Instanton gas approach to the Hubbard model

In this paper, we consider a path integral formulation of the Hubbard model based on a Hubbard-Stratonovich transformation that couples the auxiliary field to the local electronic density. This decoupling is known to have a saddle-point structure that shows a remarkable regularity: The field configuration at each saddle point can be understood in terms of a set of elementary field configurations localized in space and imaginary time which we coin instantons. The interaction between instantons is short ranged. Here, we formulate a classical partition function for the instanton gas that has predictive power. For a given set of physical parameters, we can predict the distribution of instantons and show that the instanton number is sharply defined in the thermodynamic limit, thereby defining a unique dominant saddle point. Decoupling in the charge channel conserves SU(2) spin symmetry for each field configurations. Hence, the instanton approach provides an SU(2) spin-symmetric approximation to the Hubbard model. It fails, however, to capture the magnetic transition inherent to the Hubbard model on the honeycomb lattice despite being able to describe local moment formation. In fact, the instanton itself corresponds to local moment formation and concomitant short-ranged antiferromagnetic correlations. This aspect is also seen in the single particle spectral function that shows clear signs of the upper and lower Hubbard bands. Our instanton approach bears remarkable similarities to local dynamical approaches, such as dynamical mean-field theory, in the sense that it has the unique property of allowing for local moment formation without breaking the SU(2) spin symmetry. In contrast to local approaches, it captures short-ranged magnetic fluctuations. Furthermore, it also offers possibilities for systematic improvements by taking into account fluctuations around the dominant saddle point. Finally, we show that the saddle point structure depends upon the choice of lattice geometry. For the square lattice at half filling, the saddle-point structure reflects the itinerant to localized nature of the magnetism as a function of the coupling strength. The implications of our results for Lefschetz thimble approaches to alleviate the sign problem are also discussed.

Figure 1

Spin-spin correlations, $\frac{1}{3}\langle {\widehat{\mathit{S}}}_{{\mathit{x}}_0}\left(T\right)·{\widehat{\mathit{S}}}_{{\mathit{x}}_0+\mathit{x}}\left(T\right)\rangle$, for a field configuration with one instanton at space time $(X,T)$. We consider two values of ${\mathit{x}}_0$. The left black circle corresponds to ${\mathit{x}}_0=X$. The other value of ${\mathit{x}}_0$ (right black circle) is far from the instanton. $R_1$ and $R_2$ are two Cartesian coordinates of the lattice sites, displayed in the units of the distance between nearest neighbors. These calculations were performed on a $12\times 12$ lattice at interaction strength $U=2.0\kappa$ (see Secs. 2a and 2b for the notation).

Figure 2

(a) Spectral functions in momentum space using the ALF [19] implementation of the auxiliary field QMC. (b) The same spectral functions obtained with instanton gas model. (c) The share of the lower peak in the overall spectral weight along the same profile in momentum space. Calculations were done for $12\times 12$ lattice with $N_{\tau }=256$ and $\beta \kappa =20$ (see Secs. 2a and 2b for the notation). The interaction strength is equal to $U=6.0\kappa$, which is equal to the bandwidth.

Figure 3

A comparison of the average number of instantons obtained from real QMC data with the analytical instanton gas model. The QMC data corresponds to a $6\times 6$ spatial size, with $N_{\tau }=512$ and $\beta \kappa =20$.

Figure 4

Histograms depicting the relative contributions of the various $N$-instanton saddles to the full partition function. The horizontal axis corresponds to the action of an $N$-instanton solution, offset by an amount equal to the action of the observed saddle with the least number of instantons. One can clearly see that the minimal number of observed instantons increases with increasing $U$. These calculations were performed on a $6\times 6$ lattice with $N_{\tau }=512, \beta \kappa =20$.

Figure 5

Visualization of the ${\varphi }_{\mathit{x},\tau }$ field for the saddle-point configuration with one instanton. The widths of the vertical spindles correspond to the value of $|{\varphi }_{\mathit{x},\tau }|$ at a given spatial lattice site and time step in Euclidean time. For clarity, we only draw the spindles if $|{\varphi }_{\mathit{x},\tau }|>\epsilon$, where $\epsilon$ is some suitably small threshold. To clearly illustrate the spatial positions of the spindles within the lattice, we also draw their projections on the $\tau =275$ plane. Calculations were carried out on a $6\times 6$ lattice with interaction strength $U=5.0\kappa , N_{\tau }=512$, and $\beta \kappa =20$.

