Smeared nematic quantum phase transitions due to rare-region effects in inhomogeneous systems

Tianbai Cui and Rafael M. Fernandes

Phys. Rev. B 98, 085117 – Published 8 August 2018

Abstract

The concept of a vestigial nematic order emerging from a “mother” spin or charge-density-wave state has been applied to describe the phase diagrams of several systems, including unconventional superconductors. In a perfectly clean system, the two orders appear simultaneously via a first-order quantum phase transition, implying the absence of quantum criticality. Here, we investigate how this behavior is affected by impurity-free droplets that are naturally present in inhomogeneous systems. Due to their quantum dynamics, finite-size droplets sustain long-range nematic order but not long-range density-wave order. Interestingly, rare droplets with moderately large sizes undergo a second-order nematic transition even before the first-order quantum transition of the clean system. This gives rise to an extended regime of inhomogeneous nematic order, which is followed by a density-wave quantum Griffiths phase. As a result, a smeared quantum nematic transition, separated from the density-wave quantum transition, emerges in moderately disordered systems.

Received 8 January 2018
Revised 25 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.085117

Physics Subject Headings (PhySH)

Charge order Critical phenomena Impurities Magnetic order Nematic order Nematic phase transition Quantum criticality

Unconventional superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Tianbai Cui and Rafael M. Fernandes

School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA

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Issue

Vol. 98, Iss. 8 — 15 August 2018

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Images

Figure 1
Illustration of a moderately diluted system displaying an infinite cluster (light green) and finite-size droplets (dark green) devoid of impurities. For this illustration, we consider a “mother” spin-density wave and its vestigial Ising-nematic phase. At $T = 0$ , finite-size droplets cannot sustain long-range spin order (indicated by the disordered spins in the inset), but they can support long-range nematic order (indicated by the different spin-spin correlations along the $x$ and $y$ axes, resulting in unequal $x$ and $y$ bonds). Importantly, droplets of moderate sizes undergo a second-order nematic transition even before the bulk system (and thus the infinite cluster) undergoes its simultaneous first-order density-wave and nematic transition.
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Figure 2
Schematic phase diagram illustrating our main results. Here, $r_{0}$ is a control parameter, such as doping or pressure. In the clean system at $T = 0$ , the nematic ( $ϕ$ ) and density-wave ( $Δ_{DW}$ ) order parameters appear simultaneously at $r_{0, clean}^{*}$ (dashed line). In the moderately diluted system, which still has a percolating (infinite) cluster, the first-order quantum transition is expected to be suppressed down to $r_{0, dirty}^{*}$ (dotted line). For $r_{0} > r_{0, clean}^{*}$ , moderately large droplets undergo a second-order nematic transition, giving rise to inhomogeneous nematic order. For $r_{0, dirty}^{*} < r_{0} < r_{0, clean}^{*}$ , exponentially large droplets have an exponentially large DW correlation length, resulting in a DW quantum Griffiths phase. Whether $ϕ$ jumps or continuously evolves at $r_{0, dirty}^{*}$ depends on details of the disorder distribution.
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Figure 3
Nematic order parameter $ϕ$ (in units of $Λ^{2}$ ) inside a droplet of “volume” $L^{2}$ as a function of the control parameter $r_{0}$ (in units of the value $r_{0, clean}^{*}$ for which the clean system undergoes the first-order nematic transition). $ϕ_{clean} \equiv ϕ (L \to \infty)$ is shown as a dashed line. First-order transitions are indicated by the dotted line. The inset shows the value of the tuning parameter $r_{0}^{*}$ at the nematic transition inside a droplet as a function of the droplet volume $L^{2}$ .
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Figure 4
Average nematic order parameter of all droplets $\bar{ϕ}$ (in units of $Λ^{2}$ ) as a function of the control parameter $r_{0}$ (in units of $r_{0, clean}^{*}$ ). Different curves correspond to different “critical” droplet volumes $V_{0}$ associated with the probability distribution (5).
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Figure 5
Function $F (y)$ defined in Eq. (A9).
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Figure 6
Region in the $(u, g)$ parameter space in which the droplet that orders first does so before the transition of the clean system, i.e., $r_{0}^{*} (L_{c 2}) > r_{0, clean}^{*}$ .
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