Path-integral representation for quantum spin models: Application to the quantum cavity method and Monte Carlo simulations

The cavity method is a well-established technique for solving classical spin models on sparse random graphs (mean-field models with finite connectivity). Laumann et al. [Phys. Rev. B 78, 134424 (2008)] proposed recently an extension of this method to quantum spin-1/2 models in a transverse field, using a discretized Suzuki-Trotter imaginary-time formalism. Here we show how to take analytically the continuous imaginary-time limit. Our main technical contribution is an explicit procedure to generate the spin trajectories in a path-integral representation of the imaginary-time dynamics. As a side result we also show how this procedure can be used in simple heat bath Monte Carlo simulations of generic quantum spin models. The replica symmetric continuous-time quantum cavity method is formulated for a wide class of models and applied as a simple example on the Bethe lattice ferromagnet in a transverse field. The results of the methods are confronted with various approximation schemes in this particular case. On this system we performed quantum Monte Carlo simulations that confirm the exactness of the cavity method in the thermodynamic limit.

Figure 1

A pictorial representation of Eqs. (17, 18).

Figure 2

Definition of the effective-field trajectory.

Figure 3

A pictorial representation of Eq. (23).

Figure 4

Longitudinal ($m_z$, top panel) and transverse ($m_x$, bottom panel) magnetizations as functions of the transverse field $B$ for $T=1.7,1.0,0.1$. For the lowest temperature we could not obtain reliable data for $m_z$ close to the critical field due to critical fluctuations. The dashed line is a fit to $m_z\sim {\left(B_c-B\right)}^{1/2}$ done in the interval $B∊\left[2.1,2.2\right]$ and serves as a guide for the eyes.

Figure 5

Phase diagram: the critical temperature $T_c$ as a function of the transverse field $B$.

Figure 6

Free energy as a function of $B$ for $T=0.1$. The curves for $T=0.3,0.2,0.1$ coincide within our numerical precision. Therefore we assume that this curve is representative of the ground-state energy as a function of $B$. The asymptotic values for $B\to 0$ $\left(e_{\text{GS}}=3/2\right)$ and for $B\to \infty$ $\left(e_{\text{GS}}=-B\right)$ are reported as dashed lines. The subleading correction for large $B$ is $\propto B^{-1}$ and is not reported.

Figure 7

Longitudinal magnetization $m_z$ as a function of $1/N_s$ for $T=1.0$, $B=1.5$ (triangles) and for $T=0.1$, $B=1.8$ (circles). In both cases the point at $1/N_s=0$ corresponds to the continuous-time result. Full lines are guides for the eyes, dashed lines are fit to quadratic laws, $m_z-m_z\left(N_s\to \infty \right)\propto N_s^{-2}$.

Figure 8

Phase diagram at finite $N_s$: the critical temperature $T_c$ as a function of the transverse field $B$. Note that for a fixed $N_s$, $T_c\left(B\to \infty \right)=T_c\left(B=0\right)/N_s$. Filled squares are the result of the continuous-time method, already reported in Fig. 5.

Figure 9

(a) Longitudinal magnetization $m_z$ as a function of the transverse field $B$. The full lines are the exact results reported in Fig. 4 for $T=1.7,1.0,0.1$ (from left to right). Points are the results of the static approximation: $T=1.7$ (squares), $T=1.0$ (circles), $T=0.1$ (triangles), and $T=0$ (open triangles). (b) The function $\tilde{p}\left(m\right)$ defined in the text at temperature $T=0.1$ and transverse field $B=3.2,2.2,1.2$ (from left to right). For smaller values of $B$, the maximum approaches $m=1$ and the weight of the $\delta \left(m-1\right)$ starts to increase until $\tilde{p}\left(m\right)$ vanishes at $B=0$.

Figure 10

Quantum heat bath (full symbols) vs loop algorithm (open symbols) for a random regular graph of degree 3 with ${10}^5$ spins at $B=1.6$ (circles), in the ferromagnetic phase, and $B=2.6$ (squares), in the quantum paramagnetic phase, for $T=1$. Dashed lines correspond to the values computed with the cavity method.

Figure 11

Longitudinal and transverse magnetization curves for temperatures $T=1$ (top) and $T=0.1$ (bottom). Continuous-time cavity method ($m_z$—solid line and $m_x$—dashed line) vs Monte Carlo simulations for different sizes increasing from the top to the bottom $\left(N=64,128,\dots ,2048\right)$. For the largest size the agreement with the cavity result is excellent (except close to the critical point).

Figure 12

Finite size scaling analysis of the Monte Carlo data. Top: rescaling of the longitudinal magnetization for random graphs of different sizes (increasing from the top to the bottom) $N=64,128,\dots ,2048$. Bottom: Binder parameter for the same sizes.

Figure 13

Finite size scaling analysis of the longitudinal magnetization at $T=0.01$ for sizes $N=32,64,\dots ,2048$.

Figure 14

An example of a factor graph.

Figure 15

A pictorial representation of Eq. (48).

Figure 16

(a) The phase diagram of the quantum Curie-Weiss model. (b) Longitudinal and transverse magnetizations as a function of the transverse field for $T=0.7$.