Figure 6

(a) The relative weight of the first nonvacuum thimble with respect to the full partition function as a function of the inverse temperature. The calculations were carried out on a $6\times 6$ lattice, with a Euclidean time step corresponding to $N_{\tau }=512$ at $\beta \kappa =20$. (b) The scaling of the relative weight of the first nonvacuum thimble with the spatial system size at fixed $\beta \kappa =20$ and $N_{\tau }=512$. The interaction strength is fixed at $U=2.0\kappa$ for all plots.

Figure 7

The five lowest eigenvalues of the Hessian matrices computed at the one- and two-instanton saddle points. The calculations were performed on a $6\times 6$ lattice with interaction strength $U=6.0\kappa , N_{\tau }=512$, and $\beta \kappa =20$.

Figure 8

The pairwise interaction of instantons and anti-instantons as a function of their separation in Euclidean time for a fixed spatial location. The plots in the left column [(a), (c), (e), (g)] correspond to interaction between a pair of instantons, while the plots in the right column [(b), (d), (f), (h)] correspond to the interaction between an instanton and an anti-instanton. The first row [(a), (b)] shows (anti)instantons at the same spatial lattice site, the second row [(c), (d)] corresponds to (anti)instantons located at nearest neighbors (opposite sublattices), the third row [(e), (f)] corresponds to a spatial separation of next-nearest neighbors, and the last row [(g, (h)] shows the interaction of (anti)instantons located at sites which are separated by two lattice unit vectors. In each case, we include a sketch of the corresponding spatial configuration on the hexagonal lattice in the inset. If $U_g^{\left(2\right)}$ is not shown, it means that its influence is small and the $U_S^{\left(2\right)}+U_{\lambda }^{\left(2\right)}+U_g^{\left(2\right)}$ line coincides with the $U_S^{\left(2\right)}+U_{\lambda }^{\left(2\right)}$ line. These calculations were performed on a $12\times 12$ lattice with $\beta \kappa =20$ and $N_{\tau }=512$, with interaction strength $U=4.6\kappa$.

Figure 9

(a) The action of an instanton-instanton pair as a function of the spatial separation between the instantons. In this case, zero action corresponds to the instantons being infinitely separated. (b) The same situation but for an instanton-anti-instanton pair. These calculations were performed on a $12\times 12$ lattice with $\beta \kappa =20$ and $N_{\tau }=512$, and interaction strength $U=4.6\kappa$. The (anti)instantons are placed at the same Euclidean time slice. The spatial separation is in units of the distance between nearest neighbors. In both cases, a power-law fit has been added.

Figure 10

The distribution of the number of instantons at various interaction strengths obtained from QMC configurations. The top two plots [(a), (b)] show the distributions obtained on $6\times 6$ and $12\times 12$ lattices, respectively, and the bottom plot (c) compares these data in the case of large interaction strength $U=5.0\kappa$. We note that the $x$ axis of the $12\times 12$ lattice data is rescaled by a factor of 4 to have the curves lie on top of each other and to make the comparison more straightforward. All these calculations were performed at $\beta \kappa =20$ and $N_{\tau }=512$. For each data set, a Gaussian fit of the lattice data has also been performed—it is shown with the lines of the same colors as corresponding data sets.

Figure 11

(a) A comparison of the instanton density obtained from real QMC data with various instanton gas models. (b) The same comparison but for the variance. The QMC data corresponds to $N_{\tau }=512$ and $\beta \kappa =20$.

Figure 12

(a) Double occupancy obtained from the instanton gas model and from QMC data. (b) AFM susceptibility obtained from the analytical model. In both cases, the instanton profiles and actions are obtained on the $6\times 6$ lattice with $N_{\tau }=512$ and $\beta \kappa =20$ were used as input. QMC data were obtained on the same lattice.

