Algorithmic approach to diagrammatic expansions for real-frequency evaluation of susceptibility functions

We systematically generate the perturbative expansion for the two-particle spin susceptibility in the Feynman diagrammatic formalism and apply this expansion to a model system—the single-band Hubbard model on a square lattice. We make use of algorithmic Matsubara integration (AMI) [A. Taheridehkordi, S. H. Curnoe, and J. P. F. LeBlanc, Phys. Rev. B 99, 035120 (2019)] to analytically evaluate Matsubara frequency summations, allowing us to symbolically impose analytic continuation to the real-frequency axis. We minimize our computational expense by applying graph invariant transformations [A. Taheridehkordi, S. H. Curnoe, and J. P. F. LeBlanc, Phys. Rev. B 101, 125109 (2020)]. We highlight extensions of the random-phase approximation and T-matrix methods that, due to AMI, become tractable. We present results for weak interaction strength where the direct perturbative expansion is convergent, and verify our results on the Matsubara axis by comparison to other numerical methods. By examining the spin susceptibility as a function of real frequency via an order-by-order expansion, we can identify precisely what role higher-order corrections play on spin susceptibility and demonstrate the utility and limitations of our approach.

Figure 1

Top row: Examples of ladder-like diagrams which must be excluded from the ETM series. Bottom row: Examples of non-ladder-like diagrams which must be included in the ETM series. The dashed lines identify where the ladder-like diagrams split into independent parts (there are no common independent frequency-momenta variables to the left and right). Solid and wavy lines are fermionic and interaction lines, respectively.

Figure 2

Transverse spin susceptibility ${\chi }_T^{\left(m_c\right)}$ for $m_c=0$, 1, 2, and 3 and ${\chi }_{ETM}^{\left(3\right)}$ vs Matsubara frequency $i\mathrm{\Omega }$ at $U/t=2$, $\beta t=5$, and $\mu /t=0$ for $\mathbf{q}=(\pi ,\pi )$. We also present ${\chi }_T^{\left(4\right)}(i\mathrm{\Omega }=0)$, TMA, DF [60], and fRG results from Ref. [30]. Each data point is obtained with $\approx {10}^7$ Monte Carlo samples.

Figure 3

Transverse spin susceptibility vs inverse temperature at different truncation orders $m_c=0$, 1, 2, and 3. The TMA and DF results are also shown for comparison. Data are for the top, $U/t=1$, and bottom, $U/t=2$, with $\mu /t=0$ at $\mathbf{q}=(\pi ,\pi )$ and $i\mathrm{\Omega }=i{\mathrm{\Omega }}_0=0$. Each data point is obtained with $\approx {10}^7$ Monte Carlo samples.

Figure 4

Top: Imaginary part of the $m\mathrm{th}$-order transverse spin susceptibility diagrams $\left[O\right(m\left)\right]$ vs real frequency $\mathrm{\Omega }$. Bottom: The third-order transverse spin susceptibility ${\chi }_T^{\left(3\right)}$; TMA, ${\chi }_{ETM}^{\left(2\right)}$, as well as ${\chi }_{NL}^{\left(2\right)}$ are also shown. Data are for $\beta t=5$, $U/t=2$ with $\mu /t=0$ at $\mathbf{q}=(\pi /3,\pi /2)$. We set $\mathrm{\Gamma }/t=0.02$ in the symbolic analytic continuation $i\mathrm{\Omega }\to \mathrm{\Omega }+i\mathrm{\Gamma }$. Each data point is obtained with $\approx {10}^8$ Monte Carlo samples.

Figure 5

Left: Third-order transverse susceptibility ${\chi }_T^{\left(3\right)}$ as a function of real frequency $\mathrm{\Omega }$ along the momentum cut $(0,\pi )\to (2\pi ,\pi )$. Right: Frequency cuts along $\mathrm{\Omega }=0.5$–3. Data are for $\beta t=5$, $U/t=2$, and $\mu /t=0$. We set $\mathrm{\Gamma }/t=0.02$ in the symbolic analytic continuation $i\mathrm{\Omega }\to \mathrm{\Omega }+i\mathrm{\Gamma }$. Each data point is obtained with $\approx {10}^6$ Monte Carlo samples.