Ising Nematic Quantum Critical Point in a Metal: A Monte Carlo Study

The Ising nematic quantum critical point associated with the zero-temperature transition from a symmetric to a nematic metal is an exemplar of metallic quantum criticality. We carry out a minus-sign-free quantum Monte Carlo study of this quantum critical point for a two-dimensional lattice model with sizes up to $24\times 24$ sites. For the parameters in this study, some (but not all) correlation functions exhibit scaling behavior over the accessible ranges of temperature, (imaginary) time, and distance, and the system remains nonsuperconducting down to the lowest accessible temperatures. The observed scaling behavior has remarkable similarities to recently measured properties of the Fe-based superconductors proximate to their putative nematic quantum critical point.

Popular Summary

Numerous materials exhibit quantum phase transitions at which physical properties at absolute zero temperature change dramatically with small changes of a physical parameter such as pressure or chemical doping. Here, we present the first exact theoretical study of a quantum phase transition in a model metallic system and observe behaviors that are strikingly similar to those observed in several widely studied families of high-temperature superconducting materials.

In contrast to prior work on this problem that has relied on (often uncontrolled) approximate calculations, we apply an exact numerical method called determinant quantum Monte Carlo. We study a simple model on a square lattice that describes an Ising nematic quantum phase transition, where the system spontaneously breaks the symmetry between the horizontal and vertical lattice directions. Such transitions are observed in most iron-based high-temperature superconductors. We find that the quantum phase transition in our model is continuous and that its properties are fundamentally different from those of an Ising nematic transition in an insulator.

Our work paves the way for a deeper understanding of quantum critical phenomena in metals, a frontier problem in theoretical physics with broad applications in the field of quantum materials.

Figure 1

Phase diagram of the model obtained by DQMC simulations, as a function of the transverse field $h$ and temperature $T$. Here, the other parameters entering the Hamiltonian Eq. (3) are $V=t$, $\alpha =0.5$, $\mu =-0.5t$. The solid line marks the transition temperature $T_N$ between the nematic and the symmetric phases. The line extrapolates to the $T=0$ QCP at $h_c$. The dashed line marks $T_{\text{cross}}$, where the nematic susceptibility reaches 50% of its magnitude at $h=h_c\approx 3.525$ at the same temperature. The gray lines show the corresponding temperatures for the case $\alpha =0$ (where the fermions and pseudospins are decoupled). In this case, the QCP occurs at $h_{c0}\approx 3.06$.

Figure 2

Fermion Green’s function $G(\mathbf{k},\tau =\beta /2)$ as a function of momentum $\mathbf{k}$ across the Brillouin zone for three different values of the transverse field: $h=3.35$ (in the nematic phase), $h=3.525$ (near the QCP), and $h=3.7$ (in the disordered phase). In the nematically ordered phase, the data are averaged over both orientations of the order parameter. $G(\mathbf{k},\tau =\beta /2)$ is proportional to the integral of the spectral function $A(\mathbf{k},\omega )$ over an energy window of width $T$ [see Eq. (12)]. The temperature is $T=1/8$, and the system size is $L=20$. The data are taken from systems with either periodic or antiperiodic boundary conditions. The Fermi surfaces are seen as peaks in $G(\mathbf{k},\tau =\beta /2)$. The lower right-hand panel shows the Fermi surface of the bare band structure.

Figure 3

Illustration of the lattice model Eq. (2). Spinful electrons reside on the sites, while the Ising pseudospins live on the bonds. The pseudospins interact with their neighbors antiferromagnetically, and are coupled to the fermion bond density.

Figure 4

The phase diagram for several sets of parameters. Gray circles represent the decoupled problem ($\alpha =0$), while black diamonds and blue squares represent the coupled problem ($\alpha =0.5$) at 0.6 and 0.8 fermions per site, respectively.

Figure 5

Nematic susceptibility at the critical $h$, $\chi (h_c,T)$, as a function of temperature. Panel (a) shows the susceptibility on a linear scale. The same data are displayed on a log-log scale in (b). The black line is proportional to $1/T$.

Figure 6

(a) Nematic susceptibility $\chi (h,T)$ as a function of $h-h_c$ for system sizes $L=16$, 20, and 24 at various temperatures. (b) The same data, scaled appropriately for $\lambda =1$ and $\gamma =1$, collapse on a single curve. The black curve is the function $F(x)=2.1/(1+2.9x)$.

Figure 7

Quadratic momentum dependence of the inverse nematic correlator at zero frequency. (a) $D^{-1}$ versus $|\mathbf{q}{|}^2$ for all orientations of $\mathbf{q}$, at $h=h_c$ and various temperatures. (b) The same for fixed temperature $T=0.17t$ and a variety of values of the quantum tuning parameter $h$. In each case the momentum dependence remains quadratic. Momenta from multiple system sizes (represented by squares for $L=16$, circles for $L=20$, and diamonds for $L=24$) fall on the same curve, indicating that the properties shown are in the thermodynamic limit.

