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Computational Studies of Quantum Spin Systems

Anders W. Sandvik

arXiv:1101.3281v1cond-mat.str-elhep-lat

TL;DR

These lecture notes address how computational methods can study quantum spin systems, including magnetic order, non-magnetic states, and transitions between them. They develop exact diagonalization and stochastic-series-expansion quantum Monte Carlo, then apply them to representative Heisenberg and dimerized models.

  • Problem

    Quantum spin systems rarely have exact solutions beyond one dimension, while non-magnetic states and transitions from Néel order are central problems in quantum magnetism.

  • Method

    The notes develop symmetry-based exact diagonalization and stochastic series expansion quantum Monte Carlo for ground-state and finite-temperature calculations.

  • Results

    Applications include dimerized Heisenberg models, where the ground state ranges from a singlet-product state to Néel order, and simulations identifying first-order-transition signatures.

  • Takeaways & Limitations

    These computational methods provide a framework for investigating magnetic order, non-magnetic states, and quantum phase transitions in essential quantum-magnetism models.

  • Takeaways & Limitations

    Local updating schemes develop divergent autocorrelation times near criticality, making large-system simulations difficult without cluster algorithms.

Abstract

from arXiv · show

These lecture notes introduce quantum spin systems and several computational methods for studying their ground-state and finite-temperature properties. Symmetry-breaking and critical phenomena are first discussed in the simpler setting of Monte Carlo studies of classical spin systems, to illustrate finite-size scaling at continuous and first-order phase transitions. Exact diagonalization and quantum Monte Carlo (stochastic series expansion) algorithms and their computer implementations are then discussed in detail. Applications of the methods are illustrated by results for some of the most essential models in quantum magnetism, such as the S=1/2 Heisenberg antiferromagnet in one and two dimensions, as well as extended models useful for studying quantum phase transitions between antiferromagnetic and magnetically disordered states.

5 Quantum Monte Carlo simulations and the Stochastic Series Expansion method

These notes develop quantum Monte Carlo through the stochastic series expansion representation of quantum statistical mechanics.

  • 5.1–5.2: The section covers path-integral and series-expansion formulations, including the stochastic series expansion method for the S = 1/2 Heisenberg model.The outlined topics include configuration spaces and Monte Carlo sampling procedures.

1. INTRODUCTION

The notes motivate computational studies of quantum spin systems as tools for understanding quantum many-body physics and testing analytical theories. They introduce exact diagonalization and quantum Monte Carlo, then apply them to essential spin models and quantum phase transitions.

  • 1. INTRODUCTION: Quantum spin models provide simplified settings for studying exotic many-body states and quantum phase transitions.They also help explain antiferromagnetic properties of several Mott insulators quantitatively.
  • 1. INTRODUCTION: Exact solutions are rare beyond one dimension, making unbiased computational studies essential for testing theories and continuum field descriptions.In two dimensions, analytical calculations commonly rely on approximations or assumptions lacking rigorous justification.
  • 1. INTRODUCTION: Generic quantum spin Hamiltonians pose major computational challenges, especially with frustrated interactions and strongly correlated fermions.Developing efficient numerical algorithms is therefore a central research need.
  • 1. INTRODUCTION: The notes develop exact diagonalization and stochastic-series-expansion quantum Monte Carlo, including symmetry methods, Lanczos calculations, finite-temperature studies, and implementations.Exact diagonalization is emphasized mainly for one-dimensional systems, while QMC is also applied in the T → 0 limit.
  • 1. INTRODUCTION: Applications combine computational methods with essential quantum spin models and qualitative physical descriptions, spanning established basics through ongoing exotic-transition research.The physics is presented through elementary calculations and numerical results, with references for further study.

2. QUANTUM SPIN MODELS, THEIR GROUND STATES AND QUANTUM PHASE TRANSITIONS

The notes introduce quantum spin models, their ordered and disordered ground states, and zero-temperature quantum phase transitions, emphasizing how computational methods probe these phenomena. Examples include Heisenberg, dimerized, frustrated, and J-Q systems.

