Signatures of the Mott transition in the antiferromagnetic state of the two-dimensional Hubbard model

The properties of a phase with large correlation length can be strongly influenced by the underlying normal phase. We illustrate this by studying the half-filled two-dimensional Hubbard model using cellular dynamical mean-field theory with continuous-time quantum Monte Carlo. Sharp crossovers in the mechanism that favors antiferromagnetic correlations and in the corresponding local density of states are observed. These crossovers occur at values of the interaction strength $U$ and temperature $T$ that are controlled by the underlying normal-state Mott transition.

Figure 1

(a) Néel temperature $T_N^d$ versus $U$ at $n=1$. Color corresponds to the magnitude of the staggered magnetization $m_z$ (raw data is in Appendix pp2). Magenta line indicates the maximum of $m_z$ at fixed $T$. The AF phase is defined by the loci where $m_z\ne 0$ and is delimited by $T_N^d$. At low temperature the normal state shows a first-order transition between a metal and a Mott insulator (orange triangles lines) that ends at a critical end point at $(U_{\mathrm{MIT}},T_{\mathrm{MIT}})$ (orange filled circle). (b),(c) Double occupancy $D$ versus $T$ at $U=4,5<U_{\mathrm{MIT}}$ and $U=8,12>U_{\mathrm{MIT}}$, for both AF and normal states (filled and open circles, respectively). For benchmarks, we show data from alternative methods: diagrammatic Monte Carlo from Ref. [67] (diamonds), DCA extrapolated to infinite lattice from Ref. [67] (squares), determinantal QMC on ${10}^2$ lattice from Ref. [42] (up triangles) and extrapolated to thermodynamic limit [38] (left triangles), dual boson scheme from Ref. [68] (down triangle), diagrammatic determinant Monte Carlo extrapolated to thermodynamic limit from Ref. [69] (right triangles), and dual fermion scheme from Ref. [69] (pentagons). (d) Difference in potential, kinetic, and total energies (red, blue, and green lines) between the AF and normal state versus $U$ at $T=1/20$. (e) Phase diagram $T-U$ with color map of $\mathrm{\Delta }{E}_{\mathrm{pot}}$. A change of sign in $\mathrm{\Delta }{E}_{\mathrm{pot}}$ occurs along a line connecting $U_{\mathrm{MIT}}$ and $T_N^{d,\max }$. It accompanies the loci of the largest condensation energy (green diamonds), which in turn correlates with the normal state Widom line [60, 70, 71], emanating from the Mott end point, and determined by max ${dD/dU|}_T$ (open circles). See also Fig. 10 in Appendix pp3. For benchmark, we show with a magenta circle the $T=0$ variational QMC calculation of Ref. [45] for $\mathrm{\Delta }{E}_{\mathrm{pot}}=0$.

Figure 2

Temperature evolution of (a) $N\left(\omega \right)$ and (b) its spin projections for $U=4<U_{\mathrm{MIT}}$, as obtained from analytically continued data [80]. AF (normal) state DOS are shown with color (dashed gray) lines. (c) AF gap ${\mathrm{\Delta }}_{\mathrm{AF}}$ versus $T$ for $U=4$, as measured by the distance between the Bogoliubov peaks. The closure of ${\mathrm{\Delta }}_{\mathrm{AF}}$ follows the decrease of $m_z$. (d),(e) Same as (a),(b) but for $U=12>U_{\mathrm{MIT}}$.

Figure 3

(a) $N\left(\omega \right)$ in the AF state along with (b) its two spin projections, for $T=1/20$ and different values of $U$. Normal state solutions are shown with gray lines. See also Fig. 11 in Appendix pp4. (c) Difference between the AF gap ${\mathrm{\Delta }}_{\mathrm{AF}}$ and the normal state Mott gap ${\mathrm{\Delta }}_{\mathrm{M}}$, measured by the distance between the two Hubbard bands, versus $U$ for $T=1/20$. (d) $({\mathrm{\Delta }}_{\mathrm{AF}}-{\mathrm{\Delta }}_{\mathrm{M}})/{\mathrm{\Delta }}_{\mathrm{M}}$ versus $U$ for $T=1/20$.

