Excited-state quantum phase transitions in many-body systems with infinite-range interaction: Localization, dynamics, and bifurcation

Excited-state quantum phase transitions (ESQPTs) are generalizations of quantum phase transitions to excited levels. They are associated with local divergences in the density of states. Here, we investigate how the presence of an ESQPT can be detected from the analysis of the structure of the Hamiltonian matrix, the level of localization of the eigenstates, the onset of bifurcation, and the speed of the system evolution. Our findings are illustrated for a Hamiltonian with infinite-range Ising interaction in a transverse field. This is a version of the Lipkin-Meshkov-Glick (LMG) model and the limiting case of the one-dimensional spin-$\frac{1}{2}$ system with tunable interactions realized with ion traps. From our studies for the dynamics, we uncover similarities between the LMG and the noninteracting XX models.

Figure 1

Top panels: normalized density of states for $H_s$ (4) with ${\xi }_c=0.2$ (a), $\xi =0.4$ (b), $\xi =0.6$ (c), and $\xi =0.8$ (d), $N=2000$. The corresponding classical potentials [Eq. (11)] are shown in the insets. Bottom panel (e): normalized excitation energies vs $\xi , N=100$. The separatrix [Eq. (9)] is indicated with the dashed line. All panels: even parity sector. Arbitrary units.

Figure 2

Top panels: squared coefficients $|C_{m_z}^{\left(k\right)}{|}^2$ of the eigenstates $|{\psi }_k\rangle$ written in the $\text{U}\left(1\right)$ basis vs the energies of the corresponding basis vectors; $\xi =0.6, N=2000$. The eigenstates chosen have energies $E_k^{\prime }/N=0.2515$ (a), $0.4163$ (b) (second closest to the ESQPT critical point), $0.4166$ (c) (closest one to the ESQPT critical point), and $0.5764$ (d). Vertical lines indicate the separatrix $E_{\text{ESQPT}}=0.4167$. Bottom panel (e): normalized energy of the $\text{U}\left(1\right)$ basis vectors in the total Hamiltonian vs $\xi , N=100$. The separatrix [Eq. (9)] is indicated with the dashed line and the energy of the $\text{U}\left(1\right)$ ground state $e_{-N/2}^{\prime }/N$ with the thick solid line. All panels: even parity sector. Arbitrary units.

Figure 3

Log-log plots of the largest components (filled circles) and the second largest components (empty squares) $|C_{m_z}^{\left(k\right)}{|}^2$ vs $N$ for the eigenstate that is most localized in the $\text{U}\left(1\right)$ ground state (its energy $E_{\mathrm{loc}}^{\prime }/N$ is very close to the $E_{\text{ESQPT}}$) (a) and for the eigenstate at a position $D_{\mathrm{even}}/4$ above $E_{\mathrm{loc}}^{\prime }/N$ (b). Solid lines are fittings with indicated power-law decays, $\xi =0.6$ and even parity.

Figure 4

(a)–(d) Participation ratio of all the eigenstates of the even parity sector written in the $\text{U}\left(1\right)$ basis; $N=500$ (dark curve) and 2000 (light curve). Vertical lines mark the $E_{\text{ESQPT}}$ obtained from Eq. (9). (e) Dependence on $N$ of the ratio $R_{\text{loc}}^{\text{max}}=P_{\text{U}\left(1\right)}^{\text{max}}/P_{\text{U}\left(1\right)}^{\text{loc}}$ between the participation ratio of the most delocalized state $P_{\text{U}\left(1\right)}^{\text{max}}$, and the $P$ of the most localized state in the $\text{U}\left(1\right)$ ground state $P_{\text{U}\left(1\right)}^{\text{loc}}$. Arbitrary units.

Figure 5

Normalized excitation energies vs $\xi$ for all $N+1$ eigenstates, including both parities, one parity is indicated with solid lines and the other with dashed lines; $N=100$.

Figure 6

Top panels: squared coefficients $|C_{m_x}^{\left(k\right)}{|}^2$ of the eigenstates $|{\psi }_k\rangle$ written in the $\text{SO}\left(2\right)$ basis vs the energies of the corresponding basis vectors; $\xi =0.6, N=2000$. The eigenstates shown have the energies considered in Fig. 2: two degenerate states with $E_k^{\prime }/N=0.2515$ (a), two with $E_k^{\prime }/N\sim 0.4163$ (b) (second closest energy to $E_{\text{ESQPT}}$), two with $E_k^{\prime }/N\sim 0.4166$ (c) (closest energy to $E_{\text{ESQPT}}$), and one state with $E_k^{\prime }/N=0.5764$ (d). Vertical lines mark the ESQPT energy $E_k^{\prime }/N=0.4167$. Bottom panel (e): normalized energy of the $\text{SO}\left(2\right)$ basis vectors in the total Hamiltonian vs $\xi , N=100$. The separatrix [Eq. (9)] is indicated with the dashed line. Both parities are included. Arbitrary units.

