Charge-$4e$ superconductors: A Majorana quantum Monte Carlo study

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Charge- $4 e$ superconductors: A Majorana quantum Monte Carlo study

Yi-Fan Jiang, Zi-Xiang Li, Steven A. Kivelson, and Hong Yao

Phys. Rev. B 95, 241103(R) – Published 6 June 2017

Abstract

Many features of charge- $4 e$ superconductors remain unknown because even the “mean-field Hamiltonian” describing them is an interacting model. Here we introduce an interacting model to describe a charge- $4 e$ superconductor (SC) deep in the superconducting phase and explore its properties using quantum Monte Carlo (QMC) simulations. The QMC is sign-problem-free but only when a Majorana representation is employed. As a function of the chemical potential we observe two sharply-distinct behaviors: a “strong” quarteting phase in which charge- $4 e$ quartets are tightly bound (like molecules) so that charge- $2 e$ pairing does not occur even in the temperature $T \to 0$ limit, and a “weak” quarteting phase in which a further transition to a charge- $2 e$ superconducting phase occurs at a lower critical temperature. Analogous issues arise in a putative $Z_{4}$ spin liquid with a pseudo-Fermi surface and other interacting models with composite order parameters. Under certain circumstances, we also identified a stable $T = 0$ charge- $4 e$ SC phase with gapless nodal quasiparticles. We further discuss possible relevance of our results to various experimental observations in $\frac{1}{8}$ -doped LBCO.

Received 8 August 2016
Revised 16 April 2017
Corrected 18 September 2017

DOI:https://doi.org/10.1103/PhysRevB.95.241103

Physics Subject Headings (PhySH)

Majorana fermions Quantum criticality Superconducting order parameter

Unconventional superconductors

Methods in superconductivity Quantum Monte Carlo

Condensed Matter, Materials & Applied Physics

Corrections

18 September 2017

Erratum

Publisher's Note: Charge- $4 e$ superconductors: A Majorana quantum Monte Carlo study [Phys. Rev. B 95, 241103(R) (2017)]

Yi-Fan Jiang, Zi-Xiang Li, Steven A. Kivelson, and Hong Yao

Phys. Rev. B 96, 119909 (2017)

Authors & Affiliations

Yi-Fan Jiang^1,2, Zi-Xiang Li¹, Steven A. Kivelson², and Hong Yao¹

¹Institute for Advanced Study, Tsinghua University, Beijing 100084, China
²Department of Physics, Stanford University, Stanford, California 94305, USA

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Vol. 95, Iss. 24 — 15 June 2017

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Images

Figure 1
Schematic phase diagram of a system with a high-temperature transition $T_{c}$ (the dashed line) to a charge- $4 e$ superconducting state. A “strong-quarteting” regime appears for values of the chemical potential $μ$ less than a critical value $μ_{c}$ , where charge- $4 e$ quartets are tightly-bound so that charge- $2 e$ pairing cannot occur even as $T \to 0$ . A crossover occurs at $μ_{c}$ to a “weak” quarteting regime such that for $μ > μ_{c}$ a second (Ising) transition occurs at $T_{2 e} > 0$ to a charge- $2 e$ superconductor. Note that the minimal model we studied applies only much below the dashed line.
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Figure 2
(a) The finite-size scaling of the charge- $2 e$ pairing structure factor $M_{2}$ on the square lattice of $V = 1.0 t$ for $μ = - 6.0 t$ and $- 10.0 t$ , respectively (in the “strong-quarteting” region), with size $L = 9, 10, ..., 15$ . (b) The quadratic polynomial fitting of particle density $n (L \to \infty)$ for $μ = - 10.0 t$ , with $V$ from $0.1 t$ to $0.7 t$ ; the solid line fitted through the data is $n = 0.0037 {(V / t)}^{2}$ . Here, we obtain $n (L \to \infty)$ from a finite-size scaling of $n (L)$ with $L = 8, 10, 12, 14$ . (c) The finite-size scaling of $M_{2}$ for $V = 0.6, 0.7, 0.8, 0.9, 1.0 t$ and $μ = - 1.36 t$ (finitely away from the half filling). The system size $L$ varies from 9 to 16. Here, $M_{2}$ barely changes with increasing $L$ . The colored lines are linear fits to the data points for each $V$ , which give rise to finite extracted values in the thermodynamic limit ( $L \to \infty$ ). (d) The charge- $2 e$ pairing order parameter $Δ \equiv V 〈 c_{i ↑}^{†} c_{i ↓}^{†} 〉 = V {[M_{2} (L \to \infty)]}^{1 / 2}$ (plotted on a logarithmic scale) as a function of $V^{- 1}$ for $μ = - 1.36 t$ with $V / t$ in the range 0.6 to 1.0; the fit to the solid line shows that $Δ \approx α t e^{- g t / V}$ with $α \approx 2.3$ and $g \approx 2.1$ .
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Figure 3
(a) A schematic phase diagram of the model on the honeycomb lattice at half filling ( $μ = 0$ ). The quantum phase transition is expected to occur at a finite critical value of $V = V_{c}$ and to be in the Gross-Neveu universality class of Ising symmetry breaking. Correspondingly, for $V > V_{c}$ , there should be a transition to a charge- $2 e$ SC with a finite temperature $T_{c} \sim {(V - V_{c})}^{ν}$ with $ν \approx 0.78$ obtained from MQMC simulations. (b) The Binder ratio $B (V, L)$ of the charge- $2 e$ pairing order parameter as a function of $V$ and $L$ for the case of half filling. Here, $L$ varies from 9 to 15, as plotted in different colors. The crossing point gives rise to the critical point $V_{c} \approx 0.84 t$ . (c) The critical exponent $η \approx 0.52$ and $ν \approx 0.78$ are obtained by reaching the best data collapsing.
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