Rainbow chains and numerical renormalisation group for accurate chiral conformal spectra

TL;DR

The paper develops a framework to extract detailed chiral CFT spectra from entanglement data of one-dimensional conformal critical chains by mapping reduced density matrices to thermal BCFT models through conformal transformations, and by engineering rainbow-like inhomogeneous Hamiltonians with exponentially decaying bulk terms. For free-fermion systems, entanglement spectra reproduce conformal tower structures with high resolution, enhanced by an effective chain length $L_{ ext{eff}}$ that captures finite-size effects. To address interacting cases, the authors introduce a Wilsonian numerical renormalisation group (NRG) that folds the Hamiltonian into a two-ended MPS, iteratively builds low-energy states, and unzips folded states to access entanglement spectra; applied to the three-state Potts model, it reveals fixed-point BCFT spectra consistent with the $\mathcal{M}_5$ minimal model and boundary-condition dependent towers. The approach offers a practical route to determine underlying CFT data from wave functions or unknown parent Hamiltonians and provides a flexible tool for exploring BCFT and edge theories in topological phases. The work also provides code for reproducibility and outlines avenues to extract further CFT data (fusion rules, OPE data) from NRG fixed points.

Abstract

Based on the relationship between reduced and thermal density matrices in conformal field theory (CFT), we show that the entanglement spectrum of a conformal critical chain with exponentially decaying terms consists of conformal towers of the associated chiral CFT, with only weak finite-size effects. Through free-fermion and interacting examples, we show that these entanglement spectra present a reliable method to extract detailed CFT spectra from single wave functions without access to the parent Hamiltonian. We complement our method with a Wilsonian numerical renormalisation group algorithm for solving interacting, exponentially decaying chain Hamiltonians.

Figures (11)

