Finite bond dimension at two-dimensional quantum criticality in iPEPS: Lifshitz-point convergence, operator content of the truncation geometry, and D as a continuum regulator

Corboz, Czarnik, Kapteijns and Tagliacozzo (Phys. Rev. X 8, 031031 (2018)) showed that variationally optimized infinite projected entangled-pair states (iPEPS) with finite bond dimension $D$ acquire a finite correlation length $\xi_D$ at Lorentz-invariant two-dimensional quantum critical points, argued that this has a geometric origin (a finite-$D$ landscape of towers and valleys replacing the smooth Euclidean space-time of the critical theory), and built the resulting finite correlation length scaling (FCLS) method for locating critical points and extracting critical exponents. The theory behind FCLS was left open. The open core of that outlook comprises three strands: (1) whether variational iPEPS optimizations starting from generic (random) initial states can converge to Lifshitz critical points, with dispersion $E(k)\propto k^z$ and $z\ge 2$, where the Lorentz-invariance argument fails and finite-$D$ representation may be more amenable; (2) how the finite-$D$ truncation geometry modifies the operator content (scaling operators and scaling dimensions) of the corresponding continuum field theory; (3) the quantitative law by which the required $D$ must grow as the lattice spacing decreases in continuum limits so that universal ratios remain invariant. The methodological companion issues left by the source (extrapolation of $\xi_D$ at large $D$; validation of FCLS across universality classes) lie outside this task.

Background

An infinite projected entangled-pair state (iPEPS) is a variational wave function for a two-dimensional (2D) quantum lattice model in the thermodynamic limit, built from local tensors connected by bonds whose dimension $D$ bounds the entanglement of any subregion by its boundary length (the entanglement area law). Expectation values are computed by approximately contracting an infinite 2D tensor network, typically with a corner transfer matrix renormalization group (CTMRG) scheme whose boundary dimension $\chi$ introduces a second length scale $\xi_\chi$; data must be converged or extrapolated in $\chi$ as well as $D$.

In one dimension the analogous infinite matrix product states (iMPS) provide a systematic route to criticality: although a finite-$D$ iMPS has a finite correlation length $\xi_D$ and cannot represent a critical state exactly, $\xi_D$ grows with $D$ and acts as an infrared cutoff analogous to a finite system size, so critical exponents can be extracted by finite-entanglement scaling (Pollmann, Mukerjee, Turner and Moore, Phys. Rev. Lett. 102, 255701 (2009)). Whether the same program works in 2D was clarified by Corboz, Czarnik, Kapteijns and Tagliacozzo (Phys. Rev. X 8, 031031 (2018)). For Lorentz-invariant 2D quantum critical points - those with linear low-energy dispersion $E(k)\propto k$, i.e. dynamical exponent $z=1$, where space and time enter the Euclidean path integral symmetrically - they found that the variationally optimal finite-$D$ iPEPS always acquires a finite correlation length $\xi_D$, even though 2D critical states obey the area law and are not excluded a priori. They argued this has a geometric origin: in the scaling regime the finite-$D$ iPEPS represents the smooth three-dimensional Euclidean space-time of the critical theory as a landscape of towers and valleys (quasi-one-dimensional channels with a finite correlation time along them), and this cutoff induces the observed gap and finite $\xi_D$. On this basis they proposed finite correlation length scaling (FCLS): replacing the system size $L$ by $\xi_D$ in finite-size-scaling ansatze, e.g. $m(g,D)=\xi_D^{-\beta/\nu}\mathcal{M}(g\,\xi_D^{1/\nu})$ for an order parameter $m$ at distance $g$ from criticality, together with a derivative-based $m'/m$ scheme to locate critical points without high-order moments. Both were validated on interacting spinless fermions on the honeycomb lattice against quantum Monte Carlo (Wang, Corboz and Troyer, New J. Phys. 16, 103008 (2014)).

Three notions are needed to state what remains open. First, Lifshitz critical points: critical points with quadratic-or-softer dispersion $E(k)\propto k^z$ with $z\ge 2$, where space and time play very different roles. The canonical examples are generalized Rokhsar-Kivelson (RK) Hamiltonians - fine-tuned lattice models whose ground states are superpositions of classical configurations with amplitudes given by classical Boltzmann weights - whose criticality embeds a 2D classical critical problem in a (2+1)D quantum system. Finite-$D$ iPEPS with algebraic correlations are known at such points, but they are fine-tuned, and whether a variational optimization can find them is unknown. Second, the operator content: the set of scaling operators of a conformal field theory and their scaling dimensions, which is known from boundary conformal field theory to depend on the geometry and boundary conditions of the space on which the theory is defined; the iPEPS truncation effectively imposes such a nontrivial geometry (the towers and valleys) on the emergent continuum theory. In 1D the analogous question has a precise answer - a critical iMPS with finite $D$ realizes the conformal field theory with a conformal boundary, and this picture has recently been sharpened into an effective field theory of finite-entanglement scaling - but no 2D counterpart exists. Third, continuum limits: if a critical continuum theory is approached as a lattice discretization is refined, $D$ appears to act as a regulator (the bond dimension must increase as the lattice spacing decreases to keep ratios of physical quantities to the correlation length fixed), but the quantitative scaling law is unknown.

Problem Statement

Determine how the finite bond dimension $D$ of infinite projected entangled-pair states (iPEPS) interacts with two-dimensional quantum criticality - the two-dimensional counterpart of the one-dimensional finite-entanglement-scaling theory - comprising three linked components, all left open by the source paper:

(1) Lifshitz-point convergence: whether (and how) a variational iPEPS optimization started from a generic, e.g. random, initial state can converge to a Lifshitz critical point, i.e. one with low-energy dispersion $E(k)\propto k^z$ with $z\ge 2$ such as generalized Rokhsar-Kivelson criticality, where space and time enter the path integral asymmetrically and the argument mandating a finite $\xi_D$ at Lorentz-invariant points does not apply; establish whether such points are reachable fixed points of the iPEPS variational landscape or are avoided (e.g. the flow opens a gap or defers to a nearby gapped phase), and what role fine-tuning of the initial state or of imposed tensor symmetries plays.

(2) Operator content of the truncation geometry: characterize how the finite-$D$ truncation geometry - the induced landscape of towers and valleys with its anisotropic spatial and timelike channel structure - modifies the operator content of the corresponding continuum field theory: which scaling operators survive, how their scaling dimensions are corrected, and what the resulting corrections to the finite-correlation-length scaling of observables are; the one-dimensional case, where a finite-entanglement critical state realizes a conformal field theory with a conformal boundary, is the analogous solved reference point.

(3) $D$ as a continuum regulator: establish the quantitative role of $D$ as a regulator in continuum limits of lattice-defined critical theories: how the required $D$ must scale as the lattice spacing decreases so that universal ratios - ratios of physical quantities to the correlation length, and the ratios among them - remain invariant, and what systematic errors the finite-$D$ cutoff implies at fixed lattice spacing.

The three components are facets of one objective - a controlled theory of finite-$D$ effects at two-dimensional criticality, which is what would upgrade finite correlation length scaling from an empirically successful procedure to a quantitatively reliable one, in particular for sign-problematic models, where quantum Monte Carlo fails due to the sign problem and iPEPS is the only systematic variational tool. All three components must be addressed for the task to count as resolved; a resolution of any single component is a partial result. The methodological companion issues that the source paper listed alongside these (improved extrapolation of $\xi_D$ to the infinite-boundary-dimension limit at large $D$, and breadth of FCLS validation across models) are outside this task.

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