arXiv:1803.03779 · Physical review. B./Physical review. B

Signature of a quantum dimensional transition in the spin- 12 antiferromagnetic Heisenberg model on a square lattice and space reduction in the matrix product state

Lihua Wang, Kwang S. Kim

Published Apr 29, 2019• 5 citations

Abstract

We study the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on an $\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}N$ square lattice for even $N$'s up to 14. Previously, the nonlinear sigma model perturbatively predicted that its spin-rotational symmetry breaks asymptotically with $N\ensuremath{\rightarrow}\ensuremath{\infty}$, i.e., when it becomes two dimensional (2D). However, we identify a critical width ${N}_{c}=10$ for which this symmetry breaks spontaneously. It shows the signature of a dimensional transition from one dimensional (1D) including quasi-1D to 2D. The finite-size effect differs from that of the $N\ifmmode\times\else\texttimes\fi{}N$ lattice. The ground-state (GS) energy per site approaches the thermodynamic limit value, in agreement with the previously accepted value, by one order of $1/N$ faster than when using $N\ifmmode\times\else\texttimes\fi{}N$ lattices in the literature. Methodwise, we build and variationally solve a matrix product state (MPS) on a chain, converting the $N$ sites in each rung into an effective site. We show that the area law of entanglement entropy does not apply when $N$ increases in our method and the reduced density matrix of each effective site has a saturating number of dominant diagonal elements with increasing $N$. These two characteristics make the MPS rank needed to obtain a desired energy accuracy quickly saturate when $N$ is large, making our algorithm efficient for large $N$'s. Furthermore, the latter enables space reduction in MPS. Within the framework of MPS, we prove a theorem that the spin-spin correlation at infinite separation is the square of staggered magnetization and demonstrate that the eigenvalue structure of a building MPS unit of $\ensuremath{\langle}g|g\ensuremath{\rangle},|g\ensuremath{\rangle}$ being the GS is responsible for order, disorder, and quasi-long-range order.

DOI: 10.1103/physrevb.99.134441

Related papers

Staggered flux state for rectangular-lattice spin- 12 Heisenberg antiferromagnets

N. E. Shaik, B. Dalla Piazza, Dmitri A. Ivanov +1 more

arXiv:2005.03928Physical review. B./Physical review. BDec 14, 20203 citations

Equivalence between the nonlinearσmodel and the spin-(1/2 antiferromagnetic Heisenberg model: Spin correlations in La2 CuO4

Efstratios Manousakis, Román Salvador

Physical review. B, Condensed matterAug 1, 198931 citations

Dynamic scaling of the restoration of rotational symmetry in Heisenberg quantum antiferromagnets

Phillip Weinberg, Anders W. Sandvik

arXiv:1703.04223Physical review. B./Physical review. BAug 28, 20176 citations

Entanglement entropy of the two-dimensional Heisenberg antiferromagnet

Hao Song, Nicolas Laflorencie, Stephan Rachel +1 more

arXiv:1103.1636Physical Review BJun 21, 201196 citations

Possible spin liquid state in the spin 1/2 J1-J2 antiferromagnetic Heisenberg model on square lattice: A tensor product state approach

Ling Wang, Zheng‐Cheng Gu, Frank Verstraete +1 more

Dec 15, 20112 citations

High-precision ground state parameters of the two-dimensional spin-1/2 Heisenberg model on the square lattice

Anders W. Sandvik

arXiv:2601.20189Journal of Statistical Mechanics Theory and ExperimentApr 1, 20260 citations

arXiv:1803.03779 · Physical review. B./Physical review. B

Signature of a quantum dimensional transition in the spin- 12 antiferromagnetic Heisenberg model on a square lattice and space reduction in the matrix product state

Lihua Wang, Kwang S. Kim

Published Apr 29, 2019• 5 citations

arXiv HTML PDF

Abstract

We study the spin-

\frac{1}{2}

antiferromagnetic Heisenberg model on an

\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}N

square lattice for even

N

's up to 14. Previously, the nonlinear sigma model perturbatively predicted that its spin-rotational symmetry breaks asymptotically with

N\ensuremath{\rightarrow}\ensuremath{\infty}

, i.e., when it becomes two dimensional (2D). However, we identify a critical width

{N}_{c}=10

for which this symmetry breaks spontaneously. It shows the signature of a dimensional transition from one dimensional (1D) including quasi-1D to 2D. The finite-size effect differs from that of the

