Excited states from local effective Hamiltonians of matrix product states and their entanglement spectrum transition

TL;DR

The work addresses how excited states in 1D critical quantum systems can be obtained from the local effective Hamiltonian derived from a ground-state MPS. By developing a conformal-field-theory formulation, it shows that excited-state content can be captured by a truncated basis of ground-state Schmidt vectors, with matrix elements expressed as BCFT correlators and diagonal contributions decaying exponentially with conformal weight. A key result is that the truncated diagonal sum $R'_a(D)$ approaches unity at finite $D$ for low-lying primary states, explaining numerical success in representing excitations. The paper predicts an entanglement-spectrum transition as the subsystem fraction $r$ grows, corroborated by numerical studies of the transverse-field Ising chain and the three-state clock model, where excited-state ES reorganizes into conformal towers at $r=1/2$. Overall, the findings connect MPS-based excited-state constructions to CFT, clarifying the mechanism behind their accuracy and identifying open questions about descendant states and the nature of towers at $r=1/2$.

Abstract

Solving excited states is a challenging task for interacting systems. For one-dimensional critical systems, however, excited states can be directly accessed from the eigenvectors of the local effective Hamiltonian that is constructed from the ground state obtained by variational matrix product state (MPS) optimization. Despite its numerical success, the theoretical mechanism underlying this method has remained largely unexplored. In this work, we provide a conformal field theory (CFT) perspective that helps elucidate this connection. The key insight is that this construction effectively uses a truncated basis of ground-state Schmidt vectors to represent excited states, where the contribution of each Schmidt vector can be expressed as a CFT correlation function and shown to decay with increasing Schmidt index. The CFT analysis further predicts an entanglement-spectrum transition of excited states as the ratio of the subsystem size to the total system size is varied. Our numerical results support this picture and demonstrate a reorganization of the entanglement spectrum into distinct conformal towers as this ratio changes.

Figures (4)

• Figure 1: Integration region for the path integral representation of $\langle X_A|\rho_a^A|X'_A\rangle$, before and after the conformal transformation $w=f(z)$ from an infinite cylinder to a rectangle of height $2\pi$. For the $z$ coordinate, $\sigma=0$ and $\sigma=L$ are identified, thereby being a cylinder. The ultraviolet cutoffs $\epsilon$ at the entanglement points are represented by the blue curves and lines. In $\langle X_A|\rho_a^A|X'_A\rangle$, the boundary field configurations $X_A$ and $X'_A$ are imposed on the gray lines.
• Figure 2: Entanglement spectrum $\tilde{\xi}$ of the critical Ising chain with open boundary conditions. (a) and (b) display the scaling of the ground state ($a=0$) entanglement spectrum with respect to system size $1/\log(N)$ for $N=40,80,120,160,200,240,280,320,400,480,560,640,720,800$, where the ratio $r=N_A/N$ is 0.1 for (a) and 0.5 for (b). For both (a) and (b), we shift and rescale the entanglement spectrum to $\tilde{\xi}$ such that the lowest level $\tilde{\xi}^{(a)}_0=0$ and the second-lowest level $\tilde{\xi}^{(a)}_1=1/2$. The vertical ticks on the left of the figure are the expected (or observed) conformal dimensions, where different expected conformal towers are plotted with different colors; the vertical ticks on the right are degeneracies in the thermodynamic limit. In the degeneracy label, "$2\times 1$" means the data exhibit exact two-fold degeneracy already at finite size. In (c), the system size is fixed at $N=800$, and we take $r\in[0.05,0.5]$ to illustrate the entanglement spectrum transition. We only shift the entanglement spectrum but do not rescale it. Both the vertical ticks are degeneracies. In (d), (e), and (f), we show the same analysis for the first excited state ($a=1$).
• Figure 3: Entanglement spectrum $\tilde{\xi}$ for the critical three-state clock model with open boundary conditions. The system sizes are $50 \leq N \leq 1000$, and the bond dimension is up to $D=1000$. (a) and (b) display the scaling of the ground state ($a=0$) entanglement spectrum with respect to system size $1/\log(N)$ for where the ratio $r=N_A/N$ is 0.1 for (a) and 0.5 for (b). For both (a) and (b), we shift and rescale the entanglement spectrum to $\tilde{\xi}$ such that the lowest level $\tilde{\xi}^{(a)}_0=0$ and the second-lowest level $\tilde{\xi}^{(a)}_1=2/3$. The vertical ticks on the left of the figure are the expected (or observed) conformal dimensions, where different expected conformal towers are plotted with different colors; the vertical ticks on the right are degeneracies. In (c), the system size is fixed at $N=1000$, and we take $r\in[0.05,0.5]$ to illustrate the entanglement spectrum transition. We only shift the entanglement spectrum but do not rescale it. Both the vertical ticks are degeneracies. In (d), (e), and (f), we show the same analysis for the first excited state $a=1$; and in (g),(h), and (i), we show the same analysis for the excited state $a=5$.
• Figure 4: Changes of $1-R_a(D)$ with respect to $D$, for the four lowest energy states of TFIC at the critical point $g=1$, with system size $N=40$ for (a) and $N=120$ for (b).