⇦Chinese Physics Letters, 2022, Vol. 39, No. 9, Article code 097102 Superconductivity near the (2+1)-Dimensional Ferromagnetic Quantum Critical Point Yunchao Hao (郝云超)1†, Gaopei Pan (潘高培)2,3†, Kai Sun (孙锴)4*, Zi Yang Meng (孟子杨)5*, and Yang Qi (戚扬)1,6* Affiliations 1State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China 2Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 3School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 4Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA 5Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China 6Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China Received 17 June 2022; accepted manuscript online 13 August 2022; published online 3 September 2022 †These authors contributed equally to this work.
*Corresponding authors. Email: sunkai@umich.edu; zymeng@hku.hk; qiyang@fudan.edu.cn Citation Text: Hao Y C, Pan G P, Sun K et al. 2022 Chin. Phys. Lett. 39 097102    Abstract We utilize both analytical and numerical methods to study the superconducting transition temperature $T_{\rm c}$ near a fermionic quantum critical point (QCP) using a model constructed by Xu et al. [Phys. Rev. X 7, 031059 (2017)] as an example. In this model, the bosonic critical fluctuation plays the role of pairing glue for the Cooper pairs, and we use a Bardeen–Cooper–Schrieffer-type mean-field theory to estimate $T_{\rm c}$. We further argue that the $T_{\rm c}$ computed from the BCS theory approximates a pseudogap temperature $T_{\rm PG}$, instead of the Berezinskii–Kosterlitz–Thouless transition temperature $T_{\rm KT}$, which is confirmed by our determinant quantum Monte Carlo simulation. Moreover, due to the fact that electron density of state starts to deplete at $T_{\rm PG}$, the critical scaling of the underlying QCP is also affected below $T_{\rm PG}$. Thus, when studying the critical behavior of fermionic QCPs, we need to monitor that the temperature is above $T_{\rm PG}$ instead of $T_{\rm KT}$. This was often ignored in previous studies.
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 DOI:10.1088/0256-307X/39/9/097102 © 2022 Chinese Physics Society Article Text An ubiquitous phenomenon in the vicinity of a quantum critical point (QCP) in metallic systems is superconductivity enhanced by the critical fluctuation, which provides a pairing glue for Cooper pairs.[1-6] As an important path way towards unconventional superconductivity, this scenario plays an essential role in the study of high-$T_{\rm c}$ superconductivity, including cuprates[7] and iron-based superconductors.[8] Despite of its importance, theoretical studies of this issue (including the QCP itself and the related superconductivity) are of challenge, because of the strongly correlated nature of the QCP and the lack of controlled ways of perturbative expansion.[3,9-16] On the other hand, numerical studies of this issue, in particular using determinant quantum Monte Carlo (DQMC) simulations, have been fruitful, thanks to the development of sign-problem-free “designed models” and numerical algorithms.[17,18] Compiling recent DQMC simulations on fermionic QCPs and related superconductivity, we see that the superconductivity emerging from these QCPs has drastically different transition temperatures: In some models, superconductivity appears at a relatively high temperature, and unfortunately covers the true QCP that is being studied.[2] In other models, however, the superconductivity remains absent up to the lowest temperature.