• Figure 1: (a) The reduced density matrix of an $L$-site chain at a conformal critical point is given by the partition function of the corresponding BCFT on the grey manifold (left). This can be mapped on a BCFT thermal density matrix with system size $w_0\sim \log L$ using the conformal transformation \ref{['eq: conformal chain']}. Therefore, finite-size effects scale with $\log L$, requiring immense system sizes to eliminate them. (b) If the sites are instead distributed at uniform spacings $\Delta$ in the transformed space, the entanglement spectrum corresponds to a thermal density matrix at system size $w_1\sim L$. In the original geometry, this corresponds to non-uniformly spaced lattice sites, which can be emulated using position-dependent Hamiltonian terms. (c) The rainbow chain is a free-fermion tight-binding chain with exponentially decaying couplings (top line): in the limit $\alpha\to0$, the ground state is a singlet on each arc of the rainbow, resulting in volume-law entanglement. In models with on-site couplings, the latter also decay exponentially (bottom line). In the limit $w\ll 0$, the conformally deformed chain (b) is identical to a rainbow chain with $\alpha = e^{-\Delta/2}$. (d) CFT reduced density matrices in periodic boundary conditions can also be mapped onto thermal density matrices, using the conformal transformation \ref{['eq: conformal ring']}. (Time runs radially in $z$-space.) The preimage of sites distributed uniformly in $w$-space between $\pm w_2/2$ remains symmetric under reflection across the imaginary axis. Reflection eigenvalues of Schmidt states can be used to distinguish descendant states with even and odd momentum.
• Figure 2: (a--b) Ground-state entanglement spectra of conformal \ref{['eq: deformation chain']} free-fermion chains \ref{['eq: complex fermion']} with $\Delta=1/4$ and length (a)$L=256$, (b)$L=258$. Up to 11 degenerate subspaces with the expected multiplicities $1,1,2,3,5,7,11,\dots$ are clearly seen in each charge sector. The red parabolas indicate the expected $(\Delta q)^2/2$ offset of entanglement energy between charge sectors. (c) Entanglement spectrum of the conformal \ref{['eq: deformation ring']} free-fermion ring for $\Delta=1/4$ and $L=256$. Colours indicate mirror-symmetry eigenvalues $+1$ (blue) and $-1$ (red) of the Schmidt vectors, which distinguish subspaces with even and odd momentum. (d) Part of the ground-state entanglement spectrum of the free-fermion chain with uniform $J$ and $L=256$. Unlike the rainbow chains, all but the first few CFT multiplets are too far broadened to distinguish. (e--f) Entanglement spectrum in the half-filled sector for several conformal chains \ref{['eq: deformation ring']} with (e) fixed $\Delta=1/4$, (f) fixed $L\Delta=32$ [blue in panel (e)]. (g) Schmidt values in the $n=6$ [11-fold degenerate, grey background in panels (e, f)] conformal level as a function of the effective system size $L_\mathrm{eff}$\ref{['eq: effective chain length']} for conformal chain (top) and ring (bottom) geometries. $\Delta=1/4$, $1/2$ are omitted to reduce clutter. Fits to the form $1/(aL_\mathrm{eff}^2+bL_\mathrm{eff}^3)$ are added as guides to the eye. All entanglement cuts are at the middle of the chain. The leading entanglement eigenvalue is shifted to $(\Delta q)^2/2 = 0$ or $1/8$, and the first gap within the half-filling charge sector is scaled to 1.
• Figure 3: (a) Ground-state entanglement spectrum of the conformal \ref{['eq: deformation chain']} TFI chain \ref{['eq: TFIM general']} of length $L=192$ and $\Delta=1/4$ in the even and odd fermion-parity sectors (blue and red), and after a boundary deformation to induce the ferromagnetic/SPT phase (green). The number next to each CFT multiplet indicates its multiplicity; the numbers above each spectrum identify the corresponding conformal tower of the Ising CFT: In the paramagnetic/trivial phase, we obtain the identity and fermion sectors; in the ferromagnetic/SPT one, two copies of the twist sector. (b) Entanglement spectrum of the conformal \ref{['eq: deformation ring']} TFI ring for $L=192$ and $\Delta=1/4$. Colours indicate mirror-symmetry eigenvalues $+1$ (blue) and $-1$ (red) of the Schmidt vectors, which distinguish subspaces with even and odd momentum. (c) Entanglement spectrum in the even-parity sector for several conformal chains with fixed $\Delta=1/4$. (d) Schmidt values in the $h+n=6,6.5,7,7.5$ conformal levels [gray in panel (c)] as a function of effective system size $L_\mathrm{eff}$\ref{['eq: effective chain length']}. Fits to the form $1/(aL_\mathrm{eff}^2+bL_\mathrm{eff}^3)$ are added as guides to the eye. All entanglement cuts are at the middle of the chain. The leading entanglement eigenvalue is shifted to 0 ($1/16$) and the first gap scaled to $1/2$ (1) in the trivial (topological) chain.
• Figure 4: (a) Entanglement entropy $S_\mathrm{vN}$ of the tight-binding model \ref{['eq: complex fermion']} as a function of system size for five different values of $\Delta$ [indicated by colours, see inset and panel (b)]. The data for conformal rings is shifted by $+1$ to improve readability. They obey the volume law \ref{['eq: entropy fss']} (straight lines) to a good approximation. Inset: constant volume-law offset $\ell$ as a function of $\Delta$ for the five values of $\Delta$. They follow the expected scaling \ref{['eq: entropy fss']} (black lines). (b) Entanglement gap $\delta_\mathrm{ent}$ in the dominant charge sector as a function of the effective system size $L_\mathrm{eff} = \frac{12}{c} S_\mathrm{vN}$. Data for the rainbow-chain and conformal ring geometries are shifted by $\pm0.2$ to improve readability. For large $L_\mathrm{eff}$, the inverse proportionality \ref{['eq: entropy gap']} holds (solid lines); with a $1/L_\mathrm{eff}^2$ correction (dashed lines), it fits for every system size. (c)Top: Deviation of single-particle entanglement energies $\varepsilon_\alpha$ ($\alpha=3/2$, $5/2$, $7/2$, $9/2$) from the thermodynamic-limit expectation for rainbow chains. Fits to the form $1/(aL_\mathrm{eff}^2+bL_\mathrm{eff}^3)$ are added as guides to the eye. Bottom: Logarithmic derivative of the above for $\alpha=5/2$ and $9/2$. (d--f) The same data for the transverse-field Ising model \ref{['eq: TFIM general']}. Inset of (d):$\ell(\Delta)$ for the TFI chain very closely matches $\ell(\Delta/2)$ for the tight-binding model (grey).
• Figure 5: Summary of the numerical renormalisation-group (NRG) algorithm. The MPO representing the Hamiltonian is folded at the strongest bond of the rainbow chain (a), so that sites with equal interaction strengths on the left and the right can be merged. The low-energy eigenstates of the Hamiltonian are likewise encoded in an MPS with doubled physical legs (b); using these, an environment tensor $L_i$ can be defined to capture the low-energy physics of the middle $2i$ sites of the chain (c). The NRG step now consists of diagonalising the effective Hamiltonian (d); the lowest-energy $\chi_\mathrm{NRG}$ eigenvectors build up the new MPS tensor $M^{(i+1)}$, which is used to update the environment tensor (e), now describing a segment of length $2(i+1)$. To obtain the entanglement spectrum and build an MPS wave function with the natural ordering of lattice sites, the resulting MPS must be "unzipped" (f). This is done by splitting isometries involving the left and right physical and virtual legs off $\tilde{M}^{(i)}$(g) and fusing the remainder into $M^{(i-1)}$(h), which yields $\tilde{M}^{(i-1)}$ for the next step. (i) Schmidt states of the three-state Potts model with trivial $\mathbb{Z}_3$ charge are eigenstates of the $\mathbb{Z}_2$ parity operator with eigenvalue $\pm1$, distinguishing the $0^\pm$ irreps of $S_3$; the other two charge sectors map onto each other.