Rainbow Chain: Exponential Hamiltonians

• Rainbow chain is a one-dimensional model with exponentially decaying nearest-neighbour couplings, generating concentric singlet bonds and a volume-law entangled ground state.
• It admits a continuum description via a c=1 conformal field theory on a curved background, effectively mapping to a thermofield double state with temperature h/(2π).
• Advanced techniques reveal that its entanglement spectrum organizes into chiral conformal towers and that interacting versions can be studied using a Wilsonian numerical renormalisation group.

Rainbow chain denotes a class of one-dimensional inhomogeneous local Hamiltonians whose nearest-neighbour couplings decay exponentially away from the centre. In the standard free-fermion realization, this deformation produces concentric singlet-type correlations across the central bond, a half-chain entanglement entropy that grows linearly with system size, and, in the weak-inhomogeneity regime, a continuum description in terms of a $c=1$ conformal field theory on a curved background or, equivalently, a thermofield double at temperature $T=h/(2\pi)$ (Ramírez et al., 2015, Rodríguez-Laguna et al., 2016). Recent work has used the same deformation to organize entanglement spectra into sharply resolved chiral conformal towers and to formulate a Wilsonian numerical renormalisation group for interacting exponentially decaying chains (Szabó, 2024). Closely related constructions include central-bond defect deformations, holographic descriptions based on AdS$_3$ foliations, and tensor-network realizations of rainbow-entangled phases in Motzkin and Fredkin models (Buruaga et al., 2019, MacCormack et al., 2018, Alexander et al., 2018).

1. Exponentially deformed lattice construction

The defining lattice feature is an exponential envelope of couplings away from the strongest central bond. In the free-fermion formulation emphasized by Szabó, one starts from a one-dimensional local Hamiltonian and deforms the nearest-neighbour couplings so that they fall off exponentially away from the centre,

$$H=\sum_{n=1}^{L-1}J_n\,(c_n^\dagger c_{n+1}+c_{n+1}^\dagger c_n),\qquad J_n=e^{-h(n-\tfrac12)}.$$

Here $h>0$ is the inhomogeneity, or rainbow, parameter. An equivalent notation is $\lambda\equiv e^{-h}$, so that $J_n=\lambda^{n-\tfrac12}$ (Szabó, 2024).

An equivalent mirror-symmetric open-chain parametrization uses $2L$ sites labelled by half-odd integers,

$$H=-\frac12\,J_0\,(c_{+\tfrac12}^\dagger c_{-\tfrac12}+\mathrm{h.c.}) -\sum_{n=\tfrac12}^{L-\tfrac32}\frac{J_n}{2}\, (c_n^\dagger c_{n+1}+c_{-n}^\dagger c_{-(n+1)}+\mathrm{h.c.}),$$

with $J_0=1$, $T=h/(2\pi)$0, and $T=h/(2\pi)$1 (Ramírez et al., 2015). Both parametrizations implement the same physical idea: the entanglement cut is placed at the strongest bond, while all other bonds become exponentially weaker away from the centre.

In the strong-inhomogeneity limit, this geometry produces a “rainbow” of long bonds arching across the cut. In the language used in the earlier literature, the same state is the concentric singlet phase. This suggests that the term “rainbow chain” refers not only to a particular Hamiltonian profile but also to a characteristic entanglement pattern induced by exponential inhomogeneity.

2. Ground state and entanglement structure

For $T=h/(2\pi)$2, or equivalently $T=h/(2\pi)$3, the ground state becomes a product of Bell-pair-like bonds connecting sites symmetrically placed about the centre,

$T=h/(2\pi)$4

In this limit exactly $T=h/(2\pi)$5 Bell pairs straddle the centre, and the half-chain entropy is

$T=h/(2\pi)$6

which is a strict volume law in one dimension (Ramírez et al., 2015). In the defect formulation of the model, the same strong-disorder picture is derived by a Dasgupta–Ma RG, yielding a product of concentric bonds for $T=h/(2\pi)$7 and area-law dimerized phases for $T=h/(2\pi)$8 or $T=h/(2\pi)$9 (Buruaga et al., 2019).

