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arXiv:2206.06230 · Physical review. B./Physical review. B

Symmetric projected entangled-pair states analysis of a phase transition in coupled spin- 12 ladders

Juraj Hašík, Glen Bigan Mbeng, Sylvain Capponi, Federico Becca, Andreas Martin Läuchli

Published Sep 30, 2022• 3 citations
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Abstract

Infinite projected entangled-pair states (iPEPS) have been introduced to accurately describe many-body wave functions on two-dimensional lattices. In this context, two aspects are crucial: the systematic improvement of the Ansatz by the optimization of its building blocks, i.e., tensors characterized by bond dimension $D$, and the extrapolation scheme to reach the ``thermodynamic'' limit $D\ensuremath{\rightarrow}\ensuremath{\infty}$. Recent advances in variational optimization and scaling based on correlation lengths demonstrated the ability of iPEPS to capture phases with spontaneously broken continuous symmetry such as the antiferromagnetic (N\'eel) one with high fidelity, in addition to valence-bond solids which are already well described by finite-$D$ iPEPS. In contrast, systems in the vicinity of continuous quantum phase transitions still present a challenge for iPEPS, especially when non-Abelian symmetries are involved. Here, we consider the iPEPS Ansatz to describe the continuous transition between the (gapless) antiferromagnet and the (gapped) paramagnet that exists in the $S=1/2$ Heisenberg model on coupled two-leg ladders. In particular, we show how accurate iPEPS results can be obtained down to a narrow interval around criticality and analyze the scaling of the order parameter in the N\'eel phase in a spatially anisotropic situation.