[19,20] It is desirable to understand how microscopic details of the models hosting fermionic QCPs affect the superconducting $T_{\rm c}$: Such knowledge would enable us to design models with high $T_{\rm c}$, which realizes high-temperature superconductivity, as well as models with low $T_{\rm c}$, which are necessary for studying the critical scaling of the QCP. In this work, we study the superconducting $T_{\rm c}$ near a fermionic QCP, using effective interactions mediated by bosonic critical fluctuations, and Bardeen–Cooper–Schrieffer (BCS) mean-field theory. One important difference from the original BCS theory is that the effective interaction we consider here has a strong temperature dependence because it diverges as approaching the QCP. Furthermore, we argue that the $T_{\rm c}$ computed from the BCS theory approximates a pseudogap temperature $T_{\rm PG}$, instead of the true two-dimensional (2D) superconducting $T_{\rm c}$. It is well-known that, in two dimensions, superconducting transitions are Berezinskii–Kosterlitz–Thouless (BKT) transitions,[21,22] which are driven by the lost of phase coherence of the Cooper pairs above the transition temperature. Therefore, 2D superconductivity appears in two stages: (1) The electrons form Cooper pairs at a higher temperature scale. (2) The Cooper pairs gain phase coherence at a lower temperature scale. The former corresponds to a pseudogap temperature $T_{\rm PG}$,[23] while the latter is the BKT transition temperature $T_{\rm KT}$. Because the BCS mean-field theory treats the Cooper-pairing instability of the Fermi surface while ignoring the phase fluctuation of the order parameter, what we actually estimates from the BCS theory should be the former instead of the latter. We also perform DQMC simulations on the fermionic QCP model. From the electron spectrum function computed using stochastic analytical continuation (SAC), we observe a depletion of density of state near the Fermi energy below a certain temperature, which can be identified as the pseudogap temperature. However, the BKT transition is not observed until the lowest temperature our simulation can reach. Moreover, the observed pseudogap temperature is qualitatively consistent with our estimation of the BCS mean-field theory. Combining the theoretical and numerical results, we see that the enhancement of superconductivity near fermionic QCPs can be approximated by the BCS mean-field theory, but the estimated $T_{\rm c}$ corresponds to the pseudogap temperature instead of the BKT-transition temperature. Furthermore, since the electron density of state starts to deplete at $T_{\rm PG}$, the critical scaling of the underlying QCP is also affected below $T_{\rm PG}$, which is confirmed in our DQMC simulation. Therefore, in the study of critical scaling of fermionic QCPs, it needs to be monitored that the temperature is above $T_{\rm PG}$ instead of $T_{\rm KT}$, as it was often performed in previous studies.
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Fig. 1. A schematic picture of the model defined in Eq. (1). Two layer fermions with orbits $\delta=1, 2$ moving in the square lattice with nearest hopping are coupled to Ising spins on the same site whose dynamics is governed by the ferromagnetic interaction and transverse field.
Model. We consider the following two-dimensional model constructed in Ref. [19] of itinerant fermions coupled to an Ising ferromagnet with a transverse field (see Fig. 1). The Hamiltonian consists of three parts: \begin{align} &H=H_{\rm f}+H_{\rm s}+H_{\rm sf}, \notag\\ &H_{\rm f}=-t\sum_{\langle{ij}\rangle\delta\sigma}c^†_{i\delta\sigma}c_{j\delta\sigma}+{\rm H.c.} -\mu\sum_{i\delta\sigma}c^†_{i\delta\sigma}c_{i\delta\sigma},\notag\\ &H_{\rm s}=-J\sum_{\langle{ij}\rangle}s^z_is^z_j-h\sum_i s^x_i, \notag\\ &H_{\rm sf}=-\lambda\sum_i s^z_i(\sigma^z_{i1}+\sigma^z_{i2}). \tag {1} \end{align} Here $\delta=1,2$ label orbitals, while $i,j$ and $\sigma$ are site indices and spin indices. For $H_{\rm f}$, we consider two layer fermions with intra-layer nearest neighbor hopping $t$ and chemical potential $\mu$. All energy scales are measured in units of $t$ in the following discussion and we set $t=1$. $H_{\rm s}$ is well-known transverse Ising with ferromagnetic (FM) coupling $J$ where $s_i^z=\pm1$. $H_{\rm sf}$ describes an inter-layer onsite Ising coupling between middle layer transverse Ising and two Fermion layers where $\sigma^z_{i\delta}=(n_{i\delta\uparrow}-n_{i\delta\downarrow})/2$ is the $z$-component of fermion spin. Without the fermions, the transverse-field Ising model has a ferromagnetic transition at a critical $h$. When fermions are coupled to the Ising spins, the ferromagnetic transition is shifted to a different critical field $h_{\rm c}$, and becomes a fermionic QCP with different scaling behaviors.