Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics Condensed Matter

Generalized dimer states: hierarchies

M W Long, C A Hayward

Published Dec 12, 1994• 0 citations
Publisher

Abstract

We present variational calculations for quantum spin-half models based upon modifications to nearest-neighbour dimer states. Using a sequence of projections we can attempt a sort of finite-size scaling, in the number of variational parameters, towards the exact solution. For the simple nearest-neighbour Heisenberg model we achieve an error of only 0.005% in energy for our most accurate calculation, much better than an equivalent Lanczos calculation. With which we compare. Due to the fact that we control the total spin in our wavefunctions, we can evaluate the coherence lengths associated with cyclic-exchange permutations (the spin-half sector) and for spin-spin correlations (the spin-one sector), yielding, for example, a convincing numerical confirmation that the spin-one biquadratic-exchange Hamiltonian has a finite correlation length. We perform a study of the one-dimensional J 1 -J 2 model, trying to predict the onset of the phase transition by searching directly for the divergence of the correlation length. Our calculations predict the phase transition near J 2 /J 1 approximately 1/4, as expected. We also perform a study of the ladder geometry, giving further evidence that infinitesimal coupling between ladders yields an immediate gap in the spectrum. The only complication is that the minimization over our space of variational parameters is subtle: surprisingly complicated and "difficult to find" structure is observed in our low-energy solutions which might make our approach of limited use.

DOI
10.1088/0953-8984/6/50/018

Related papers

Generalized dimer states

M W Long, C A Hayward

We present a new class of variational wavefunctions that can be used to investigate quantum antiferromagnetic systems. The states are a natural generalization of the nearest-neighbour dimer phases which have been thoroughly studied in the literature. We describe a one-dimensional class of total-spin-singlet wavefunctions, which include as special cases the two nearest-neighbour dimer phases and the exact solution to the total-spin-two projector Hamiltonian. These different solutions are 'smoothly' interpolated by a range of singlets with short-range correlations. Although the states have only short-range order, we can obtain over 98% of the ground-state energy of the nearest-neighbour Heisenberg model. We apply our basis to the J 1 -J 2 Heisenberg model and the single hole in the t 1 -t 2 infinite-U Hubbard model, successfully predicting the behaviour observed in finite-size scaling calculations.

Journal of Physics Condensed Matter•Apr 4, 1994•2 citations

Ring exchange and the Heisenberg and Hubbard models

M W Long, C. W. M. Castleton, C A Hayward

We study ring-exchange or cyclic-permutation correlations in one-dimensional quantum spin-half systems. For the Heisenberg model we show numerically that these correlations decay as R n approximately 1/ square root n, although we can deduce nothing about any possible important logarithmic corrections. As such, ring-exchange correlations are much longer range than the more commonly considered spin-spin correlation functions. By considering the relationship between solitonic excitations and cyclic permutations, we suggest a way to predict the value of J 2 /J 1 at which the phase transition between a gapped and gapless phase occurs in the next-nearest-neighbour Heisenberg model, suggesting J 2 =4J 1 as the exact transition point. For the Hubbard model with a spin-charge-separated solution, we show that the occupation number, n k , is a 'convolution' of the cyclic-permutation correlations of the spin ground state with the anyonic occupation number of the charge ground state, with the integration being over statistical phase. We deduce the one-eighth singularity previously found for the U= infinity Hubbard model using this new route. We show that for the limit where nearest-neighbour hopping dominates longer-range hopping in the U= infinity Hubbard model, the single-particle correlation function, n k , for an infinitesimal concentration of holes in a half-filled system, is identical to the Fourier transform of the cyclic-exchange correlations of the corresponding spin wavefunction. For the elementary t 1 -t 2 model, we show a relationship between the singularity which occurs at the Fermi surface, the so-called Luttinger-liquid singularity, and the long-range Heisenberg-model cyclic-permutation correlations.

