Theory of the finite-entanglement perturbation and the encoding of conformal operator content in matrix product state transfer operators

Variational matrix product states (MPS, and their continuum limit cMPS) approximate critical one-dimensional ground states with a finite correlation length $\xi(D)$ that grows as a power law in the bond dimension $D$; finite-entanglement scaling (FES) treats $\xi(D)$ as a stand-in for system size and thereby extracts conformal field theory (CFT) data — the central charge and scaling dimensions — from thermodynamic-limit simulations. The program's foundational formulation by Stojevic, Haegeman, McCulloch, Tagliacozzo and Verstraete (PRB 91, 035120 (2015)) explicitly left its theoretical basis open: the renormalization-group and geometric interpretation of the perturbation induced by finite bond dimension, the mapping between the CFT's primary/descendant structure and the eigen-decomposition of the (c)MPS transfer operator, and the analytic status of the relation $\xi(D)\sim D^{\kappa}$ with $\kappa(c)=6/(c(\sqrt{12/c}+1))$, which deviates from measured values at accessible $D$. Recent work has clarified major pieces — an effective field theory in which finite $D$ acts as a relevant perturbation whose flow halts at a self-congruent point (Schneider et al.), a boundary-CFT description of the entanglement Hamiltonian (Huang et al.), and direct computation of the induced perturbations, which shows subleading relevant operators appear beyond the simplest models (Hodgkiss et al.) — but a complete predictive theory of the induced perturbation, an operator-independent dictionary from transfer-operator spectra to conformal data, and a first-principles derivation of the $\xi$–$D$ relation are all still missing. This problem asks for that theory.

Background

One-dimensional quantum systems at continuous phase transitions are described at long distances and times by two-dimensional conformal field theories (CFTs). A CFT is characterized by its central charge $c$, which counts the universal degrees of freedom, and by its scaling operators: primary operators label the independent conformal families, while their Virasoro descendants — the operators generated from a primary by the action of the conformal symmetry algebra — fill out each family. Through the operator-state correspondence, a scaling operator of dimension $\Delta$ corresponds to an eigenstate of the CFT on a circle of circumference $L$ with energy $2\pi v(\Delta+n)/L$ above the ground state, where $v$ is the sound velocity and $n\ge 0$ the descendant level. Finite-size scaling, i.e. extrapolating lattice spectra and correlators in $L$, is the classical route to this conformal data.

The density matrix renormalization group (DMRG) instead represents ground states as matrix product states: wavefunctions built from a chain of $D\times D$ matrices, where the bond dimension $D$ bounds the bipartite entanglement entropy by $\log D$. Its continuum generalization, the continuous matrix product state (cMPS), applies directly to quantum field theories. Because the half-chain entanglement entropy of a critical ground state grows as $(c/6)\log\xi$ with the correlation length $\xi$, an exact critical state requires $D\to\infty$; at any finite $D$ the variationally optimal (c)MPS has a finite $\xi(D)$ and behaves as if the critical point were perturbed by a relevant, gap-opening perturbation whose strength decreases as $D$ grows.

Finite-entanglement scaling (FES), developed by Tagliacozzo et al., Pollmann et al. and Pirvu et al., turns this truncation into a tool: $\xi(D)$ plays the role that $L$ plays in finite-size scaling. Two ingredients are central. First, $\xi(D)\sim D^{\kappa}$, with the exponent predicted by Pollmann et al. on the basis of the Calabrese–Lefevre universal decay of the Schmidt (entanglement) spectrum to be $\kappa(c)=6/(c(\sqrt{12/c}+1))$. Second, the (c)MPS transfer operator $E$ — the contraction of the state with its complex conjugate, a map on a $D^2$-dimensional space whose subleading eigenvalues $\lambda_i$ govern correlation decay with lengths $\mu_i^{-1}=-1/\log(\lambda_i/\lambda_1)$ — plays the role of the finite-size Hamiltonian: its low-lying level spacing should encode the scaling dimensions, one-point functions of a scaling operator should decay as $\xi^{-\Delta}$, and the entanglement entropy should grow as $(\kappa c/6)\log D$. Stojevic et al. (the source paper) combined these into a scaling algorithm, using $\xi(D)$ itself rather than $D$ as the scaling parameter, and extracted central charges and scaling dimensions of local and non-local operators for the Lieb–Liniger model, the massless relativistic boson, and the critical quantum Ising chain with accuracy competitive with the multi-scale entanglement renormalization ansatz (MERA) at far lower cost.