Figure 13

The distribution of the squared spin across the hexagonal lattice for background field configuration consisting of a single instanton (a) or two instantons (b). What is plotted is the difference between the squared spin at a given lattice site $x$ in the presence of semiclassical objects and its value for the vacuum configuration: $\langle {\widehat{S}}_x^2\rangle {{|}_{N_{\text{inst}}}-\langle {\widehat{S}}_x^2\rangle |}_{\text{vac}}$. A base-10 logarithmic scale is used for the color scale. Both instantons are located at the same Euclidean time slice where the spin operator is measured. This calculation refers to a $12\times 12$ lattice with $\beta \kappa =20$ and $N_{\tau }=512$, with interaction strength $U=2.0\kappa$. $R_1$ and $R_2$ are Cartesian coordinates of the lattice sites, displayed in the units of the distance between nearest neighbors.

Figure 14

(a) Charge-charge correlations, $\frac{1}{4}\langle {\widehat{q}}_{{\mathit{x}}_0}\left(T\right)·{\widehat{q}}_{{\mathit{x}}_0+\mathit{x}}\left(T\right)\rangle$, for a field configuration with one instanton at space-time $(X,T)$. We consider two values of ${\mathit{x}}_0$. The left black circle corresponds to ${\mathit{x}}_0=X$. The other value of ${\mathit{x}}_0$ (right black circle) is far from the instanton. $R_1$ and $R_2$ are two Cartesian coordinates of the lattice sites, displayed in the units of the distance between nearest neighbors. The corresponding plot for the spin-spin fluctuations can be found in Fig. 1. (b) Ratio of spin-spin and charge-charge correlators on the same one-instanton saddle $4\langle {\widehat{S}}_{{\mathit{x}}_0}^{\left(3\right)}\left(T\right){\widehat{S}}_{{\mathit{x}}_0+\mathit{x}}^{\left(3\right)}\left(T\right)\rangle /\langle {\widehat{q}}_{{\mathit{x}}_0}\left(T\right){\widehat{q}}_{{\mathit{x}}_0+\mathit{x}}\left(T\right)\rangle$ centered at the location of the instanton ${\mathit{x}}_0=X$. The calculation was done on the same lattice as for Fig. 13.

Figure 15

A comparison of the density of states at the Dirac point obtained from QMC data (left column) with that obtained from the instanton gas model (right column). All calculations correspond to a spatial volume of $12\times 12$ with $\beta \kappa =20$ and $N_{\tau }=256$, while the corresponding interaction strength in units of $\kappa$ is displayed on each plot. The analytical continuation has been performed with stochastic MEM.

Figure 16

(a) The Euclidean-time fermion propagator at the Dirac point in momentum space, obtained from the instanton gas model with hardcore repulsion. Spatial volumes of $6\times 6, 12\times 12$, and $18\times 18$ are compared at $U=6.0\kappa$, with $N_{\tau }=256$ in each case. (b) The electron density of states obtained after analytical continuation performed with stochastic MEM.

Figure 17

Spectral functions at the $\mathrm{\Gamma }$ point obtained from QMC data with Ising fields (left column) and from the instanton gas model with hardcore repulsion (right column). The calculations were performed on a $12\times 12$ lattice with $\beta \kappa =20$ and $N_{\tau }=256$, while the corresponding interaction strength is shown on each plot.

Figure 18

Comparison of the spectral functions at the $\mathrm{\Gamma }$ point for different spatial volumes in the instanton gas model with hardcore repulsion. The interaction strength is equal to $U=6.0\kappa$ and inverse temperature $\beta \kappa =20$.

Figure 19

The energies corresponding to the upper and lower peaks in the single electron spectral function at the $\mathrm{\Gamma }$ point in the strong coupling limit $U>3.5$, where the two-peak structure becomes apparent (see Fig. 17 for the full profiles). All data were produced on a $12\times 12$ lattice at $N_{\tau }=256$ and $\beta \kappa =20$. The hardcore repulsion model was used for the instanton gas approach.

Figure 20

(a) The relative weight of instantons, domain walls, and vacuum saddle in the full partition function for the square lattice Hubbard model. These were obtained from the configurations generated in QMC on the square lattice, followed by the application of GF. The data was generated on a $8\times 8$ square lattice with $\beta \kappa =20, N_{\tau }=512$. (b) An example of a domain-wall configuration at $U=1.5\kappa$. The color scale shows the value of the auxiliary field $\varphi$.