Journal of Physics Condensed Matter•Oct 31, 1994•2 citations

Clusterentwicklungen für dimerisierte Spinsysteme

Martin Müller

The subject of the present work are dimerized spin 1/2 systems in one, two and three dimensions.The models are investigated perturbatively by means of cluster expansion.This method makes it possible to study systems even in two and three dimension up to high orders.Starting with a generic HAMILTON-Operator which includes both bilinear and biquadratic interaction terms, the groundstate energies and the low lying excitations of the alternating spin chain und the spin ladder are calculated.Well-known results from BETHE ansatz and bosonization are used to verify the series.Based on this model, two speci£c one-dimensional problems are investigated in more detail: the £rst one is the incommensurate regime of the generalized spin ladder, which was previously analyzed by other approximation techniques.This regime is characterized by a minimum of the dispersion between 0 and π.The second problem deals with a quantum phase transition in the spin ladder with additional cyclic four-spin interaction (ring exchange).The excitation gap which is always present in the regular spin ladder can be reduced by taking the ring exchange into account.Within the region which is accessible by perturbation theory a transition line is presented on which the gap vanishes.For further characterization of the phase transition critical exponents are calculated which give a connection between exchange strength and gap size.Within the range of the higher-dimensional spin dimer systems the compounds KCuCl 3 , TlCuCl 3 (3d) and (C 4 H 12 N 2 )Cu 2 Cl 6 (2d) are studied.To clarify the microscopic interactions, dispersion relations which take all reasonable (i.e.evident from the chemical structure) interactions into account are computed.In the case of KCuCl 3 and TlCuCl 3 there are nine and in the case of (C 4 H 12 N 2 )Cu 2 Cl 6 there are ten spin spin interactions considered.The strength of the exchange parameters is determined by comparison with inelastic neutron scattering data.For the above-mentioned substances the interaction constants are evaluated to match the dispersion relation with the experimental data to a high accuracy.Finally, the dynamical structure factor which is limited to the singlet-triplet transition is calculated perturbatively for the compounds KCuCl 3 and TlCuCl 3 .Since it is proportional to the differential cross section, one is able to establish a relationship to inelastic neutron scattering experiments.

•Jan 1, 2002•0 citations

SU(N) quantum spin models: a variational wavefunction study

Arun Paramekanti, John Bradley Marston

SU ( N ) quantum spin systems may be realized in a variety of physical systems including ultracold atoms in optical lattices. The study of such models also leads to insights into possible novel quantum phases and phase transitions of SU (2) spin models. Here we use Gutzwiller projected fermionic variational wavefunctions to explore the phase diagram and correlation functions of SU ( N ) quantum spin models in the self-conjugate representation. In one dimension, the ground state of the SU (4) spin chain with Heisenberg bilinear and biquadratic interactions is studied by examining instabilities of the Gutzwiller projected free fermion ground state to various broken symmetries. The variational phase diagram so obtained agrees well with exact results. The spin–spin and dimer–dimer correlation functions of the Gutzwiller projected free fermion state with N flavours of fermions are in good agreement with exact and 1/ N calculations for the critical points of SU ( N ) spin chains. In two dimensions, the phase diagram of the antiferromagnetic Heisenberg model on the square lattice is obtained by finding instabilities of the Gutzwiller projected π-flux state. In the absence of biquadratic interactions, the model exhibits long-range Néel order for N = 2 and 4, and spin Peierls (columnar dimer) order for N >4. Upon including biquadratic interactions in the SU (4) model (with a sign appropriate to a fermionic Hubbard model), the Néel order diminishes and eventually disappears, giving way to an extended valence bond crystal. In the case of the SU (6) model, the dimerized ground state melts at sufficiently large biquadratic interaction, yielding a projected π-flux spin liquid phase which in turn undergoes a transition into an extended valence bond crystal at even larger biquadratic interaction. The spin correlations of the projected π-flux state at N = 4 are in good agreement with 1/ N calculations. We find that the state shows strongly enhanced dimer correlations, in qualitative agreement with recent theoretical predictions. We also compare our results with a recent quantum Monte Carlo study of the SU (4) Heisenberg model.

arXiv:cond-mat/0608691•Journal of Physics Condensed Matter•Mar 6, 2007•54 citations

Dynamical structure factor of the J1−J2 Heisenberg model in one dimension: The variational Monte Carlo approach

Francesco Ferrari, Alberto Parola, Sandro Sorella +1 more

The dynamical spin structure factor is computed within a variational framework to study the one-dimensional ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. Starting from Gutzwiller-projected fermionic wave functions, the low-energy spectrum is constructed from two-spinon excitations. The direct comparison with Lanczos calculations on small clusters demonstrates the excellent description of both gapless and gapped (dimerized) phases, including incommensurate structures for ${J}_{2}/{J}_{1}>0.5$. Calculations on large clusters show how the intensity evolves when increasing the frustrating ratio and give an unprecedented accurate characterization of the dynamical properties of (nonintegrable) frustrated spin models.

