liberates itself from the spinon due to the strictly 1D nature of its motion. Here we investigate this mechanism in detail by: (i) deriving the microscopic spin-orbital superexchange model …
It is known that the separation of electrons into spinons and chargons, the spin-charge separation, plays a decisive role when describing strongly correlated density distributions in one dimension. In this manuscript, we extend the investigation by considering a model for the third electron fractionalization: the separation into spinons, chargons and orbitons -- the last associated with the electronic orbital degree of freedom. Specifically, we deal with two exact constraints of exchange-correlation (XC) density-functionals: (i) The constancy of the highest occupied (HO) Kohn-Sham (KS) eigenva…
The SU(2) slave-boson mean-field theory for the tt'J model is analyzed. The role of next-nearest-neighbor hopping t' on the phase-diagram is studied. We find a pseudogap phase in hole-doped materials (where t'<0). The pseudo-gap phase is a U(1) spin liquid (the staggered-flux phase) with a U(1) gauge interaction and no fractionalization. This agrees with experiments on hole doped samples. The same calculation also indicates that a positive t' favors a Z2 state with true spin-charge separation. The Z2 state that exists when t' > 0.5J can be a candidate for the pseudo-gap phase of electron-doped…
This article surveys the physics of systems proximate to Mott insulators, and presents a classification using conventional and topological order parameters. This classification offers a valuable perspective on a variety of conducting correlated electron systems, from the cuprate superconductors to the heavy fermion compounds. Connections are drawn, and distinctions made, between collinear/non-collinear magnetic order, bond order, neutral spin 1/2 excitations in insulators, electron Fermi surfaces which violate Luttinger's theorem, fractionalization of the electron, and the fractionalization of…
Spin liquids may host emergent quasiparticles, collective excitations of the spin degrees of freedom with characteristic features of Majorana fermions, which experimentally are detectable by broad excitation continua due to spin fractionalization. The latter is predicted for the Kitaev spin liquid, an exactly solvable model with bond-dependent interactions on a two-dimensional honeycomb lattice. Here we report on detailed THz experiments in α-RuCl3, identifying these characteristic fingerprints of Majorana fermions. The continuum intensity decreases and finally vanishes on increasing temperatu…
Spin-charge separation is known to be broken in many physically interesting one-dimensional (1D) and quasi-1D systems with spin-orbit interaction because of which spin and charge degrees of freedom are mixed in collective excitations. Mixed spin-charge modes carry an electric charge and therefore can be investigated by electrical means. We explore this possibility by studying the dynamic conductance of a 1D electron system with image-potential-induced spin-orbit interaction. The real part of the admittance reveals an oscillatory behavior versus frequency that reflects the collective excitation…
Electron fractionalization into spinons and chargeons plays a crucial role in 2D models of strongly correlated electrons. In this paper we show that spin-charge separation is not a phenomenon confined to lower dimensions but, rather, we present a field-theoretic model in which it is realized in 3D. The model involves two gauge fields, a standard one and a two-form gauge field. The physical picture is that of a two-fluid model of chargeons and spinons interacting by the topological BF term. When a Higgs mechanism of the second kind for the two-form gauge field takes place, chargeons and spinons…
Using time-dependent density-matrix renormalization group, we study the time evolution of electronic wave packets in the one-dimensional extended Hubbard model with on-site and nearest neighbor repulsion, U and V, respectively. As expected, the wave packets separate into spin-only and charge-only excitations (spin-charge separation). Charge and spin velocities exhibit non-monotonic dependence on V. For small and intermediate values of V, both velocities increase with V. However, the charge velocity exhibits a stronger dependence than that of the spin, leading to a more pronounced spin-charge s…