N\ifmmode\times\else\texttimes\fi{}N

lattice. The ground-state (GS) energy per site approaches the thermodynamic limit value, in agreement with the previously accepted value, by one order of

1/N

faster than when using

N\ifmmode\times\else\texttimes\fi{}N

lattices in the literature. Methodwise, we build and variationally solve a matrix product state (MPS) on a chain, converting the

N

sites in each rung into an effective site. We show that the area law of entanglement entropy does not apply when

N

increases in our method and the reduced density matrix of each effective site has a saturating number of dominant diagonal elements with increasing

N

. These two characteristics make the MPS rank needed to obtain a desired energy accuracy quickly saturate when

N

is large, making our algorithm efficient for large

N

's. Furthermore, the latter enables space reduction in MPS. Within the framework of MPS, we prove a theorem that the spin-spin correlation at infinite separation is the square of staggered magnetization and demonstrate that the eigenvalue structure of a building MPS unit of

\ensuremath{\langle}g|g\ensuremath{\rangle},|g\ensuremath{\rangle}

being the GS is responsible for order, disorder, and quasi-long-range order.

DOI: 10.1103/physrevb.99.134441

Related papers

Staggered flux state for rectangular-lattice spin- 12 Heisenberg antiferromagnets

N. E. Shaik, B. Dalla Piazza, Dmitri A. Ivanov +1 more

We investigate the spin- $\frac{1}{2}$ Heisenberg model on a rectangular lattice, using the Gutzwiller projected variational wave function known as the staggered flux state. Using Monte Carlo techniques, the variational parameters and instantaneous spin-spin correlation function for different coupling anisotropies $\ensuremath{\gamma}={J}_{y}/{J}_{x}$ are calculated. We observe a gradual evolution of the ground state energy towards a value which is very close to the one-dimensional (1D) estimate provided by the Bethe ansatz and a good agreement between the finite-size scaling of the energies. The spin-spin correlation functions exhibit a power-law decay with varying exponents for different anisotropies. Though the lack of N\'eel order makes the staggered flux state energetically unfavorable in the symmetric case $\ensuremath{\gamma}=1$, it appears to capture the essence of the system close to 1D. Hence we believe that the staggered flux state provides an interesting starting point to explore the crossover from quantum disordered chains to the N\'eel ordered two-dimensional square lattices.We investigate the spin-$\frac{1}{2}$ Heisenberg model on a rectangular lattice, using the Gutzwiller projected variational wave function known as the staggered flux state. Using Monte Carlo techniques, the variational parameters and instantaneous spin-spin correlation function for different coupling anisotropies $\ensuremath{\gamma}={J}_{y}/{J}_{x}$ are calculated. We observe a gradual evolution of the ground state energy towards a value which is very close to the one-dimensional (1D) estimate provided by the Bethe ansatz and a good agreement between the finite-size scaling of the energies. The spin-spin correlation functions exhibit a power-law decay with varying exponents for different anisotropies. Though the lack of N\'eel order makes the staggered flux state energetically unfavorable in the symmetric case $\ensuremath{\gamma}=1$, it appears to capture the essence of the system close to 1D. Hence we believe that the staggered flux state provides an interesting starting point to explore the crossover from quantum disordered chains to the N\'eel ordered two-dimensional square lattices.

arXiv:2005.03928Physical review. B./Physical review. BDec 14, 20203 citations

Equivalence between the nonlinearσmodel and the spin-(1/2 antiferromagnetic Heisenberg model: Spin correlations in La2 CuO4

