Completing the self-congruent-point effective field theory for finite-entanglement scaling in critical matrix product states

Variationally optimized matrix product states (MPS) of finite bond dimension $\chi$ (the size of the tensors' internal indices) approximate critical ground states with a finite correlation length $\xi(\chi)\propto\chi^{\kappa}$, producing finite-entanglement scaling. Schneider, Ueda, Liu, Läuchli, Oshikawa and Tagliacozzo [SciPost Phys. 18, 142 (2025)] resolved a long-standing puzzle by showing that the gap ratios of the MPS transfer matrix (TM) are independent of $\chi$ yet differ from the conformal field theory (CFT) values because finite $\chi$ acts as a relevant perturbation whose renormalization-group flow halts at a 'self-congruent point' at finite distance from the critical fixed point. Their Gaussian (free-fermion) effective lattice model reproduces only roughly the first four TM gap ratios. The problem recorded here is the completion of this effective field theory along the two directions stated in the authors' outlook: (i) a systematic survey of which universality classes and microscopic symmetry constraints produce self-congruent points and $\chi$-independent TM spectra, and which behave qualitatively differently; and (ii) identification of the models and mechanisms — the non-Gaussian operator content — governing the dense higher-energy TM spectrum. A 2026 preprint by Hodgkiss et al. determines the induced perturbation numerically for the Ising case and shows that the naive CFT-relevance picture fails for two decoupled critical Ising chains represented by a single MPS, but neither facet is closed.

Background

Matrix product states (MPS) are one-dimensional tensor-network wavefunctions whose expressive power is set by the bond dimension $\chi$, the size of the internal tensor indices; for infinite chains they are optimized by the infinite-system density matrix renormalization group (iDMRG). At a continuous quantum phase transition the exact ground state has unbounded entanglement, so a finite-$\chi$ MPS necessarily has a finite correlation length $\xi(\chi)$, which grows algebraically with the bond dimension, $\xi(\chi)\propto\chi^{\kappa}$ with $\kappa=6/[c(1+\sqrt{12/c})]$, where $c$ is the central charge — the number labeling the degrees of freedom of the conformal field theory (CFT) describing the critical point. Physical observables then obey finite-entanglement scaling, an exact analogue of finite-size scaling in which $\xi(\chi)$ replaces the system size (Tagliacozzo et al. 2008; Pollmann et al. 2009; Pirvu et al. 2012).

A sharp diagnostic of a critical MPS is the spectrum of its transfer matrix (TM) — the double-layer tensor obtained by contracting the state with its complex conjugate, whose eigenvalue gaps govern correlation functions. For an iMPS optimized at a critical point, the ratios of the TM gaps are observed to be scale invariant, i.e. independent of $\chi$ even as $\chi\to\infty$, yet their numerical values differ from the conformal weights (the level spectrum) of the underlying CFT and of the boundary CFT (the CFT on a half-plane, which governs entanglement spectra). Explaining both the scale invariance and the actual values of these ratios was a standing puzzle of the finite-entanglement-scaling program.

Schneider, Ueda, Liu, Läuchli, Oshikawa and Tagliacozzo (SciPost Phys. 18, 142 (2025)) proposed an effective-field-theory (EFT) resolution. Truncating to finite $\chi$ is equivalent to perturbing the critical Hamiltonian by a relevant operator (one whose coupling grows under renormalization-group coarse-graining): the magnetic operator $\sigma$ when the microscopic $\mathbb{Z}_2$ symmetry is not enforced on the tensors, and the thermal operator $\varepsilon$ when it is. Under the RG the perturbing coupling $g_r$ would normally flow away from criticality, but in the MPS setting the initial (lattice-scale) coupling $g_r(1,\chi)$ decreases with $\chi$ while the infrared cutoff $\xi(\chi)$ increases; the two effects compensate, and the flow halts at a finite distance from the critical fixed point — a self-congruent point — at which the coupling is stationary in $\chi$. This simultaneously explains the $\chi$-independence of the TM gap ratios and why they differ from the CFT values, while leaving the finite-entanglement scaling exponents intact. The authors corroborate the picture with the exact solution of the critical Ising chain, with iDMRG spectra, and with an effective lattice model — a critical transverse-field Ising chain of length $L=2\xi(\chi)$ carrying a scale-dependent thermal perturbation $\delta=-m/L$ with $m$ fixed to $m=1$ by an explicit argument — which, being Gaussian (mapping to free fermions), is exactly solvable and reproduces roughly the first four TM gap ratios, the finite-entanglement critical exponent, and the convergence of the entanglement spectrum toward the boundary-CFT prediction (the latter also established independently by Huang et al. 2024). The same TM spectrum is found at a critical point of the Alcaraz self-dual extended Ising model (same universality class), and in both the critical Ising and 3-state Potts models the induced perturbation points toward the spontaneously symmetry-broken (SSB) phase, counter to the naive expectation that a fixed-strength perturbation should minimize the bipartite entanglement.

Two limitations of this EFT, singled out as open directions in the authors' conclusion, define the open problem recorded here: the mechanism has been verified only for the Ising universality class and the 3-state Potts model, and the Gaussian effective model misses the higher-energy part of the TM spectrum, which the authors attribute to non-Gaussian effects. Further limitations noted in the paper remain as background: the selection of the perturbation's operator content and sign is argued only case by case, and the RG arguments establish scaling exponents but not the nonuniversal prefactors.

Problem Statement

Complete the effective field theory of finite-entanglement scaling so that, given the conformal field theory data and the microscopic symmetry constraints of a critical point, it establishes where its self-congruent-point mechanism applies and reproduces the resulting transfer-matrix (TM) spectrum of the optimized matrix product state at all energy scales. Two linked questions delimit this objective:

1. (Domain of validity) Determine, by systematic survey across universality classes and across enforced versus non-enforced tensor symmetry constraints, which critical points exhibit a self-congruent point — an RG halting point at which the induced relevant coupling stops flowing as $\chi$ grows — with $\chi$-independent TM gap ratios distinct from the CFT values, and which instead show qualitatively different behavior. Qualitatively different behavior here means concretely that the TM gap ratios drift with $\chi$ or that the induced relevant coupling does not halt at a finite distance from the critical fixed point; the required output is a general criterion that, given the conformal-field-theory data and the microscopic symmetry constraints, decides which case a critical point exhibits, with the models named below serving as anchors of the survey. Beyond the transverse-field Ising chain, the Alcaraz self-dual extended Ising model (same universality class), the 3-state Potts model, and the two-copy Ising case (two decoupled critical Ising chains represented by a single MPS), no such classification exists.

2. (Spectral completeness) Identify the models and mechanisms — in EFT language, the non-Gaussian operators or interactions beyond the Gaussian/free-fermion effective lattice model — that reproduce the higher-energy part of the TM spectrum (above roughly the first four gap ratios), and characterize which of its features are universal and which are nonuniversal.

Facet 1 fixes the EFT's domain of validity (where it applies); facet 2 completes its output (the full TM spectrum). Target quantities for the classification and the spectral analysis include the extrapolated $\chi\to\infty$ TM gap ratios of the critical transverse-field Ising chain in both the $\mathbb{Z}_2$-enforced and non-enforced tensor sectors, and the 3-state Potts ratios $\Delta_i/\Delta_3$ (where $\Delta_i$ denotes the $i$-th transfer-matrix gap) with their shifts relative to the CFT values $[0, 1/3, 5/6, 1, \ldots]$.

References

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