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Entropic Renormalization of Crossing Layers in a One-Dimensional Block-Spin Chain

S. Pan

Published Apr 22, 2026• 0 citations
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Abstract

We study finite-temperature corrections to the crossing-layer theory of a one-dimensional interacting block-spin chain. Previous static work shows that the minimal phase-crossing energy scales as K*sqrt(J) and that any optimal crossing layer has width of order sqrt(J). The present paper identifies the corresponding entropic correction at the block level. We show that the dominant entropy of a mesoscopic crossing layer is not of order sqrt(J)*log(K), but of order K*sqrt(J), because each block magnetization carries an internal binomial multiplicity. This leads naturally to an effective free-energy density V_{beta,K}(m) = U(m) - beta^{-1}s_K(m), where s_K(m) is the block entropy density. We formulate the corresponding free-energy crossing problem, derive the renormalized central-band gap a_{beta,delta,K}, and obtain the associated free-energy tradeoff and mesoscopic layer scale. The paper should be read as a finite-temperature renormalization of the static crossing-layer theory rather than a complete dynamical slowdown theorem. Related earlier works by S. Pan are available at DOI: 10.5281/zenodo.19673404 and DOI: 10.5281/zenodo.19689210.

DOI
10.5281/zenodo.19691920

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We study finite-temperature corrections to the crossing-layer theory of a one-dimensional interacting block-spin chain. Previous static work shows that the minimal phase-crossing energy scales as K*sqrt(J) and that any optimal crossing layer has width of order sqrt(J). The present paper identifies the corresponding entropic correction at the block level. We show that the dominant entropy of a mesoscopic crossing layer is not of order sqrt(J)log(K), but of order Ksqrt(J), because each block magnetization carries an internal binomial multiplicity. This leads naturally to an effective free-energy density V_{beta,K}(m) = U(m) - beta^{-1}s_K(m), where s_K(m) is the block entropy density. We formulate the corresponding free-energy crossing problem, derive the renormalized central-band gap a_{beta,delta,K}, and obtain the associated free-energy tradeoff and mesoscopic layer scale. The paper should be read as a finite-temperature renormalization of the static crossing-layer theory rather than a complete dynamical slowdown theorem. Related earlier works by S. Pan are available at DOI: 10.5281/zenodo.19673404 and DOI: 10.5281/zenodo.19689210.
We study the critical behavior and the ground-state entanglement of a large class of su(1|1) supersymmetric spin chains with a general (not necessarily monotonic) dispersion relation. We show that this class includes several relevant models, with both short- and long-range interactions of a simple form. We determine the low temperature behavior of the free energy per spin, and deduce that the models considered have a critical phase in the same universality class as a (1+1)-dimensional conformal field theory (CFT) with central charge equal to the number of connected components of the Fermi sea. We also study the Rényi entanglement entropy of the ground state, deriving its asymptotic behavior as the block size tends to infinity. In particular, we show that this entropy exhibits the logarithmic growth characteristic of (1+1)-dimensional CFTs and one-dimensional (fermionic) critical lattice models, with a central charge consistent with the low-temperature behavior of the free energy. Our results confirm the widely believed conjecture that the critical behavior of fermionic lattice models is completely determined by the topology of their Fermi surface.
We show that the leading finite-size correction to lnZ for a two-dimensional system at a conformally invariant critical point on a strip of length L width \ensuremath{\beta}(\ensuremath{\beta}\ensuremath{\ll}L) is (\ensuremath{\pi}/6)c(L/\ensuremath{\beta}), where c is the conformal anomaly. Equivalently, the leading low-temperature correction to the free energy of a one-dimensional quantum system is -(\ensuremath{\pi}/6)cL(kT)2{)}^{2})2\ensuremath{\Elzxh}v, where v is the effective ``velocity of light.'' The latter formula is used to check recently derived critical theories of spin-s quantum chains against Bethe-Ansatz solutions.
We consider the Rényi entropies in the one-dimensional spin-1/2 Heisenberg XX chain in a magnetic field. The case n = 1 corresponds to the von Neumann ‘entanglement’ entropy. Using a combination of methods based on the generalized Fisher–Hartwig conjecture and a recurrence relation connected to the Painlevé VI differential equation we obtain the asymptotic behaviour, accurate to order , of the Rényi entropies for large block lengths . For n = 1, 2, 3, 10 this constitutes the 3, 6, 10, 48 leading terms respectively. The o(1) contributions are found to exhibit a rich structure of oscillatory behaviour, which we analyse in some detail both for finite n and in the limit .
Combination of the finite-temperature density-matrix renormalization group and the maximum entropy presents a new method to calculate dynamic quantities of one-dimensional many-body systems. In the present paper, density of states, local dynamic spin, and charge correlation functions of the one-dimensional Kondo insulator are discussed. Excitation gaps open with decreasing temperature and the gaps take different values depending on channels. The excitation spectra change qualitatively around the characteristic temperature corresponding to the spin gap which is the lowest energy scale of the Kondo insulator.
We study single-spin-flip switching paths in a one-dimensional interacting block-spin chain. Previous static work shows that the minimal phase-crossing energy scales as K*sqrt(J) and that any optimal crossing layer has width of order sqrt(J). The missing step toward metastable slowdown is dynamical: one must show that near-optimal switching paths cannot avoid such an extended layer. In this paper we isolate that bridge. We prove that any minus-to-plus path whose maximal energy remains on the near-optimal K*sqrt(J) scale must contain an intermediate configuration with a transition corridor of width at least c*sqrt(J). Thus the static mesoscopic layer is not merely an optimizer of a boundary-value problem: it is dynamically unavoidable along low-barrier switching paths. The present result is a mechanism theorem rather than a full mixing-time theorem, and it prepares the corridor-shell bottleneck analysis needed for later conductance and spectral-gap estimates. Related earlier works by S. Pan are available at DOI: 10.5281/zenodo.19673404 and DOI: 10.5281/zenodo.19689210.
We study the von Neumann block entropy in the Kondo necklace model for different anisotropies $\ensuremath{\eta}$ in the XY\mathit{XY}XY interaction between conduction spins using the density matrix renormalization group method. It was found that the block entropy presents a maximum for each $\ensuremath{\eta}$ considered, and, comparing it with the results of the quantum criticality of the model based on the behavior of the energy gap, we observe that the maximum block entropy occurs at the quantum critical point between an antiferromagnetic and a Kondo singlet state, so this measure of entanglement is useful for giving information about where a quantum phase transition occurs in this model. We observe that the block entropy also presents a maximum at the quantum critical points that are obtained when an anisotropy $\ensuremath{\Delta}$ is included in the Kondo exchange between localized and conduction spins; when $\ensuremath{\Delta}$ diminishes for a fixed value of $\ensuremath{\eta}$, the critical point increases, favoring the antiferromagnetic phase.