Efstratios Manousakis, Román Salvador

We study the continuum limit of the quantum nonlinear \ensuremath{\sigma} model in 2+1 dimensions and at finite temperature T using both Monte Carlo simulation on large-size lattices ( ${100}^{2}$ \ifmmode\times\else\texttimes\fi{}8 is our largest-size lattice) and saddle-point approximation. At zero temperature, we find the critical point ${g}_{c}$ that separates the quantum disordered phase from the phase with spontaneous symmetry breaking (nonzero staggered magnetization). We calculate the model's renormalization group \ensuremath{\beta} function close to the critical point. Using the \ensuremath{\beta} function, we rescale the correlation lengths calculated at various values of the coupling constant (spin stiffness) and temperature and find that they all collapse on the same curve \ensuremath{\xi}/${a}_{\ensuremath{\sigma}}$=f(T/${T}_{\ensuremath{\sigma}}$). Even though the lattice spacing vanishes, a finite unit of length ${a}_{\ensuremath{\sigma}}$ and a temperature scale ${T}_{\ensuremath{\sigma}}$ is generated via dimensional transmutation. Assuming that the nonlinear \ensuremath{\sigma} model and the spin-(1/2 antiferromagnetic (AF) Heisenberg model are equivalent at low temperature, we relate the units ${a}_{\ensuremath{\sigma}}$ and ${T}_{\ensuremath{\sigma}}$ to the lattice spacing ${a}_{H}$ and the AF coupling J of the Heisenberg model so that the correlation lengths obtained from the simulation of the two models agree. In order to achieve this agreement we find that (a) the spin-(1/2 AF Heisenberg model should order at T=0 and (b) the relationship between the scales ${a}_{\ensuremath{\sigma}}$,${T}_{\ensuremath{\sigma}}$ and ${a}_{H}$ , J is obtained, and f(T/${T}_{\ensuremath{\sigma}}$) can be accurately approximated by an exponential of ${T}_{\ensuremath{\sigma}}$/T below ${g}_{c}$ . We obtain a reasonable fit to the neutron scattering data of the insulator ${\mathrm{La}}_{2}$ ${\mathrm{CuO}}_{4}$ by taking J=1270 K, a value close to that reported by Raman scattering experiments.We study the continuum limit of the quantum nonlinear \ensuremath{\sigma} model in 2+1 dimensions and at finite temperature T using both Monte Carlo simulation on large-size lattices (${100}^{2}$\ifmmode\times\else\texttimes\fi{}8 is our largest-size lattice) and saddle-point approximation. At zero temperature, we find the critical point ${g}_{c}$ that separates the quantum disordered phase from the phase with spontaneous symmetry breaking (nonzero staggered magnetization). We calculate the model's renormalization group \ensuremath{\beta} function close to the critical point. Using the \ensuremath{\beta} function, we rescale the correlation lengths calculated at various values of the coupling constant (spin stiffness) and temperature and find that they all collapse on the same curve \ensuremath{\xi}/${a}_{\ensuremath{\sigma}}$=f(T/${T}_{\ensuremath{\sigma}}$). Even though the lattice spacing vanishes, a finite unit of length ${a}_{\ensuremath{\sigma}}$ and a temperature scale ${T}_{\ensuremath{\sigma}}$ is generated via dimensional transmutation. Assuming that the nonlinear \ensuremath{\sigma} model and the spin-(1/2 antiferromagnetic (AF) Heisenberg model are equivalent at low temperature, we relate the units ${a}_{\ensuremath{\sigma}}$ and ${T}_{\ensuremath{\sigma}}$ to the lattice spacing ${a}_{H}$ and the AF coupling J of the Heisenberg model so that the correlation lengths obtained from the simulation of the two models agree. In order to achieve this agreement we find that (a) the spin-(1/2 AF Heisenberg model should order at T=0 and (b) the relationship between the scales ${a}_{\ensuremath{\sigma}}$,${T}_{\ensuremath{\sigma}}$ and ${a}_{H}$, J is obtained, and f(T/${T}_{\ensuremath{\sigma}}$) can be accurately approximated by an exponential of ${T}_{\ensuremath{\sigma}}$/T below ${g}_{c}$. We obtain a reasonable fit to the neutron scattering data of the insulator ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$ by taking J=1270 K, a value close to that reported by Raman scattering experiments.

Physical review. B, Condensed matterAug 1, 198931 citations

Dynamic scaling of the restoration of rotational symmetry in Heisenberg quantum antiferromagnets

