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Spin Charge Separation in the Quantum Spin Hall State SLAC-PUB-13922 Spin Charge Separation in the Quantum Spin Hall State Xiao-Liang Qi and Shou-Cheng Zhang Department of Physics, McCullough Building, Stanford University, Stanford, CA 94305-4045 (Dated: February 2, 2008) The quantum spin Hall state is a topologically non-trivial insulator state protected by the time reversal symmetry. We show that such a state always leads to spin-charge separation in the presence of a π flux. Our result is generally valid for any interacting system. We present a proposal to ex- perimentally observe the phenomenon of spin-charge separation in the recently discovered quantum spin Hall system. PACS numbers: 72.25.Dc, 73.43.-f, 05.30.Pr, 71.10.Pm Spin-charge separation is one of the deepest concepts tal missing links in the QSH systems. Needed is a general in condensed matter physics. In the Su-Schrieffer-Heeger classification of time reversal invariant (TRI) topological model of polyacetylene[1], a domain wall induces two insulators in two dimensions which is valid in the presence mid-gap states, one for each spin orientation of the elec- of arbitrary interactions. Such a general classification be- tron. If both states are unoccupied, or both states are yond the single particle band picture is especially called occupied, the domain wall soliton has charge ±e but no for since the concept of a topological Mott insulator has spin. If only one of the state is occupied, the domain recently been introduced[14]. More importantly, we need wall soliton has spin Sz = ±1/2 but no charge. In this to find experimentally measurable properties which di- remarkable way, the two fundamental degrees of free- rectly demonstrate the topological non-triviality of the dom of an electron is split apart. Since then, the con- QSH state. cept of spin-charge separation has become a corner stone In this paper, we solve both problems by providing a in condensed matter physics. This phenomenon occurs deep connection between the concept of spin-charge sep- generally in interacting quantum many-body systems in aration and the QSH effect. Following Laughlin’s argu- one dimension, and can be demonstrated convincingly ment for the QH effect, we consider the adiabatic inser- by the bosonization techniques. The concept has also tion of a pure gauge flux in the QSH state. We show been generalized to two dimensions. In particular, it is that there are four different ways of reaching the final conjectured that such a phenomenon occurs in the high flux of π, and that these four processes create the spin- temperature superconductors[2, 3]. However, this phe- charge separated holon, chargeon and two spinon states nomenon has not yet been convincingly observed in any which are exponentially localized near the flux. We then two dimensional systems. prove two general theorems providing a Z2 classification Recently, a new two dimensional quantum state of of TRI insulators in two dimensions. This new classifi- matter has been theoretically proposed[4, 5, 6]. The cation scheme is generally valid in the presence of many- quantum spin Hall (QSH) state is a topologically non- body interactions, and leads to spin-charge separation as trivial state of matter protected by the time reversal sym- its direct physical consequence. Finally, we propose an metry. It has a bulk insulating gap, but has helical edge experimental setting to observe the phenomenon of spin- states on the sample boundary, where electron states with charge separation in the recently discovered QSH system. opposite spins counter-propagate at a given edge. This We first present an argument which is physically intu- novel quantum state of matter has recently been theo- itive, but only valid when there is at least a Us(1) spin retically predicted[6] and experimentally observed[7] in rotation symmetry. In this case, the QSH effect is simply the HgT e quantum wells. The topological property of defined as two copies of QH, with opposite Hall conduc- the quantum Hall (QH) state is described by an inte- tances of ±e2/h for opposite spin orientations. Without ger Chern number[8], defined over the single particle mo- loss of generality, we first consider a disk geometry with mentum space, and this integer is directly related to the a gauge flux of φ↑ = φ↓ = hc/2e, or simply π in units experimentally observed quantum of Hall conductance. of ~ = c = e = 1, through a hole at the center, see This construction can also be generalized to an interact- Fig. 1. The gauge flux acts on both spin orientations, ing system, where the Chern number is defined over the and the π flux