Parameter selection for real-space renormalization-group transformations in finite-lattice approximations

Real-space renormalization-group (RG) transformations of lattice spin systems carry free parameters (block-spin weight parameters, scale conventions). Bell and Wilson (Phys. Rev. B 11, 3431 (1975)) showed, for a solvable linear RG on the three-dimensional Gaussian model, that the exact transformation admits nontrivial fixed points only at $b = 2^{-5/2}$, whereas its finite-lattice approximation admits fixed points for a range of $b$, and the eigenvalues of the linearized transformation — which determine the computed critical exponents — depend on the value chosen. Of the selection criteria they tested, the exact magnetic-eigenvalue relation $Λ_h = b^{-1}$ was found to hold identically for every admissible $b$ under periodic boundary conditions and so carries no information, while the appearance of a nearly marginal eigenoperator (a vestige of the infinite-lattice line of fixed points) succeeded for the Gaussian model, but only at lattice size $N = 10$ and not at $N = 6$. Successor literature proposed magnetic-eigenvalue matching, variational bound optimization, group-structure matching conditions, Monte Carlo RG with empirically tuned transformations, and information-theoretic (machine-learned) transformations, each with a restricted domain of success; parameter tuning is still performed empirically and per observable. The open problem, in the sense of Bell and Wilson, is a criterion for identifying the correct values of the transformation parameters in a finite-lattice (truncated) approximation to a real-space RG transformation — a rule, applicable within the truncated computation itself, that selects the parameter values for which the computed critical exponents are correct.

Background

Near a continuous phase transition, the correlation length of a statistical system diverges, and the renormalization group (RG) is the standard framework for universality: one repeatedly coarsens the microscopic model and tracks how its effective couplings flow. In real-space (position-space) RG the coarsening is performed directly on the lattice by a block-spin transformation: the spins on a block of sites (a $2×2×2$ cube in the simplest case) are combined into a single block variable, and the Hamiltonian is rewritten as a function of the block variables [Niemeijer and van Leeuwen 1973; Kadanoff and Houghton 1975]. Iteration defines a map on the space of Hamiltonians whose fixed points (Hamiltonians mapped to themselves) govern the critical behavior. Linearizing the map at a fixed point yields eigenoperators with eigenvalues $Λ$; the thermal eigenvalue $Λ_t$ (the eigenvalue of the temperature-like perturbation) and the magnetic eigenvalue $Λ_h$ (that of the field-like perturbation) determine the critical exponents through $ν = \ln 2/\ln Λ_t$ and $η = d + 2 − 2\ln Λ_h/\ln 2$ for spatial dimension $d$ and a linear block-size factor of 2, where $ν$ controls the divergence of the correlation length and $η$ the algebraic decay of the critical spin-spin correlation function. An eigenoperator with eigenvalue exactly 1 is called marginal: perturbations along it neither grow nor shrink under iteration, and in exactly solvable cases such operators typically parameterize a line of fixed points.

Bell and Wilson studied this structure on the Gaussian model of Berlin and Kac (1952), an exactly solvable lattice model with real-valued spins and a quadratic (Gaussian) Hamiltonian. For this model the RG map acts linearly on the coupling space, so its fixed points and eigenvalues can be computed essentially exactly. They introduced the finite-lattice approximation, which is the paper's central object: the RG equations are truncated to the interactions that can be represented on a finite $N×N×N$ lattice with periodic boundary conditions, exploiting that sufficiently short-range (local) interactions decouple distant regions of the lattice. The approximation becomes exact as $N→∞$, but at finite $N$ the linear map is represented by a finite matrix and the fixed-point problem becomes finite-dimensional — and its solutions change character.

Bell and Wilson analyzed a two-parameter family $T_{a,b}$ of linear RG transformations for the three-dimensional Gaussian model, where $a$ is the width of the block-spin weight and $b$ is the transformation parameter. In the exact infinite-lattice theory, nontrivial fixed points exist only for $b = b^* = 2^{-5/2}$ (in general $b = 2^{-(d+2+ε)/2}$, where $ε$ measures the long-distance tail of the block-spin weight); at this value there is a line of fixed points, the thermal eigenvalue is exactly 4 (so $ν = 1/2$), the marginal eigenvalue is exactly 1, and $Λ_h = 2^{5/2}$ (so $η = 0$), consistent with the exact relation $Λ_h = b^{-1}$ that follows from linearity. On the finite lattice with periodic boundary conditions, by contrast, fixed points exist for a whole range of $b$ near $b^*$, and the computed eigenvalues vary continuously with the chosen $b$ — so the finite-lattice calculation of critical exponents requires a rule for choosing $b$.