Phillip Weinberg, Anders W. Sandvik

We apply imaginary-time evolution with the operator ${e}^{\ensuremath{-}\ensuremath{\tau}H}$ to study relaxation dynamics of gapless quantum antiferromagnets described by the spin-rotation-invariant Heisenberg Hamiltonian $H$ . Using quantum Monte Carlo simulations to obtain unbiased results, we propagate an initial state with maximal order parameter ${m}_{s}^{z}$ (the staggered magnetization) in the $z$ spin direction and monitor the expectation value $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$ as a function of imaginary time $\ensuremath{\tau}$. Results for different system sizes (lengths) $L$ exhibit an initial essentially size independent relaxation of $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$ toward its value in the infinite-size spontaneously symmetry broken state, followed by a strongly size dependent final decay to zero when the $\mathrm{O}(3)$ rotational symmetry of the order parameter is restored. We develop a generic finite-size scaling theory that shows the relaxation time diverges asymptotically as ${L}^{z}$ , where $z$ is the dynamic exponent of the low-energy excitations. We use the scaling theory to develop a practical way of extracting the dynamic exponent from the numerical finite-size data, systematically eliminating scaling corrections. We apply the method to spin- $1/2$ Heisenberg antiferromagnets on two different lattice geometries: the standard two-dimensional (2D) square lattice and a site-diluted 2D square lattice at the percolation threshold. In the 2D case we obtain $z=2.001(5)$ , which is consistent with the known value $z=2$ , while for the site-diluted lattice we find $z=3.90(1)$ or $z=2.056(8){D}_{f}$ , where ${D}_{f}=91/48$ is the fractal dimensionality of the percolating system. This is an improvement on previous estimates of $z\ensuremath{\approx}3.7$. The scaling results also show a fundamental difference between the two cases; for the 2D square lattice, the data can be collapsed onto a common scaling function even when $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$ is relatively large, reflecting the Anderson tower of quantum rotor states with a common dynamic exponent $z=2$ . For the diluted 2D square lattice, the scaling works well only for small $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$, indicating a mixture of different relaxation-time scalings between the low-energy states. Nevertheless, the low-energy dynamic here also corresponds to a tower of excitations.We apply imaginary-time evolution with the operator ${e}^{\ensuremath{-}\ensuremath{\tau}H}$ to study relaxation dynamics of gapless quantum antiferromagnets described by the spin-rotation-invariant Heisenberg Hamiltonian $H$. Using quantum Monte Carlo simulations to obtain unbiased results, we propagate an initial state with maximal order parameter ${m}_{s}^{z}$ (the staggered magnetization) in the $z$ spin direction and monitor the expectation value $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$ as a function of imaginary time $\ensuremath{\tau}$. Results for different system sizes (lengths) $L$ exhibit an initial essentially size independent relaxation of $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$ toward its value in the infinite-size spontaneously symmetry broken state, followed by a strongly size dependent final decay to zero when the $\mathrm{O}(3)$ rotational symmetry of the order parameter is restored. We develop a generic finite-size scaling theory that shows the relaxation time diverges asymptotically as ${L}^{z}$, where $z$ is the dynamic exponent of the low-energy excitations. We use the scaling theory to develop a practical way of extracting the dynamic exponent from the numerical finite-size data, systematically eliminating scaling corrections. We apply the method to spin-$1/2$ Heisenberg antiferromagnets on two different lattice geometries: the standard two-dimensional (2D) square lattice and a site-diluted 2D square lattice at the percolation threshold. In the 2D case we obtain $z=2.001(5)$, which is consistent with the known value $z=2$, while for the site-diluted lattice we find $z=3.90(1)$ or $z=2.056(8){D}_{f}$, where ${D}_{f}=91/48$ is the fractal dimensionality of the percolating system. This is an improvement on previous estimates of $z\ensuremath{\approx}3.7$. The scaling results also show a fundamental difference between the two cases; for the 2D square lattice, the data can be collapsed onto a common scaling function even when $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$ is relatively large, reflecting the Anderson tower of quantum rotor states with a common dynamic exponent $z=2$. For the diluted 2D square lattice, the scaling works well only for small $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$, indicating a mixture of different relaxation-time scalings between the low-energy states. Nevertheless, the low-energy dynamic here also corresponds to a tower of excitations.

