Semiclassical excited-state signatures of quantum phase transitions in spin chains with variable-range interactions

We study the excitation spectrum of a family of transverse-field spin chain models with variable interaction range and arbitrary spin $S$, which in the case of $S=1/2$ interpolates between the Lipkin-Meshkov-Glick and the Ising model. For any finite number $N$ of spins, a semiclassical energy manifold is derived in the large-$S$ limit employing bosonization methods, and its geometry is shown to determine not only the leading-order term but also the higher-order quantum fluctuations. Based on a multiconfigurational mean-field ansatz, we obtain the semiclassical backbone of the quantum spectrum through the extremal points of a series of one-dimensional energy landscapes—each one exhibiting a bifurcation when the external magnetic field drops below a threshold value. The obtained spectra become exact in the limit of vanishing or very strong external, transverse magnetic fields. Further analysis of the higher-order corrections in $1/\sqrt{2S}$ enables us to analytically study the dispersion relations of spin-wave excitations around the semiclassical energy levels. Within the same model, we are able to investigate quantum bifurcations, which occur in the semiclassical $\left(S\gg 1\right)$ limit, and quantum phase transitions, which are observed in the thermodynamic $\left(N\to \infty \right)$ limit.

Figure 1

Sketch of the model for a chain with $N$ sites and $M=3$. The upper part of the sketch depicts the spin chain in terms of collective angular momentum operators $S_{\beta }^{\left(i\right)}$ at the $i\mathrm{th}$ site. Correspondingly, the lower part shows the representation in terms of the elementary spins ${\sigma }_{\beta }^{(i,k)}$ with $\beta \in \{x,y,z\}$ and $k=1,2,3$.

Figure 2

Spin configurations ${\mathbf{\varphi }}_{\mathrm{e}}$ and ${\mathbf{\varphi }}_{\mathrm{a}}$ associated with the largest and the smallest energy eigenvalue of (26), for $N=6,S=1/2$, and $\varphi =\pi /2$. The spin coherent state (top), where all spins align in equal directions, generates the semiclassical ground state configuration when $J_0>0$. In this case, the alternating configuration (bottom) corresponds to the semiclassical configuration which yields the largest energy eigenvalue. In the special case of the Ising model $\left(\alpha \to \infty \right)$, the configurations ${\mathbf{\varphi }}_{\mathrm{e}}$ (equal directions) and ${\mathbf{\varphi }}_{\mathrm{a}}$ (alternating directions) yield energies $E_{\mathrm{e}}=-J_0(N-1)$ and $E_{\mathrm{a}}=J_0(N-1)$, respectively [see Eq. (31)].

Figure 3

The counting function (33) in the ferromagnetic regime $(J_0>0,B=0)$ for $S=1/2$ reflects a strongly degenerate, quadratically spaced energy spectrum of the Lipkin-Meshkov-Glick model at $\alpha =0$, described by Eq. (32). As the range of the interaction decreases, i.e., as $\alpha$ increases, the spectrum evolves into a broadly distributed energy distribution, especially at values of $\alpha \approx 1$, and finally approaches the equally spaced, and, again, strongly degenerate spectrum of the Ising model at $\alpha =\infty$, described by Eq. (31), which is symmetric around zero. The spectra are obtained using the exact semiclassical result (28) for $N=15$ spins.

Figure 4

(a) In the thermodynamic limit $\left(N\to \infty \right)$, the effective spin-spin coupling ${\mathsf{J}}_{\mathrm{a}} \left[{\mathsf{J}}_{\mathrm{e}}\right]$ of the ground state configuration is given by $\eta \left(\alpha \right) \left[\zeta \right(\alpha \left)\right]$ if $J_0<0 [J_0>0]$, which coincide for $\alpha \to \infty$. (b) Finite-size effects (see dashed lines for $N=10$ and $N=50)$ can be compensated with the correction term $c_N=1-1/N$. The rescaled effective spin-spin couplings ${\mathsf{J}}_{\mathrm{a}}/\left(J_0{c}_N\right)$ and ${\mathsf{J}}_{\mathrm{e}}/\left(J_0{c}_N\right)$ collapse onto the thermodynamic limit, except for deviations at very small values of $\alpha$.