  • 2. Quantum spin models: Heisenberg interactions, lattice structure, and dimensionality determine the ground states, excitations, and finite-temperature properties of quantum spin systems.Continuous SU(2) symmetry cannot break in one dimension at T ≥ 0 or in two dimensions at T > 0.
  • 2. Quantum spin models: The notes focus primarily on antiferromagnetic S = 1/2 systems, where quantum fluctuations are strongest relative to the classical S →∞ limit.Only antiferromagnetic couplings Ji j > 0 are considered.
  • 2.1–2.3: In two dimensions, the nearest-neighbor Heisenberg model has Néel order, while quantum fluctuations can produce non-magnetic spin-liquid and valence-bond-solid states.One-dimensional Heisenberg chains instead have critical correlations decaying as (−1)^r/r with logarithmic corrections.
  • 2.1.1. The Néel state and its quantum fluctuations: Linear spin-wave theory gives E0/JN = −0.65795 and ⟨ms⟩ = 0.3034 for the square-lattice S = 1/2 Heisenberg model.The sublattice magnetization is approximately 61% of the classical value, so quantum fluctuations reduce but do not destroy long-range order.
  • 2.4.1. Dimerized systems: Dimerization interpolates between a singlet-product state at J1 = 0 and Néel order at J2 = J1, creating a quantum phase-transition problem.For columnar dimers, QMC finds the Néel order vanishes at g ≈ 1.9; at g = 1, the extrapolated ⟨ms⟩ is 0.3074.

3. CLASSICAL PHASE TRANSITIONS, MONTE CARLO SIMULATIONS, AND FINITE-SIZE SCALING

The section uses classical Ising systems to introduce Monte Carlo sampling, symmetry breaking, critical behavior, and finite-size scaling before quantum calculations. Mean-field theory provides a starting point, while simulations reveal finite-size ordering, critical scaling, and algorithmic limitations near criticality.

  • Mean-field theory: Mean-field theory identifies a transition at Tc = Js, with nonzero magnetization possible below Tc.For the 2D nearest-neighbor Ising model, this gives Tc = 4J, above the exact Tc/J ≈2.269 because fluctuations are neglected.
  • Monte Carlo simulations: At low temperature, double-peaked magnetization distributions signal practical ordering in finite systems, while reversals become rare and sampling can become non-ergodic.The reversal barrier grows with system size because configurations near m ≈0 have increasingly high energy.
  • Monte Carlo simulations: Autocorrelation times control statistical precision and diverge for local updates as T →Tc and N →∞, making large near-critical simulations difficult.Cluster algorithms can largely solve this problem by flipping collectively constructed spin clusters.
  • Finite-size scaling: Finite-size scaling distinguishes phases through ⟨m2⟩: it approaches a nonzero value below Tc, decays as L−2 above Tc, and follows ⟨m2⟩∝L−2β/ν at criticality.Binder-ratio curves for different sizes intersect near Tc because leading finite-size corrections partly cancel.
  • Finite-size scaling: Finite-size estimates of critical temperatures and exponents carry systematic uncertainty from scaling corrections, and fitted error bars are not always reliable.Size-dependent features shift proportionally to L−1/ν, with a quantity-dependent prefactor.

3.4. First-order transitions

First-order transitions retain finite correlation length but develop discontinuous jumps and phase coexistence in the thermodynamic limit. Finite-size scaling and order-parameter distributions provide practical diagnostics, though weak and strong transitions pose distinct Monte Carlo challenges.

  • First-order transitions exhibit finite correlation length and discontinuous jumps in the order parameter and other observables as system size increases.These discontinuities can be studied using finite-size scaling.
  • Weak first-order transitions can resemble continuous transitions because large corrections obscure the expected leading finite-size scaling.Accessible system sizes may not clearly distinguish slowly developing discontinuities from weaker continuous-transition singularities.
  • Monte Carlo simulations of strong first-order transitions may become trapped in metastable states, producing incorrect thermal averages and hindering transition-point estimates.Tempering, parallel tempering, and non-Boltzmann extended-ensemble methods improve configuration-space exploration.
  • Phase coexistence appears as two configuration types near the transition, provided the full configuration space is sampled ergodically.The coexistence window narrows toward a single temperature as system size grows.
  • A negative Binder-cumulant window and diverging peak can diagnose first-order behavior through coexistence distributions.The idealized distribution transfers weight between a disordered Gaussian component and an ordered delta peak.
  • Frustrated Ising model: At g = 0.55, the specific-heat peak follows an apparent exponent near 1.2 up to L ≈64 before tending toward the expected Cmax ∼L2 scaling.At g = 0.51, the expected L2 scaling is already visible for the largest lattices.

4. EXACT DIAGONALIZATION METHODS

Exact diagonalization provides complete finite-system eigenstate information for static and dynamic observables, but its exponential Hilbert-space growth restricts accessible lattices. Symmetry-based block diagonalization reduces computational cost and classifies excitations.

