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arXiv:1202.3118 · Physical Review B

Finite-size scaling and boundary effects in two-dimensional valence-bond solids

Anders W. Sandvik

Published Apr 3, 2012• 73 citations
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Abstract

Various lattice geometries and boundary conditions are used to investigate valence-bond-solid (VBS) ordering in the ground state of an $S=1/2$ square-lattice quantum spin model---the $J$-$Q$ model, in which four- or six-spin interactions $Q$ are added to the standard Heisenberg exchange $J$. Ground-state results for finite systems (with up to thousands of spins) are obtained using an unbiased projector quantum Monte Carlo method. It is found that great care has to be taken when extrapolating the order parameter to infinite lattice size, in particular, in cylinder geometry. Even though strong VBS order exists in two dimensions, and is established clearly with increasing system size on $L\ifmmode\times\else\texttimes\fi{}L$ lattices (or ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ lattices with a fixed aspect ratio ${L}_{x}/{L}_{y}$ of order 1), only short-range VBS correlations are observed on long cylinders (when ${L}_{x}\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed ${L}_{y}$). The correlation length increases with the cylinder width, until long-range order sets in at a ``critical'' width. This width is very large even when the 2D order is relatively strong. For example, for a system in which the order parameter is $70%$ of the largest possible value, ${L}_{y}=8$ is required for ordering. Extrapolations of the VBS order parameter based on correlation functions (the square of the order parameter) for small $L\ifmmode\times\else\texttimes\fi{}L$ lattices can also be misleading. For a $20%$-ordered system, results for $L$ up to $\ensuremath{\approx}$20 appear to extrapolate clearly to a vanishing order parameter, while for larger lattices the scaling behavior crosses over and extrapolates to a nonzero value (with exponentially small finite-size corrections). The VBS order parameter also exhibits interesting edge effects related the known emergent U(1) symmetry close to a ``deconfined'' critical point, which, if not considered properly, can lead to wrong conclusions for the thermodynamic limit. The observed finite-size behavior for small $L\ifmmode\times\else\texttimes\fi{}L$ lattices and long cylinders is very similar to that predicted for a ${Z}_{2}$ spin liquid. The results therefore raise concerns about recent numerical work claiming ${Z}_{2}$ spin-liquid ground states in 2D frustrated quantum spin systems, in particular, the Heisenberg model with nearest and next-nearest-neighbor couplings. Based on the results presented so far, a VBS state in this system cannot be ruled out.

