What makes the Tc of monolayer FeSe on SrTiO3 so high: a sign-problem-free quantum Monte Carlo study

Monolayer FeSe films grown on SrTiO3 (STO) substrate show superconducting gap-opening temperatures (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\mathrm{c}}$$\end{document}Tc) which are almost an order of magnitude higher than those of the bulk FeSe and are highest among all known Fe-based superconductors. Angle-resolved photoemission spectroscopy observed “replica bands” suggesting the importance of the interaction between FeSe electrons and STO phonons. These facts rejuvenated the quest for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{{\mathrm{c}}}$$\end{document}Tc enhancement mechanisms in iron-based, especially iron-chalcogenide, superconductors. Here, we perform the first numerically-exact sign-problem-free quantum Monte Carlo simulations to iron-based superconductors. We (1) study the electronic pairing mechanism intrinsic to heavily electron doped FeSe films, and (2) examine the effects of electron–phonon interaction between FeSe and STO as well as nematic fluctuations on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{{\mathrm{c}}}$$\end{document}Tc. Armed with these results, we return to the question “what makes the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{{\mathrm{c}}}$$\end{document}Tc of monolayer FeSe on SrTiO3 so high?” in the conclusion and discussions. Electronic supplementary material The online version of this article (doi:10.1007/s11434-016-1087-x) contains supplementary material, which is available to authorized users.


I. The effective action for the J1 and J2 types of spin fluctuations
The effective action, based on a two band model describing band structure of single-layer (FeSe) 1 /STO, is given by S = S F + S s where S s = S B + S FB and where for J 1 -type of spin fluctuations. If the spin fluctuation is J 2 type In the above equations, j, k labels the sites of a square lattice, α = x, y labels the two orbitals (which transform into each other under the 90 • rotation) from which the red and blue Fermi surfaces in Fig. 1a of the main text are derived from, τ denotes the imaginary time and β is the inverse temperature. In Eq. (S2), ⃗ φ s is the AFM collective mode and the operator ψ iα is a spinor operator which annihilates an electron in orbital α and on site i. The three ⃗ σ are the spin Pauli matrices. It is important to note that in Eq. (S4) the fermion-boson coupling has an extra factor i.
The parameters in this effective action include r s which tunes the system across the AFM phase transition, c s is the spin-wave velocity and u s is the self-interactions of ⃗ φ s . λ s is the "Yukawa" coupling between electrons and AFM order parameter. In our computation, we fix c s = u s = λ s = 1.0 and vary the value of r s to control the severity of AFM fluctuation. In the fermion action the hopping integral t ij is chosen to be among nearest neighbor sites and equal to t ∥ = 1.0 for x(y)-orbital along x(y) direction and t ⊥ = −0.5 for y(x) orbital along x(y) direction. We fix the occupancy in our computation to be 0.1 such that the Fermi surface is shown in Fig. 1a of the main text.

II. The effective action for antiferro-orbital fluctuations
The AFO effective action is given by In Eq. (S6), φ o is the AFO order parameter and σ 0 is the identity matrix in the spin space.

III. The electron-phonon effective action
The electron-phonon effective action is given by Here φ ph is the phonon field, r ph is the frequency of the optical phonon at ⃗ q = 0 and c ph is velocity of phonon. We fix parameters r ph = 0.5, c ph = 1.0 and vary the value of λ ep to tune the strength of electron-phonon coupling.

IV. The effective action for nematic fluctuations
In Fig. S1, we show how does the nematic order parameter distort the Fermi surfaces. The nematic effective action is given by S = S F + S n where S n = S B + S FB and In Eq. (S10), φ n is the nematic order parameter. The parameters in this effective action include the velocity c n , and r n which tunes the system across the nematic phase transition, and u n is the self-interactions of the φ n field. λ n is the Yukawa coupling between electrons and nematic order parameter. In our computation, we fix c n = u n = λ n = 1.0 and vary the value of r n to control the severity of nematic fluctuation.

V. The superconducting pair correlation function
To investigate superconductivity we calculate the equal time pair-pair correlation functions are the s (+ sign) and d (− sign) wave Cooper pair operators, respectively. To determine whether there is long-range order we put ⃗ r i to the maximum separation ⃗ x max of the pair fields (for a system with linear dimension L the value of ⃗ x max is (L/2, L/2). Moreover, in order to minimize statistical errors, we average the correlation function over 25 sites around ⃗ x max . Thus the actual pair correlation we study is (S13)

VI. Effective actions that are amenable to sign-problem-free QMC simulation
The actions that are amenable to sign-problem-free QMC simulations are any mixture of S F + c 1 S s + c 2 S ph + c 3 S n and S F + c 1 S o + c 2 S ph + c 3 S n where c 1,2,3 = 0, 1. Note, however, for S s we can use either J 1 or J 2 type spin fluctuation actions but not both.
Aside from the square lattice spatial symmetries the action S F + c 1 S s + c 2 S ph + c 3 S n is invariant under the antiunitary transformation U = τ z (iσ y )K and the action Here τ z is the third Pauli matrix acting in orbital (x, y) space and K denotes complex conjugation. It can be shown that because of these symmetries, the fermion determinant for arbitrary Bose fields configuration is positive hence the QMC simulation is free of minus-sign. This enables us to perform large-scale projector QMC simulation. The anti-unitary symmetry U = τ z (iσ y )K is the same as that in Ref. [1]. In this section, we examine the enhancement of the SC order parameters triggered by the J 1 type spin and AFO fluctuations as a function of the dimensionless electron-phonon coupling strength λ.
In Fig. S2a and b, we plot the enhancement of spin fluctuation induced P d (L/2, L/2) and orbital fluctuation induced P s (L/2, L/2) as a function of λ for L = 14. Apparently the enhancement of superconductivity peaks at λ ∼ 1.5 for the J 1 -type spin fluctuation triggered d-wave pairing. For the AFO induced s-wave pairing the pair correlation increases monotonously with the amplitude of electron-phonon coupling up to the maximum value of λ we studied (=2.2). * yaohong@tsinghua.edu.cn, dunghai@berkeley.edu.