arXiv:1703.04223Physical review. B./Physical review. BAug 28, 20176 citations

Entanglement entropy of the two-dimensional Heisenberg antiferromagnet

Hao Song, Nicolas Laflorencie, Stephan Rachel +1 more

We compute the von Neumann and generalized R\'enyi entanglement entropies in the ground-state of the spin- $1/2$ antiferromagnetic Heisenberg model on the square lattice using the modified spin-wave theory for finite lattices. The addition of a staggered magnetic field to regularize the zero modes associated with symmetry breaking is shown to be essential for obtaining well-behaved values for the entanglement entropy. The von Neumann and R\'enyi entropies obey an area law with additive logarithmic corrections, and are in good quantitative agreement with numerical results from valence bond quantum Monte Carlo and density matrix renormalization group calculations. We also compute the spin fluctuations and observe a multiplicative logarithmic correction to the area law in excellent agreement with quantum Monte Carlo calculations.We compute the von Neumann and generalized R\'enyi entanglement entropies in the ground-state of the spin-$1/2$ antiferromagnetic Heisenberg model on the square lattice using the modified spin-wave theory for finite lattices. The addition of a staggered magnetic field to regularize the zero modes associated with symmetry breaking is shown to be essential for obtaining well-behaved values for the entanglement entropy. The von Neumann and R\'enyi entropies obey an area law with additive logarithmic corrections, and are in good quantitative agreement with numerical results from valence bond quantum Monte Carlo and density matrix renormalization group calculations. We also compute the spin fluctuations and observe a multiplicative logarithmic correction to the area law in excellent agreement with quantum Monte Carlo calculations.

arXiv:1103.1636Physical Review BJun 21, 201196 citations

Possible spin liquid state in the spin 1/2 J1-J2 antiferromagnetic Heisenberg model on square lattice: A tensor product state approach

Ling Wang, Zheng‐Cheng Gu, Frank Verstraete +1 more

We study the spin 1/2 J1-J2 antiferromagnetic Heisenberg model on square lattice by using a recently proposed cluster update method for tensor product states (TPSs). The ground state energies in the thermodynamic limit are in good agreement with the state of art exact diagonalization study, and the energy differences between these two studies are of the order of 10J1 per site. For small bond dimension D (D < 5), we find a paramagnetic ground state with mixed columnar and staggered dimer orders of a magnitude 10−2 in the range of 0.47 < J2/J1 < 0.6, which agrees with previous results. For large bond dimension D (D ≥ 5), we observe all these orders vanish, implying the emergence of a spin liquid phase.We study the spin 1/2 J1-J2 antiferromagnetic Heisenberg model on square lattice by using a recently proposed cluster update method for tensor product states (TPSs). The ground state energies in the thermodynamic limit are in good agreement with the state of art exact diagonalization study, and the energy differences between these two studies are of the order of 10J1 per site. For small bond dimension D (D < 5), we find a paramagnetic ground state with mixed columnar and staggered dimer orders of a magnitude 10−2 in the range of 0.47 < J2/J1 < 0.6, which agrees with previous results. For large bond dimension D (D ≥ 5), we observe all these orders vanish, implying the emergence of a spin liquid phase.

Dec 15, 20112 citations

High-precision ground state parameters of the two-dimensional spin-1/2 Heisenberg model on the square lattice

Anders W. Sandvik

Abstract Motivated by the need for high-precision benchmark results for basic quantum many-body models, several ground state properties of the square lattice S = 1 / 2 Heisenberg antiferromagnet are computed (the energy, order parameter, spin stiffness, spinwave velocity, long-wavelength susceptibility and staggered susceptibility) using extensive quantum Monte Carlo simulations with the stochastic series expansion method. Moderately sized lattices are studied at temperatures T sufficiently low to realize the T → 0 limit, with significantly reduced statistical errors relative to previous works. Results for periodic L × L lattices with L ∈ [ 6 , 96 ] are tabulated versus L and extrapolations to infinite system size are carried out. The relative statistical errors of finite-size energies are below 10 −7 . The extrapolated ground state energy density is e 0 = − 0.669 441 857 ( 7 ) , where the number within parentheses is the statistical error (one standard deviation) of the preceding digit. This value represents an improvement in precision of three orders of magnitude over the previous best result. The leading and subleading finite-size corrections to e 0 are in full quantitative agreement with predictions from chiral perturbation theory, thus further supporting the soundness of both the extrapolations and the theory. The extrapolated sublattice magnetization is m s = 0.307 447 ( 2 ) , which agrees well with previous estimates but with a much smaller statistical error. The coefficient of the linear in L −1 correction to m s 2 agrees with the value from chiral perturbation theory and the presence of a factor ln γ ⁡ ( L ) in the second-order correction is also confirmed, with the previously unknown value of the exponent being γ = 0.82 ( 4 ) . The finite-size corrections to the staggered susceptibility point to logarithmic corrections in this quantity as well. To facilitate benchmarking of methods for which periodic boundary conditions are challenging, results for systems with open and cylindrical boundaries are also listed and their spatially inhomogeneous order parameters are analyzed.Abstract Motivated by the need for high-precision benchmark results for basic quantum many-body models, several ground state properties of the square lattice S = 1 / 2 Heisenberg antiferromagnet are computed (the energy, order parameter, spin stiffness, spinwave velocity, long-wavelength susceptibility and staggered susceptibility) using extensive quantum Monte Carlo simulations with the stochastic series expansion method. Moderately sized lattices are studied at temperatures T sufficiently low to realize the T → 0 limit, with significantly reduced statistical errors relative to previous works. Results for periodic L × L lattices with L ∈ [ 6 , 96 ] are tabulated versus L and extrapolations to infinite system size are carried out. The relative statistical errors of finite-size energies are below 10 −7 . The extrapolated ground state energy density is e 0 = − 0.669 441 857 ( 7 ) , where the number within parentheses is the statistical error (one standard deviation) of the preceding digit. This value represents an improvement in precision of three orders of magnitude over the previous best result. The leading and subleading finite-size corrections to e 0 are in full quantitative agreement with predictions from chiral perturbation theory, thus further supporting the soundness of both the extrapolations and the theory. The extrapolated sublattice magnetization is m s = 0.307 447 ( 2 ) , which agrees well with previous estimates but with a much smaller statistical error. The coefficient of the linear in L −1 correction to m s 2 agrees with the value from chiral perturbation theory and the presence of a factor ln γ ⁡ ( L ) in the second-order correction is also confirmed, with the previously unknown value of the exponent being γ = 0.82 ( 4 ) . The finite-size corrections to the staggered susceptibility point to logarithmic corrections in this quantity as well. To facilitate benchmarking of methods for which periodic boundary conditions are challenging, results for systems with open and cylindrical boundaries are also listed and their spatially inhomogeneous order parameters are analyzed.