The weak-inhomogeneity regime $_3$0 admits a continuum treatment. The inhomogeneous chain is mapped to a uniform CFT on the stretched coordinate

$_3$1

and the half-chain von Neumann entropy becomes

$_3$2

For the rainbow chain $_3$3. When $_3$4, this reduces to

$_3$5

so the entropy again obeys a volume law, now derived from conformal methods rather than from the explicit Bell-pair limit (Ramírez et al., 2015, Rodríguez-Laguna et al., 2016).

The same curved-CFT framework yields closed forms for more general bipartitions. For an interval anchored at the edge, $_3$6, the $_3$7-th Rényi entropy can be written as

$_3$8

with

$_3$9

and the von Neumann entropy follows from the limit $$H=\sum_{n=1}^{L-1}J_n\,(c_n^\dagger c_{n+1}+c_{n+1}^\dagger c_n),\qquad J_n=e^{-h(n-\tfrac12)}.$$0 (Rodríguez-Laguna et al., 2016). These formulas show that the volume-law behavior is not restricted to the half-chain cut.

3. Curved-space, thermofield-double, and holographic descriptions

The continuum limit of the free rainbow chain is a massless Dirac fermion on a static curved background. After linearizing around the two Fermi points, the resulting Lorentzian metric is

$$H=\sum_{n=1}^{L-1}J_n\,(c_n^\dagger c_{n+1}+c_{n+1}^\dagger c_n),\qquad J_n=e^{-h(n-\tfrac12)}.$$1

or, in Euclidean signature,

$$H=\sum_{n=1}^{L-1}J_n\,(c_n^\dagger c_{n+1}+c_{n+1}^\dagger c_n),\qquad J_n=e^{-h(n-\tfrac12)}.$$2

In the zweibein formalism the scalar curvature is

$$H=\sum_{n=1}^{L-1}J_n\,(c_n^\dagger c_{n+1}+c_{n+1}^\dagger c_n),\qquad J_n=e^{-h(n-\tfrac12)}.$$3

so the background has constant negative curvature $$H=\sum_{n=1}^{L-1}J_n\,(c_n^\dagger c_{n+1}+c_{n+1}^\dagger c_n),\qquad J_n=e^{-h(n-\tfrac12)}.$$4 away from the origin (Rodríguez-Laguna et al., 2016).

This geometric reformulation leads directly to an effective temperature. The analytic continuation of the coordinate map extends the ground state to a strip of width $$H=\sum_{n=1}^{L-1}J_n\,(c_n^\dagger c_{n+1}+c_{n+1}^\dagger c_n),\qquad J_n=e^{-h(n-\tfrac12)}.$$5, and the state can be written in thermofield-double form,

$$H=\sum_{n=1}^{L-1}J_n\,(c_n^\dagger c_{n+1}+c_{n+1}^\dagger c_n),\qquad J_n=e^{-h(n-\tfrac12)}.$$6

Equivalently, the reduced density matrix of one half-chain is approximately thermal,

$$H=\sum_{n=1}^{L-1}J_n\,(c_n^\dagger c_{n+1}+c_{n+1}^\dagger c_n),\qquad J_n=e^{-h(n-\tfrac12)}.$$7

(Ramírez et al., 2015). The appearance of a thermofield double and an effective temperature $$H=\sum_{n=1}^{L-1}J_n\,(c_n^\dagger c_{n+1}+c_{n+1}^\dagger c_n),\qquad J_n=e^{-h(n-\tfrac12)}.$$8 is described as strongly reminiscent of Hartle–Hawking and Unruh/Rindler constructions in the original discussion (Ramírez et al., 2015).

A holographic realization arrives at the same boundary geometry. By foliating AdS$$H=\sum_{n=1}^{L-1}J_n\,(c_n^\dagger c_{n+1}+c_{n+1}^\dagger c_n),\qquad J_n=e^{-h(n-\tfrac12)}.$$9 with AdS$h>0$0 slices, MacCormack et al. obtained the induced boundary metric

$h>0$1

with envelope $h>0$2, in exact agreement with the rainbow-chain hopping amplitudes. The Ryu–Takayanagi prescription then reproduces the entanglement structure, including the half-chain result

$h>0$3

which crosses over to linear growth with slope $h>0$4 at $h>0$5, and the centre-cut result

$h>0$6

which is independent of subsystem size (MacCormack et al., 2018). A plausible implication is that the same exponential deformation admits mutually consistent lattice, curved-CFT, and holographic descriptions.

