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Dirac-type nodal spin liquid revealed by refined quantum many-body solver using neural-network wave function, correlation ratio, and level spectroscopy

Yusuke Nomura, Masatoshi Imada

arXiv:2005.14142v6cond-mat.str-elcond-mat.dis-nncond-mat.stat-mechcond-mat.supr-conquant-ph

TL;DR

The paper tackles the unresolved nature of fractionalized excitations and the controversial intermediate phase of the square-lattice J1-J2 Heisenberg model. It combines RBM+PP wave functions with correlation ratio and level spectroscopy methods, finding a finite QSL region with gapless Dirac-type spinon signatures and dual algebraic correlations.

  • Problem

    The nature of fractionalized particles and the intermediate phase of the frustrated J1-J2 Heisenberg model remain unresolved because competing scenarios and finite-size difficulties limit reliable identification.

    Source: §I · p. 1, §II · p. 2

  • Method

    The study combines RBM+PP representations of ground and excited states with correlation ratio, level spectroscopy, and finite-size analyses.

    Source: A. Machine learning for quantum… · p. 3, B. Strategy to overcome numerical… · p. 4

  • Results

    A finite QSL region 0.49 ≲ J2/J1 ≲ 0.54 is supported by independent analyses, with gapless singlet and triplet Dirac-type excitations and coexisting algebraic Néel and dimer correlations.

    Source: §V · p. 10, §VI · p. 11

  • Takeaways & Limitations

    The QSL is characterized by fractionalized spin-1/2 spinons and a dual algebraic correlation structure linking Néel and VBS tendencies.

    Source: §V · p. 10, §VI · p. 11

Abstract

from arXiv · show

Pursuing fractionalized particles that do not bear properties of conventional measurable objects, exemplified by bare particles in the vacuum such as electrons and elementary excitations such as magnons, is a challenge in physics. Here we show that a machine-learning method for quantum many-body systems that has achieved state-of-the-art accuracy reveals the existence of a quantum spin liquid (QSL) phase in the region $0.49\lesssim J_2/J_1\lesssim0.54$ convincingly in spin-1/2 frustrated Heisenberg model with the nearest and next-nearest neighbor exchanges, $J_1$ and $J_2$, respectively, on the square lattice. This is achieved by combining with the cutting-edge computational schemes known as the correlation ratio and level spectroscopy methods to mitigate the finite-size effects. The quantitative one-to-one correspondence between the correlations in the ground state and the excitation spectra enables the reliable identification and estimation of the QSL and its nature. The spin excitation spectra containing both singlet and triplet gapless Dirac-like dispersions signal the emergence of gapless fractionalized spin-1/2 Dirac-type spinons in the distinctive QSL phase. Unexplored critical behavior with coexisting and dual power-law decays of N\'{e}el antiferromagnetic and dimer correlations is revealed. The power-law decay exponents of the two correlations differently vary with $J_2/J_1$ in the QSL phase and thus have different values except for a single point satisfying the symmetry of the two correlations. The isomorph of excitations with the cuprate $d$-wave superconductors implies a tight connection between the present QSL and superconductivity. This achievement demonstrates that the quantum-state representation using machine learning techniques, which had mostly been limited to benchmarks, is a promising tool for investigating grand challenges in quantum many-body physics.

I. INTRODUCTION

The paper addresses the unresolved nature of fractionalized excitations and the controversial intermediate phase of the frustrated square-lattice J1-J2 Heisenberg model. It combines accurate wave functions with complementary ground- and excited-state analyses to identify the QSL regime and its excitation structure.

I. INTRODUCTION

  • The QSL's fractionalized particles remain experimentally and theoretically difficult to identify because their nature is hidden and several QSL types have been proposed.Source: §I · pp. 1–2
  • The study uses RBM+PP wave functions for ground and momentum-resolved excited states, supplemented by correlation ratio and level spectroscopy analyses.Source: §I · p. 2, §II · p. 3
  • The reported QSL spans 0.49 ≲ J2/J1 ≲ 0.54, with gapless singlet and triplet excitations at four symmetric momenta and coexisting spin and dimer power-law correlations.Source: §I · p. 2

II. J1-J2 HEISENBERG MODEL ON SQUARE LATTICE

  • Competing J1 and J2 interactions favor Néel and stripe antiferromagnetism, leaving the intermediate region near J2/J1 = 0.5 controversial among QSL, VBS, and critical-point scenarios.Source: §II · p. 2
  • Reliable resolution requires a finely sampled J2 survey, high accuracy for near-degenerate states, and controlled extrapolation to the thermodynamic limit.Source: §II · p. 2

III. METHODS

The method combines RBM and pair-product wave functions to represent quantum many-body ground and excited states. Its variational energy is optimized as a machine-learning loss over a systematically improvable parameterization.

