The Shastry-Sutherland model is an effective model of the layered material SrCu_{2}(BO_{3})_{2}, which exhibits an extremely rich phase diagram as a function of pressure and magnetic field. Motivated by the recent controversy regarding its phase diagram at zero magnetic field, we perform large-scale simulations based on infinite projected entangled-pair states, a two-dimensional tensor network Ansatz to represent the ground state directly in the thermodynamic limit. By employing the latest optimization techniques, we obtain variational states with lower energy than previous results obtained from other methods. Using systematic extrapolations to the exact infinite bond dimension limit, our simulations reveal a narrow quantum spin liquid phase between the plaquette and antiferromagnetic phases in the range 0.785(5)≤J^{'}/J≤0.82(1).
Infinite Projected entangled-pair state (iPEPS) is a two-dimensional tensor network (TN) ansatz targeting the ground states of lattice models directly in the thermodynamic limit. It is the most successful among the true two-dimensional TN methods, challenging the more common density matrix renormalization group computations conducted on a finite-width cylinders. iPEPS, being a variational method does not suffer from the infamous sign problem, and hence could be readily applied to both fermionic problems off the half-filling and frustrated spin system. Using the imaginary-time optimization, the so called Full Update, led to the celebrated results of iPEPS such as the analysis of stripes in doped t-J and Hubbard models or the insight into the magnetization plateaus in Shastry-Sutherland model. However, the physics of the continuous phase transitions from the symmetry-broken Neel phase into the valence-bond solids or spin-liquids in frustrated magnets is still eluding iPEPS. The main difficulty lies in the optimization, since Full Update is just not precise enough. In the pursuit of the best variational iPEPS states, the gradient methods were put forward in 2016 by Corboz and Vanderstraeten. However, due to their complex nature, they have not been widely adopted especially for iPEPS ansatze with larger unit cells. We will introduce an alternative approach first applied in the context of iPEPS by Liao et al., based on the well established method of Algorithmic differentiation (AD) which allows us to evaluate gradients for iPEPS in a robust and conceptually simple way. Finally, we will present the results of our investigation into the physics of frustrated magnets putting AD to work.
We use the recently developed tensor network algorithm based on infinite projected entangled pair states (iPEPS) to study the phase diagram of frustrated antiferromagnetic
J1-
J2 Heisenberg model on a checkerboard lattice. The simulation indicates a N\'eel ordered phase when
J2<0.88J1, a plaquette valence bond solid state when
0.88<J2/J1<1.11, and a stripe phase when
J2>1.11J1, with two first-order transitions across the phase boundaries. The calculation shows that the cross dimer state proposed before is unlikely to be the ground state of the model, although such a state indeed arises as a metastable state in some parameter region.
The two-dimensional infinite projected entangled pair state tensor network is evolved in imaginary time with the full update (FU) algorithm to simulate the Shastry-Sutherland model in a magnetic field at finite temperature directly in the thermodynamic limit. We focus on the phase transition into the
m=21 magnetization plateau, which was observed in experiments on
SrCu2(BO3)2. For the largest simulated bond dimension, the early evolution in the high-temperature regime is simulated with the simple update (SU) scheme and then, as the correlation length increases, continued with the FU scheme towards the critical regime. We apply a small symmetry-breaking bias field and then extrapolate towards zero bias using a simple scaling theory in the bias field. The combined SU
+ FU scheme provides an accurate estimate of the critical temperature, even though the results could not be fully converged in the bond dimension in the vicinity of the transition. The critical temperature estimate is improved with a generalized scaling theory that combines two divergent length scales: One due to the bias, and the other due to the finite bond dimension. The obtained results are consistent with the transition being in the universality class of the two-dimensional classical Ising model. The estimated critical temperature is 3.5(2) K, which is well above the temperature 2.1 K used in the experiments.
We develop an improved variant of
U(1)-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin-
1/2 square ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the
U(1)-symmetric iPEPS ansatz (for a fixed bond dimension
D) set a better upper bound, improving previous tensor-network-based results. By studying the finite-
D scaling of the magnetically order parameter, we find a N\'eel phase for
J2/J1<0.53. For
0.53<J2/J1<0.61, a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at
J2c1/J1=0.530(5), where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with
J2/J1 and finally a first-order quantum phase transition takes place at
J2c2/J1=0.610(2) to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.
We develop an improved variant of
U(1)-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground state phase diagram of the spin-
1/2 square
J1-
J2 Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the
U(1)-symmetric iPEPS ansatz (for a fixed bond dimension
D) set a better upper bound, improving previous tensor-network-based results. By studying N\'{e}el order parameter and correlation function in the large-
D limit, we find that a deconfined quantum-critical point occurs at
J2c1=0.530(5), where the correlation length of the system diverges as
ξ∼D1.2. The estimated critical anomalous exponents
ηs=0.65(5) and
ηd=1.9(1) for spin and dimer correlations respectively show strong deviation from those of the well-known
J-
Q model, suggesting possibly a different universality class. The pattern of local bond energy for the intermediate phase is compatible with the columnar valence bond solid (VBS) state, with a quite competitive variational ground-state energy in comparison with plaquette VBS one predicted by the density matrix renormalization group. We show that the associated VBS order parameter first monotonically increases with
J2 and finally a first-order quantum phase transition takes place at
J2c2=0.610(3) to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.