4. Entanglement Hamiltonian and chiral conformal towers

A central development is the use of rainbow chains to read off detailed chiral CFT spectra from a single ground-state wavefunction. For a long uniform chain at a $h>0$7-dimensional conformal critical point, the reduced density matrix of the left half is

$h>0$8

which defines an entanglement Hamiltonian. In free-fermion chains, $h>0$9 is quadratic,

$\lambda\equiv e^{-h}$0

so the $\lambda\equiv e^{-h}$1 form the single-particle entanglement spectrum (Szabó, 2024).

Cardy, Läuchli, and Szabó’s key observation is that in continuum CFT the same $\lambda\equiv e^{-h}$2 can be viewed as a thermal density matrix of a chiral half-chain of length $\lambda\equiv e^{-h}$3 at inverse temperature $\lambda\equiv e^{-h}$4,

$\lambda\equiv e^{-h}$5

On a segment with suitable boundary conditions, this $\lambda\equiv e^{-h}$6 has a purely chiral Virasoro spectrum, so the entanglement spectrum arranges into chiral conformal towers (Szabó, 2024).

The conformal map underlying this construction is

$\lambda\equiv e^{-h}$7

and, for a ring,

$\lambda\equiv e^{-h}$8

Uniform discretization in $\lambda\equiv e^{-h}$9-space induces physical couplings in $J_n=\lambda^{n-\tfrac12}$0-space proportional to $J_n=\lambda^{n-\tfrac12}$1, hence exponentially varying couplings and the rainbow Hamiltonian (Szabó, 2024).

In the quadratic rainbow chain, the many-body entanglement energies

$J_n=\lambda^{n-\tfrac12}$2

form multiplets whose degeneracies match the chiral CFT degeneracies $J_n=\lambda^{n-\tfrac12}$3, and the zero-mode shift is the continuum primary dimension. Finite-size corrections scale empirically as $J_n=\lambda^{n-\tfrac12}$4, in contrast to the $J_n=\lambda^{n-\tfrac12}$5 behavior of a uniform chain (Szabó, 2024).

Model	Entanglement-spectrum tower structure	Distinctive observation
Complex fermion, $J_n=\lambda^{n-\tfrac12}$6	Degeneracies $J_n=\lambda^{n-\tfrac12}$7 in each charge sector $J_n=\lambda^{n-\tfrac12}$8; charge offsets $J_n=\lambda^{n-\tfrac12}$9	Multiplets resolved up to the 7th level
Majorana / critical Ising, $2L$0	Even parity: identity tower; odd parity: $2L$1-tower; with SPT boundary conditions: $2L$2-tower in both parities	Mirror eigenvalue on a periodic rainbow ring distinguishes even vs. odd Virasoro level
Three-state Potts, $2L$3	Trivial $2L$4 sector: $2L$5; irrep $2L$6: two copies of $2L$7	Strong boundary fields induce $2L$8 or $2L$9

These results show that the rainbow deformation suppresses finite-size broadening enough to resolve eleven or more conformal levels in some examples, and that the entanglement spectrum can serve as a practical probe of chiral BCFT data even when the parent Hamiltonian is inaccessible (Szabó, 2024).

