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Perturbation Theory by Flow Equations: Dimerized and Frustrated S=1/2 Chain

Christian Knetter, Goetz S. Uhrig

arXiv:cond-mat/9906243v1cond-mat.str-elcond-mat.stat-mech

TL;DR

The paper develops a perturbative approach for constructing effective Hamiltonians that conserve energy-quanta number. It applies the approach to dimerized, frustrated S = 1/2 chains, reporting ground-state and magnon results as polynomials that can be used in data fits.

  • Problem

    The paper seeks a computer-implementable perturbation scheme for problems with a suitable unperturbed spectrum and perturbations that change the energy-quanta number by a bounded amount.

    Source: §I · p. 1

  • Method

    A flow-equation-based continuous unitary transformation maps the problem to an effective Hamiltonian conserving energy-quanta number, illustrated using dimerized, frustrated chains.

    Source: §V · p. 13

  • Results

    The spin-chain results include ground-state energies and magnon dispersions, given as polynomials in frustration for use in data fits.

    Source: §V · p. 13

  • Takeaways & Limitations

    The scheme separates sectors with different excitation numbers and provides reusable polynomial results for analyzing spin-chain data.

    Source: §V · p. 13

Abstract

from arXiv · show

The flow equation method (Wegner 1994) is used as continuous unitary transformation to construct perturbatively effective Hamiltonians. The method is illustrated in detail for dimerized and frustrated antiferromagnetic S=1/2 chains. The effective Hamiltonians conserve the number of elementary excitations which are S=1 magnons for the dimerized chains. The sectors of different number of excitations are clearly separated. Easy-to-use results for the gap, the dispersion and the ground state energies of the chains are provided.

I. INTRODUCTION

The paper proposes a computer-implementable perturbation scheme that uses flow equations to map suitable problems to effective Hamiltonians conserving the number of energy quanta. It develops the method for dimerized, frustrated spin chains, where triplets are the energy quanta.

I. INTRODUCTION

  • The method separates into a general mapping to a number-conserving effective Hamiltonian and a model-specific calculation of that Hamiltonian.Both steps can be implemented directly on a computer.Source: §I · p. 1
  • The scheme requires an unperturbed equidistant spectrum labeled by energy quanta and perturbations that change their number by at most a finite N.The paper specializes to N = 2.Source: §I · p. 1
  • For dimerized chains, the isolated-dimer limit supplies the equidistant spectrum: singlet dimers form the ground state and local triplets are the energy quanta.The weak inter-dimer couplings can create or annihilate at most two triplets.Source: §I · pp. 1–2

II. PERTURBATION BY FLOW EQUATIONS

  • The flow evolves the bare Hamiltonian continuously into an effective Hamiltonian whose surviving terms conserve energy-quanta number.In the chain example, this separates sectors with different triplet counts.Source: §I · p. 1, A. General Formalism · pp. 2, 4

A. General Formalism

  • A modified infinitesimal generator preserves the finite band-block structure during the flow, unlike the naive use of Wegner’s generator described here.For the illustrated problem, the band width remains N = 2.Source: A. General Formalism · pp. 2–3
  • The recursive coefficient relations can be implemented symbolically, yielding polynomial results that are readily used in data fits.Only terms conserving energy-quanta number survive in the effective Hamiltonian.Source: A. General Formalism · p. 4

B. Computer Aided Evaluation

The coefficient calculation is implemented as a recursive symbolic computation over perturbation order and operator sequences. Its memory requirement grows exponentially with order.

B. Computer Aided Evaluation

  • Exact rational coefficients are retained, with simplification and sorting used to combine like terms and limit the number of addends.The reported sorting procedure has computation time of order n ln(n) in the number of addends n.Source: B. Computer Aided Evaluation · p. 4
  • The coefficient recursion is evaluated by multiplying, adding, and integrating expressions represented as sums of terms in the flow parameter.The implementation enumerates operator sequences and their nontrivial breakups order by order.Source: B. Computer Aided Evaluation · p. 4

DIMERIZED AND FRUSTRATED S = 1/2 CHAIN

The method is applied to dimerized, frustrated S = 1/2 chains by constructing the triplet-changing operators and evaluating the effective Hamiltonian in the zero- and one-triplet sectors. It yields ground-state and magnon-dispersion results, while the two-triplet sector is left for future calculation.