arXiv:1803.02359•Physical review. B./Physical review. B•Jun 5, 2018•51 citations

Scaling of excitations in dimerized and frustrated spin- 12 chains

Davide Controzzi, C. Degli Esposti Boschi, F. Ortolani +1 more

We study the finite-size behavior of the low-lying excitations of spin-$\frac{1}{2}$ Heisenberg chains with dimerization and next-to-nearest neighbors interaction ${J}_{2}$. The numerical analysis, performed using density-matrix renormalization group, confirms previous exact diagonalization results and shows that, for different values of the dimerization parameter $\ensuremath{\delta}$, the elementary triplet and singlet excitations present a clear scaling behavior in a wide range of $l=L∕\ensuremath{\xi}$ (where $L$ is the length of the chain and $\ensuremath{\xi}$ is the correlation length). At ${J}_{2}={J}_{2c}$, where no logarithmic corrections are present, we compare the numerical results with finite-size predictions for the sine-Gordon model obtained using L\"uscher's theory. For small $\ensuremath{\delta}$ we find a very good agreement for $l\ensuremath{\gtrsim}4$ or 7 depending on the excitation considered.

arXiv:cond-mat/0503539•Physical Review B•Nov 17, 2005•4 citations
We present variational calculations for quantum spin-half models based upon modifications to nearest-neighbour dimer states. Using a sequence of projections we can attempt a sort of finite-size scaling, in the number of variational parameters, towards the exact solution. For the simple nearest-neighbour Heisenberg model we achieve an error of only 0.005% in energy for our most accurate calculation, much better than an equivalent Lanczos calculation. With which we compare. Due to the fact that we control the total spin in our wavefunctions, we can evaluate the coherence lengths associated with cyclic-exchange permutations (the spin-half sector) and for spin-spin correlations (the spin-one sector), yielding, for example, a convincing numerical confirmation that the spin-one biquadratic-exchange Hamiltonian has a finite correlation length. We perform a study of the one-dimensional J 1 -J 2 model, trying to predict the onset of the phase transition by searching directly for the divergence of the correlation length. Our calculations predict the phase transition near J 2 /J 1 approximately 1/4, as expected. We also perform a study of the ladder geometry, giving further evidence that infinitesimal coupling between ladders yields an immediate gap in the spectrum. The only complication is that the minimization over our space of variational parameters is subtle: surprisingly complicated and "difficult to find" structure is observed in our low-energy solutions which might make our approach of limited use.
We present a new class of variational wavefunctions that can be used to investigate quantum antiferromagnetic systems. The states are a natural generalization of the nearest-neighbour dimer phases which have been thoroughly studied in the literature. We describe a one-dimensional class of total-spin-singlet wavefunctions, which include as special cases the two nearest-neighbour dimer phases and the exact solution to the total-spin-two projector Hamiltonian. These different solutions are 'smoothly' interpolated by a range of singlets with short-range correlations. Although the states have only short-range order, we can obtain over 98% of the ground-state energy of the nearest-neighbour Heisenberg model. We apply our basis to the J 1 -J 2 Heisenberg model and the single hole in the t 1 -t 2 infinite-U Hubbard model, successfully predicting the behaviour observed in finite-size scaling calculations.
We study ring-exchange or cyclic-permutation correlations in one-dimensional quantum spin-half systems. For the Heisenberg model we show numerically that these correlations decay as R n approximately 1/ square root n, although we can deduce nothing about any possible important logarithmic corrections. As such, ring-exchange correlations are much longer range than the more commonly considered spin-spin correlation functions. By considering the relationship between solitonic excitations and cyclic permutations, we suggest a way to predict the value of J 2 /J 1 at which the phase transition between a gapped and gapless phase occurs in the next-nearest-neighbour Heisenberg model, suggesting J 2 =4J 1 as the exact transition point. For the Hubbard model with a spin-charge-separated solution, we show that the occupation number, n k , is a 'convolution' of the cyclic-permutation correlations of the spin ground state with the anyonic occupation number of the charge ground state, with the integration being over statistical phase. We deduce the one-eighth singularity previously found for the U= infinity Hubbard model using this new route. We show that for the limit where nearest-neighbour hopping dominates longer-range hopping in the U= infinity Hubbard model, the single-particle correlation function, n k , for an infinitesimal concentration of holes in a half-filled system, is identical to the Fourier transform of the cyclic-exchange correlations of the corresponding spin wavefunction. For the elementary t 1 -t 2 model, we show a relationship between the singularity which occurs at the Fermi surface, the so-called Luttinger-liquid singularity, and the long-range Heisenberg-model cyclic-permutation correlations.