5. Interacting rainbow chains and Wilsonian numerical renormalisation group

For non-free theories, the entanglement Hamiltonian cannot be reconstructed from a correlation matrix. Szabó therefore complements the rainbow-chain construction with a Wilsonian numerical renormalisation group tailored to exponentially decaying Hamiltonians. The algorithm folds the MPO of the rainbow Hamiltonian about the strongest central bond, turning sites $$H=-\frac12\,J_0\,(c_{+\tfrac12}^\dagger c_{-\tfrac12}+\mathrm{h.c.}) -\sum_{n=\tfrac12}^{L-\tfrac32}\frac{J_n}{2}\, (c_n^\dagger c_{n+1}+c_{-n}^\dagger c_{-(n+1)}+\mathrm{h.c.}),$$0 and $$H=-\frac12\,J_0\,(c_{+\tfrac12}^\dagger c_{-\tfrac12}+\mathrm{h.c.}) -\sum_{n=\tfrac12}^{L-\tfrac32}\frac{J_n}{2}\, (c_n^\dagger c_{n+1}+c_{-n}^\dagger c_{-(n+1)}+\mathrm{h.c.}),$$1 into a single supersite. The low-energy eigenstates of the first $$H=-\frac12\,J_0\,(c_{+\tfrac12}^\dagger c_{-\tfrac12}+\mathrm{h.c.}) -\sum_{n=\tfrac12}^{L-\tfrac32}\frac{J_n}{2}\, (c_n^\dagger c_{n+1}+c_{-n}^\dagger c_{-(n+1)}+\mathrm{h.c.}),$$2 physical sites are represented by an MPS tensor $$H=-\frac12\,J_0\,(c_{+\tfrac12}^\dagger c_{-\tfrac12}+\mathrm{h.c.}) -\sum_{n=\tfrac12}^{L-\tfrac32}\frac{J_n}{2}\, (c_n^\dagger c_{n+1}+c_{-n}^\dagger c_{-(n+1)}+\mathrm{h.c.}),$$3 with two physical legs and one virtual leg. At step $$H=-\frac12\,J_0\,(c_{+\tfrac12}^\dagger c_{-\tfrac12}+\mathrm{h.c.}) -\sum_{n=\tfrac12}^{L-\tfrac32}\frac{J_n}{2}\, (c_n^\dagger c_{n+1}+c_{-n}^\dagger c_{-(n+1)}+\mathrm{h.c.}),$$4, one forms an effective Hamiltonian $$H=-\frac12\,J_0\,(c_{+\tfrac12}^\dagger c_{-\tfrac12}+\mathrm{h.c.}) -\sum_{n=\tfrac12}^{L-\tfrac32}\frac{J_n}{2}\, (c_n^\dagger c_{n+1}+c_{-n}^\dagger c_{-(n+1)}+\mathrm{h.c.}),$$5, diagonalizes it, retains only the $$H=-\frac12\,J_0\,(c_{+\tfrac12}^\dagger c_{-\tfrac12}+\mathrm{h.c.}) -\sum_{n=\tfrac12}^{L-\tfrac32}\frac{J_n}{2}\, (c_n^\dagger c_{n+1}+c_{-n}^\dagger c_{-(n+1)}+\mathrm{h.c.}),$$6 eigenstates, stores them as $$H=-\frac12\,J_0\,(c_{+\tfrac12}^\dagger c_{-\tfrac12}+\mathrm{h.c.}) -\sum_{n=\tfrac12}^{L-\tfrac32}\frac{J_n}{2}\, (c_n^\dagger c_{n+1}+c_{-n}^\dagger c_{-(n+1)}+\mathrm{h.c.}),$$7, repeats until the flow converges to a fixed-point spectrum, and finally “unzips” the folded MPS by successive SVDs to recover the usual ordering and the entanglement spectrum (Szabó, 2024).

Convergence is monitored by the stability of the low-energy levels as $$H=-\frac12\,J_0\,(c_{+\tfrac12}^\dagger c_{-\tfrac12}+\mathrm{h.c.}) -\sum_{n=\tfrac12}^{L-\tfrac32}\frac{J_n}{2}\, (c_n^\dagger c_{n+1}+c_{-n}^\dagger c_{-(n+1)}+\mathrm{h.c.}),$$8 and by the decay of discarded weights. In the three-state Potts example,

$$H=-\frac12\,J_0\,(c_{+\tfrac12}^\dagger c_{-\tfrac12}+\mathrm{h.c.}) -\sum_{n=\tfrac12}^{L-\tfrac32}\frac{J_n}{2}\, (c_n^\dagger c_{n+1}+c_{-n}^\dagger c_{-(n+1)}+\mathrm{h.c.}),$$9