Figure 8

Comparison between the thermodynamic (${\omega }_n=0$) functional approximant $\mathcal{A}$, from Eq. (6), and the nematic correlator $D$. Data shown represent $L=16$, 20, and 24, $h=3.525$, 3.7, 3.9, and 4.1, and temperatures $T=1.0$, 0.67, 0.5, 0.33, 0.25, 0.17, 0.13, 0.1, 0.05, and 0.025. To exhibit the momentum dependence, we use blue circles for $\mathbf{q}=\mathbf{0}$ (71 data points), black squares for $|\mathbf{q}|=(2\pi /L)$ and $(2\pi /L)\sqrt{2}$ (568 data points), and red diamonds for all other $\mathbf{q}$ (1916 data points).

Figure 9

Nematic and quadrupolar susceptibilities versus inverse temperature at $h=h_c$. Multiple system sizes are shown with squares for $L=16$, circles for $L=20$, and diamonds for $L=24$. The error bars reflect uncertainties due to finite system sizes, estimated from the differences between the measured susceptibilities of systems with different boundary conditions.

Figure 10

Inverse nematic and quadrupolar susceptibilities versus $h-h_c$ at various temperatures. Multiple system sizes are shown with squares for $L=16$, circles for $L=20$, and diamonds for $L=24$. The error bars reflect uncertainties due to finite system sizes, estimated from the differences between the measured susceptibilities of systems with different boundary conditions.

Figure 11

Inverse nematic and quadrupolar correlators versus $|\mathbf{q}{|}^2$ for all orientations of $\mathbf{q}$ at $h=h_c$ and $T$ from 0.1 to 1.0. Momenta from multiple system sizes (represented by squares for $L=16$, circles for $L=20$, and diamonds for $L=24$) fall on the same curve, indicating that the properties shown are in the thermodynamic limit. Error bars are smaller than the symbol size.

Figure 12

Frequency dependence of the inverse nematic correlator at $h\approx h_c$ for a variety of small momenta $\mathbf{q}$ in an $L=20$ system at temperatures $T=0.025$, 0.033, 0.05, 0.1 (shown as squares, stars, circles, and diamonds, respectively). Different colors represent different values of $\mathbf{q}$, and filled symbols mark frequencies $|{\omega }_n|<v_F|\mathbf{q}|$, where $v_F$ is the minimum value of the bare Fermi velocity. The frequency dependence is essentially linear except at the few smallest frequencies and momenta. The inset shows the subset of the data with $T=0.025$ and momenta along the (10) direction shifted by their zero-frequency value, with the same scale as the main figure.

Figure 13

The inverse of the equal time nematic correlator at $h=h_c$ and $T=0.1$, plotted versus the square of the spatial separation $\mathbf{r}$ for various system sizes. The narrow spread in the data (which have not been chosen to lie along any high-symmetry direction) indicates an emergent isotropy of the correlations in this regime. The apparent $1/|\mathbf{r}{|}^2$ behavior in the thermodynamic limit is what one would get by Fourier transform of $\mathcal{A}$ from Eq. (6).

Figure 14

Frequency dependence of the inverse quadrupolar correlator at $h\approx h_c$ for a variety of small momenta in an $L=20$ system at temperatures $T=0.025$, 0.033, 0.05, 0.1 (shown as squares, stars, circles, and diamonds, respectively). Different colors represent different values of $\mathbf{q}$, with the same color scale as in Fig. 12. Filled symbols mark frequencies $|{\omega }_n|<v_F|\mathbf{q}|$, where $v_F$ is the minimum value of the bare Fermi velocity. The inset shows the subset of the data with $T=0.025$ and momenta along the (10) direction shifted by their zero-frequency value, with the same scale as the main figure. The data for $\mathbf{q}=0$ are scaled down by a factor of 0.02 to appear on the same scale.

Figure 15

$K_{xx}(q_x=0,q_y)$ defined in Eq. (8) for $T=0.05t$, $h=h_c$, and various values of system size, normalized by the universal value ${\rho }_s^{\mathrm{BKT}}$. Momenta from different system sizes lie on the same curve, illustrating the thermodynamic limit. At small values of $q_y$, $K_{xx}(q_x=0,q_y)$ lies well below ${\rho }_s^{\mathrm{BKT}}$, indicating the system is not superconducting.

Figure 16

(a),(b) Pairing susceptibility in the $s$-wave and $d$-wave channels, $P_{s,d}$ [Eq. (11)], as a function of $h$, at $T=0.083$. Different system sizes are displayed. (c),(d) $P_{s,d}$ as a function of temperature at the QCP, $h=3.525$.

Figure 17

$\tilde{N}(T)$, defined in Eq. (13), as a function of $h$ for system size $L=20$ at different temperatures. $\tilde{N}$ equals the density of states at the Fermi energy in the limit $T\to 0$.

