Valence-bond solid to antiferromagnet transition in the two-dimensional Su-Schrieffer-Heeger model by Langevin dynamics

The two-dimensional Su-Schrieffer-Heeger model of electrons coupled to quantum phonons is investigated using Langevin dynamics within the framework of auxiliary-field quantum Monte Carlo. Based on an explicit determination of the density of zeros of the fermion determinant, it is argued that the method is efficient in the challenging adiabatic limit. Large-scale simulations at the O(4)-symmetric point establish that the ground state of the 2D SSH model undergoes a transition from a $(\pi ,\pi )$ valence bond solid to an antiferromagnet with increasing phonon frequency, yet still in the adiabatic regime. The single-particle spectrum illustrates the renormalization of the electronic band and suggests the existence of a gapped polaronic band, whereas the particle-hole channels show gapless modes associated with long-range bond and magnetic order, respectively. The simulations are supplemented with a mean-field analysis and a self-consistent Born approximation.

Figure 1

 Schematic phase diagram of the O(4)-symmetric, square-lattice SSH model as a function of phonon frequency. The left inset depicts the VBS phase, showing one of four equivalent $(\pi ,\pi )$ patterns of strong and weak bonds. The right inset illustrates the AFM phase which, due to the O(4) symmetry, is equivalent to a phase with CDW or SC order.

Figure 2

Comparison of Langevin, CT-INT, and HQMC results for (a) the local imaginary-time Green function and (b) the spin correlation function on an $L=4$ lattice for $\beta =1.0, \mathrm{\Delta }\tau =0.1, g=1.0, k=2, {\omega }_0=1.0$, and $\delta t_l=0.0005$. HQMC and CT-INT data taken from Ref. [26].

Figure 3

(a) Average sign of the Pfaffian for the $t\text{−}V$ model on the $\pi$-flux square lattice with $L=4, \beta t=4$, and $\delta t_l=0.005$. This model exhibits a Gross-Neveu phase transition at $V_c=1.279\left(3\right)t$ [50]. (b) Average sign of the Pfaffian for the SSH model with antiperiodic-periodic (a-p) or periodic-periodic (p-p) boundary conditions and $\beta =5.0, \mathrm{\Delta }\tau =0.1$, and $\delta t_l=0.0005$.

Figure 4

(a) Equal-time spin correlation function $S_S(\mathit{q},0)$ and (b) corresponding autocorrelation function $C_S\left(t_l\right)$ at wave vector $\mathit{Q}$ as a function of Langevin time $t_l$ with and without the use of Fourier acceleration (FA). The parameter sets for both runs are exactly the same (including number of sweeps/bins): ${\omega }_0=0.4, L=6, \beta =7.0$, and $\delta t_l=0.001$.

Figure 5

(a) Unit cell used in mean-field theory, indicated by the colored sites and bonds. (b) Resulting minimal-energy $(\pi ,\pi )$ VBS pattern. Strong bonds are colored while weak bonds are represented by black lines.

Figure 6

(a) Bond susceptibility as a function of temperature for different system sizes. (b) Energy as a function of temperature for $L=12$. Simulations were done using $\delta t_l=0.01$ and starting from the mean-field configuration.

Figure 7

(a) Finite-size scaling of the bond-kinetic susceptibility at $T=1/40$. (b) Temperature dependence of the bond-kinetic structure factor and (c) of the bond-kinetic susceptibility for $L=12$.

Figure 8

(a) Finite-size scaling of the spin susceptibility at $T=1/40$. (b) Temperature dependence of the spin structure factor for $L=10$. (c) Temperature dependence of the spin susceptibility for $L=10$.

Figure 9

Correlation ratio based on the spin susceptibility as defined in Eq. (45). Here, $T=1/40$, which is representative of the ground state for this quantity.

Figure 10

Parity susceptibility [Eq. (47)] at $T=0.1$.

Figure 11

 Single-particle spectral function $A(\mathit{k},\omega )$ for different phonon frequencies. Here, $L=12, \beta =40$.

Figure 12

 (a) Single-particle spectral function $A(\mathit{k},\omega )$ from a self-consistent Born approximation for $\beta =40, t=1.85, g=1.5, {\omega }_0=1.0$. (b) As in (a) but with broken O(4) symmetry via addition of ${\widehat{H}}_{\lambda }$ [Eq. (49)] with $\lambda =0.5$.

Figure 13

 VBS dynamical structure factor $S_K(\mathit{q},\omega )$ for different phonon frequencies. Here, $L=12, \beta =40$.

Figure 14

 VBS dynamical structure factor for (a) $\mathit{q}=\mathit{Q}$ and (b) $\mathit{q}=(0,\pi )$ for different ${\omega }_0$. Here, $L=12, \beta =40$.

Figure 15

 Spin dynamical structure factor $S_S(\mathit{q},\omega )$ for (a) ${\omega }_0=0.4$, (b) ${\omega }_0=2.0$, (c) fixed $\mathit{q}=\mathit{Q}$ and different ${\omega }_0$. Here, $L=12, \beta =40$.

Figure 16

 A vortex of the $(\pi ,\pi )$ VBS state. Arrows represent the four degenerate VBS patterns that break the ${\mathrm{C}}_4$ symmetry. By crossing a domain wall (orange lines), the angle changes by $\pi /2$. A full circle around the core yields $2\pi$. This trivial vortex does not carry a spin-1/2 degree of freedom, in contrast to the case of $(0,\pi )$ or $(\pi ,0)$ VBS orders [59].