Formal theory and domain of validity of the transfer-matrix–effective-Hamiltonian spectral correspondence in finite-entanglement matrix product states

In infinite matrix product state (MPS) simulations of critical one-dimensional quantum systems, the low-energy spectra of the truncated effective Hamiltonian and of the truncated transfer matrix were shown numerically to coincide up to a single global rescaling factor — the emergent velocity ("speed of light") of the governing conformal field theory — yielding the most accurate method available for extracting velocities of non-integrable critical chains (Eberharter, Vanderstraeten, Verstraete and Läuchli, Phys. Rev. Lett. 131, 226502 (2023)). This correspondence is, however, an empirical observation supported only by heuristic path-integral and corner-transfer-matrix arguments, expected to hold only for single-velocity conformal critical points. The open problem is to turn it into a derived, validated tool with a mapped domain of validity: a formal characterization of the conditions under which finite-entanglement truncation acts isotropically in space and time; its fate in systems with two or more distinct low-energy velocities (doped Hubbard chains, multi-component Luttinger liquids); its behavior where the velocity vanishes (dynamical exponent $z\neq 1$, e.g. $z=2$); and an analogous correspondence for 2+1D critical tensor-network simulations.

Background

Simulating one-dimensional quantum systems at a continuous quantum phase transition is a central use of tensor networks. The low-energy physics there is governed by a conformal field theory (CFT) — a scale-invariant relativistic field theory in which excitations disperse linearly, $E\approx v|k|$, with a single non-universal velocity $v$, the "speed of light" of the emergent theory. This velocity enters many downstream quantities (spinon velocities, Casimir-amplitude checks of the ground-state energy, dynamic structure factors, and Luttinger parameters), yet it is non-universal and was notoriously hard to compute accurately for non-integrable lattice models.

The workhorse method is the density-matrix renormalization group (DMRG), which variationally optimizes a matrix product state (MPS): a wavefunction built from a chain of tensors contracted through a virtual index of bounded dimension $D$ (the bond dimension), which caps the entanglement the ansatz can carry. In the infinite-system (uniform MPS) formulation two auxiliary operators appear. First, the effective Hamiltonian $\tilde H$: the physical Hamiltonian projected onto the tangent space of the MPS, whose low-lying eigenvectors approximate excited states of the infinite chain. Second, the transfer matrix $\tilde T$: the operator formed by contracting the MPS tensor with its complex conjugate over the physical index, whose subleading eigenvalues $\epsilon_i$ define correlation lengths $\xi_i=-1/\ln|\epsilon_i|$.

Because a critical ground state carries unbounded entanglement, a finite-$D$ MPS always describes a slightly gapped, finite-correlation-length system. The resulting "finite-entanglement scaling" (Tagliacozzo, de Oliveira, Iblisdir and Latorre, 2008; Pollmann, Mukerjee, Turner and Moore, 2009; Pirvu, Vidal, Verstraete and Tagliacozzo, 2012) — the analogue of finite-size scaling with the effective length set by $\xi(D)$ — underlies the extraction of universal CFT data from infinite MPS simulations, and the transfer-matrix spectrum was known to encode the low-energy excitation structure (Zauner et al., 2015). In the finite-size setting the analogous statement is that the local effective Hamiltonian of finite-size DMRG reproduces the CFT finite-size spectrum (Chepiga and Mila, 2017).