  • Exact diagonalization yields all eigenstates of a finite quantum spin system, allowing static and dynamic quantities to be computed.For S = 1/2, the basis contains 2^N states, limiting practical studies to a few tens of spins.
  • Symmetries convert the Hamiltonian into independently diagonalizable blocks labeled by conserved quantum numbers.Magnetization and crystal momentum are examples that reduce computational effort and classify excitations.
  • Symmetry implementation involves a trade-off: magnetization conservation is relatively easy, whereas momentum and total-spin conservation require more complicated basis constructions.Some symmetries are omitted when implementation costs outweigh computational benefits.
  • The methods are developed using the S = 1/2 Heisenberg chain, progressing from bit-based basis states to unsymmetrized and symmetry-block-diagonalized Hamiltonians.Applications include complete diagonalization and the Lanczos method for ground states and low-energy excitations.

4.1. Diagonalization of the Heisenberg chain

The Heisenberg-chain implementation represents spin states as integer bits and exploits spin, translation, reflection, and inversion symmetries. These conserved quantities partition the Hamiltonian into smaller sectors that can be constructed and diagonalized efficiently.

  • The S = 1/2 antiferromagnetic Heisenberg chain is formulated with arbitrary boundaries initially and periodic boundaries when momentum states are used.The periodic condition is SN = S0.
  • Lattice symmetries: Lattice translations and reflections act as permutations of spin indices, while translation invariance produces momentum blocks with k = 2nπ/N.Different momenta define independently diagonalizable Hamiltonian blocks.
  • Spin symmetries: Spin-rotational invariance supplies total-spin quantum numbers, while magnetization conservation and spin inversion further subdivide the Hamiltonian.Spin inversion has eigenvalues z = ±1 and is especially useful in the mz = 0 sector.
  • Bit representation: For S = 1/2 models, an integer encodes each basis state through bit values 0 and 1 for down and up spins.The Heisenberg off-diagonal terms can be implemented by flipping two corresponding bits with a bitwise operation.
  • Computational scope: Exact diagonalization is practically restricted by exponential basis growth, although sufficiently polarized magnetization sectors can remain manageable.Single-integer bit representations extend to N = 32 with standard integers and N = 64 with long integers.
  • Hamiltonian construction: The unsymmetrized Hamiltonian is a single 2^N × 2^N matrix whose diagonal terms inspect neighboring bits and whose off-diagonal terms flip antiparallel pairs.The matrix element for an allowed two-spin flip is 1/2.

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Momentum bases are constructed from translated representatives, with periodicity determining normalization and momentum compatibility. The resulting symmetry sectors support exact finite-chain thermodynamics, while low-temperature thermodynamic-limit extrapolation remains unreliable.

  • Momentum states are formed from a reference spin state and all its translations, producing translation eigenstates with eigenvalue e^ik.Periodic boundaries allow k = 2nπ/N.
  • Only one representative from each translation orbit is retained to obtain orthogonal momentum-basis states.Representatives with periodicity below N require modified normalization and may be excluded when incompatible with the chosen momentum.
  • Semi-momentum states combine translation and reflection information while remaining real-valued, unlike standard complex momentum states.Parity and translation can be used simultaneously as ordinary momentum states only at k = 0 or π.
  • Finite-chain results: For a 16-site chain, the lowest state in the k = 0 sector is a singlet, while the first excited state is a k = π triplet at E = −6.87210668.The triplet has symmetry labels (p = −1,z = −1).
  • Finite-chain results: Spin-inversion quantum number z provides limited spin information, helping distinguish low-energy singlet and triplet states even without directly calculating total spin.The lowest state in a symmetry sector need not be the system’s global ground state.
  • Thermodynamics: For N = 16, specific heat and susceptibility are well converged to the thermodynamic limit down to T/J ≈0.25, with high-temperature behavior approaching analytic limits.At low temperature, finite systems show exponentially vanishing C and χ because of the finite excitation gap.
  • Thermodynamics: Exact diagonalization cannot reliably extrapolate Heisenberg-chain properties to the thermodynamic limit at low temperatures.Long-chain methods and field-theoretical results are needed for more reliable access to this regime.

4.2. The Lanczos method

The Lanczos method approximates low-lying eigenstates in a Krylov subspace, replacing prohibitively large full diagonalization with a small tridiagonal problem. Its convergence can be monitored iteratively, with ground-state and several excited-state energies converging accurately for studied Heisenberg systems.