[19] BCS Superconductivity—Mean-Field Analysis of the Transition Temperature. We now use the BCS theory to estimate the superconducting critical temperature for this model. To study the enhancement of superconductivity from the QCP, we replace the phonon modes in the BCS theory by the critical fluctuation of the Ising spins, and study the effective electron interaction it induces. Close to the FM-QCP, the low-frequency and low-momentum critical fluctuation of the Ising spins is described by the following effective Hamiltonian $\tilde{H}_{\rm s}$ for the Ising spin:[24] \begin{eqnarray} \tilde{H}_{\rm s}=\sum_i \alpha(h-h_{\rm c}) m^2_i, \tag {2} \end{eqnarray} where $\alpha(h-h_{\rm c})$ is the inverse of spin susceptibility and $m_i$ is the effective spin order parameter on site $i$. Combining this with the spin-fermion interaction and integrating out $m_i$, we obtain the following four-fermion interaction: \begin{align} H_{{\rm eff}}=\,&-\frac{\lambda^2}{8\alpha(h-h_{\rm c})}\sum_i(c^†_{i1\uparrow}c^†_{i2\uparrow}c_{i2\uparrow}c_{i1\uparrow}\notag\\ &+c^†_{i1\downarrow}c^†_{i2\downarrow}c_{i2\downarrow}c_{i1\downarrow}).\tag {3} \end{align} Here, we only keep the attractive terms, which are relevant for the superconductivity. We also notice that this interaction is based on the Ginzburg–Landau-type effective Hamiltonian $\tilde H_{\rm s}$ in Eq. (2), which will fail if it is too close to the QCP. In fact, in the following, we provide a more accurate form of interaction, based on critical scaling form of the spin susceptibility measured from DQMC simulations. The spin fluctuations induce an attractive interaction in the spin-triplet channel with the order parameter, \begin{eqnarray} \varDelta_{i,\sigma}=c_{i2\sigma}c_{i1\sigma}. \tag {4} \end{eqnarray} Then, we can perform the BCS mean field theory to this effective Hamiltonian and reach the self-consistent equation: \begin{eqnarray} 1=\frac{\lambda^2}{8\alpha(h\!-\!h_{\rm c})}\frac{1}{L^2}\sum_{\boldsymbol{k}\in\frac12{\rm BZ}}\frac{\tanh[\sqrt{\epsilon(\boldsymbol k)^2\!+\!|\varDelta_\sigma|^2}/(2\,T)]}{\sqrt{\epsilon(\boldsymbol k)^2\!+\!|\varDelta_\sigma|^2}}, \tag {5} \end{eqnarray} where $\epsilon(\boldsymbol{k})=-t(2\cos k_x + 2\cos k_y)-\mu$ is the energy dispersion for the fermions and $\varDelta_\sigma=\frac{\lambda^2}{8\alpha(h-h_{\rm c})L^2}\sum_{\boldsymbol{k}}\langle{c_{\boldsymbol{k}2\sigma}c_{-\boldsymbol{k}1\sigma}}\rangle$. The momentum summation is restricted to one half of the Brillouin zone to avoid double counting. The detailed derivation of the gap Eq. (5) can be found in the Supplementary Materials. Since the coefficient $\alpha(h-h_{\rm c})$ is inverse Ising spin susceptibility at QCP [$\chi(h_{\rm c},T,\boldsymbol{0},0$)] as a function of temperature $T$, which can be measured via determinantal quantum Monte Carlo technique described in the following, we can solve this self-consistent equation numerically. We rewrite this self-consistent equation into the following integral form: \begin{align} 1&=\frac{\lambda^2}{8\alpha(h\!-\!h_{\rm c})}\int^{\pi}_0\frac{dk_x}{2\pi} \int^{\pi}_{-\pi}\frac{dk_y}{2\pi} \frac{\tanh[\sqrt{\epsilon^2\!+\!|\varDelta_\sigma|^2}/(2\,T)]}{\sqrt{\epsilon^2\!+\!|\varDelta_\sigma|^2}} \nonumber \\ &=\frac{\lambda^2}{8\alpha(h-h_{\rm c})}\int d\epsilon g(\epsilon) \frac{\tanh[\sqrt{\epsilon^2+|\varDelta_\sigma|^2}/(2\,T)]}{\sqrt{\epsilon^2+|\varDelta_\sigma|^2}},\tag {6} \end{align} where $g(\epsilon)$ is the density of state. Unlike conventional BCS theories where the energy integral is cut off by the Debye frequency, the pairing interaction mediated by critical fluctuation appears in a wide energy scale and thus does not provide an energy cutoff. The integral is instead naturally cut off by the band width $\varLambda\simeq t$. To compute $T_{\rm c}$, we set $\varDelta_\sigma=0$ and