preserves time reversal symmetry. We space of twisted boundary conditions[9]. The topological consider adiabatic processes of φ↑(t) and φ↓(t), where property of the QSH state is currently described by a Z2 φ↑(t) = φ↓(t) = 0 at t = 0, and φ↑(t) = φ↓(t) = ±π topological number[10, 11, 12, 13], which is also defined at t = 1. Since the flux of π is equivalent to the flux over the single particle momentum space. This Z2 clas- of −π, there are four different adiabatic processes all sification has provided an important insight on the topo- reaching the same final flux configuration. In process logical non-triviality of the QSH state. However, unlike (a), φ↑(t) = −φ↓(t) and φ↑(t = 1) = π. In process the situation in QH systems, there are several fundamen- (b), φ↑(t) = −φ↓(t) and φ↑(t = 1) = −π. In process SIMES, SLAC National Accelerator Center, 2575 Sand Hill Road, Menlo Park, CA 94309 Work supported in part by US Department of Energy contract DE-AC02-76SF00515. 2 (c), φ↑(t) = φ↓(t) and φ↑(t = 1) = π. In process (d), and the charge quantum numbers are sharply defined φ↑(t) = φ↓(t) and φ↑(t = 1) = −π. These four pro- quantum numbers[15]. The insulating state has a bulk cesses are illustrated in Fig 1. Note that process (a) and gap ∆, and an associated coherence length ξ = ~vF /∆. (b) preserves time reversal symmetry at all intermediate As long as the radius of the Gaussian loop rG far ex- stages, while process (c) and (d) only preserves the time ceeds the coherence length, i.e., rG ≫ ξ, the spin and reversal symmetry at the final stage. the charge quantum numbers are sharply defined within We consider a Gaussian loop surrounding the flux. As exponential accuracy. Recently, similar proposals of frac- the flux φ↑(t) is turned on adiabatically, Faraday’s law tionalization phenomena in two-dimensions induced by induction states that a tangential electric field E↑ is in- topological defects have been studied in several other duced along the Gaussian loop. The quantized Hall con- systems.[16, 17, 18, 19] 2 e While the argument above is intuitive and generally ductance implies a radial current j↑ = h z×E↑, resulting in a net charge flow ∆Q↑ through the Gaussian loop: valid in the presence of both interaction and disorder, it has a serious shortcoming. It relies on the Us(1) spin ro- 1 e2 1 tation symmetry which is not generic in the presence of ∆Q↑ = − dt dn · j↑ = − dt dl · E↑ Z0 Z h Z0 Z spin-orbit interactions. Therefore, we first need a general e2 1 ∂φ e2 hc e definition of the concept of spin-charge separation rely- = − dt = − = − (1) ing only on the generic time-reversal symmetry. In the hc Z ∂t hc 2e 2 0 absence of spin rotational symmetry, we can still use the Identical argument applied to the down spin component generic time reversal symmetry and the Kramers theorem shows that ∆Q↓ = −e/2. Therefore, this adiabatic pro- to classify integer versus half-integer spin states. Denot- cess creates the holon state with ∆Q = ∆Q↑ +∆Q↓ = −e ing the time reversal operator as T , and the charge op- and ∆Sz = ∆Q↑ − ∆Q↓ = 0. erator as N, we give the following general definition of spin-charge separation: (a) (b) Definition I φ φ • : A generalized chargeon (or π π holon) state is a quantum state |ψci satisfying N (−1) |ψci = −|ψci, and T |ψci = |ψci. A gener- 0 t t 1 0 1 alized spinon state is a doublet of quantum states −π ∆ −π ∆ + − N ± ± Q=-e Q=e |ψs i and |ψs i, satisfying (−1) |ψs i = |ψs i, T |ψ+i = |ψ−i and T |ψ−i = −|ψ+i. (c) (d) s s s s φ φ π = π The Kramers degeneracy is generally lifted in the presence of a magnetic field, and the resulting energy 0 t t 1 0 1 splitting of the doublet is linear in magnetic field, ı.e. −π −π ∗ ∗ = ∆E = g µB|B|. The constant of proportionality g can ∆ S =-1/2 ∆ S =1/2 z z be defined as the effective g factor of the spinon. We now consider a TRI insulator without any additional spin ro- tational symmetry. We consider the generalizations of FIG. 1: Four different adiabatic processes from φ↑ = φ↓ = 0 processes (a) and (b), by replacing the hopping matrix iθ(t)Γ to φ↑ = φ↓ = π. The red (blue) curve stands for the flux elements tij with tij e [20, 21] on all links along a ± φ↑(↓)(t), respectively. The symbol “ ” (“ ”) represents in- string extending from the flux tube to infinity. Here Γ is ⊙ ⊗ creasing (decreasing) fluxes, and the arrows show the current a matrix in the spin space, and the intuitive discussion into and out of the Gaussian loop, induced by the changing given above corresponds to the choice of Γ = Sz. All the flux. Charge is pumped in the processes with φ↑(t)= φ↓(t), − following discussions are valid even if Γ is not conserved.
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