  • Motivation: Full diagonalization becomes prohibitively costly beyond approximately 20 S = 1/2 spins, while Krylov methods reach systems roughly twice as large for low-lying states.The method targets the ground state and possibly several low-lying excitations rather than the entire spectrum.
  • Krylov-space construction: The Krylov space is generated by repeatedly applying H to an initial state and spans the vectors H^m|Ψ⟩ for m = 0,...,Λ.The initial state must have nonzero overlap with the target ground state; a random state generally satisfies this.
  • Lanczos basis: Lanczos orthogonalizes Krylov vectors so that H becomes tridiagonal in the resulting basis, enabling rapid construction and diagonalization of the reduced matrix.The reduced matrix has nonzero elements only on the diagonal and adjacent off-diagonals.
  • Lanczos basis: The basis is generated iteratively with coefficients chosen to enforce orthogonality to preceding vectors, and computation can stop when target eigenvalues change by less than a tolerance ε.The implementation may use normalized vectors when unnormalized normalization constants become excessively large.
  • Convergence of Lanczos calculations: At Λ = 60, four displayed levels converged to better than 10 decimal places, while the ground state reached that accuracy at Λ = 30 for the studied model.Energies converged monotonically in this example, whereas other quantities did not necessarily do so.

4.3. 1D states and quantum phase transitions

The 1D Heisenberg chain has gapless low-energy excitations with finite-size gaps scaling as 1/N, while frustration drives a transition from a critical state to a dimerized VBS phase. With longer-range interactions, this continuous transition can evolve into a first-order Néel–VBS transition.

  • 4.3.1. Ground state and excitations of the Heisenberg chain: The Heisenberg-chain excitation spectrum is close to the exact infinite-size triplet dispersion derived using the Bethe ansatz.The lowest edge of the spectrum arises from weakly interacting spinon excitations.
  • 4.3.1. Ground state and excitations of the Heisenberg chain: The lowest triplet gaps scale to zero as 1/N, corresponding to dynamic exponent z = 1, with weak logarithmic corrections.The gap prefactor is related to the spinon velocity, but logarithmic corrections make extraction from finite-size data difficult.
  • 4.3.2. Frustration-driven quantum phase transition: For g > gc, long-range dimer order coexists with exponentially decaying spin correlations and a triplet gap that remains finite as N →∞.The dimer modulation becomes nonzero at gc, while the spin-correlation peak shifts from k = π toward k = π/2 near g ≈0.52 and may evolve continuously for g > 1.
  • 4.3.2. Frustration-driven quantum phase transition: At the Majumdar–Ghosh point g = 1/2, the even-length ring has an exactly demonstrable two-fold degenerate singlet-product ground state.This provides an exact example of VBS order in the frustrated chain.
  • 4.3.2. Frustration-driven quantum phase transition: Dimer correlations follow 1/r for g < gc but extrapolate to a nonzero value inside the VBS phase, although finite-size extrapolation becomes difficult near and beyond the transition.Near gc, apparent 1/N behavior and non-monotonic finite-size trends can obscure the expected exponential convergence.
  • 4.3.3. Chains with long-range interactions: Increasing long-range interactions changes the dimerization transition from continuous near α ≈2 to first-order for smaller α, separating Néel order from VBS-related states.The first-order regime is identified through sharpening energy maxima, level crossings, and discontinuous order parameters; finite-size corrections depend on α.

4.4. Two-dimensional systems

Two-dimensional exact diagonalization exploits translation and lattice symmetries to block-diagonalize Heisenberg-model Hamiltonians. Lanczos calculations expose finite-size corrections to quantum-rotor behavior and support long-range order through comparison with QMC results.

  • 4.4.1. Momentum states in two dimensions: Momentum conservation block-diagonalizes the square-lattice Heisenberg Hamiltonian within fixed-magnetization sectors.Additional lattice symmetries can split these blocks further, though their use depends on momentum.
  • 4.4.1. Momentum states in two dimensions: For generic momenta, reflections cannot provide further blocking, whereas high-symmetry momenta allow selected axial or diagonal reflection quantum numbers.The usable reflection operators depend on whether kx, ky, or their magnitudes satisfy special relations.
  • 4.4.2. The Néel state and its quantum rotor excitations: For L = 4 and L = 6, χ−1(S,N) decreases significantly with S and increases with N, with roughly 10% separating S = 1 and S = N/2 at fixed N.The small lattices cannot reliably reach the S ≪ L regime; larger-lattice QMC gives χ−1(S,N) → 22.8 for small S and large N.