obtain \begin{align} 1=\,&\frac{\lambda^2}{8\alpha(h-h_{\rm c})}g(0)\int^{\varLambda}_{_{\scriptstyle -\varLambda}} d\epsilon\frac{\tanh[|\epsilon|/(2T_{\rm c})]}{|\epsilon|} \notag \\ =\,&\frac{\lambda^2}{4\alpha(h-h_{\rm c})}g(0)\Big(\ln\Big(\frac{\varLambda}{2T_{\rm c}}\Big)-\ln\Big(\frac{\pi}{4}\Big)\notag\\ &+\gamma+\mathcal{O}\Big(\frac{T_{\rm c}}{\varLambda}\Big)\Big). \tag {7} \end{align} Here, in the last line we assume $\varLambda\gg T_{\rm c}$, and $\gamma$ is Euler's constant. Thus we obtain the expression for the critical temperature as follows: \begin{eqnarray} T_{\rm c}=\frac{2}{\pi}e^\gamma\varLambda e^{-\frac{4\alpha(h-h_{\rm c})}{\lambda^2g(0)}} \simeq1.13\,\varLambda e^{-\frac{4\alpha(h-h_{\rm c})}{\lambda^2g(0)}}. \tag {8} \end{eqnarray} We can further simplify this equation if we assume that the contribution to the susceptibility only comes from bosons, i.e., $\alpha=\chi^{-1}=a T^2$ at the QCP. Near a QCP in a fermionic system, the critical behavior may first exhibit universality of a bosonic QCP at high temperatures, and crossover to the true universality of the fermionic QCP at lower temperatures, closer to the true zero-temperature QCP. This assumption applies if $T_{\rm c}$ is higher than the crossover temperature scale and the critical fluctuation still follows the bosonic criticality. Assuming the scaling form of the Landau theory, we can then solve the expression for $T_{\rm c}$ as \begin{eqnarray} \frac{T_{\rm c}}{\varLambda}=0.7071\sqrt{\frac{{W}(2.554\varTheta)}{\varTheta}}, \tag {9} \end{eqnarray} where $\varTheta=\frac{4a \varLambda^2}{g(0)\lambda^2}$ a dimensionless parameter and $W$ is Lambert $W$ function which is the inverse function of the form $we^w$. We can use asymptotic expansion of the Lambert $W$ function to analyze the expression since for large values of $x$, ${\rm W}(x)\sim \ln x$. Thus, we have $\frac{T_{\rm c}}{\varLambda}\sim\lambda$ up to logarithmic corrections. If the critical fluctuation already follows scaling behaviors of the fermionic QCP around $T_{\rm c}$, $T_{\rm c}$ should be solved using $\chi$ of the fermionic QCP instead. This will be discussed later. Relation to Kosterlitz–Thouless Transition. In the BCS superconductivity, i.e., weakly correlated superconductivity in three dimensions, due to the fact that Cooper pairs are always formulating and condensing simultaneously, we believe that the temperature predicted by the BCS theory is the superconductivity transition temperature. However, intrinsically, BCS theory as a mean-field theory should give the temperature for Cooper pair formation. The problem of interest in this study is concerned with a system in two spatial dimensions. Since continuous $U(1)$ symmetry cannot be spontaneously broken at finite temperature as a result of the well-known Mermin–Wagner theorem, what really happened in our system should be the Kosterlitz–Thouless (KT) transition. Above the KT transition critical temperature, a large amount of unpaired vortices and anti-vortices formation makes the Cooper pairs lose phase coherence. Hence, we now have two energy scales: One is a higher temperature scale corresponding to the temperature for Cooper formation. This temperature scale should be predicted by the BCS theory. The other is a lower temperature scale corresponding to the temperature for Cooper pairs restoring phase coherence at which the KT transition happens. Between these two temperature scales, the phase fluctuations may cause the formation of pseudogap, i.e., a gap-like phenomenon in electron spectrum function. In particular, we expect that the pseudogap will modify the behavior of low energy fermions to affect the universal behavior at the QCP. Thus, in the following determinantal quantum Monte Carlo (DQMC) study, we will not only focus on the scaling behavior of superconducting susceptibility to determine the transition temperature but more importantly, the change in electron spectrum function. Solution of the Gap Equation. We can use the Ising spin susceptibility obtained by the DQMC to solve