Figure 4

$2\times 2$ plaquette for the CDMFT.

Figure 5

(a) Average sign in CTQMC AF simulations versus $U$. (b) Angle $\theta$ in Eqs. (A21), (A22) versus $U$. Data are shown for temperatures $T=1/20,1/10,1/5$, and $1/4$. The angle $\pi /4$ appears only in the normal phase, namely when $m_z=0$.

Figure 6

Staggered magnetization $m_z$ versus $U$ for different values of $T$. $T_N^d$ is obtained from the mean of the two temperatures where $m_z$ changes from finite to a small value (here $m_z=0.045$).

Figure 7

Contributions to the difference in kinetic energy between AF and normal state, as a function of $U$. Top (bottom) panels show $\mathrm{\Delta }{E}_{\mathrm{kin}}^{\left(1\right)}$ ($\mathrm{\Delta }{E}_{\mathrm{kin}}^{\left(2\right)}$). Data are shown for $T=1/20<T_{\mathrm{MIT}}, T=1/10>T_{\mathrm{MIT}}$, and $T=1/4$ (left, central, and right columns, respectively). For $T<T_{\mathrm{MIT}}$ two normal state solutions coexist at the Mott transition.

Figure 8

Difference in potential energy $\mathrm{\Delta }{E}_{\mathrm{pot}}=E_{\mathrm{pot}}^{\mathrm{AF}}-E_{\mathrm{pot}}^{\mathrm{NS}}$ (top panels), in kinetic energy $\mathrm{\Delta }{E}_{\mathrm{kin}}$ (central panels), and in total energy $\mathrm{\Delta }{E}_{\mathrm{tot}}$ (bottom panels) as a function of $U$. As in Fig. 7, data are shown for $T=1/20<T_{\mathrm{MIT}}, T=1/10>T_{\mathrm{MIT}}$, and $T=1/4$ (left, central, and right columns, respectively). For benchmark, at our lowest temperature [panels (a),(d),(g)] we show the $T=0$ variational QMC calculation of Ref. [45].

Figure 9

Ratio between kinetic energy gain and potential energy gain versus $U$, at $T=1/10$ (red triangles) and $T=1/20$ (black circles). At large $U, \mathrm{\Delta }{E}_{\mathrm{kin}}$ is approaching minus twice $\mathrm{\Delta }{E}_{\mathrm{pot}}$.

Figure 10

Phase diagram $T-U$ of the two-dimensional half-filled Hubbard model solved with plaquette CDMFT. This figure extends Fig. 1 of the main text. In the normal state, the first-order Mott metal-insulator transition is delimited by the spinodal lines $U_{c1}\left(T\right)$ and $U_{c2}\left(T\right)$ (up and down triangles, respectively), and terminates at the critical end point (filled orange circle). The Widom line $T_{\mathrm{W}}$ (open circles with dotted orange line) is a crossover that extends the first-order transition in the supercritical region and is determined by the inflection points along paths at constant temperature in the double occupancy $D$ (i.e., by max ${dD/dU|}_T$). At values of $U$ and $T$ where the properties of the underlying normal state are governed by the Mott transition we find sharp crossovers between weakly and strongly correlated AF. The loci where the potential (kinetic) energy changes sign are shown by red triangles (blue squares). The line where $\mathrm{\Delta }{E}_{\mathrm{pot}}$ changes sign extends up to the maximum of the AF dome. It parallels the loci of the largest condensation energy (green diamonds), which in turn correlates with the Widom line $T_{\mathrm{W}}$. The line where $\mathrm{\Delta }{E}_{\mathrm{kin}}$ changes sign still emerges from the Mott end point, but extends down to $U\approx 5$ and up to $T\approx 1/5$.

Figure 11

(a) $N\left(\omega \right)$ in the AF state along with (b) its two spin projections, $N_{\uparrow }\left(\omega \right)$ and $N_{\downarrow }\left(\omega \right)$, for $T=1/20$ and different values of $U$. Normal state solutions are shown with gray lines. This figure extends Figs. 3 and 3 of the main text, where fewer values of $U$ are shown.