5. QUANTUM MONTE CARLO SIMULATIONS AND THE STOCHASTIC SERIES EXPANSION METHOD

Quantum Monte Carlo methods are presented through path-integral and series-expansion formulations, with emphasis on the exact stochastic series expansion. Loop, worm, and directed-loop updates broaden the range of efficiently studied quantum spin and boson models, while frustration causes low-temperature sign problems.

  • The exact stochastic series expansion samples traces directly, avoiding the earlier reliance on permutation-operator algebra.
  • Loop-cluster updates efficiently sample Heisenberg-model configurations, while worm and directed-loop generalizations extend applicability to external fields and broader spin and boson models.These methods enable large-scale studies approaching the detail achievable for classical systems.
  • Frustrated spin systems generally produce sign problems because the path-integral or series-expansion weights are not positive definite.Mixed-sign statistical errors become uncontrollable at low temperatures, although progress exists at high temperatures.

5.1. Path integral and series expansion formulations of quantum statistical mechanics

Path-integral and series-expansion formulations transform quantum statistical mechanics into representations suitable for Monte Carlo sampling. The notes connect imaginary-time world lines, discretization choices, estimators, and sign constraints to the practical SSE method.

  • 5.1. Path integral and series expansion formulations of quantum statistical mechanics: Directly constructing exp(−βH) becomes infeasible beyond a few tens of spins, motivating transformations of the partition function into Monte Carlo-sampleable forms.
  • 5.1.1. The imaginary-time path integral: The imaginary-time path integral writes the exponential operator as a product of L factors with time step ∆τ = β/L, inserts complete states, and sums products of matrix elements.
  • 5.1.1. The imaginary-time path integral: Imaginary-time evolution maps a d-dimensional quantum system to an equivalent d + 1-dimensional representation with periodic boundary conditions in time.The mapping is physically useful for QMC only when the resulting path weights are suitable for sampling.
  • 5.1.1. The imaginary-time path integral: The linear time-slice approximation has relative partition-function error of order ∆τ at fixed β, whereas better approximations reduce the required number of slices.The notes explain that roughly Nβ slices may be needed with the linear approximation to accommodate world-line jumps.
  • 5.1.1. The imaginary-time path integral: World-line configurations encode boson motion through space-time, and their winding number measures nonlocal wrapping that corresponds to superfluidity or spin stiffness.
  • 5.1.1. The imaginary-time path integral: The kinetic-energy estimator is ⟨K⟩= −⟨nK⟩/β, where nK counts kinetic jumps in the world-line configuration.
  • 5.1.2. The Suzuki-Trotter decomposition: On bipartite lattices, periodic world-line constraints cancel the signs of allowed configurations, but frustrated systems retain both positive and negative weights, creating a sign problem.
  • 5.1.3. The series expansion representation: In the SSE representation, the sampled expansion-order distribution has variance relation C = ⟨n²⟩−⟨n⟩²−⟨n⟩ and is automatically sampled according to sector weights.

5.2. SSE method for the S = 1/2 Heisenberg model

The SSE method represents the S = 1/2 Heisenberg model with fixed-length operator strings and propagated spin states, enabling positive-definite sampling on bipartite lattices. Diagonal and loop updates generate configurations, while loop symmetries support improved estimators and observable calculations.

  • Configuration space: SSE applies to bipartite lattices, where the sign factor remains positive and configurations can be sampled without a sign problem.The initial implementation considers uniform antiferromagnetic coupling, with simple modifications for non-uniform systems.
  • Configuration space: The Heisenberg interaction is split into diagonal and off-diagonal bond operators, with a constant shift making the series expansion positive-definite.Parallel-spin operations annihilate states, so contributing configurations contain only operations on antiparallel spins.
  • Configuration space: Allowed configurations require periodic propagated states, |α(n)⟩ = |α(0)⟩, in addition to the local antiparallel-spin constraints.The fixed-length operator string contains n non-[0,0] operators, with its cutoff L chosen to exceed sampled expansion orders.
  • Configuration space: The operator string stores diagonal and off-diagonal operators as even and odd integers, while spins are encoded as σ(i) = ±1.Propagated states can be generated as needed from the stored initial state rather than retained simultaneously.
  • Improved estimators: Loop flips preserve the operator count n, creating 2^m equal-weight cluster orientations whose analytic averaging can reduce estimator noise.Specific heat is difficult to estimate reliably at low temperatures because it subtracts two large numbers.

