Exact excited states of nonintegrable models

We discuss a method of numerically identifying exact energy eigenstates for a finite system, whose form can then be obtained analytically. We demonstrate our method by identifying and deriving exact analytic expressions for several excited states, including an infinite tower, of the one-dimensional spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) model, a celebrated nonintegrable model. The states thus obtained for the AKLT model can be interpreted as from one to an extensive number of quasiparticles on the ground state or on the highest excited state when written in terms of dimers. Included in these exact states is a tower of states spanning energies from the ground state to the highest excited state. Some of the states of the tower appear to be in the bulk of the energy spectrum, allowing us to make conjectures on the strong eigenstate thermalization hypothesis. We also generalize these exact states including the tower of states to the generalized integer spin AKLT models. Furthermore, we establish a correspondence between some of our states and those of the Majumdar-Ghosh model, yet another nonintegrable model, and extend our construction to the generalized integer spin AKLT models.

Figure 1

Energy level spacing statistics for $L=16$ with periodic boundary conditions in a typical quantum number sector $(s,S_z,k,I,P_z)=(4,0,0,-1,1)$ that has a Hilbert space dimension 26 429. $\mathrm{\Delta }E$ is the level spacing between adjacent energy levels after a mapping of the energy spectrum to produce a constant density of states. $P\left(\mathrm{\Delta }E\right)$ is the distribution of the level spacings. The peak of the distribution at nonzero $\mathrm{\Delta }E$ indicates level repulsion. The green curve is the GOE distribution. The mean ratio of adjacent level spacings is $\langle r\rangle \approx 0.5316$, close to the GOE value of $\langle r\rangle \approx 0.5295$ [52, 53].

Figure 2

AKLT ground state. The big circles are physical spin-1's and the smaller circles within the spin-1's are spin-1/2 Schwinger bosons. Symmetric combinations of the Schwinger bosons on each site form the physical spin-1. The lines joining the Schwinger bosons represent singlets. $\left|G\right.〉$ with periodic boundary conditions.

Figure 3

An example of the action of $P_{ij}^{(2,1)}$ on a dimer configuration. The configurations of the filled small circles are not relevant for the scattering equation and are the same on all terms in the equation. The directions of the arrows are crucial; reversing an arrow contributes a factor of $(-1)$.

Figure 4

(a) Arovas $A$ configuration $\left|A_n\right.〉$. (b) Scattering term configuration $\left|B_n\right.〉$.

Figure 5

(a) Arovas $B$ configuration $\left|B_n\right.〉$. (b) Scattering term $\left|A_n\right.〉$. (c) Scattering term $\left|C_n\right.〉$. (d) Scattering term $\left|D_n\right.〉$. (e) Scattering term $\left|E_n\right.〉$.

Figure 6

The two configurations that appear in the derivation of the spin-2 magnon state. (a) The spin-2 magnon state $\left|M_n\right.〉$. (b) Scattering state $\left|N_n\right.〉$.

Figure 7

(a) Two-magnon configuration $\left|M_n,M_{n+4}\right.〉$. (b) Scattering state $\left|N_n,M_{n+4}\right.〉$. (c) Two-magnon configuration $\left|M_n,M_{n+2}\right.〉$. (d) Two-magnon scattering configuration $\left|M_n,N_{n+2}\right.〉$.

Figure 8

(a) A configuration with magnons on sites $n$ and $n+2$. (b) A configuration with magnons on sites $n$ and $n+3$. (c) Common scattering configuration.

Figure 9

Positions of the $S_z=0$ states of the tower of states within the energy spectrum of their own quantum number sector plotted against the energy density $\epsilon =E/L$ for $L=8,10,12,14,16$ with periodic boundary conditions. Each dot corresponds to a state with $N$ spin-2 magnons with $N=0,1,\cdots ,L/2-2$. The inset shows the density of states for $L=16$ in the quantum number sector $(s,S_z,k,I,P_z)=(8,0,0,-1,1)$. The vertical green line in the inset indicates the position of $\left|S_8\right.〉$.

Figure 10

(a) Ferromagnetic state $\left|F\right.〉$. (b) One-dimer configuration $\left|n\right.〉$.