arXiv:2601.20189Journal of Statistical Mechanics Theory and ExperimentApr 1, 20260 citations

We study the spin-

\frac{1}{2}

antiferromagnetic Heisenberg model on an

\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}N

square lattice for even

N

's up to 14. Previously, the nonlinear sigma model perturbatively predicted that its spin-rotational symmetry breaks asymptotically with

N\ensuremath{\rightarrow}\ensuremath{\infty}

, i.e., when it becomes two dimensional (2D). However, we identify a critical width

{N}_{c}=10

for which this symmetry breaks spontaneously. It shows the signature of a dimensional transition from one dimensional (1D) including quasi-1D to 2D. The finite-size effect differs from that of the

N\ifmmode\times\else\texttimes\fi{}N

lattice. The ground-state (GS) energy per site approaches the thermodynamic limit value, in agreement with the previously accepted value, by one order of

1/N

faster than when using

N\ifmmode\times\else\texttimes\fi{}N

lattices in the literature. Methodwise, we build and variationally solve a matrix product state (MPS) on a chain, converting the

N

sites in each rung into an effective site. We show that the area law of entanglement entropy does not apply when

N

increases in our method and the reduced density matrix of each effective site has a saturating number of dominant diagonal elements with increasing

N

. These two characteristics make the MPS rank needed to obtain a desired energy accuracy quickly saturate when

N

is large, making our algorithm efficient for large

N

\ensuremath{\langle}g|g\ensuremath{\rangle},|g\ensuremath{\rangle}

being the GS is responsible for order, disorder, and quasi-long-range order.

We investigate the spin-

\frac{1}{2}

Heisenberg model on a rectangular lattice, using the Gutzwiller projected variational wave function known as the staggered flux state. Using Monte Carlo techniques, the variational parameters and instantaneous spin-spin correlation function for different coupling anisotropies $\ensuremath{\gamma}={J}_{y}/{J}_{x}$ are calculated. We observe a gradual evolution of the ground state energy towards a value which is very close to the one-dimensional (1D) estimate provided by the Bethe ansatz and a good agreement between the finite-size scaling of the energies. The spin-spin correlation functions exhibit a power-law decay with varying exponents for different anisotropies. Though the lack of N\'eel order makes the staggered flux state energetically unfavorable in the symmetric case $\ensuremath{\gamma}=1$, it appears to capture the essence of the system close to 1D. Hence we believe that the staggered flux state provides an interesting starting point to explore the crossover from quantum disordered chains to the N\'eel ordered two-dimensional square lattices.