Figure 5

(a) Energy landscape for positive ${\mathsf{J}}_{\mu }$ as a function of $\varphi$ and ${\mathsf{B}}_{\mu }$. The energy minimum (red line) represents a semiclassical energy level as a function of ${\mathsf{B}}_{\mu }\propto B$ [see Eq. (45)]. The change of the dependence of the minimum energy on $B$ from quadratic to linear is a consequence of the underlying bifurcation into two degenerate but distinct solutions at weak magnetic fields. In the case of the ground state, such a bifurcation represents the semiclassical analog of the tipping point between the symmetric paramagnetic state, where the symmetry of the Hamiltonian is dictated by the $B$-dependent term, and the symmetry-broken (anti)ferromagnetic state, where the symmetry of the Hamiltonian is dictated by the $J_0$ term. (b) When ${\mathsf{J}}_{\mu }$ is negative, the energy maximum describes a semiclassical energy level and exhibits analogous behavior.

Figure 6

Comparison between the exact quantum spectrum (black lines) and the semiclassical energy values, depicted as green and red lines, depending on whether the latter are given by a minimum [see Fig. 5] or a maximum [Fig. 5] of the semiclassical energy landscape, respectively. The semiclassical energy levels yield exact results at $B/|J_0|=0$ and $B/|J_0|=\infty$. The dots indicate the points at which the corresponding extremum of the semiclassical energy landscape exhibits a bifurcation, together with a change of its magnetic field dependence from quadratic to linear. Parameters in (1) are $N=11,S=1/2$, and $J_0>0$, for different interaction ranges $\alpha$ as indicated. Changing signs on all energy levels yields the corresponding spectra for $-J_0$. The insets zoom into the parameter region where levels reorganize to mediate the transition from the ferro- to the paramagnetic phase.

Figure 7

Exact quantum spectrum (black lines) and semiclassical energy values (red lines) for the positive energy sector of $H$ with $N=3,S=5/2,J_0>0$, and $\alpha =1$, as a function of $B/J_0$, on a double-logarithmic scale.

Figure 8

Relative deviation $d\left(B\right)$, Eq. (48), between the semiclassical and the numerically exact ground state energy for $S=1/2$ and different interaction ranges. In most cases, the exact solution at small and large $B$ is approached as $(B/|J_0{\left|\right)}^{\pm 2}$. Exceptions are found for very long range interactions, which converge faster towards the (anti)ferromagnetic solution, unless $J_0<0$ and $N$ odd. The strongest deviation is found in the vicinity of the critical point.

Figure 9

Logarithmically binned histograms of stable bifurcation points defined by the extremal points of the effective (semiclassical) energy landscapes (see Fig. 5), for the special cases of $\alpha =0$ (Lipkin-Meshkov-Glick) and $\alpha =\infty$ (Ising).

Figure 10

Energy dispersion relation ${\varepsilon }_{\mathrm{e}}\left(k\right)$ and finite-size effects for spin-wave quantum corrections to the ferromagnetic ground state. The plot displays the parameters $\alpha =2$ and $B/J_0=2$ (red), $B/J_0=2\zeta \left(2\right)$ (blue), and $B/J_0=4$ (green) for $N=17$ (squares), $N=51$ (dots), and $N=\infty$ (lines).

Figure 11

Closing of the excitation gap for the ferromagnetic ground state and finite-size effects at $k=0$ for $\alpha =2$ and $N=2$ (dashed-dotted line), $N=17$ (dotted line), $N=51$ (dashed line) and $N=\infty$ (continuous line). The quantum phase transition from ferromagnet to paramagnet occurs when the excitation gap closes at $B_{\mathrm{e}}^c=2{\mathsf{J}}_{\mathrm{e}}^{\mathrm{p}}$.

Figure 12

(a) Energy dispersion relation ${\varepsilon }_{\mathrm{e}}\left(k\right)$ for spin-wave excitations to the ferromagnetic ground state, as a function of $B$ for $\alpha =3$ in the thermodynamic limit $N\to \infty$. The closing of the excitation gap can be observed at $k=0$ (blue line) at $B=2J_0\zeta \left(3\right)\approx 2.4J_0$. (b) Energy of the $k=0$ mode as a function of $B$ and $\alpha$. The gap closes at $B=2J_0\zeta \left(\alpha \right)$ (red line) for $\alpha >1$.

Figure 13

(a) Energy dispersion relation ${\varepsilon }_{\mathrm{a}}\left(k\right)$ for spin-wave excitations around the highest excited ferromagnetic state, as a function of $B$ for $\alpha =3$ in the thermodynamic limit $N\to \infty$. The closing of the excitation gap now occurs at $k=\pm \pi$ (blue lines). (b) Energy of the $k=\pi$ mode as a function of $B$ and $\alpha$. The gap closes at $B=-2J_0\eta \left(\alpha \right)$ (red line).