the gap Eq. (5). Here we use the set of parameter $\{\lambda=3.0, J=0.3, \mu=-2.0\}$ as an example, where the temperature dependence of Ising spin susceptibility is obtained by fitting to DQMC simulations and has the form $\chi^{-1}(h_{\rm c},T,\boldsymbol{0},0)=2.386T^{0.9474}$. Setting $\varDelta_\sigma=0$, we can reach an integral equation about the transition temperature $T_{\rm c}$: \begin{align} 0=\frac{3^2}{8}\int^{\pi}_0\frac{dk_x}{2\pi}\int^{\pi}_{-\pi}\frac{dk_y}{2\pi} \frac{\tanh[\sqrt{\epsilon^2+|\varDelta_\sigma|^2}/(2T_{\rm c})]}{\sqrt{\epsilon^2+|\varDelta_\sigma|^2}}-\chi^{-1}, \tag {10} \end{align} where $\epsilon(\boldsymbol{k})=-(2\cos k_x + 2\cos k_y)+2$. We can set a series of temperatures to the right-hand side of the above equation and calculate their integral values to find the location of the crossing of the curve and the $x$ axis, which is the transition temperature. In the set of parameter we present, the crossing is at $T_{\rm c}=0.165$, as shown in Fig. 2. This is the transition temperature $T_{\rm c}=0.165$, i.e., $\beta_{\rm c}=6.06$.
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Fig. 2. In the parameter $\{\lambda=3.0, J=0.3, \mu=-2.0\}$, the solution of the self-consistent gap Eq. (5), i.e., the crossing at the $x$ axis $T_{\rm c}=0.165$.
DQMC Results. In this work, we use the standard DQMC method[25-27] to perform our simulations. For large-scale correlated electron systems, DQMC is a good unbiased method of calculations, especially when there are no so-called sign problems. Since in our problem as shown in Eq. (1), the fermion part is already bi-linear form coupled to bosonic field, so we do not need to do the Hubbard–Stratonovich (HS) transformation[26,28] to decouple. As mentioned above, we expect that with increasing the coupling between fermions and Ising ferromagnet, the transition temperature will be increased. Comparing with the results in Ref. [19] where the largest coupling constant the authors simulated is the parameter set $\{\lambda=3.0, \mu=-2.0, J=0.5\}$, we decide to perform DQMC simulations in the parameter set $\{\lambda=3.0, \mu=-2.0, J=0.3\}$, where we decrease the ferromagnetic coupling $J$ to effectively increase the coupling constant $\lambda$ to maintain the numerical stability in the simulation. Applying the standard data collapses of Binder ratios, we find the QCP locates at $h_{\rm c}=1.3952(\pm 0.09)$ for this parameter set. Search for KT Transition. We present the results of superconducting order parameter based on DQMC simulations in the canonical ensemble. Define order parameter $C$: \begin{eqnarray} C=\frac{1}{L^2}\sum_{ij} \langle{\varDelta_i^† \varDelta_j}\rangle. \tag {11} \end{eqnarray} Here we consider onsite, orbital singlet, spin triplet, s-wave pairing, which means $\varDelta_i=c_{i1\uparrow}c_{i2 \uparrow}$. In each configuration, one can derive Wick's theorem to obtain \begin{align} C &=\frac{1}{L^2}\sum_{i,j} \langle{(c_{i 1 \uparrow} c_{i 2 \uparrow})^†(c_{j 1 \uparrow} c_{j 2 \uparrow})}\rangle_{C} \nonumber \\ &= \frac{1}{L^2}\sum_{i,j}\langle{ c_{i 2 \uparrow}^† c_{i 1 \uparrow}^† c_{j 1 \uparrow} c_{j 2 \uparrow}}\rangle_{C} \nonumber \\ &= \frac{1}{L^2}\sum_{i,j} \langle{c_{i 2 \uparrow}^† c_{j 2 \uparrow}}\rangle \langle{ c_{i 1 \uparrow}^† c_{j 1 \uparrow}}\rangle_{C}. \tag {12} \end{align} For superconducting (SC) susceptibility: \begin{align} \chi_{\scriptscriptstyle{\rm SC}} &=\frac{1}{L^2}\sum_{i,j}\langle{ c_{i 2 \uparrow}^†(\tau) c_{i 1 \uparrow}^†(\tau) c_{j 1 \uparrow}(0) c_{j 2 \uparrow}(0)}\rangle \nonumber \\ &= \frac{1}{L^2}\sum_{i,j} \langle{c_{i 2 \uparrow}^†(\tau) c_{j 2 \uparrow}(0)}\rangle \langle{ c_{i 1 \uparrow}^†(\tau) c_{j 1 \uparrow}(0)}\rangle . \tag {13} \end{align} Close to the phase boundary, the SC susceptibility is expected to obey the finite size scaling forms:[29] \begin{eqnarray} \chi_{\scriptscriptstyle{\rm SC}}(T, L)=L^{2-\eta} f[L^{-1}\exp (a t^{-1/2})], \tag {14} \end{eqnarray} where $a$ is a nonuniversal constant, $\eta=1/4$ is critical exponent of 2D KT phase transition and $t=(T-T_{\rm c})/T_{\rm c}$. Here $T_{\rm c}$ is the critical temperature corresponding to the KT phase transition associated with superconductivity. If the temperature is low enough in our calculations, then the curve of $\chi_{\scriptscriptstyle{\rm SC}} L^{\eta-2}$ should cross at the critical inverse temperature $\beta_{\rm c}$. As shown in Fig. 3, if superconductivity exists, then the critical temperature for KT phase transitions should be lower than what we are currently able to achieve in this work.
