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Critical behavior of quantum magnets with long-range interactions in the thermodynamic limit
S. Fey, K. P. Schmidt
TL;DR
The paper investigates critical behavior in the one-dimensional transverse-field Ising model with algebraically decaying long-range interactions. It uses gap-series extrapolation and scaling, finding that antiferromagnetic coefficients converge faster than ferromagnetic ones and better match Wynn extrapolations.
Problem
The paper investigates the critical behavior of the one-dimensional transverse-field Ising model with algebraically decaying long-range interactions.
Source: White-graph expansion of the… · p. 5Method
The authors extrapolate finite-cutoff gap-series sums with Wynn’s method and scaling, and apply Dlog-Padé extrapolation to the gap series.
Source: Wynn algorithm · p. 7, Scaling · p. 7, Extrapolation of high-order series · p. 8Results
Antiferromagnetic prefactors converge faster than ferromagnetic ones and agree better with Wynn extrapolations.
Source: Wynn extrapolation and scaling analysis · p. 8Takeaways & Limitations
Differences between the two extrapolation and scaling schemes affect the final critical values and exponents only marginally.
Source: Wynn extrapolation and scaling analysis · p. 8The scaling argument assumes that nested conditions in the sums do not alter the scaling behavior.
Source: ferromagnetic case · p. 7
Abstract
from arXiv · showhide
Quasiparticle properties of quantum magnets with long-range interactions are investigated by high-order linked-cluster expansions in the thermodynamic limit. It is established that perturbative continuous unitary transformations on white graphs are a promising and flexible approach to treat long-range interactions in quantum many-body systems. We exemplify this scheme for the one-dimensional transverse-field Ising chain with long-range interactions. For this model the elementary Quasiparticle gap is determined allowing to access the quantum-critical regime including critical exponents and multiplicative logarithmic corrections for the ferro- and antiferromagnetic case.
White-graph expansion of the long-range TFIM
The section applies perturbative continuous unitary transformations to the long-range transverse-field Ising chain to derive its quasiparticle gap in the thermodynamic limit. White-graph expansion handles the long-range interactions by embedding graph contributions into the infinite lattice.
White-graph expansion of the long-range TFIM
- pCUTs yield a quasiparticle-number-conserving effective Hamiltonian whose one-quasiparticle sector is a hopping model with amplitudes derived to high perturbative orders.The gap is the minimum of the resulting dispersion, at q = 0 for ferromagnetic and q = π for antiferromagnetic interactions.Source: White-graph expansion of the… · p. 5
- The quasiparticle gap ∆ was calculated as a series in perturbation parameter λ through order 8; the order-8 term sums contributions from 358 graphs in total.Source: White-graph expansion of the… · p. 5
- White-graph calculations assign independent couplings to graph links, then embed graph contributions by summing over their lattice realizations.Long-range interactions allow infinitely many embeddings; graph symmetry factors correct overcounting.Source: White-graph expansion of the… · pp. 5–6
- At q = 0 and q = π, the illustrated embedding sums can be evaluated analytically as products of Riemann zeta functions.These momenta correspond to the ferromagnetic and antiferromagnetic gap minima, respectively.Source: White-graph expansion of the… · pp. 5–6
Extrapolation of data sequences
The gap-series sums are truncated at finite limits and extrapolated to the infinite-sum limit. Ferromagnetic and antiferromagnetic sequences have distinct convergence behavior.
Extrapolation of data sequences
- The infinite sums are truncated at N and the gap coefficients are extrapolated using the Wynn algorithm and scaling in 1/N.The authors extrapolate the coefficients to limit the number of extrapolations needed to obtain the gap.Source: Extrapolation of data sequences · pp. 6–7
- Antiferromagnetic coefficients converge faster with increasing cutoff N than ferromagnetic coefficients, whose sequences converge monotonically only for sufficiently large N.The antiferromagnetic sequences alternate about the infinite-limit value, and the two cases require different scaling behavior.Source: Extrapolation of data sequences · p. 7
Wynn algorithm
Wynn’s epsilon method accelerates slowly converging series. For antiferromagnetic partial sums, the authors first select every second point to form a monotonically converging sequence.
Wynn algorithm
- For antiferromagnetic interactions, only every second partial sum is used before extrapolation because the original partial sums alternate.Several Wynn extrapolations are averaged using the best-converged data points.Source: Wynn algorithm · p. 7
- Wynn’s epsilon method accelerates slowly converging series, using partial sums as its initial values.The recurrence is initialized with ϵ0(Sn) = Sn and ϵ−1(Sn) = 0.Source: Wynn algorithm · p. 7
Scaling
The section motivates scaling finite-cutoff gap-series sums by comparing their terms with products of Riemann zeta functions.
