Effects of dynamical paths on the energy gap and the corrections to the free energy in path integrals of mean-field quantum spin systems

In current studies of mean-field quantum spin systems, much attention is placed on the calculation of the ground-state energy and the excitation gap, especially the latter, which plays an important role in quantum annealing. In pure systems, the finite gap can be obtained by various existing methods such as the Holstein-Primakoff transform, while the tunneling splitting at first-order phase transitions has also been studied in detail using instantons in many previous works. In disordered systems, however, it remains challenging to compute the gap of large-size systems with specific realization of disorder. Hitherto, only quantum Monte Carlo techniques are practical for such studies. Recently, Knysh [Nature Comm. 7, 12370 (2016)] proposed a method where the exponentially large dimensionality of such systems is condensed onto a random potential of much lower dimension, enabling efficient study of such systems. Here we propose a slightly different approach, building upon the method of static approximation of the partition function widely used for analyzing mean-field models. Quantum effects giving rise to the excitation gap and nonextensive corrections to the free energy are accounted for by incorporating dynamical paths into the path integral. The time-dependence of the trace of the time-ordered exponential of the effective Hamiltonian is calculated by solving a differential equation perturbatively, yielding a finite-size series expansion of the path integral. Formulae for the first excited-state energy are proposed to aid in computing the gap. We illustrate our approach using the infinite-range ferromagnetic Ising model and the Hopfield model, both in the presence of a transverse field.

Figure 1

Results of numerical diagonalization of the ferromagnetic model in the sector with total angular momentum $N/2$. (a) Difference between the ground-state energy $E_0$ and the free energy $N{f}_s$ for various $N$. The $N^0$ term (black solid line) is the first correction to the free energy obtained by incorporating dynamical paths into the path integral [Eq. (36)]. (b) The energy gap for various $N$. For each $N$, the upper curve is $E_2-E_1$ while the lower one is $E_1-E_0$, as indicated for the case of $N=50$. In the ferromagnetic phase ($\mathrm{\Gamma }<2J$), the ground-state is doubly degnerate and $E_2$ is the energy of the first-excited state. The curve labeled “path integral” (black solid line) is the gap obtained using our path integral formulation of the energy gap.

Figure 2

(a) The $O\left(N^{-1}\right)$ part of $E_0$ of the ferromagnetic model. For the curves of various $N$, the free energy and $N^0$ term are subtracted from the numerically computed $E_0$ and the result multiplied by $N$. The $N^{-1}$ term (black solid line) is the second correction to the free energy obtained by incorporating dynamical paths into the path integral. (b) Individual terms of the $N^{-1}$ term in the ferromagnetic phase.

Figure 3

Graphs showing the nonvanishing of ${\sum }_{a,b}{\left|〈E_0^a\left|A_{\mu }\right|E_1^b〉\right|}^2$ for the ferromagnetic model. The matrix elements are computed numerically and their absolute values plotted for various $N$. (a) $|\langle E_0|A_z|E_1\rangle |$ in the paramagnetic phase. (b) $|\langle E_0|A_y|E_1\rangle |$. In the ferromagnetic phase ($\mathrm{\Gamma }<2J$), the matrix element vanishes because it becomes $\langle E_0^{+}\left|A_y\right|E_0^{-}\rangle$. (c) $|\langle E_0^{-}|A_y|E_1^{+}\rangle |$ in the ferromagnetic phase. (d) $|\langle E_0^{+}|A_y|E_1^{-}\rangle |$ in the ferromagnetic phase.

Figure 4

Numerical solutions of Eqs. (79) for a particular realization of $p=5$ patterns ($N=1000$). Solid (red) lines show the variables $m_s^{\mu }$ that are uncondensed in both phases. The dashed (blue) line shows the one that magnetizes macroscopically upon entering the condensed phase. At ${\mathrm{\Gamma }}_{\mathrm{b}}$ the zero solution becomes unstable and nonzero solutions appear.

Figure 5

Comparing various energies of the Hopfield model with that of a pure (i.e., ferromagnetic) system. Results for the system with the magnetization curves of Fig. 4 are shown. (a) Paramagnetic phase. Solid (black) line with circles shows $N(f_s^{\mathrm{HM}}-f_s^{\mathrm{pure}})$. Dashed (red) line shows the Gaussian correction ${\mathsf{g}}_d^{\mathrm{HM}}-{\mathsf{g}}_d^{\mathrm{pure}}$. Solid (blue) line shows the shift of the ground-state energy of the Hopfield model given by Eq. (82) from that of a pure system. Inset: Graphs of ${\mathsf{g}}_d^{\mathrm{HM}}$ (red dashed line) and ${\mathsf{g}}_d^{\mathrm{pure}}$ (green solid line). (b) Condensed phase. Graphs of $f_s^{\mathrm{HM}}-f_s^{\mathrm{pure}}$ for all five memory states. The memory state from Fig. 4 is plotted using solid (black) line with circles. The interpattern energy gap ${\mathrm{\Delta }}_{\mathrm{inter}}$ is defined as the energy difference between the two memory states with the lowest energies, as shown.

Figure 6

Interpattern energy gap ${\mathrm{\Delta }}_{\mathrm{inter}}$ of the Hopfield model in the condensed phase. (a) Graphs of ${\mathrm{\Delta }}_{\mathrm{inter}}$ of particular realizations of patterns with $p=5$ and $N=1000$. Main plot shows the ${\mathrm{\Delta }}_{\mathrm{inter}}$ obtained from the realization of Fig. 5. Dashed (blue) line shows the ${\mathrm{\Delta }}_{\mathrm{inter}}$ when the energy of the memory states is approximated by $N{f}_s^{\mathrm{HM}}$; solid (red) line shows the case when the energy is approximated by $N{f}_s^{\mathrm{HM}}+{\mathsf{g}}_d^{\mathrm{HM}}$. Insets (i)–(iii): Other examples of ${\mathrm{\Delta }}_{\mathrm{inter}}$ with different realizations of patterns (same $p$ and $N$). (b) Graphs of the mean gap, $\langle {\mathrm{\Delta }}_{\mathrm{inter}}\rangle$, for various $N$ ($p=5$). Each particular ${\mathrm{\Delta }}_{\mathrm{inter}}$ is calculated by approximating the energy of the memory states by $N{f}_s^{\mathrm{HM}}$. The curve of each $N$ is obtained by averaging over 5000 different realizations. Error bars indicate the standard deviation associated with the mean.

Figure 7

Results for ${\mathrm{\Delta }}_{\mathrm{para}}$, the excitation gap of the Hopfield model in the paramagnetic phase, approximated by Eq. (83), for various $N$ ($p=5$). Each curve represents the gap of one particular realization of patterns. Five different realizations are plotted for each $N$. The gap of a pure system is also shown for comparison. The solid (red) curve with circles corresponds to the system with the magnetization curves of Fig. 4.

Figure 8

Example of an anomalous transition occurring within the basin a particular memory state ($p=5$ and $N=250$). The main plot shows that two minima exist on the energy surface of $f_s^{\mathrm{HM}}$, labeled “min. 1” (green dashed line) and “min. 2” (red solid line). As $\mathrm{\Gamma }$ decreases from 1, min. 1 is the ground state. At ${\mathrm{\Gamma }}_{\mathrm{a}}$ an anomalous transition occurs, after which min. 2 becomes the ground state. Inset: Evolution of one of the magnetizations $m_s^{\mu }$ as $\mathrm{\Gamma }$ decreases. At ${\mathrm{\Gamma }}_{\mathrm{a}}$, the magnetization change discontinuously as it jumps from min. 1 to min. 2.