Eberharter, Vanderstraeten, Verstraete and Läuchli (Phys. Rev. Lett. 131, 226502 (2023)) discovered that in the finite-entanglement regime these two spectra coincide: for a critical chain governed by a single-velocity CFT,

$$(E_i-E_0)\,\xi_i = v,\qquad i>0,$$

with $E_i$ the gaps of the truncated effective Hamiltonian and $\xi_i$ the correlation lengths from the truncated transfer matrix. This yields highly accurate velocities (e.g. for the non-integrable half-integer-spin Heisenberg chains) and, combined with the compressibility, Luttinger parameters $K$ (the dimensionless stiffness controlling power-law correlations in one-dimensional gapless "Luttinger liquid" systems). The evidence is numerical — critical Ising and $Q$-state Potts chains (the Ising model generalized from two to $Q$ states per site), the $S=1/2$ Heisenberg chain, doped Hubbard chains and ladders — and the justification heuristic. Viewing the MPS approximation of the ground state as a two-dimensional (space–imaginary-time) tensor network, the spatial and temporal transfer matrices must share their low-energy spectrum when the network is isotropic: there the variational fixed-point equations are manifestly rotation invariant, consistent with the corner-transfer-matrix renormalization group (CTMRG), an iterative contraction method that truncates the singular values of a corner tensor of the network. For anisotropic (quantum-Hamiltonian) networks, the corner-tensor truncation only "suggests" isotropy in the low-energy sector; no derivation exists. The authors stress that the property is emergent, expected so far only for a single-velocity CFT, and that the detailed structure of both the transfer matrix and the effective Hamiltonian in the finite-entanglement regime is not understood.

Problem Statement

Establish a rigorous theory of the finite-entanglement spectral correspondence and map its domain of validity, turning the observed relation $(E_i-E_0)\,\xi_i=v$ between the truncated effective-Hamiltonian gaps and transfer-matrix correlation lengths in infinite MPS simulations of critical chains into a derived, validated tool. The linked questions, which jointly define the completion boundary, are:

1. Formal characterization. Under which precise conditions — on the universality class of the critical point, on the variational MPS ensemble and its imposed symmetries, and on the bond-dimension regime — do the low-energy spectra of $\tilde H$ and $\tilde T$ coincide up to a single global factor $v$ (the relation $(E_i-E_0)\,\xi_i = v$ displayed above, Eq. (1) of the source paper)? Derive the isotropy of the finite-entanglement truncation in space and time from first principles (renormalization-group/effective-field-theory or tensor-network fixed-point arguments), replacing the present path-integral and corner-transfer-matrix heuristics, and characterize the range of levels over which the proportionality holds and the mechanism for the deviations beyond it.

2. Multiple velocities. Determine whether the correspondence generalizes to critical points with two or more distinct low-energy velocities — for example the doped Hubbard chain with central charge $c=2$ (the central charge counts the effective number of gapless degrees of freedom of the critical theory) and different charge and spin velocities, or multi-component Luttinger liquids — e.g. through a sector-wise rescaling of the spectra with one velocity per conformal sector, or whether it breaks down; validate against velocities known exactly from the Bethe ansatz, the exact solution technique available for integrable one-dimensional models.

3. Vanishing velocity and $z\neq 1$. Characterize the behavior of the spectral correspondence at and near transitions where the velocity vanishes or the dynamical exponent (relating gaps to inverse length, $\Delta\sim L^{-z}$) differs from the Lorentz-invariant value $z=1$ — for instance the $z=2$ transitions expected to terminate the incommensurate floating phases of Rydberg-blockaded chains (chains of atoms driven into high-lying Rydberg states, whose strong mutual repulsion enforces a minimum separation between excitations) — and identify what replaces $v$ in the spectral matching there.

4. 2+1D analogue. Investigate whether an analogous correspondence between transfer-operator spectra and parent-Hamiltonian (the local Hamiltonian for which the tensor network is an exact ground state) or effective-Hamiltonian spectra can be established for tensor-network simulations of 2+1D critical systems and 2+1D CFTs, and how it interacts with finite-size torus spectroscopy in situations where interaction effects distort the finite-size spectrum.

Items 1–3 form the theoretical core of the correspondence in one spatial dimension; item 4 extends the same correspondence to the next dimension. A complete answer establishes the finite-entanglement spectral correspondence as a derived tool whose validity conditions, failure modes, and extension range are known.

References

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