Linked-cluster expansions for quantum magnets on the hypercubic lattice

For arbitrary space dimension $d$, we investigate the quantum phase transitions of two paradigmatic spin models defined on a hypercubic lattice, the coupled-dimer Heisenberg model and the transverse-field Ising model. To this end, high-order linked-cluster expansions for the ground-state energy and the one-particle gap are performed up to order 9 about the decoupled-dimer and high-field limits, respectively. Extrapolations of the high-order series yield the location of the quantum phase transition and the correlation-length exponent $\nu$ as a function of space dimension $d$. The results are complemented by $1/d$ expansions to next-to-leading order of observables across the phase diagrams. Remarkably, our analysis of the extrapolated linked-cluster expansion allows to extract the coefficients of the full $1/d$ expansion for the phase-boundary location in both models exactly in leading order and quantitatively for subleading corrections.

Figure 1

Critical point vs inverse dimension $1/d$ for the CDHM on the hypercubic lattice. Red diamond (green triangle) corresponds to the critical value obtained from quantum Monte Carlo simulations in Ref. [10] (Ref. [11]) for $d=2$ ($d=3$). Green dashed lines represent the estimated full $1/d$ expansion Eq. (52) up to order $\mathcal{O}\left(n\right)$ with $n\in \{1,2,3\}$.

Figure 2

Critical exponent $\nu$ vs inverse dimension $1/d$ for the CDHM on the hypercubic lattice. Dot-dashed line indicates the mean-field exponent $\nu =1/2$ and the red diamond corresponds to the critical exponent obtained from quantum Monte Carlo simulations in Ref. [32] for $d=2$.

Figure 3

Multiplicative logarithmic correction $p_{\mathrm{gap}}^{\mathrm{CDHM}}$ vs perturbative order $L+M$ used in the Dlog-Padé extrapolation Eq. (27) with $q_c^{\mathrm{CDHM}}=0.6206$ for the CDHM on the hypercubic lattice. Extrapolants with constant $c=L-M$ with $c\in \{-1,0,1,2\}$ are shown with the same black symbols. Dashed line indicates the exact exponent [33] $(-5/22)$.

Figure 4

Critical point vs inverse dimension $1/d$ for the TFIM on the hypercubic lattice. Red diamond (green triangle) corresponds to the critical value obtained from series expansion (SE) in Ref. [14] (Ref. [19]) for $d=2$ ($d=3$) and quantum Monte Carlo (QMC) simulations from Ref. [17] (the two techniques fully agree on the displayed scale and are therefore shown together as a single symbol). Blue square corresponds to the exact solution for $d=1$ [27]. Green dashed lines represent the estimated full $1/d$ expansion Eq. (53) up to order $\mathcal{O}\left(n\right)$ with $n\in \{1,2,3\}$.

Figure 5

Critical exponent $\nu$ vs dimension for the TFIM on the hypercubic lattice. Dot-dashed line indicates the mean-field exponent $\nu =1/2$. The red diamond corresponds to the critical exponent obtained from quantum Monte Carlo simulations in Ref. [16] for $d=2$. The blue square illustrates the exact value for $d=1$ [27]. In the shaded regions, no consistent extrapolation has been achieved due to spurious poles in the Dlog-Padé extrapolation.