5.3. Applications of SSE to 1D and 2D systems

SSE calculations benchmark one- and two-dimensional Heisenberg systems and locate quantum critical behavior through finite-size scaling. The results reproduce known asymptotic forms, quantify ground-state properties, and distinguish continuous from first-order transitions.

  • The Heisenberg chain: The S = 1/2 Heisenberg chain’s spin correlations reproduce the logarithmically corrected 1/r form through r = 29 for N = 4096.Allowing the logarithmic exponent to vary gives a value near 0.5, but with an uncertainty of roughly ±0.1.
  • The Heisenberg chain: Low-temperature chain results agree well with the asymptotic form for T/J below 0.05, while logarithmic corrections make extrapolation to T = 0 slow.The T →0 susceptibility-related value agrees with the Bethe ansatz through the spinon velocity c = Jπ/2.
  • Two-dimensional systems: QMC extrapolation gives the square-lattice Heisenberg model’s sublattice magnetization as ms = 0.30743(1), about 1% above the linear spin-wave value 0.3034.Higher-order spin-wave calculations give ms ≈ 0.3070, and related ground-state quantities show similarly good agreement.
  • Two-dimensional systems: Finite-temperature square-lattice calculations require large systems because the correlation length diverges rapidly; L = 256 is well converged for T/J ≥ 0.25.The asymptotic correlation-length form describes the data well in this converged temperature range.
  • Quantum phase transition in a dimerized system: For the dimerized system, finite-size crossings extrapolate consistently across observables, with critical estimates in the range [1.9094,1.9096] and correction exponent ω ≈ 2–2.5.The crossing points for the Binder cumulant and scaled stiffness approach the critical coupling from opposite directions, aiding bracketing.

6. SURVEY OF RELATED COMPUTATIONAL METHODS

The notes survey computational methods beyond the main exact-diagonalization and SSE treatments, emphasizing their applicability, limitations, and extensions. Valence-bond projector QMC provides an alternative sampling framework with access to additional observables.

  • Other computational methods: The notes identify series expansions and DMRG as important methods outside their scope, while noting controllability issues when extrapolating series expansions.For unfrustrated systems, series expansions generally do not reach the precision of existing QMC methods.
  • Research strategy: The author advocates a two-pronged strategy: improve methods for models beyond current QMC reach while studying sign-problem-free models with state-of-the-art QMC.This recommendation reflects the differing advantages and disadvantages of available computational approaches.
  • Directed loop QMC algorithms: Directed-loop algorithms generalize SSE operator loops by allowing multiple paths through vertices and enable efficient simulations in cases where standard loops do not apply.They were applied to anisotropic S = 1/2 Heisenberg systems, external magnetic fields, and higher-S models.
  • QMC algorithms in the valence-bond basis: Valence-bond basis states are products of two-spin singlets, but the basis is overcomplete and non-orthogonal, making expansions non-unique.Amplitude product states assign coefficients from products of bond-shape amplitudes h(rα,i).
  • QMC algorithms in the valence-bond basis: Amplitude product states closely reproduce many bipartite Heisenberg ground states; for the 2D Heisenberg model, optimized states achieve energy within 0.1% and sublattice magnetization within 1% of QMC values.For frustrated systems, appropriate sign rules are not known.
  • QMC algorithms in the valence-bond basis: A spin-explicit valence-bond projector formulation cuts open finite-temperature time boundaries, places valence-bond states at the boundaries, and samples loop configurations similarly to SSE and world-line methods.Loops can be flipped without changing configuration weight, and spin variables make sampling more efficient while preserving the pure valence-bond formulation.
  • QMC algorithms in the valence-bond basis: Projector QMC applies a high power of the Hamiltonian so the ground-state component survives as the power tends to infinity, with expectation values sampled by Monte Carlo.The method samples strings of singlet projectors together with valence-bond configurations.
  • QMC algorithms in the valence-bond basis: Valence-bond methods can access observables otherwise difficult to calculate, including excited-state properties and magnetization distributions with unpaired spins.Extended valence-bond bases include triplet sectors and one or several unpaired spins.

non-universal (lattice effects)

The supplied passages identify non-universal quantities associated with lattice effects, including quantum-disordered regimes and an order-parameter expression.

  • non-universal (lattice effects): The listed non-universal labels include renormalized, classical, and quantum-disordered regimes.The passage provides these terms without further explanation.
  • non-universal (lattice effects): An order parameter is represented as [Ss,d(Q)/N].The supplied passage gives the expression without defining its symbols.
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