A. Machine learning for quantum many-body systems

B. Strategy to overcome numerical challenges

Machine learning alone does not resolve finite-size and excitation-spectrum challenges, so the study combines independent correlation-ratio and level-spectroscopy analyses. The correlation ratio estimates ordering transitions from structure-factor peak sharpness.

B. Strategy to overcome numerical challenges

1. Correlation ratio

  • The correlation ratio R = 1 − S(Q + δq)/S(Q) measures structure-factor peak sharpness, approaching 1 in ordered phases and 0 in disordered phases.Source: §1 · p. 4
  • Crossings of correlation-ratio curves for different system sizes provide phase-boundary estimates with weak size dependence.Source: §1 · p. 4
  • Separate ratios from spin-spin and dimer-dimer structure factors identify Néel-AF and VBS transitions at their respective ordering momenta.Source: §1 · p. 4

2. Level spectroscopy

Level spectroscopy locates transitions through crossings of low-lying excitation energies, while quantum-number projection resolves momentum and spin-parity sectors. The resulting accurate calculations support phase-diagram determination across system sizes.

2. Level spectroscopy

  • Level spectroscopy estimates transitions by extrapolating crossings of low-lying excitation energies with small system-size dependence.Source: §2 · p. 4

3. Quantum number projection

  • Quantum-number projection targets total momentum and spin parity, enabling separate optimization of singlet and triplet states.Source: §3 · pp. 4–5
  • Energy differences from the K = (0, 0), even-spin ground state yield momentum-resolved singlet and triplet excitation spectra.Source: §3 · p. 5

C. Finite-size scaling method

  • Finite-size scaling relates susceptibility and structure-factor data to critical exponents and universal scaling functions near transition points.Source: C. Finite-size scaling method · p. 5

IV. RESULTS

  • RBM+PP achieves best compared ground-state precision on 8 × 8 and 10 × 10 lattices and unprecedented excited-state accuracy, enabling extensive large-size calculations.Source: §IV · p. 5
  • The resulting phase diagram contains Néel order at small J2, stripe order at large J2, and intervening nonmagnetic regions identified using the RBM+PP calculations.Source: Figure 1 · p. 5, §IV · p. 5

A. Ground-state phase diagram

The J1-J2 model contains a finite QSL region between Néel-AF and VBS phases, consistently located by correlation-ratio crossings and level spectroscopy.

A. Ground-state phase diagram

  • The VBS-to-stripe-AF transition is first order, whereas the Néel-AF and QSL-related transitions are continuous.The first-order transition is characterized by a kink in the ground-state energy.Source: A. Ground-state phase diagram · p. 5

1. Phase boundary determined by correlation ratio

  • Correlation-ratio curves cross near J2 ≈ 0.49 for Néel correlations and J2 ≈ 0.54 for VBS correlations, supporting distinct phase boundaries.The crossings are nearly system-size independent and indicate an intervening phase without long-range ordering.Source: §1 · p. 6, Figure 2 · p. 6

A. Ground-state phase diagram

2. Phase boundary determined by level spectroscopy

  • Level spectroscopy extrapolates singlet-quintuplet and singlet-triplet crossings to J2 = 0.493(2) and 0.532(2), respectively.The crossings signal the AF-QSL and QSL-VBS transitions and quantitatively agree with the correlation-ratio estimates.Source: §2 · p. 6
  • At both level crossings, the excitation gap scales as Δ ∝ 1/L and vanishes in the thermodynamic limit.This identifies both transition points as gapless critical points.Source: §2 · p. 6

B. Excitation spectrum in quantum spin liquid phase

Within the QSL phase, singlet and triplet excitations become gapless at four high-symmetry momenta with Dirac-type dispersions. The vanishing lowest-branch spectral weight at q = (π, 0) further supports fractionalization into spinons.

B. Excitation spectrum in quantum spin liquid phase

  • Singlet and triplet excitations are both gapless at the four high-symmetry momenta in the QSL phase.The construction combines gapless singlet and triplet excitations at distinct momenta to obtain additional gapless modes.Source: B. Excitation spectrum in quantum spin… · pp. 6–7
  • The thermodynamic excitation spectra vanish only at (0, 0), (π, 0), (0, π), and (π, π), suggesting Dirac-type linear dispersion around four nodes.Gaps decrease with system size at high-symmetry points but remain nearly constant at intermediate momenta.Source: B. Excitation spectrum in quantum spin… · pp. 6–7, Figure 5 · p. 7
  • Momentum-resolved calculations at additional points support four Dirac-type nodes but do not exclude quadratic touching or tiny extended gapless regions.The conclusion is based on a limited set of sampled momenta.Source: B. Excitation spectrum in quantum spin… · pp. 7–8