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Fig. 3. The data collapse of the scaled superconducting susceptibility $\chi_{\scriptscriptstyle{\rm SC}}L^{-2+\eta}$ against the inverse temperature $\beta$, where $\eta=\frac{1}{4}$. The corresponding parameter is $\{\lambda=3.0, J=0.3, \mu=-2.0\}$. We can see that there is no sign of crossing and as we increase the lattice size $L$, the $\chi_{\scriptscriptstyle{\rm SC}}L^{-2+\eta}$ decreases, suggesting that if the superconductivity exists, the KT transition temperature will be below the lowest temperature we reached.
Spin Susceptibility. We study the critical fluctuation of the fermionic QCP from measuring the spin susceptibility at different temperatures. Following the procedure in Ref. [19], we fit the Ising spin susceptibility $\chi_{_{\scriptstyle \rm B}} =\frac{1}{L^2}\sum_{ij} \int_{0}^{\beta} \mathrm{d}\tau \, s_i (\tau) s_j(0) $ at zero momentum and frequency to the following form: \begin{eqnarray} \chi_{_{\scriptstyle \rm B}}(h, T, \boldsymbol{0}, 0)=\frac{1}{c_{t} T^{a_{t}}+c_{h}|h-h_{\rm c}|^{\gamma}}. \tag {15} \end{eqnarray} From fitting of $\chi_{_{\scriptstyle \rm B}}(h=h_{\rm c}, T, \boldsymbol{0}, 0)^{-1}=c_{t} T^{a_{t}}$, we plot the data in log-log scale, the red dotted line as shown in Fig. 4. Then, we can get $a_t= 0.947(4)$ and $c_t = 2.38(6)$. This form of $\chi_{_{\scriptstyle \rm B}}(T)$ is used above to compute the $T_{\rm c}$. The green dotted line is the similar fitting in Ref. [19] ($\lambda=1,J=1$), with a different $a_t= 1.48(4)$. Pseudogap. We can use the stochastic analytic continuation (SAC) of quantum Monte Carlo data[30-33] to get the local density of states. By calculating the imaginary time Green's function $G_{i,j}(0,\tau)=\langle{T_{\tau} c_i(0)c_{j}^†(\tau)}\rangle=-\langle{c_{j}^†(\tau) c_i(0)}\rangle$ and utilizing the stochastic analytic continuation technique, we can calculate the spectrum function $A(k,\omega)$ and then get the density of states of the electrons $N(\omega)$, as shown in Fig. 5. Since the Laplace transform is irreversible, it is difficult to obtain the spectral function from the data of imaginary time green's function by using analytic continuation method. At least it requires that we have good enough Monte Carlo data. Because the computational complexity of DQMC is $O(\beta N^3)$, and imaginary time measurement is too heavy, we only measured imaginary time green function for $L=8$ at quantum critical point $h_{\rm c}=1.3952$ and at different temperatures, and obtain the spectrum through stochastic analytic continuation, as shown in Fig. 5. As temperature lowers or $\beta$ increases, the density of states starts to show a dip at zero frequency, which becomes deeper as $\beta$ increases. This indicates the appearance of a pseudogap,[34] and the onset temperature of the pseudogap can be estimated as $\beta=3$. Compared to the BCS result of $\beta_{\rm c}=6.06$ given above, the two temperature scales agree qualitatively. This is consistent with our expectation that the BCS theory estimates the onset temperature of the formation of Cooper pairs, instead of the condensation temperature of the Cooper pairs. This qualitative agreement also suggests that the pseudogap observed in SAC mainly comes from thermal fluctuations of superconducting order parameters, although it does not rule out contributions from other factors such as the fluctuation of the Ising order parameter. We shall leave a detailed study of possible contributions to the pseudogap to future studies.