Scaling
- Each term in the nested sums is assumed to scale like a product of Riemann zeta functions, providing the scaling used for the numerical data sequences.The proposed scaling is derived analytically for the zeta-function products.Source: Scaling · p. 7
ferromagnetic case
For ferromagnetic interactions, the coefficients’ finite-sum corrections are treated as scaling like products of harmonic sums. This gives a common large-N exponent, independent of the number of sums.
ferromagnetic case
- A single ferromagnetic harmonic sum converges to ζ(α) as N → ∞.Its large-N correction is analyzed for α > 1 by replacing the sum with an integral.Source: ferromagnetic case · p. 7
- The shared scaling is assumed to persist when nested-sum conditions prevent direct factorization.The argument invokes rewriting nested products as unnested sums and the factorized sums from the longest chain graph.Source: ferromagnetic case · p. 7
- Products of independent harmonic sums scale with exponent 1 − α, independent of the number of sums.This scaling is used for the ferromagnetic gap coefficients.Source: ferromagnetic case · p. 7
antiferromagnetic case
For antiferromagnetic interactions, the finite sums alternate, so odd and even terms are separated before estimating their large-N behavior. Products of sums then scale with exponent −α, independent of their number.
antiferromagnetic case
- The alternating sums are split into odd and even terms, with N taken even, before estimating their large-N correction.This separation produces a monotonic sum that is approximated by an integral.Source: antiferromagnetic case · pp. 7–8
- The scaling is generalized from one sum to products of independent sums.The limiting value of a single sum is denoted ϵ(α) as N → ∞.Source: antiferromagnetic case · p. 8
- Products of antiferromagnetic sums scale with exponent −α, independent of the number of sums.The paper uses this scaling for the antiferromagnetic gap coefficients.Source: antiferromagnetic case · p. 8
Wynn extrapolation and scaling analysis
The section compares finite-cutoff scaling fits with Wynn extrapolations for high-order prefactors. Antiferromagnetic prefactors converge faster and agree better with Wynn extrapolations than their ferromagnetic counterparts.
Wynn extrapolation and scaling analysis
- The prefactors are estimated at N → ∞ by fitting the last two partial sums against 1/N^(α−1) for ferromagnets and 1/N^α for antiferromagnets.The partial sums are evaluated to an upper boundary limited by computation time.Source: Wynn extrapolation and scaling analysis · p. 8
- At small α in the ferromagnetic case, Wynn extrapolants deviate considerably from scaled results, but the resulting critical values and exponents differ only marginally.For antiferromagnetic alternating series, every second value is used to obtain a monotonically converging sequence.Source: Wynn extrapolation and scaling analysis · p. 8
- Wynn results are averaged only after they appear converged, with the chosen cutoff and spread shown in Figs. 6–9.The figure captions identify the cutoff as a vertical line and the standard deviation as a gray area.Source: Wynn extrapolation and scaling analysis · p. 8
- Antiferromagnetic prefactors converge much faster than ferromagnetic ones and agree better with Wynn extrapolations.This comparison is reported for the representative high-order prefactors shown in Figs. 6–9.Source: Wynn extrapolation and scaling analysis · p. 8
Extrapolation of high-order series
The gap series is extrapolated with Dlog-Padé methods to estimate critical points and exponents. At the ferromagnetic upper critical α = 5/3, estimating the multiplicative logarithmic correction requires fixing the critical exponent to its mean-field value.
Extrapolation of high-order series
- Dlog-Padé extrapolations of the gap series are used to estimate critical points and critical exponents.The method is presented as applicable to second-order transitions, where physical poles indicate critical values.Source: Extrapolation of high-order series · pp. 8–11
- The method begins with a power-series representation and uses its logarithmic derivative, which is expected to have a pole at λc for power-law behavior.The pole is the basis for locating the critical point.Source: Extrapolation of high-order series · pp. 8–9
- When λc is known or estimated independently, fixing it can give a better critical-exponent estimate, though the estimate is biased.The procedure defines an extrapolation using the specified critical point.Source: Extrapolation of high-order series · p. 11
- At the ferromagnetic upper critical α = 5/3, multiplicative logarithmic corrections are expected near the quantum critical point.The correction exponent p is extracted by biasing the extrapolation with θ fixed to the mean-field value 1/2.Source: Extrapolation of high-order series · p. 11