C. Signature of fractionalization in quantum spin liquid

  • The dynamic spin structure factor is used to test fractionalization because vanishing lowest-branch weight indicates that spectral weight shifts to an incoherent spinon continuum.The continuum is interpreted as two-spinon particle-particle, hole-hole, or particle-hole excitations.Source: C. Signature of fractionalization in… · p. 8
  • At J2/J1 = 0.5, the lowest-branch weight at q = (π, 0) decreases rapidly toward zero with increasing system size.The spectral weight is therefore dominated by an incoherent continuum rather than a conventional magnon branch.Source: C. Signature of fractionalization in… · p. 8, Figure 6 · p. 8

D. Real-space correlation functions in the quantum spin liquid phase

In the QSL phase, spin-spin and dimer-dimer correlations both decay algebraically, while finite-size scaling yields distinct unconventional critical exponents for Néel-AF and VBS order parameters.

D. Real-space correlation functions in the quantum spin liquid phase

E. Finite-size scaling and size dependence of order parameter

  • Finite-size scaling gives z + η = 1.410(4) and ν = 1.21(5) for Néel-AF, versus z + η = 1.436(6) and ν = 0.67(2) for VBS.The analysis assumes critical points at J_Néel 2/J1 = 0.49 and J_VBS 2/J1 = 0.54.Source: E. Finite-size scaling and size… · p. 9, Figure 8 · p. 9
  • The extracted exponents do not belong to known universality classes, suggesting unconventional criticality.Conventional finite-size scaling also supports the correlation-ratio and level-spectroscopy results.Source: E. Finite-size scaling and size… · p. 9

V. DISCUSSION

The discussion consolidates evidence for a finite QSL region and relates its ground-state correlations to gapless excitation structures. It further characterizes the phase as a Z2 nodal QSL and connects its spectrum to cuprate d-wave superconductivity.

V. DISCUSSION

  • Agreement between correlation-ratio and level-spectroscopy analyses supports a finite QSL region 0.49 ≲ J2/J1 ≲ 0.54.The authors describe this quantitative consistency as evidence obtained from systematic, high-accuracy calculations and thermodynamic-limit extrapolation.Source: §V · p. 9, §VI · p. 11
  • The analysis supports a Z2 nodal QSL with gapless fractionalized spin-1/2 spinons at (±π/2, ±π/2), while not excluding U(1) QSL solely from spin spectra.The conclusion is obtained without assuming a variational wave-function form or using the cited approximations.Source: §V · p. 10
  • Singlet and triplet excitations are gapless at (0, 0), (π, 0), (0, π), and (π, π), consistently with algebraic VBS and Néel-AF correlations.The observable spin excitations are interpreted as two-spinon excitations.Source: §V · p. 10, §VI · p. 11
  • The spin-spin and dimer-dimer power-law exponents vary oppositely across the QSL region and coincide only near J2 ≈ 0.52.This behavior supports an extended Z2 QSL rather than an extended U(1)-symmetric critical phase, with U(1) symmetry inferred at a single point.Source: §V · p. 10
  • The excitation structure is isomorphic with charge and spin excitations of cuprate d-wave superconductors, suggesting a connection when carriers are doped.The authors also relate the calculated incoherent spin-excitation continuum to unresolved cuprate transport and charge-dynamics puzzles.Source: §V · p. 10

VI. SUMMARY

  • The study establishes a one-to-one correspondence between ground-state phase evolution and excitation-structure changes through level crossings.Coexisting real-space power-law correlations corroborate the corresponding gapless singlet and triplet excitations.Source: §VI · p. 11
  • Combining RBM+PP with quantum-number projection and finite-size-correction methods enabled accurate, tractable calculations on large systems.The paper presents this combination as extending machine-learning wave functions beyond benchmark problems toward difficult many-body systems.Source: §VI · p. 11

Appendix A: Methods –Detail

The RBM+PP wave function is optimized variationally with stochastic reconfiguration, structured pair-product parameters, and multiple randomized initializations to control cost and avoid bias.

Optimization of RBM+PP wave function

  • Stochastic reconfiguration minimizes the RBM+PP energy and is equivalent to imaginary-time evolution within the variational Hilbert space.This evolution stably targets the lowest-energy state in each quantum-number sector when the initial state has nonzero overlap.Source: Optimization of RBM+PP wave function · p. 12
  • A 4×4 or 6×6 sublattice structure reduces independent pair-product parameters through spatial translations for larger lattices.The 4×4 structure reduces the count to 16N_site, while onsite parameters are redundant after Gutzwiller projection.Source: Optimization of RBM+PP wave function · p. 12
  • Random initial parameters and several independent optimizations are used to avoid imposing bias and to test optimization stability across J2.The study uses at least three independent optimizations for each J2 point.Source: Optimization of RBM+PP wave function · p. 12
  • The pair-product component dominates computational cost, while the RBM component provides an efficient route to improve accuracy.The sublattice structure is introduced to reduce the computational burden for larger systems.Source: Optimization of RBM+PP wave function · p. 12