A. General Equations

  • The dimerized-chain calculation constructs the operators T_n by sorting local matrix elements according to their net change in triplet number.Frustration is incorporated using the different spin-exchange behavior of singlets and triplets.Source: A. General Equations · p. 5
  • The ground-state energy is evaluated in the product of singlets, while the one-magnon calculation uses the sector containing exactly one triplet.Because triplet number is conserved, the triplet can only be shifted by the effective Hamiltonian.Source: A. General Equations · p. 6
  • The real-space construction does not require eigenstates to be known in advance and does not require translational invariance.Translational invariance makes the hopping coefficients independent of dimer position; a Fourier transform then gives the dispersion.Source: A. General Equations · p. 6
  • The paper leaves the two-triplet effective Hamiltonian uncomputed, though its matrix can be constructed and diagonalized to obtain properties such as bound states.The proposed route is to determine the two-triplet matrix coefficients and diagonalize the resulting matrix.Source: A. General Equations · p. 6
  • The same unitary transformation can be applied to observables to obtain their transformed matrix elements, although this calculation is more laborious than transforming the Hamiltonian.The authors report preliminary studies indicating that the perturbative treatment is feasible.Source: A. General Equations · p. 6

B. Computer Aided Evaluation

The computer-aided evaluation encodes dimer-chain states as bit patterns, applies the operators T_i to them, and combines the resulting states and polynomial prefactors. The procedure yields ground-state energy and dispersion coefficients through tenth order.

B. Computer Aided Evaluation

  • Two bits encode each dimer, so four bytes represent a 32-site chain state.The states are represented by basic data elements (BDEs).Source: B. Computer Aided Evaluation · p. 6
  • For each adjacent-dimer pair, the algorithm masks the state, dispatches on the extracted four-bit value to apply T_i, and multiplies the resulting prefactor.The pair is addressed with periodic boundary conditions.Source: B. Computer Aided Evaluation · p. 7
  • After applying T_i, identical chain states are sorted together and their prefactors added, reducing memory use.Linear combinations are stored as lists of states encoded by unsigned integers.Source: B. Computer Aided Evaluation · p. 7
  • Products of T_i are evaluated by repeating the operator procedure on each term in the current linear combination.Source: B. Computer Aided Evaluation · p. 7
  • For the ground-state energy, the coefficient-weighted operators act on the triplet vacuum; M(m)=0 terms reproduce it with polynomial prefactors, which Maple sums.The reported ground-state energy per spin is expanded through tenth order in λ.Source: B. Computer Aided Evaluation · p. 7
  • Dispersion hopping coefficients were calculated on suitable chain segments through tenth order in λ; runs took 10–30 hours and used 100–500 MB on a Sun Ultra.The operator terms shift a triplet by at most |m| dimers.Source: B. Computer Aided Evaluation · p. 7

IV. RESULTS AND COMPARISON TO OTHER METHODS

For results on Hamiltonian (3), the authors substitute λ according to Eq. (5) before further manipulation, while keeping frustration α fixed until the end.

IV. RESULTS AND COMPARISON TO OTHER METHODS

  • The results are reformulated for Hamiltonian (3) by substituting according to Eq. (5).Source: §IV · p. 7
  • Because λ is the expansion parameter, it is substituted before further manipulation of the equations.Source: §IV · p. 7
  • Frustration α is held fixed during the further manipulations and substituted only at the end.Source: §IV · p. 8

A. Ground State Energy

This section reports ground-state energy and gap calculations for dimerized and frustrated chains, comparing perturbative results with numerical data. The energy estimates agree closely across much of the studied range, while gap extrapolations remain difficult near critical frustration.

A. Ground State Energy

  • For finite frustration, the ground-state energy analysis describes a crossover from δ^(4/3) behavior below critical frustration to linear behavior above it.The reported finite-frustration dimerization range is δ∈[0,0.2].Source: A. Ground State Energy · pp. 8–9
  • The ground-state energy extrapolation agrees excellently with DMRG, and the continuum-biased estimates are quantitatively reliable down to 1–2%.The continuum-biased extrapolation improves on the plain series.Source: A. Ground State Energy · p. 8
  • At α0=0.241 and 0.35, the continuum-informed energy extrapolation agrees excellently with DMRG down to very low dimerization.The extrapolation is designed to capture the crossover from δ^(4/3) to linear behavior.Source: A. Ground State Energy · pp. 8–9

B. Energy Gap

  • Biased gap extrapolations agree extremely precisely with DMRG for most parameter choices.The solid curves incorporate continuum predictions for the gap’s dimerization dependence.Source: B. Energy Gap · pp. 9–10
  • Without frustration, the Dlog Padé estimate is λc=1.002 and γ=0.74, close to the predicted critical position but less accurate for the exponent.Setting λc=1 in advance does not significantly change the exponent estimate.Source: B. Energy Gap · p. 11
  • A biased Dlog Padé analysis estimates γ=0.68 and h=5.65 at zero frustration, and γ=0.56 and h=2.50 at critical frustration.The approximation uses a polynomial in δ² and estimates γ from its value at zero.Source: B. Energy Gap · p. 11

C. Dispersions

The dispersion section develops extrapolation procedures for dimerized and frustrated chains and explains how the spin excitations differ across parameter regimes. The conclusions present a flow-equation scheme that conserves excitation number and provides usable results for spin chains.