Figure 11

Several configurations relevant for the $\left|2_n\right.〉$ states. (a) Two-dimer configuration $\left|m,m+4\right.〉$. (b) Scattering term $\left|m+1,m+4\right.〉$. (c) Two-dimer configuration $\left|m,m+1\right.〉$. (d) Scattering term $\left|m,m+2\right.〉$. (e) Two-dimer configuration $\left|m,m\right.〉$.

Figure 12

Typical configurations involved in the derivation of the $\left|2_k\right.〉$ states. Two-dimer configurations (a) $\left|n,n+1\right.〉$, (b) $\left|n,n+3\right.〉$, (c) $\left|n,n+5\right.〉$. Scattering terms (d) $\left|n,n+2\right.〉$, (e) $\left|n,n+4\right.〉$.

Figure 13

The only two possible dimer configurations up to translations in $\left|S_4\right.〉$ for $L=6$.

Figure 14

$S=2$ AKLT ground state with two dimers between nearest neighbors. Spin-$S$ AKLT would have $S$ dimers. $\left|2G\right.〉$ with periodic boundary conditions.

Figure 15

The spin-2 AKLT configurations (a) $\left|B{G}_n\right.〉$ and (b) $\left|B_n^2\right.〉$. Such configurations form the generalization of the Arovas $B$ state in the spin-$S$ AKLT model. (c) An example of a symmetric configuration that appears only in ${\mathcal{C}}_{B^2}^S$. (d) An example of a symmetric configuration that appears in both ${\mathcal{C}}_{BG}^S$ and ${\mathcal{C}}_{B^2}^S$. (e) An example of a nonsymmetric scattering configuration $\left|\zeta \right.〉$.

Figure 16

(a) Spin-4 magnon $\left|2M_n\right.〉$ and (b) scattering state $\left|2N_n\right.〉$ for spin-2 AKLT model. A similar picture holds for spin-$S$ AKLT model.

Figure 17

Upper spectrum state configurations for $S=2$. (a) The ferromagnetic state with $s=2L$ and $E=L$. (b) An example of a nonscattering configuration with $s=2L-1,E=L$. (c) A configuration that forms an exact state for $S=2$ with $E=L-2,s=2L-4$. (d) A dressed configuration that forms an exact state with $E=L-2,s=2L-5$.

Figure 18

Two types of singlet configurations around a bond $\{i,j\}$.

Figure 19

Diagrammatic representation of the action of the projector $P_{ij}^{(2,1)}$ on various configurations of dimers around bond $\{i,j\}$. The configurations of the filled small circles are not relevant to the scattering equation and are the same (in terms of the Schwinger bosons) on both sides of a given equation. The directions of the arrows are crucial; reversing an arrow contributes a factor of $(-1)$.

Figure 20

Types of nonsinglet configurations around a bond $\{i,j\}$.

Figure 21

Scattering examples of $S=2$ dimer configurations under the action of $c_{ij}$. Configurations of type (a) scatter to configurations of the type (b). Configurations such as (c) and (d) are annihilated by ${\left(c_{ij}\right)}^4$. Configuration (e) scatters to (f) under the action of ${\left(c_{ij}\right)}^2$. The configurations of the filled small circles are irrelevant for the scattering due to $c_{ij}$.

Figure 22

Symmetric scattering configurations that appear in the scattering equations of the $S=2$ Arovas configurations. All these configurations are obtained gluing two $S=1$ Arovas scattering configurations, shown in Figs. 4 and 5.

Figure 23

Nonsymmetric scattering configurations that appear in the scattering equation of the $S=2$ Arovas configurations.

Figure 24

Cells in the Majumdar-Ghosh model. (a) “Even” ground state. (b) “Odd” ground state.

Figure 25

Triplet ground state $\left|G^O\right.〉$ with open boundary conditions. The spins at the edge represent the two dangling spin-1/2's with $S_z=1/2$ each.

Figure 26

(a) Spin-2 magnon state $\left|M_2^O\right.〉$ for the chain with open boundary conditions. (b) $\left|N_2^O\right.〉$, the only scattering state of $\left|M_2^O\right.〉$.

Figure 27

The ground state $\left|2G^O\right.〉$ of the spin-2 AKLT Hamiltonian with open boundary conditions. The two free spin-1/2's at each edge depict the dangling spin-1's. The ground state of the spin-$S$ AKLT model would have $S$ free Schwinger bosons on the edge spins, equivalent to one spin-$S/2$.