DOI
10.1103/physrevb.106.125154

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Infinite projected entangled-pair states (iPEPS) have been introduced to accurately describe many-body wave functions on two-dimensional lattices. In this context, two aspects are crucial: the systematic improvement of the Ansatz by the optimization of its building blocks, i.e., tensors characterized by bond dimension DDD, and the extrapolation scheme to reach the ``thermodynamic'' limit D\ensuremath→\ensuremath∞D\ensuremath{\rightarrow}\ensuremath{\infty}D\ensuremath→\ensuremath∞. Recent advances in variational optimization and scaling based on correlation lengths demonstrated the ability of iPEPS to capture phases with spontaneously broken continuous symmetry such as the antiferromagnetic (N'eel) one with high fidelity, in addition to valence-bond solids which are already well described by finite-DDD iPEPS. In contrast, systems in the vicinity of continuous quantum phase transitions still present a challenge for iPEPS, especially when non-Abelian symmetries are involved. Here, we consider the iPEPS Ansatz to describe the continuous transition between the (gapless) antiferromagnet and the (gapped) paramagnet that exists in the S=1/2S=1/2S=1/2 Heisenberg model on coupled two-leg ladders. In particular, we show how accurate iPEPS results can be obtained down to a narrow interval around criticality and analyze the scaling of the order parameter in the N'eel phase in a spatially anisotropic situation.
We develop an improved variant of U(1)U(1)U(1)-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin-1/21/21/2 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)U(1)U(1)-symmetric iPEPS ansatz (for a fixed bond dimension D)D)D) set a better upper bound, improving previous tensor-network-based results. By studying the finite-DDD scaling of the magnetically order parameter, we find a N\'eel phase for J2/J1<0.53{J}_{2}/{J}_{1}<0.53J2​/J1​<0.53. For 0.53<J2/J1<0.610.53<{J}_{2}/{J}_{1}<0.610.53<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){J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)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{J}_{2}/{J}_{1}J2​/J1​ and finally a first-order quantum phase transition takes place at J2c2/J1=0.610(2){J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)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.
The U(1)U(1)U(1)-symmetric iPEPS study of the spin-1/21/21/2 square J1J_{1}J1​-J2J_{2}J2​ Heisenberg model
We develop an improved variant of U(1)U(1)U(1)-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground state phase diagram of the spin-1/21/21/2 square J1J_{1}J1​-J2J_{2}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)U(1)U(1)-symmetric iPEPS ansatz (for a fixed bond dimension DDD) set a better upper bound, improving previous tensor-network-based results. By studying N\'{e}el order parameter and correlation function in the large-DDD limit, we find that a deconfined quantum-critical point occurs at J2c1=0.530(5)J^{c_1}_{2}=0.530(5)J2c1​​=0.530(5), where the correlation length of the system diverges as ξ∼D1.2\xi \sim D^{1.2}ξ∼D1.2. The estimated critical anomalous exponents ηs=0.65(5)\eta_{s}=0.65(5)ηs​=0.65(5) and ηd=1.9(1)\eta_{d}=1.9(1)ηd​=1.9(1) for spin and dimer correlations respectively show strong deviation from those of the well-known J\mathcal{J}J-Q\mathcal{Q}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 J2J_2J2​ and finally a first-order quantum phase transition takes place at J2c2=0.610(3)J^{c_2}_{2}=0.610(3)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.
The recent progress in the optimization of two-dimensional tensor networks [H.-J. Liao, J.-G. Liu, L. Wang, and T. Xiang, Phys. Rev. X 9, 031041 (2019)] based on automatic differentiation opened the way towards precise and fast optimization of such states and, in particular, infinite projected entangled-pair states (iPEPS) that constitute a generic-purpose Ansatz for lattice problems governed by local Hamiltonians. In this work, we perform an extensive study of a paradigmatic model of frustrated magnetism, the J_1-J_2 J 1 − J 2 Heisenberg antiferromagnet on the square lattice. By using advances in both optimization and subsequent data analysis, through finite correlation-length scaling, we report accurate estimations of the magnetization curve in the N'eel phase for J_2/J_1 \le 0.45 J 2 / J 1 ≤ 0.45 . The unrestricted iPEPS simulations reveal an U(1) U ( 1 ) symmetric structure, which we identify and impose on tensors, resulting in a clean and consistent picture of antiferromagnetic order vanishing at the phase transition with a quantum paramagnet at J_2/J_1 \approx 0.46(1) J 2 / J 1 ≈ 0.46 ( 1 ) . The present methodology can be extended beyond this model to study generic order-to-disorder transitions in magnetic systems.
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.
An infinite projected entangled-pair state (iPEPS) is a variational tensor network ansatz for two-dimensional wave functions in the thermodynamic limit where the accuracy can be systematically controlled by the bond dimension DDD. We show that for the doped Hubbard model in the strongly correlated regime (U/t=8,n=0.875)(U/t=8,n=0.875)(U/t=8,n=0.875) the iPEPS yields lower variational energies than state-of-the-art variational methods in the large two-dimensional limit, which demonstrates the competitiveness of the method. In order to obtain an accurate estimate of the energy in the exact infinite-DDD limit we introduce and test an extrapolation technique based on a truncation error computed in the iPEPS imaginary time evolution algorithm. The extrapolated energies are compared with accurate quantum Monte Carlo results at half-filling and with various other methods in the doped, strongly correlated regime.
The infinite projected entangled pair states (iPEPS) technique [J. Jordan et al., Phys. Rev. Lett. 101, 250602 (2008)] has been widely used in the recent years to assess the properties of two-dimensional quantum systems, working directly in the thermodynamic limit. This formalism, which is based upon a tensor-network representation of the ground-state wave function, has several appealing features, e.g., encoding the so-called area law of entanglement entropy by construction; still, the method presents critical issues when dealing with the optimization of tensors in order to find the best possible approximation to the exact ground state of a given Hamiltonian. Here, we discuss the obstacles that arise in the optimization by imaginary-time evolution within the so-called simple and full updates and connect them to the emergence of a sharp multiplet structure in the ``virtual'' indices of tensors. In this case, a generic choice of the bond dimension DDD is not compatible with the multiplets and leads to a symmetry breaking (e.g., generating a finite magnetic order). In addition, varying the initial guess, different final states may be reached, with very large deviations in the magnetization value. In order to exemplify this behavior, we show the results of the S=1/2S=1/2S=1/2 Heisenberg model on an array of coupled ladders, for which a vanishing magnetization below the critical interladder coupling is recovered only for selected values of DDD, while a blind optimization with a generic DDD gives rise to a finite magnetization down to the limit of decoupled ladders.