Figure 21

(a) The profile of the action during the downward flow to the saddle points. This procedure essentially amounts to the solution of the gradient flow equation in Eq. (13). Four different configurations are shown, each ending at a different saddle. (b) The distribution for the final actions obtained after the application of the downward gradient flow to the set of configurations generated with HMC. These calculations were performed on a $12\times 12$ lattice with $\beta \kappa =20$ and $N_{\tau }=512$, with interaction strength $U=2.0\kappa$ and $\alpha =0.99$.

Figure 22

(a) Relative weight of the thimble attached to the first nontrivial saddle with respect to the full partition function, $Z_1/Z$, as a function of the $\alpha$ parameter. These calculations were performed on a $6\times 6$ lattice with $N_{\tau }=512, \beta \kappa =20$. (b) A comparison of the histograms obtained for $N_{\tau }=256$ and $N_{\tau }=512$ with other parameters fixed ($6\times 6$ lattice, $\beta \kappa =20, \alpha =0.99$). The interaction strength is fixed at $U=2.0\kappa$ for all plots.

Figure 23

Analytical profiles for instantons obtained from Eq. (B25) for the case of single-instanton [(a), (b)] and instanton-anti-instanton [(c), (d)] solutions. (a) and (c) show the derivative $\dot{\theta }$, while the plots (b) and (d) show the $\theta$ angle itself.

Figure 24

(a) Absolute values of the elements of the reduced Hessian for the one-instanton saddle. The red rectangle denotes the region of the Hessian matrix which is sufficient to calculate the full determinant with high precision. (b) The elements of the reduced Hessian for the two-instanton saddle. To highlight the most important part of the matrix, we only plot the points where the element of the matrix is larger than 0.01. The conversion of the $2+1\mathrm{D}$ coordinates to a linear index is done according to the rule Eq. (C4). These calculations were performed on a $6\times 6$ lattice with $N_{\tau }=256, \beta \kappa =20, U=5.0\kappa$.

Figure 25

Numerical proof for the Eq. (C8). $lndet{\mathcal{H}}^{\left(0\right)}$ is shown for the vacuum, and $lndet{\mathcal{H}}_{\perp }^{\left(N\right)}$ is shown for the $N$-instanton saddle. The calculations were performed on a $6\times 6$ lattice with $\beta \kappa =20, U=2\kappa$ and $N_{\tau }=256$.

Figure 26

The distribution of the number of instantons obtained from classical grand canonical Monte Carlo for instantons. The first two plots [(a), (b)] show the results for the model which incorporates the full interaction profile (obtained from the $6\times 6$ and $12\times 12$ lattices), while the last plot (c) shows the distribution for the case where only a hardcore repulsion between the instantons is taken into account. For all these calculations, $\beta \kappa =20$. Gaussian fits are also included (shown with lines of the same colors as the corresponding data sets).

Figure 27

(a) The average number of instantons, taken as the center of the distribution, from the classical grand canonical Monte Carlo simulations of the instanton gas model, taking into account only hardcore repulsion. (b) The variance of the distribution for the number of instantons from the same simulations. All data are obtained at $\beta \kappa =20$. Note the rescaling of the data points for the $12\times 12$ and $18\times 18$ lattices.

Figure 28

The ratio of frequencies ${\mathcal{R}}_{\mathit{x}}$ from Eq. (E15). These calculations were performed on the one instanton saddle on a $6\times 6$ lattice with $\beta \kappa =20, U=6\kappa$, and $N_{\tau }=512$. The instanton is located at the origin.

Figure 29

(a) Dependence of the $\theta$ angle in the Gutzwiller projection for the optimal description of the $N$ instanton saddle point. All instantons are located at the same spatial site and placed equidistantly in Euclidean time. (b) Overlap between two variants of probe wave function and the wave function following from the $N$-instanton saddle. These calculations were performed on a $6\times 6$ lattice with $\beta \kappa =20, U=6\kappa$, and $N_{\tau }=512$.