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Fig. 4. The red line shows inverse Ising spin susceptibility at QCP $[\chi_{_{\scriptstyle \rm B}}^{-1}(h_{\rm c},T,\boldsymbol{0},0)]$ as a function of temperature $T$, the slope of the log-log plot reveals the power at $a_t= 0.947(4)$ and the intercept gives rise to the prefactor $c_t = 2.38(6)$. As a comparison, the green line shows the temperature dependence obtained in Ref. [19] where $a_t= 1.48(4)$. Obviously, the scaling behavior is different when we increase the coupling constant.
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Fig. 5. Density of states data obtained form SAC. Above the quantum critical point $h_{\rm c}=1.3952$, as the temperature decreases, the density of states gradually decreases at zero frequency, which means that as the temperature drops we will see a pseudogap.
In summary, we have studied the superconductivity near a fermionic QCP using BCS theory and DQMC simulations. Using the model in Ref. [19] as an example, we compute the $T_{\rm c}$ at the QCP using a BCS-type mean-field theory, replacing the phonon by the critical fluctuations. The mean-field result agrees qualitatively with the onset temperature of pseudogap behavior observed in DQMC simulations. This temperature scale is much larger than the KT-transition temperature, which is below the lowest temperature our simulation can reach. Our finding supports the scenario of preformed Cooper pairs in unconventional superconductivity, which is a possible explanation for the pseudogap phenomenon in cuprate superconductors.[35-38] It will be interesting to apply the same method to other models[20] of fermionic QCPs, to see if the mean-field theory can predict $T_{\rm c}$ qualitatively. If so, it would be useful for designing models and searching for materials realizing high-$T_{\rm c}$ superconductivity or fermionic QCPs. Furthermore, it will be interesting to consider the impact of corrections beyond mean-field theory, and use the full momentum and frequency dependence of $\chi$ and Eliashberg theory[39,40] to give better estimation of $T_{\rm c}$. We will leave these to future works. In the numerical study of fermionic QCPs, it is necessary to monitor the onset of superconductivity, because the formation of superconducting gap depletes the fermionic density of states near the Fermi energy, and therefore changing the critical scaling of the QCP. In our simulation, Fig. 5 shows that this process begins at the pseudogap temperature, instead of the KT-transition temperature. In fact, in Fig. 4, the scaling behavior we observe below the pseudogap temperature is different from the previous work,[19] which may be caused by this effect. This implies that, in the study of fermionic QCPs, to avoid the impact of superconductivity, one should monitor the pseudogap temperature instead of the KT-transition temperature, which was performed in most of the previous studies on this subject.[19,20] Acknowledgements. We thank Xiao Yan Xu and Chuang Chen for invaluable discussions. Y.Q. is supported by the National Natural Science Foundation of China (Grant No. 11874115). GPP and ZYM acknowledge the support from the Research Grants Council of Hong Kong SAR of China (Grant Nos. 17303019, 17301420, 17301721, 17309822, and AoE/P-701/20), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB33000000), the K. C. Wong Education Foundation (Grant No. GJTD-2020-01), and the Seed Funding “QuantumInspired explainable-AI” at the HKU-TCL Joint Research Centre for Artificial Intelligence. We thank the Information Technology Services at the University of Hong Kong and the Tianhe platforms at the National Supercomputer Center in Guangzhou for their technical support and generous allocation of CPU time. The authors acknowledge Beijing PARATERA Tech CO., Ltd. (https://www.paratera.com/) for providing HPC resources that have contributed to the research results reported within this study. 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