The subject of the present work are dimerized spin 1/2 systems in one, two and three dimensions.The models are investigated perturbatively by means of cluster expansion.This method makes it possible to study systems even in two and three dimension up to high orders.Starting with a generic HAMILTON-Operator which includes both bilinear and biquadratic interaction terms, the groundstate energies and the low lying excitations of the alternating spin chain und the spin ladder are calculated.Well-known results from BETHE ansatz and bosonization are used to verify the series.Based on this model, two speci£c one-dimensional problems are investigated in more detail: the £rst one is the incommensurate regime of the generalized spin ladder, which was previously analyzed by other approximation techniques.This regime is characterized by a minimum of the dispersion between 0 and π.The second problem deals with a quantum phase transition in the spin ladder with additional cyclic four-spin interaction (ring exchange).The excitation gap which is always present in the regular spin ladder can be reduced by taking the ring exchange into account.Within the region which is accessible by perturbation theory a transition line is presented on which the gap vanishes.For further characterization of the phase transition critical exponents are calculated which give a connection between exchange strength and gap size.Within the range of the higher-dimensional spin dimer systems the compounds KCuCl 3 , TlCuCl 3 (3d) and (C 4 H 12 N 2 )Cu 2 Cl 6 (2d) are studied.To clarify the microscopic interactions, dispersion relations which take all reasonable (i.e.evident from the chemical structure) interactions into account are computed.In the case of KCuCl 3 and TlCuCl 3 there are nine and in the case of (C 4 H 12 N 2 )Cu 2 Cl 6 there are ten spin spin interactions considered.The strength of the exchange parameters is determined by comparison with inelastic neutron scattering data.For the above-mentioned substances the interaction constants are evaluated to match the dispersion relation with the experimental data to a high accuracy.Finally, the dynamical structure factor which is limited to the singlet-triplet transition is calculated perturbatively for the compounds KCuCl 3 and TlCuCl 3 .Since it is proportional to the differential cross section, one is able to establish a relationship to inelastic neutron scattering experiments.
SU ( N ) quantum spin systems may be realized in a variety of physical systems including ultracold atoms in optical lattices. The study of such models also leads to insights into possible novel quantum phases and phase transitions of SU (2) spin models. Here we use Gutzwiller projected fermionic variational wavefunctions to explore the phase diagram and correlation functions of SU ( N ) quantum spin models in the self-conjugate representation. In one dimension, the ground state of the SU (4) spin chain with Heisenberg bilinear and biquadratic interactions is studied by examining instabilities of the Gutzwiller projected free fermion ground state to various broken symmetries. The variational phase diagram so obtained agrees well with exact results. The spin–spin and dimer–dimer correlation functions of the Gutzwiller projected free fermion state with N flavours of fermions are in good agreement with exact and 1/ N calculations for the critical points of SU ( N ) spin chains. In two dimensions, the phase diagram of the antiferromagnetic Heisenberg model on the square lattice is obtained by finding instabilities of the Gutzwiller projected π-flux state. In the absence of biquadratic interactions, the model exhibits long-range Néel order for N = 2 and 4, and spin Peierls (columnar dimer) order for N >4. Upon including biquadratic interactions in the SU (4) model (with a sign appropriate to a fermionic Hubbard model), the Néel order diminishes and eventually disappears, giving way to an extended valence bond crystal. In the case of the SU (6) model, the dimerized ground state melts at sufficiently large biquadratic interaction, yielding a projected π-flux spin liquid phase which in turn undergoes a transition into an extended valence bond crystal at even larger biquadratic interaction. The spin correlations of the projected π-flux state at N = 4 are in good agreement with 1/ N calculations. We find that the state shows strongly enhanced dimer correlations, in qualitative agreement with recent theoretical predictions. We also compare our results with a recent quantum Monte Carlo study of the SU (4) Heisenberg model.
The dynamical spin structure factor is computed within a variational framework to study the one-dimensional ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. Starting from Gutzwiller-projected fermionic wave functions, the low-energy spectrum is constructed from two-spinon excitations. The direct comparison with Lanczos calculations on small clusters demonstrates the excellent description of both gapless and gapped (dimerized) phases, including incommensurate structures for J2/J1>0.5{J}_{2}/{J}_{1}>0.5J2​/J1​>0.5. Calculations on large clusters show how the intensity evolves when increasing the frustrating ratio and give an unprecedented accurate characterization of the dynamical properties of (nonintegrable) frustrated spin models.
We study the finite-size behavior of the low-lying excitations of spin-12\frac{1}{2}21​ Heisenberg chains with dimerization and next-to-nearest neighbors interaction J2{J}_{2}J2​. The numerical analysis, performed using density-matrix renormalization group, confirms previous exact diagonalization results and shows that, for different values of the dimerization parameter $\ensuremath{\delta}$, the elementary triplet and singlet excitations present a clear scaling behavior in a wide range of $l=L∕\ensuremath{\xi}$ (where LLL is the length of the chain and $\ensuremath{\xi}$ is the correlation length). At J2=J2c{J}_{2}={J}_{2c}J2​=J2c​, where no logarithmic corrections are present, we compare the numerical results with finite-size predictions for the sine-Gordon model obtained using L\"uscher's theory. For small $\ensuremath{\delta}$ we find a very good agreement for $l\ensuremath{\gtrsim}4$ or 7 depending on the excitation considered.