C. Dispersions

  • The dispersion procedure starts from the plain series, expands its difference from a Lorentz-inspired square-root form about δ = 1, then applies the quadratic mean.Biased extrapolations are used to improve gap estimates; the Taylor expansion is carried to tenth order with α fixed.Source: C. Dispersions · p. 11
  • At δ = α0 = 0, the quadratic mean changes only low-energy dispersion; for δ ≥0.05, its difference from the plain series is no longer discernible.The results are compared with the rigorous Bethe-ansatz result.Source: C. Dispersions · p. 11, Figure 10 · p. 11, Figure 11 · p. 12
  • The Padé interpolation combines the square-root form’s accurate gap with improved agreement near the dispersion maximum at α0 = 0.5.The square-root form alone reproduces the gap but worsens agreement at the maximum; the Taylor form is inaccurate at low wave vectors.Source: C. Dispersions · p. 12, Figure 14 · p. 12
  • For α0 ≤0.4, the Padé representation reproduces the Fig. 13 results within 1%, while Taylor expansion performs equally well.In this regime, the more involved Padé procedure is unnecessary.Source: C. Dispersions · p. 12, Figure 13 · p. 12
  • The dispersion near k = 0 can mark the lower edge of a two-spinon continuum rather than a well-defined magnon branch as δ → 0 at strong frustration.At α0 = 1/2, the low-lying excitations are asymptotically free spinons; near k = π, bound triplets help explain why a local-triplet expansion works more easily.Source: C. Dispersions · p. 13

V. CONCLUSIONS

  • The flow-equation transformation generalizes Stein’s N = 1 scheme to finite N and maps the problem to an effective Hamiltonian that conserves energy-quanta number.This separates Hilbert-space sectors and can be useful when perturbed-state structure is unknown; symbolic polynomial results distinguish the approach from conventional implementations.Source: §V · p. 13
  • The general coefficients are provided through order 6 in print, with higher orders available electronically; the approach is also proposed for higher-dimensional systems and two-magnon spectra.The authors note existing application to (VO)2P2O7 and describe further investigations and observable-based spectral functions as prospects.Source: §V · p. 13
  • For dimerized frustrated S = 1/2 chains, weak inter-dimer couplings change triplet number by at most two, and the transformed Hamiltonian conserves it.The paper reports ground-state and one-magnon results as polynomials in frustration, usable for fits; results are reliable to about 6% without extra information and about 2% with information such as the critical exponent.Source: §V · p. 13

APPENDIX A: PROOF OF BLOCK DIAGONALITY

The appendix proves that the chosen infinitesimal generator yields a block-diagonal effective Hamiltonian under stated conditions. It also specifies the boundedness and gap-continuity boundaries relevant to applying the expansion.

APPENDIX A: PROOF OF BLOCK DIAGONALITY

  • The generator argument requires the whole Hamiltonian to be bounded; without that condition, this choice may be problematic even though Wegner’s choice still works.For finite-order expansions with short-range interactions, a finite cluster suffices and gives a finite-dimensional, bounded Hamiltonian.Source: Appendix A · pp. 14–15
  • At infinite flow, matrix elements between nondegenerate eigenstates of H0 vanish, so the effective Hamiltonian is block diagonal and conserves H0 energy-quanta number.The proof uses the choice of infinitesimal generator in Eq. (12).Source: Appendix A · pp. 14–15
  • For dimerized chains, the expansion applies while the gap stays open along a path from the dimer limit; gapless cases are accessible only as limits of such gapped systems.Source: Appendix A · p. 15

APPENDIX B: COEFFICIENTS OF THE EFFECTIVE HAMILTONIAN

The appendix tabulates coefficients of the effective Hamiltonian through sixth order and identifies the availability of higher-order coefficients.

APPENDIX B: COEFFICIENTS OF THE EFFECTIVE HAMILTONIAN

  • Coefficients C(m) of effective Hamiltonian (25) are given through order 6, inclusive.Source: Appendix B · p. 16
  • Orders 7 through 10 are stated to be available electronically.Source: Appendix B · p. 16
  • The tabulated quantities are the C(m) coefficients of the effective Hamiltonian identified as Eq. (25).Source: Appendix B · p. 16

APPENDIX C: EFFECTIVE HOPPING ELEMENTS FOR THE SPIN CHAIN

The appendix presents effective hopping elements for the spin chain through sixth order and states how higher-order elements will be supplied.

APPENDIX C: EFFECTIVE HOPPING ELEMENTS FOR THE SPIN CHAIN

  • Effective hopping elements appearing in Eq. (38) are given through order 6.Source: Appendix C · p. 18
  • Effective hopping elements through order 10 will be provided electronically.Source: Appendix C · p. 18
  • The appendix’s hopping elements are those used in Eq. (38).Source: Appendix C · p. 18
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