Bell and Wilson tested four candidate criteria. (1) Requiring that fixed points exist only at the correct $b$ fails: in the finite-lattice equations fixed points persist over a range of $b$. (2) Requiring locality of the fixed-point Hamiltonian is qualitative and, when made quantitative, arbitrary. (3) Subbarao's criterion — that the exact relation $Λ_h = b^{-1}$ hold — fails decisively: the relation is satisfied for every $b$ that admits a nontrivial fixed point of the finite-lattice equations, a consequence of the periodic boundary conditions, and therefore completely fails to distinguish one value of $b$ from another. (4) Requiring that a nearly marginal eigenoperator (eigenvalue near 1) appear in the truncated spectrum — a vestige of the infinite-lattice line of fixed points — was 'eminently successful for the Gaussian model': at $N = 6$ (four independent interactions) no marginal eigenoperator appears near the correct $b$, but at $N = 10$ (ten independent interactions) the fixed point first appears at exactly the correct $b$, the computed spectrum contains an exactly marginal eigenvalue 1.00000 alongside the thermal eigenvalue 4.00000, and 'the appearance of the marginal eigenoperator selects $b$', and therewith $Λ_t$ and $Λ_h$.

Two further features of their discussion frame the open problem. First, for nonlinear RG transformations — the type used directly for the Ising model, whose parameters enter only implicitly — the exact linear relations need not hold; Bell and Wilson proposed choosing parameters to minimize the dependence of the thermal eigenvalue on the parameters, or better on the fixed point, while noting such criteria need lattice sizes of the same order as those required for a marginal eigenoperator to appear. Second, the magnetic-eigenvalue-matching criterion introduced by Kadanoff and Houghton (1975) for the two-dimensional Ising model, in which two independent evaluations of $Λ_h$ are required to agree, was shown by Bell and Wilson to fail for a linear RG in the same way as Subbarao's criterion. Later frameworks in which the same choice-of-transformation problem recurs include the Monte Carlo renormalization group (MCRG — Swendsen's replacement of the exact truncated RG map by Monte Carlo estimates of block correlation functions) and modern coarse-graining schemes whose transformations are optimized by variational, information-theoretic, or machine-learning objectives.

Problem Statement

Establish a general criterion for selecting the values of the free transformation parameters of a truncated real-space renormalization-group (RG) transformation.

Setting. A real-space RG transformation $T$ with parameters $θ$ maps the space of Hamiltonians of a lattice spin system to itself (block-spin coarsening); linearization of $T_θ$ at a fixed point yields the thermal and magnetic eigenvalues from which the critical exponents are computed. In the exact infinite-lattice theory the parameters are pinned: for the Bell–Wilson linear RG on the three-dimensional Gaussian model, nontrivial fixed points exist only at $b = 2^{-5/2}$, where a line of fixed points produces an exactly marginal eigenoperator, and the exact eigenvalues are 4 (thermal), 1 (marginal), and $2^{5/2}$ (magnetic). In any truncation to a finite lattice (a finite retained interaction set on an $N×N×N$ lattice with chosen boundary conditions), fixed points exist for a range of parameter values, and the computed exponents depend on the value chosen.

Required result. A criterion that, given the model, the transformation family, the truncation (retained interaction set and lattice size), and the boundary conditions, determines the parameter values — or determines that none are admissible — such that the eigenvalues of the linearized transformation at the selected fixed point reproduce the true critical exponents of the model. The criterion must return a determination for an arbitrary such input, not only for the exemplar below, and the negative branch — concluding that no parameter values are admissible — is a substantive outcome the criterion must establish for the given input, not a default that discharges the task. The criterion is to be applicable within the truncated computation itself; the known exact and high-precision exponent values serve as the benchmarks against which a proposed criterion is tested, not as inputs to it.

The Gaussian-model case, in the truncation that retains the ten independent interactions on the $N = 10$ lattice with periodic boundary conditions, where the correct value $b^* = 2^{-5/2}$ and the exact eigenvalues are known, is a necessary check that any criterion must pass. It is not sufficient: a criterion that restates the marginal-eigenoperator selection observed in this one truncation reproduces the exemplar but supplies no general rule for other models or truncations, and therefore does not by itself satisfy the required result.

References

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