Intrinsic translational symmetry breaking in a doped Mott insulator

Zheng Zhu, D. N. Sheng, and Zheng-Yu Weng

Phys. Rev. B 98, 035129 – Published 19 July 2018

Abstract

A central issue of Mott physics, with symmetries being fully retained in the spin background, concerns the charge excitation. In a two-leg spin ladder with a spin gap, an injected hole can exhibit either a Bloch wave or a density wave by tuning the ladder anisotropy through a “quantum critical point” (QCP). The nature of such a QCP has been a subject of recent studies by density matrix renormalization group. In this paper, we reexamine the ground state of the one doped hole, and show that a two-component structure is present in the density wave regime, in contrast to the single component in the Bloch-wave regime. In the former, the density wave itself is still contributed by a standing-wave-like component characterized by a quasiparticle spectral weight $Z$ in a finite-size system. But there is an additional charge incoherent component emerging, which intrinsically breaks the translational symmetry associated with the density wave. The partial momentum is carried away by neutral spin excitations. Such an incoherent part does not manifest in the single-particle spectral function, directly probed by the angle-resolved photoemission spectroscopy measurement, however, it is demonstrated in the momentum distribution function. Landau's one-to-one correspondence hypothesis for a Fermi liquid breaks down here. The microscopic origin of this density wave state as an intrinsic manifestation of the doped Mott physics will be also discussed.

Received 11 January 2016
Revised 2 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.035129

Physics Subject Headings (PhySH)

Mott insulators

Density matrix renormalization group

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Zheng Zhu¹, D. N. Sheng², and Zheng-Yu Weng^3,4

¹Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²Department of Physics and Astronomy, California State University, Northridge, California 91330, USA
³Institute for Advanced Study, Tsinghua University, Beijing 100084, China
⁴Collaborative Innovation Center of Quantum Matter, Tsinghua University, Beijing 100084, China

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Vol. 98, Iss. 3 — 15 July 2018

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Images

Figure 1
Illustration of the density profile $n_{j}^{h}$ , which exhibits a charge modulation for a single hole injected into a two-leg spin ladder governed by the $t − J$ model. Inset: The wave vector $Q_{0}$ as a function of the anisotropic parameter $α$ , which vanishes at $α < α_{c}$ . The nature of such a one-hole state will be carefully reexamined $α > α_{c}$ by DMRG and analytic analysis in this work. (Here, the ladder size $N = 200 \times 2$ with open boundaries, in which $j$ denoting the site along the chain direction as the density is the same on each rung, with $α_{c} ≃ 0.7$ at $t / J = 3$ based on the DMRG results [34].)
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Figure 2
The quasiparticle spectral weight $Z_{k}$ can determine the total momentum $k_{0} \equiv (k_{0}, k_{0 y})$ via a finite-size analysis. Insets: The original $Z_{k}$ 's. (a) A well-quantized Bloch wave, with $k_{0} = π$ and $k_{0 y} = 0$ in the $t − J$ case at $α = 0.4 < α_{c}$ , is characterized by the scaling law with the $k_{x}$ axis replaced by $(k_{x} - k_{0}) N_{x} / π$ . (b) A well-quantized Bloch wave with $k_{0} = 0$ in the $σ \cdot t − J$ ladder at $α = 5$ . (c) and (d) The $t − J$ case at $α = 5 > α_{c}$ : The quantization in a finite-size sample breaks down even at small variations of the sample length, for example, $N_{x} = 192$ , 200, 202, and 206 [cf. (d)]. Here, the total momentum $k_{0}$ is split into two $k_{0}^{\pm}$ separated by an incommensurate $Q_{0}$ defined by Eq. (9) with $k_{0 y} = 0$ .
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Figure 3
$Z_{tot}$ measures the overlap of the true ground state with a bare hole state, but it does not solely decide the coherence of the quasiparticle should the one-to-one correspondence principle fail (see text). Here, $α$ denotes an anisotropic parameter for the two-leg $t − J$ ladder. A critical point $α_{c}$ is marked by the vertical dashed line, which is previously determined [33] for the $t − J$ case at $t / J = 3$ with $α_{c} \approx 0.7$ (no critical $α_{c}$ for the $σ \cdot t − J$ model). Inset: The convergence of $Z_{tot}$ vs the sample size $N = N_{x} \times 2$ at $α = 5$ .
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Figure 4
The origin of charge density modulation in $n_{j}^{h}$ can be fully attributed to that of the standing-wave component in the ground state Eq. (3), whose probability is measured by $Z_{j}$ . (a) A fast oscillation in $Z_{j}$ is modulated by a slower variation at a length scale of $λ$ at $α = 5 > α_{c}$ . (b) The Fourier transformation of $Z_{j}, F_{q}$ reveals the characteristic wave vector $Q_{0}$ with a continuous spread $\sim 2 π / λ$ . (c) and (d) The corresponding hole density distribution $n_{j}^{h}$ and its Fourier transformation. Inset of (c): The length scale $λ$ vs $N_{x}$ . (e) and (f) The contribution from $| Ψ_{inc} 〉$ estimated by $n_{j}^{inc} \equiv n_{j}^{h} - Z_{j} - (β Z_{j} + Z_{tot} / N)$ ( $β ≃ - 1 / 2$ ) and its Fourier transformation $N_{q}^{inc} = N_{q} - (1 + β) F_{q}$ , which show negligible traces of the wave vector $Q_{0}$ (see text).
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Figure 5
Distinctive momentum distributions of the hole in the two phases separated by $α_{c} = 0.7$ . (a) At $α = 0.4 < α_{c}$ , $n^{h} (k)$ shows a finite-size scaling similar to $Z_{k}$ in Fig. 2, in consistency with a Bloch-wave description. (b) At $α = 5 > α_{c}$ , besides two sharp peaks located around the total momentum $k_{0}^{\pm}$ represented by $Z_{k}$ (marked by the vertical dashed lines), an additional continuous background (blue curves), respectively, at both $k_{y} = 0$ and $k_{y} = π$ (the inset), satisfies a different scaling law, i.e., $n^{h} (k) \cdot N_{x}$ , which ensures a finite weight in the sum rule Eq. (13). (c) The broad feature at $k_{y} = 0$ in the main panel of (b) is redrawn as $[n^{h} (k) - Z_{k}] \cdot N_{x}$ .
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Figure 6
The energy change due to the charge response to an inserted flux $Φ = π$ into the ring geometry of the ladder (see text). (a) Typical Bloch-wave behavior ( $\propto 1 / N_{x}^{2}$ ) for the $σ \cdot t − J$ case and the $t − J$ ladder at $α < α_{c}$ (the inset, in which the slope for the $t − J$ ladder changes sign but not amplitude at $N_{x} =$ odd). (b) The non-Bloch-wave response at $α > α_{c}$ in the $t − J$ case. The energy change can be well fitted by Eq. (25), which directly relates the phase-string effect according to Eq. (22) as the underlying cause for charge incoherence and incommensurate momentum splitting, as shown in the inset (the Fourier transformation of the energy change vs $N_{x}$ , which gives rise to the wave vector splitting precisely at $Q_{0}$ with some intrinsic broadening).
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