DOI
10.1103/physrevb.85.134407

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Various lattice geometries and boundary conditions are used to investigate valence-bond-solid (VBS) ordering in the ground state of an S=1/2S=1/2S=1/2 square-lattice quantum spin model---the JJJ-QQQ model, in which four- or six-spin interactions QQQ are added to the standard Heisenberg exchange JJJ. Ground-state results for finite systems (with up to thousands of spins) are obtained using an unbiased projector quantum Monte Carlo method. It is found that great care has to be taken when extrapolating the order parameter to infinite lattice size, in particular, in cylinder geometry. Even though strong VBS order exists in two dimensions, and is established clearly with increasing system size on L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL lattices (or Lx\ifmmode×\else\texttimes\fiLy{L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}Lx​\ifmmode×\else\texttimes\fiLy​ lattices with a fixed aspect ratio Lx/Ly{L}_{x}/{L}_{y}Lx​/Ly​ of order 1), only short-range VBS correlations are observed on long cylinders (when Lx\ensuremath→\ensuremath∞{L}_{x}\ensuremath{\rightarrow}\ensuremath{\infty}Lx​\ensuremath→\ensuremath∞ at fixed Ly{L}_{y}Ly​). The correlation length increases with the cylinder width, until long-range order sets in at a critical'' width. This width is very large even when the 2D order is relatively strong. For example, for a system in which the order parameter is $70%$ of the largest possible value, ${L}_{y}=8$ is required for ordering. Extrapolations of the VBS order parameter based on correlation functions (the square of the order parameter) for small $L\ifmmode\times\else\texttimes\fi{}L$ lattices can also be misleading. For a $20%$-ordered system, results for $L$ up to $\ensuremath{\approx}$20 appear to extrapolate clearly to a vanishing order parameter, while for larger lattices the scaling behavior crosses over and extrapolates to a nonzero value (with exponentially small finite-size corrections). The VBS order parameter also exhibits interesting edge effects related the known emergent U(1) symmetry close to a deconfined'' critical point, which, if not considered properly, can lead to wrong conclusions for the thermodynamic limit. The observed finite-size behavior for small L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL lattices and long cylinders is very similar to that predicted for a Z2{Z}_{2}Z2​ spin liquid. The results therefore raise concerns about recent numerical work claiming Z2{Z}_{2}Z2​ spin-liquid ground states in 2D frustrated quantum spin systems, in particular, the Heisenberg model with nearest and next-nearest-neighbor couplings. Based on the results presented so far, a VBS state in this system cannot be ruled out.
We study the S=1/2S=1/2S=1/2 Heisenberg (JJJ) model on the two-dimensional (2D) square lattice in the presence of additional higher-order spin interactions (QQQ) which lead to a valence-bond-solid (VBS) ground state. Using quantum Monte Carlo simulations, we analyze the thermal VBS transition. We find continuously varying exponents, with the correlation-length exponent $\ensuremath{\nu}$ close to the Ising value for large Q/JQ/JQ/J and diverging when Q/JQ/JQ/J approaches the quantum-critical point (the critical temperature ${T}_{c}\ensuremath{\rightarrow}0$). We identify the transition with a class of conformal field theories with charge c=1c=1c=1 and critical exponents varying between those of the 2D Ising model and the Kosterlitz-Thouless (KT) fixed point. We find explicit evidence for KT physics by studying the emergence of U(1)U(1)U(1) symmetry of the order parameter at T=TcT={T}_{c}T=Tc​ when ${T}_{c}\ensuremath{\rightarrow}0$.
Using quantum Monte Carlo simulations, we study effects of disorder on the S=1/2S=1/2S=1/2 Heisenberg model with exchange constant JJJ on the square lattice supplemented by multispin interactions QQQ. It was found recently [L. Liu et al., Phys. Rev. X 8, 041040 (2018)] that the ground state of this $J\ensuremath{-}Q$ model with random couplings undergoes a quantum phase transition from the N\'eel antiferromagnetic state into a randomness-induced spin-liquid-like state that is a close analog to the well known random-singlet (RS) state of the Heisenberg chain with random couplings. This 2D RS state arises from a spontaneously symmetry-broken fourfold degenerate columnar valence-bond solid that is broken up by the disorder into finite domains, with spinons localized at topological defects. The interacting spinons form a critical collective many-body state without magnetic long range order but with the mean spin-spin correlations decaying with distance rrr as ${r}^{\ensuremath{-}2}$, as in the one-dimensional RS state. The dynamic exponent $z\ensuremath{\ge}2$, varying continuously with the model parameters. In this work, we further investigate the properties of the RS state in the $J\ensuremath{-}Q$ model with random QQQ couplings. We study the temperature dependence of the specific heat and various susceptibilities for large enough systems to reach the thermodynamic limit. We also analyze the size dependence of the critical magnetic order parameter and its susceptibility in the ground state. For all these quantities, we find consistency with the conventional quantum-critical scaling laws when the condition implied by the ${r}^{\ensuremath{-}2}$ form of the spin correlations is imposed. In particular, all the different quantities can be explained by the same value of the dynamic exponent zzz at fixed model parameters. We argue that the RS state identified in the $J\ensuremath{-}Q$ model corresponds to a generic renormalization group fixed point that can be reached in many quantum magnets with random couplings and that it has already been observed experimentally.
We report large-scale quantum Monte Carlo calculations at the T=0 antiferromagnetic to valence-bond-solid (VBS) transition of a two-dimensional S=1/2 XY model with four-spin interactions. Finite-size scaling suggests a discontinuous spin stiffness, but a continuous VBS order-parameter. We propose that this is a continuous VBS transition without critical spin fluctuations. We also argue that the system is close to a point, in an extended parameter space, where both the magnetic and VBS fluctuations are critical--possibly a deconfined quantum-critical point.
We study the spin-1/2 Heisenberg model on the square lattice with first- and second-neighbor antiferromagnetic interactions J(1) and J(2), which possesses a nonmagnetic region that has been debated for many years and might realize the interesting Z(2) spin liquid. We use the density matrix renormalization group approach with explicit implementation of SU(2) spin rotation symmetry and study the model accurately on open cylinders with different boundary conditions. With increasing J(2), we find a Néel phase and a plaquette valence-bond (PVB) phase with a finite spin gap. From the finite-size scaling of the magnetic order parameter, we estimate that the Néel order vanishes at J(2)/J(1)≃0.44. For 0.5<J(2)/J(1)<0.61, we find dimer correlations and PVB textures whose decay lengths grow strongly with increasing system width, consistent with a long-range PVB order in the two-dimensional limit. The dimer-dimer correlations reveal the s-wave character of the PVB order. For 0.44<J(2)/J(1)<0.5, spin order, dimer order, and spin gap are small on finite-size systems, which is consistent with a near-critical behavior. The critical exponents obtained from the finite-size spin and dimer correlations could be compatible with the deconfined criticality in this small region. We compare and contrast our results with earlier numerical studies.
Using a loop–cluster algorithm we investigate the spin- Heisenberg antiferromagnet on a square lattice with exchange coupling J and an additional four-spin interaction of strength Q . We confirm the existence of a phase transition separating antiferromagnetism at J / Q > J c / Q from a valence bond solid (VBS) state at J / Q < J c / Q . Although our Monte Carlo data are consistent with those of previous studies, we do not confirm the existence of a deconfined quantum critical point. Instead, using a flowgram method on lattices as large as 80 2 , we find evidence for a weak first-order phase transition. We also present a detailed study of the antiferromagnetic phase. For J / Q > J c / Q the staggered magnetization, the spin stiffness and the spinwave velocity of the antiferromagnet are determined by fitting Monte Carlo data to analytic results from the systematic low-energy effective field theory for magnons. Finally, we also investigate the physics of the VBS state at J / Q < J c / Q and we show that long but finite antiferromagnetic correlations are still present.
We utilize density matrix renormalization group and fully augmented matrix product states (FAMPS) methods to investigate the valence bond solid (VBS) phase in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square lattice Heisenberg model. To differentiate between the columnar valence bond solid and plaquette valence bond solid (PVBS) phases, we introduce an anisotropy ${\mathrm{\ensuremath{\Delta}}}_{y}$ in the nearest-neighboring coupling in the yyy direction, aiming at detecting the possible spontaneous rotational symmetry breaking in the VBS phase. In the calculations, we push the bond dimension to as large as D=25000D=25\phantom{\rule{0.16em}{0ex}}000D=25000 in FAMPS, simulating systems at a maximum size of $14\ifmmode\times\else\texttimes\fi{}14$. With a careful extrapolation of the truncation errors and appropriate finite-size scaling, followed by finite ${\mathrm{\ensuremath{\Delta}}}_{y}$ scaling analysis of the VBS dimer order parameters, we identify the VBS phase as a PVBS type, meaning there is no spontaneous rotational symmetry breaking in the VBS phase. This paper not only supports the presence of PVBS order in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square lattice Heisenberg model, but also highlights the capabilities of FAMPS in the study of two-dimensional quantum many-body systems.