Figure 7

Participation ratio of all $N+1$ eigenstates of both parity sectors written in the $\text{SO}\left(2\right)$ basis; $N=500$ (dark curve) and 2000 (light curve). Vertical lines mark the $E_{\text{ESQPT}}$ obtained from Eq. (9). Arbitrary units.

Figure 8

Top: normalized total magnetization in the $z$ direction for all eigenstates with even parity. Bottom: normalized total magnetization in the $x$ direction for all eigenstates of both parities. Vertical lines indicate the separatrix [Eq. (9)]; $N=2000$.

Figure 9

Structure of the $\text{U}\left(1\right)$ basis vectors projected onto the eigenstates of the total Hamiltonian $H_s$: even parity, $N=2000, \xi =0.6$. The values of $m_z$ are $-1000$ (a), $-960$ (b), $-900$ (c), $-800$ (d), $-600$ (e), $-400$ (f), $-200$ (g), 334 (h), 600 (i), 800 (j), 900 (k), 980 (l). Vertical dashed lines mark $E_{\text{ESQPT}}$. The states with energy closest to the separatrix are (a), with $e_{-1000}/N=0.4164$, and (h), with $e_{334}/N=0.4166$.

Figure 10

LDOS for initial states corresponding to $\text{U}\left(1\right)$ basis vectors with $m_z=-N/2$ (a), $m_z=-N/2+2$ (b), and the one with the second closest $e_{m_z}^{\prime }/N$ to $E_{\text{ESQPT}}$ (c); $N={10}^4, \xi =0.6$. In (c), the dashed line represents Eq. (17) with ${\mathcal{A}}^2\sim 0.27$ and $e_{m_z}^{\prime }/N=E_{\text{ESQPT}}$.

Figure 11

Survival probability vs time. In (a) from top to bottom: initial states corresponding to $\text{U}\left(1\right)$ basis vectors with $m_z=-N/2, m_z=-N/2+2$, and the one with the second closest $e_{m_z}^{\prime }/N$ to $E_{\text{ESQPT}}$; $N={10}^4$. (b) $m_z=-N/2$ for $N={10}^3$ (first curve to show revival) and $N={10}^4$. (c) Initial state with the second closest $e_{m_z}^{\prime }/N$ to $E_{\text{ESQPT}}$; $N={10}^3$ (top) and $N={10}^4$ (bottom); dashed lines give $F\left(t\right)\propto 1/t$. All panels: $\xi =0.6$. Arbitrary units.

Figure 12

Structure of the Hamiltonian matrix of the LMG model $H_{\text{s}}$ [Eq. (4)], written in the $\text{U}\left(1\right)$ basis; only even parity is considered. (a) Coupling strength between two neighboring levels vs spacing between those levels; from top to bottom: $\xi =0.2,0.4,0.6,0.8,1.0$. (b)–(f) Ratio of the spacing between neighboring levels and their coupling strength; the value of $\xi$ is indicated in the panels. Absolute ratio $>1$ indicates ineffective coupling. Arbitrary units.

Figure 13

Evolution of the total magnetization in the $z$ direction. The values of $m_z\left(0\right)$ are indicated in the panels; they are $m_z\left(0\right)=-N/2, m_z\left(0\right)=-N/2+2$, and the one with the second closest $e_{m_z}^{\prime }/N$ to $E_{\text{ESQPT}}$. (a) $N={10}^3$ and (b) $N={10}^4$. Both panels: $\xi =0.6$. Arbitrary units.

Figure 14

Structure of the $\text{SO}\left(2\right)$ basis vectors projected onto the eigenstates of the total Hamiltonian $H_s$; $N=200, \xi =0.6$. The values of $m_x$ are $-100$ (a), $-75$ (b), $-58$ (c), $-39$ (d), $-10$ (e), $-5$ (f), $-3$ (g), $-1$ (h), 0 (i). Circles are numerical results and thin black lines are guides for the eye. Vertical dashed lines mark $E_{\text{ESQPT}}$. Arbitrary units.

Figure 15

Evolution of the total magnetization in the $x$ direction (left) and of the survival probability (right); $N={10}^3$. The values of $m_x\left(0\right)$ are indicated in the panels. Dashed lines on the right panels correspond to $F\left(t\right)\propto 1/t$. All panels: $\xi =0.6$. Arbitrary units.