Figure 18

Left-hand panels show $Z_{{\mathbf{k}}_{\mathbf{F}}}$ [defined in Eq. (15)] as a function of angle $\theta$ along the Fermi surface, for different values of $h$. The QCP is at $h_c\approx 3.525$. Right-hand panels show the temperature dependence of $Z_{{\mathbf{k}}_{\mathbf{F}}}(T)$ for high-symmetry directions of $k_F$. $Z_{{\mathbf{k}}_{\mathbf{F}}}$ approaches the conventionally defined quasiparticle weight as $T\to 0$.

Figure 19

(a) $Z_{{\mathbf{k}}_{\mathbf{F}}}$ as a function of $h$ for different temperatures. Circles (diamonds) correspond to $\theta =45°$ ($\theta =0°$) (where $\theta$ is defined in the inset of Fig. 18). (b) The Fermi velocity renormalization $v_F/v_{F,0}$ [where $v_F$ is defined in Eq. (14) and $v_{F,0}$ is the noninteracting Fermi velocity] as a function of $h$ for $\theta =0°$, 45°.

Figure 20

An example of the procedure used to calculate the critical value of $h$ for a given temperature. Here, for $T=0.33$, the best data collapse is obtained for $h_N\approx 3.25$. The exponents take the two-dimensional Ising values $\gamma =7/4$, $\nu =1$ for this thermal transition. The nematic susceptibility and quadrupolar susceptibility have similar behavior.

Figure 21

Left-hand panels: The imaginary part of the inverse of the Matsubara Green function $\mathrm{Im}[G^{-1}({\mathbf{k}}_{\mathbf{F}},{\omega }_n)]$ for $h=3.525$. Data points from different temperatures mostly lie on the same curve. Right-hand panels: The imaginary part of the self-energy. The dashed line in the bottom left-hand panel corresponds to $Z=0.93$.

Figure 22

Inverse nematic and quadrupolar correlators versus $|\mathbf{q}|$ for all orientations of $\mathbf{q}$ at $h=h_c$ and temperatures $T=0.1$, 0.05, 0.025 (curves are offset for clarity). Momenta from multiple system sizes (represented by squares for $L=16$, circles for $L=20$, and diamonds for $L=24$) fall on the same curve, indicating that the properties shown are in the thermodynamic limit. On the plot of $D^{-1}$ at the left, the dotted lines are the functional approximant, Eq. (6). On the plot of $Q^{-1}$ at the right, the dotted lines are a linear fit to the data at nonzero momentum, with a temperature-independent slope. The apparent linear momentum dependence of $Q^{-1}$ suggests a critical power law, though with an apparently different exponent than the one found in $D^{-1}$.

Figure 23

Frequency dependence of the inverse nematic correlator $D^{-1}$ (top) and inverse quadrupolar correlator $Q^{-a}$ (bottom) at $h=4.5\approx 1.3h_c$ for a variety of small momenta $\mathbf{q}$ in an $L=20$ system at temperatures $T=0.025$, 0.033, 0.05, 0.1 (shown as squares, stars, circles, and diamonds, respectively). Here, there is a nonvanishing energy scale associated with nematic fluctuations since we are deep in the disordered phase, but the frequency dependence is similar to that at $h=h_c$ (see Fig. 12). Different colors represent different values of $\mathbf{q}$, and filled symbols mark frequencies $|{\omega }_n|<v_F|\mathbf{q}|$. The insets show the subset of the data with $T=0.025$ and momenta along the (10) direction shifted by their zero-frequency values. For the inset of the bottom panel, the data at $\mathbf{q}=0$ are scaled down by a factor of 0.01.

Figure 24

Comparison between the inverse of the full dynamical functional approximant from Eq. (6), ${\mathcal{A}}^{-1}$, and the inverse of the nematic correlator $D^{-1}$. Data shown represent $L=16$, 20, and 24, $h=3.525$, 3.7, 3.9, and 4.1, and temperatures $T=1.0$, 0.67, 0.5, 0.33, 0.25, 0.17, 0.13, 0.1, 0.05, and 0.025. To exhibit the momentum dependence, we use blue circles for $\mathbf{q}=\mathbf{0}$ (403 data points), black squares for $|\mathbf{q}|=(2\pi /L)$ and $(2\pi /L)\sqrt{2}$ (2,840 data points), and red diamonds for all other $\mathbf{q}$ (5,332 data points).

Figure 25

The inverse of the equal time quadrupolar correlator at $h=h_c$ and $T=0.1$, plotted versus the square of the spatial separation $\mathbf{r}$ along high-symmetry directions for several system sizes. In contrast to the data of Fig. 13, substantial tetragonal anisotropy can be seen by comparing the left-hand panel (horizontal displacements) and the right-hand panel (diagonal displacements). However, the data apparently represent the thermodynamic limit only for $|\mathbf{r}|\lesssim 4$, and therefore we cannot reliably extract long-distance behavior.