Fractionalization is a ubiquitous phenomenon in topological states of matter. In this work, we study the collective behavior of fractionalized topological charges and their instabilities, through the $J_1$-$J_2$-$J_3$ Ising model on a kagome lattice, which can be mapped to a model of interacting topological charges under the constraint of Gauss' law. We find that the recombination of topological charges gives rise to a yet unexplored classical spin liquid. This spin liquid is characterized by an extensive residual entropy, as well as the formation of hexamers of same-sign topological charges. …
I discuss origin and possible experimental manifestations of charge-spin separation in 1D Luttinger and Luther-Emery liquids, the latter describing 1D Mott and Peierls insulators and superconductors. Emphasis is on photoemission where the spectral function generically shows two dispersing peaks associated with the collective charge and spin excitations, and on transport. I analyse the temperature dependences of the charge and spin conductivities of two organic conductors and conclude that most likely, charge-spin separation is not realized there and that they can be described as fluctuating Pe…
Beyond spin-charge separation: Helical modes and topological quantum phase transitions in one-dimensional Fermi gases with spin-orbit and Rabi couplings
We have determined the dynamical dielectric response of a one-dimensional, correlated insulator by carrying out electron energy-loss spectroscopy on Sr2CuO3 single crystals. The observed momentum and energy dependence of the low-energy features, which correspond to collective transitions across the gap, are well described by an extended one-band Hubbard model with moderate nearest neighbor Coulomb interaction strength. An exciton-like peak appears with increasing momentum transfer. These observations provide experimental evidence for spin-charge separation in the relevant excitations of this c…
We investigate tunneling between two spinful Tomonaga-Luttinger liquids (TLLs) realized, e.g., as two crossed nanowires or quantum Hall edge states. When injecting into each TLL one electron of opposite spin, the dc current measured after the crossing differs for singlet, triplet, or product states. This is a striking new non-Fermi liquid feature because the (mean) current in a noninteracting beam splitter is insensitive to spin entanglement. It can be understood in terms of collective excitations subject to spin-charge separation. This behavior may offer an easier alternative to traditional e…
The spectrum of collective excitations in Weyl materials is studied by using consistent hydrodynamics. The corresponding framework includes the vortical and chiral anomaly effects, as well as the dependence on the separations between the Weyl nodes in energy $b_0$ and momentum $\mathbf{b}$. The latter are introduced via the Chern-Simons contributions to the electric current and charge densities in Maxwell's equations. It is found that, even in the absence of a background magnetic field, certain collective excitations (e.g. the helicon-like modes and the anomalous Hall waves) are strongly affec…
A unified theory is developed to explain various types of electronic collective behaviors in doped manganites R$_{1-x}$X$_x$MnO$_3$ (R = La, Pr,Nd etc. and X = Ca, Sr, Ba etc.). Starting from a realistic electronic model, we derive an effective Hamiltonianis by ultilizing the projection perturbation techniques and develop a spin-charge-orbital coherent state theory, in which the Jahn-Teller effect and the orbital degeneracy of e$_g$ electrons in Mn ions are taken into account. Physically, the experimentally observed charge ordering state and electronic phase separation are two macroscopic quan…
Hill et al. have recently measured both the thermal and charge conductivities in the normal state of a high temperature superconductor. Based on the vanishing of the Wiedemann-Franz ratio in the extrapolated zero temperature limit, they conclude that the charge carriers in this material are not fermionic. Here I make a simple observation that the prefactor in the temperature dependence of the measured thermal conductivity is unusually large, corresponding to an extremely small energy scale $T_0 \approx 0.15$ K. I argue that $T_0$ should be interpreted as a collective scale. Based on model-inde…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Pennsylvania State University（宾夕法尼亚州立大学）；University of Texas at Dallas（德克萨斯大学达拉斯分校）；Oden Institute for Computational Engineering and Sciences, University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学研究所）；National Yang Ming Chiao Tung University (NYCU)（国立阳明交通大学）
Intertwined spin and charge dynamics in one-dimensional supersymmetric t-J model
Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交大李政道研究所）；School of Physics Science and Engineering, Tongji University（同济大学物理科学与工程学院）；Shanghai Branch, Hefei National Laboratory（合肥国家实验室上海分部）