Special treatments to obtain some specific excited

The study uses symmetry projections and RBM+PP benchmarks to obtain momentum-resolved excited states and establish high accuracy for ground- and excited-state calculations.

states

  • Point-group projection separates the ground state from a distinct irreducible representation to obtain the S = 0 excited state at K = (0, 0).The ground state belongs to the A representation, while the targeted excited state belongs to the B representation under C4v.Source: states · p. 12
  • Full spin projection with mVMC is used for S = 2 states, at roughly five times the computational cost of spin-parity projection.The resulting mVMC accuracy is comparable to RBM+PP.Source: states · p. 12

Calculation conditions

Accuracy of the RBM+PP wave function

  • The RBM+PP wave function gives the best ground-state energy accuracy among compared variational methods for 8×8 and 10×10 lattices.Systematic benchmarks on 6×6, 8×8, and 10×10 lattices support the state-of-the-art accuracy claim.Source: Accuracy of the RBM+PP wave function · p. 13, Table II · p. 14
  • A sublattice structure reduces computational cost but slightly worsens accuracy; at 8×8 and J2 = 0.5, the energy difference is less than 0.1%.The structured result is −0.498460(6), compared with −0.498886(1) without the structure.Source: Accuracy of the RBM+PP wave function · pp. 13–14
  • Excitation energies for singlet, triplet, and quintuplet states agree with exact diagonalization to within 0.01 on the 6×6 lattice.Momentum resolution enables level spectroscopy and phase-boundary estimation.Source: Accuracy of the RBM+PP wave function · p. 14, Figure 11 · p. 14

Initial-parameter dependence of the RBM+PP

Optimization is stable at J2 = 0.5 but can encounter competing local minima near the first-order VBS–stripe-AF transition around J2 ≈ 0.61.

Initial-parameter dependence of the RBM+PP

  • At J2 = 0.5, four independent optimizations converge stably to the same energy, whereas at J2 = 0.6 some runs become trapped in local minima.The instability is attributed to competing same-sector states near the transition.Source: optimization · p. 15
  • The optimized wave functions are essentially identical despite possible dependence of variational parameters on initialization.This was confirmed through Monte Carlo overlap calculations.Source: optimization · p. 15
  • Several independent optimizations are required to avoid local minima and maintain smooth physical quantities across J2.The procedure is especially important near competing low-energy solutions.Source: optimization · p. 15

Ground state energy

The ground-state energy identifies a first-order VBS–stripe-AF transition near J2 ≈ 0.61, while correlation-ratio crossings estimate the neighboring AF–QSL and QSL–VBS boundaries.

Ground state energy

  • A clear energy kink develops near J2 ≈ 0.61 with increasing system size, providing evidence for a first-order VBS–stripe-AF transition.The kink is shown in the ground-state energy as a function of J2.Source: Ground state energy · p. 15

Structure factors

the AF and VBS correlation ratios

dependence in the PP part

The PP-state analysis tests momentum- and system-size dependence of excitations while using correlation-ratio crossings to locate phase boundaries. Agreement across sublattice resolutions supports Dirac-type nodal behavior, and supplementary excitation features track the AF–QSL and QSL–VBS boundaries.

dependence in the PP part

  • The 4×4 PP sublattice structure provides momentum resolution at K = (mπ/2, nπ/2), while 6×6 sampling gives K = (mπ/3, nπ/3).The comparison is performed on a 12×12 lattice.Source: dependence in the PP part · p. 16
  • Excitation gaps at high-symmetry momenta agree between 4×4 and 6×6 sublattice structures, supporting the Dirac-type nodal QSL scenario.At intermediate momenta, excitation energies remain larger than at (0, 0), (π, 0), and (π, π).Source: dependence in the PP part · p. 16, Figure 17 · p. 16
  • Correlation-ratio crossing points are compared across specified lattice-size pairs to determine the N´eel-AF and VBS phase boundaries.The crossings are plotted as a function of Lmid = (L1 + L2)/2.Source: Figure 16 · p. 16

Appendix D: Supplementary Notes

  • Near J2/J1 ≈ 0.49, a singlet excitation kink at K = (π, π) reflects a level crossing and a change in the lowest state's point-group irreducible representation.These features occur around the AF-QSL boundary.Source: Appendix D · p. 16
  • Near J2/J1 ≈ 0.54, the triplet excitation at K = (π, 0) turns upward with increasing J2, consistent with a triplet gap in the VBS phase.The observation is reported for the 16×16 lattice and is suggestive of a phase-transition connection.Source: Appendix D · p. 17
  • The S = 0, K = (0, 0) excitation remains nearly size-independent because the method accesses a different irreducible representation from the ground state.The corresponding same-representation excited states may behave like those at other high-symmetry momenta.Source: Appendix D · p. 17
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