We study the continuum limit of the quantum nonlinear \ensuremath{\sigma} model in 2+1 dimensions and at finite temperature T using both Monte Carlo simulation on large-size lattices (

{100}^{2}

\ifmmode\times\else\texttimes\fi{}8 is our largest-size lattice) and saddle-point approximation. At zero temperature, we find the critical point

{g}_{c}

that separates the quantum disordered phase from the phase with spontaneous symmetry breaking (nonzero staggered magnetization). We calculate the model's renormalization group \ensuremath{\beta} function close to the critical point. Using the \ensuremath{\beta} function, we rescale the correlation lengths calculated at various values of the coupling constant (spin stiffness) and temperature and find that they all collapse on the same curve \ensuremath{\xi}/${a}_{\ensuremath{\sigma}}$=f(T/${T}_{\ensuremath{\sigma}}$). Even though the lattice spacing vanishes, a finite unit of length ${a}_{\ensuremath{\sigma}}$ and a temperature scale ${T}_{\ensuremath{\sigma}}$ is generated via dimensional transmutation. Assuming that the nonlinear \ensuremath{\sigma} model and the spin-(1/2 antiferromagnetic (AF) Heisenberg model are equivalent at low temperature, we relate the units ${a}_{\ensuremath{\sigma}}$ and ${T}_{\ensuremath{\sigma}}$ to the lattice spacing

{a}_{H}

and the AF coupling J of the Heisenberg model so that the correlation lengths obtained from the simulation of the two models agree. In order to achieve this agreement we find that (a) the spin-(1/2 AF Heisenberg model should order at T=0 and (b) the relationship between the scales ${a}_{\ensuremath{\sigma}}$,${T}_{\ensuremath{\sigma}}$ and

{a}_{H}

, J is obtained, and f(T/${T}_{\ensuremath{\sigma}}$) can be accurately approximated by an exponential of ${T}_{\ensuremath{\sigma}}$/T below

{g}_{c}

. We obtain a reasonable fit to the neutron scattering data of the insulator

{\mathrm{La}}_{2}

{\mathrm{CuO}}_{4}

by taking J=1270 K, a value close to that reported by Raman scattering experiments.

H

. Using quantum Monte Carlo simulations to obtain unbiased results, we propagate an initial state with maximal order parameter

{m}_{s}^{z}

(the staggered magnetization) in the

z

spin direction and monitor the expectation value $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$ as a function of imaginary time $\ensuremath{\tau}$. Results for different system sizes (lengths)

L

exhibit an initial essentially size independent relaxation of $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$ toward its value in the infinite-size spontaneously symmetry broken state, followed by a strongly size dependent final decay to zero when the

\mathrm{O}(3)

rotational symmetry of the order parameter is restored. We develop a generic finite-size scaling theory that shows the relaxation time diverges asymptotically as

{L}^{z}

, where

z

is the dynamic exponent of the low-energy excitations. We use the scaling theory to develop a practical way of extracting the dynamic exponent from the numerical finite-size data, systematically eliminating scaling corrections. We apply the method to spin-

1/2

Heisenberg antiferromagnets on two different lattice geometries: the standard two-dimensional (2D) square lattice and a site-diluted 2D square lattice at the percolation threshold. In the 2D case we obtain

z=2.001(5)

, which is consistent with the known value

z=2

, while for the site-diluted lattice we find

z=3.90(1)

z=2.056(8){D}_{f}

, where

{D}_{f}=91/48

is the fractal dimensionality of the percolating system. This is an improvement on previous estimates of $z\ensuremath{\approx}3.7$. The scaling results also show a fundamental difference between the two cases; for the 2D square lattice, the data can be collapsed onto a common scaling function even when $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$ is relatively large, reflecting the Anderson tower of quantum rotor states with a common dynamic exponent

z=2

. For the diluted 2D square lattice, the scaling works well only for small $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$, indicating a mixture of different relaxation-time scalings between the low-energy states. Nevertheless, the low-energy dynamic here also corresponds to a tower of excitations.

We compute the von Neumann and generalized R\'enyi entanglement entropies in the ground-state of the spin-

1/2

antiferromagnetic Heisenberg model on the square lattice using the modified spin-wave theory for finite lattices. The addition of a staggered magnetic field to regularize the zero modes associated with symmetry breaking is shown to be essential for obtaining well-behaved values for the entanglement entropy. The von Neumann and R\'enyi entropies obey an area law with additive logarithmic corrections, and are in good quantitative agreement with numerical results from valence bond quantum Monte Carlo and density matrix renormalization group calculations. We also compute the spin fluctuations and observe a multiplicative logarithmic correction to the area law in excellent agreement with quantum Monte Carlo calculations.