the critical point $J_0=1$0 realizes the $J_0=1$1 minimal model. Using Wilson-NRG with $J_0=1$2, the folded-chain fixed-point spectrum exhibits the towers $J_0=1$3 in the trivial $J_0=1$4 sector and two copies of $J_0=1$5 in the two-dimensional irrep $J_0=1$6. After unzipping, the ground-state entanglement spectrum shows the same exact degeneracies. Strong boundary fields can be used to force the $J_0=1$7 or $J_0=1$8 towers, in agreement with BCFT predictions. The finite-size and fixed-$J_0=1$9 corrections scale as $T=h/(2\pi)$00 and $T=h/(2\pi)$01 (Szabó, 2024).

This interacting extension is significant because it detaches the conformal-spectrum extraction problem from free-fermion integrability. A plausible implication is that rainbow deformation functions as a spectroscopy device for chiral CFT data across both quadratic and interacting critical chains.

6. Defects, symmetry variants, and related rainbow phases

A natural deformation introduces a tunable central defect $T=h/(2\pi)$02. In an open spin-$T=h/(2\pi)$03 chain of $T=h/(2\pi)$04 sites, the inhomogeneous XX Hamiltonian with bond-centred defect is

$T=h/(2\pi)$05

with

$T=h/(2\pi)$06

For $T=h/(2\pi)$07 one recovers the usual rainbow couplings (Buruaga et al., 2019).

The strong-inhomogeneity phase diagram has three regimes. For $T=h/(2\pi)$08, the ground state remains a rainbow valence-bond state generated by the RG as a sequence of concentric bonds between sites $T=h/(2\pi)$09. For $T=h/(2\pi)$10 or $T=h/(2\pi)$11, the ground state becomes a trivial dimer covering, with area-law entanglement bounded by $T=h/(2\pi)$12. At $T=h/(2\pi)$13, the three central couplings tie and the RG yields local multi-site modes and a valence-bond structure built from 4-site operators; cuts carry $T=h/(2\pi)$14 entanglement per block but do not grow with system size (Buruaga et al., 2019).

In the weak-inhomogeneity regime, the defect generates a scattering amplitude

$T=h/(2\pi)$15

and the half-chain entropy becomes

$T=h/(2\pi)$16

with

$T=h/(2\pi)$17

The entanglement properties are invariant under $T=h/(2\pi)$18, and the self-dual point $T=h/(2\pi)$19 is the defect-free rainbow chain. For $T=h/(2\pi)$20, the single-particle entanglement spectrum is gapped, with the gap increasing with defect strength; at $T=h/(2\pi)$21, one recovers the $T=h/(2\pi)$22 spacing characteristic of CFT on a strip (Buruaga et al., 2019).

A site-centred version maps, after folding, to an inhomogeneous Su–Schrieffer–Heeger ladder. In the strong-coupling limit and for $T=h/(2\pi)$23, a trivial dimerized region forms around the defect, while an SSH SPT region persists toward the ends. The entanglement spectrum of a central block $T=h/(2\pi)$24 shows three zero modes for $T=h/(2\pi)$25, then one topological zero mode for $T=h/(2\pi)$26 (Buruaga et al., 2019).

Related rainbow-entangled phases also arise outside the exponentially inhomogeneous free-fermion chain. In area-deformed Motzkin and Fredkin models, the deformation parameter $T=h/(2\pi)$27 drives a transition from an area-law phase ($T=h/(2\pi)$28) to a rainbow or volume phase ($T=h/(2\pi)$29), where

$T=h/(2\pi)$30

for the left half-chain in the colorful case (Alexander et al., 2018). The associated “step-pyramid” tensor network exactly contracts to the ground state and interpolates between low-entanglement and rainbow-pairing regimes. A higher-dimensional free-fermion extension on a $T=h/(2\pi)$31 square lattice gives

$T=h/(2\pi)$32

which is a genuine two-dimensional volume law for $T=h/(2\pi)$33 (Ramírez et al., 2015). These related constructions indicate that “rainbow” has become a broader label for concentric, maximally nonlocal entanglement structures generated either by explicit inhomogeneous couplings or by weighted superpositions of extended paths.