The source paper also showed that the ratios of the transfer-operator levels converge to a $D$-independent universal value — evidence for a CFT interpretation of FES — but not to the values expected from the CFT on a strip, and it proposed, without verification, a more sophisticated geometric picture: a strip bisected by a line of impurities arising where the MPS is glued to its conjugate. Its conclusions list the missing theoretical foundation: no renormalization-group or geometric interpretation of the finite-$D$ perturbation; no mapping between the primary/descendant structure of the CFT and the eigen-decomposition of the transfer operator (so operator-level input, such as a lattice realization of the scaling operator, is still needed to extract conformal data); and significant deviations of the measured $\kappa$ from $\kappa(c)$ at the accessible bond dimensions $32\le D\le 64$.

Three later lines sharpen the problem. Schneider et al. constructed an effective field theory in which finite bond dimension is equivalent to a relevant perturbation of the CFT whose coupling stops flowing at a self-congruent point — a finite distance from the CFT fixed point but well inside its scaling regime — because increasing the infrared cutoff $\xi(\chi)$ and decreasing the perturbation strength at the lattice scale balance each other; this explains the universal, $\chi$-independent transfer-operator gap ratios and why they differ from CFT spectra even as $\chi\to\infty$. Huang et al. showed that the bipartite entanglement Hamiltonian of the finite-$D$ MPS is described by a boundary CFT (a CFT on a domain with conformally invariant boundary conditions) with a physical and an entanglement boundary. Hodgkiss et al. computed the actual induced perturbations by implicit differentiation of the defining tensors: for the critical Ising model it is, to overlap $\approx 0.999$, the most relevant operator $\sigma$, but for two decoupled critical Ising chains subleading relevant, entangling, symmetry-breaking operators contribute on the same footing — so FES is not simply 'the CFT plus its most relevant operator'. In parallel, Zou, Milsted and Vidal established a complementary route: periodic uniform MPS (puMPS) Bloch states — momentum eigenstates built from matrix product states on a ring — combined with the Koo–Saleur formula (a lattice construction interpolating between lattice operators and CFT scaling operators) identify scaling operators, conformal data and even operator product expansion (OPE) coefficients — but this route requires operator-level input and works at finite size rather than in the thermodynamic FES regime.

Problem Statement

Construct the conformal-field-theory interpretation of finite-entanglement scaling for translation-invariant (c)MPS approximations of critical one-dimensional systems: a theory of the perturbation that finite bond dimension induces, and of how the CFT's operator content is encoded in transfer-operator data. This problem has three jointly required components that together close the theoretical gap left by Stojevic et al. (PRB 91, 035120 (2015)) and only partially filled by subsequent work.

(i) Perturbation theory. Determine, for a general critical system, which relevant perturbation(s) of the CFT a finite bond dimension $D$ effectively induces — the operator content, the $D$-dependence of the couplings, and the renormalization-group interpretation (including any geometric/boundary-CFT picture of the transfer operator, such as the strip-with-impurity-line proposal). The characterization must cover cases beyond the simplest models: it must account for the demonstrated fact that for two decoupled critical Ising chains subleading relevant, entangling operators contribute on the same footing as the most relevant one, and must specify when the single-most-relevant-operator picture holds and when, and how, it fails.

(ii) Operator dictionary. Establish the mapping between the eigen-decomposition of the (c)MPS transfer operator and the primary/descendant structure of the CFT, such that conformal data — the identification of scaling operators and their scaling dimensions — can be extracted from (c)MPS data alone, without operator-level input such as a lattice realization of the scaling operator. The dictionary must account for the observed universal, $D$-independent transfer-operator gap ratios and explain their deviation from the CFT spectrum, which the self-congruent-point effective theory attributes to a renormalization-group flow halted at finite distance from the fixed point.

(iii) Scaling exponent. Settle the analytic status of the bond-dimension–correlation-length relation $\xi(D)\sim D^{\kappa}$, $\kappa(c)=6/(c(\sqrt{12/c}+1))$: either derive this relation (or its correct replacement) from first principles, and characterize the finite-$D$ corrections responsible for the observed deviations of measured $\kappa$ from $\kappa(c)$ at accessible bond dimensions.

These components form one objective: (i) supplies the perturbed theory whose scale-invariant transfer-operator spectrum (ii) decodes into conformal data, and (iii) fixes the $D$-scaling that links the two. An answer is complete when the framework predicts, from the CFT and the variational ansatz alone, the induced perturbation, the resulting transfer-operator spectrum, and the $\xi$–$D$ relation, in agreement with existing numerical evidence. The source paper's further extensions — three-point (OPE) data, and critical exponents of boundary CFTs describing edges — are part of the same research agenda but are applications of the framework rather than components of the theoretical core: the OPE-coefficient goal has been substantially achieved by operator-input methods (Zou et al. 2020), and the boundary-CFT (edge) extension remains open in the literature; neither is restated as a requirement here, and the completion condition is fixed by (i)–(iii) above.

References

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