We consider a family of SU(2)-symmetric Projected Entangled Paired States (PEPS) on the square lattice, defining colored-Resonating Valence Bond (RVB) states, to describe the quantum disordered phase of the J_1-J_2 J 1 − J 2 frustrated Heisenberg model. For J_2/J_1\sim 0.55 J 2 / J 1 ∼ 0.55 we show the emergence of critical (algebraic) dimer-dimer correlations – typical of Rokhsar-Kivelson (RK) points of quantum dimer models on bipartite lattices – while, simultaneously, the spin-spin correlation length remains short. Our findings are consistent with a spin liquid or a weak Valence Bond Crystal in the neighborhood of an RK point.
The nature of the quantum spin liquid (QSL) phase in the frustrated
J1-
J2 Heisenberg model on the triangular lattice remains an open and actively debated problem. In this work, we employ the infinite projected entangled-pair state (PEPS) to systematically investigate the model under different symmetry constraints. Our simulations reveal a direct transition from the
120∘ Néel state to a putative QSL at
J2/J1≈0.08, signaled by the collapse of magnetic order. We further show that, through either an appropriate unitary rotation or spontaneous spin long-range order, the stripe antiferromagnetic phase can also be accurately captured within the infinite PEPS framework. A central focus of our study is the role played by the PEPS symmetry in approximating the QSL ground-state sandwiched between the two magnetic phases. We first found that a fully-symmetric topological
Z2 Resonating Valence Bond state, which can be written as a simple PEPS with bond dimension
D=3, exhibits a reasonably good variational energy. Motivated by this finding, we have further constructed generic
Z2-symmetric PEPS of larger bond dimension (up to
D=7). We found that, under wavefunction optimization, spinons condense and, simultaneously, topological vison excitations get confined, hence precluding
Z2 topological order. This strongly indicates the gapless (or critical) nature of the QSL phase, which is most naturally consistent with a U(1) Dirac spin liquid scenario.
The presence of strong-interactions and entanglement in a quantum many-body system can lead to several exotic states of matter. Quantum Spin liquids are one such set of exotic states that have attracted significant interest due to their interesting properties and possible technological applications. The rapid progress in cold atom experiment setups, particularly the Rydberg atom simulators, has opened up new avenues to study these spin liquids in one and two-dimensions. This calls for advanced theoretical techniques to investigate the non-equilibrium properties of isolated pure quantum states. This thesis aims to develop numerical algorithms to study the non-equilibrium properties of spin liquids on a square lattice following a quantum quench using the framework of tensor networks. A variational form of Projected Entangled Pair State (PEPS) ansatz is used to describe the spin liquids. A new symmetric form of the Corner Transfer Matrix Renormalization group (CTMRG), the algorithm necessary to construct the reduced density matrix of the wavefunction in the thermodynamic limit for a complex-valued PEPS wavefunction, is described. To study the quench dynamics, two algorithms called simple-update and full update, along with the necessary modifications to adapt them to study spin liquid wavefunctions, are explained. The tools are then used to study the non-equilibrium dynamics of the resonating valence bond (RVB) state with nearest-neighbor (NN) valence bond configurations, a spin-liquid state that is known to exhibit critical dimer-dimer correlations, within the PEPS manifold.
Within the Projected Entangled Pair State (PEPS) tensor network formalism, a simple update (SU) method has been used to investigate the time evolution of a two-dimensional U(1) critical spin-1/2 spin liquid under Hamiltonian quench [Phys. Rev. B 106, 195132 (2022)]. Here we introduce two different variational frameworks to describe the time dynamics of SU(2)-symmetric translationally-invariant PEPS, aiming to improve the accuracy. In one approach, after using a Trotter-Suzuki decomposition of the time evolution operator in term of two-site elementary gates, one considers a single bond embedded in an environment approximated by a Corner Transfer Matrix Renormalization Group (CTMRG). A variational update of the two tensors on the bond is performed under the application of the elementary gate and then, after symmetrization of the site tensors, the environment is updated. In the second approach, a cluster optimization is performed on a finite (periodic) cluster, maximizing the overlap of the exact time-evolved state with a symmetric finite-size PEPS ansatz. Observables are then computed on the infinite lattice contracting the infinite-PEPS (iPEPS) by CTMRG. We show that the variational schemes outperform the SU method and remain accurate over a significant time interval before hitting the entanglement barrier. Studying the spectrum of the transfer matrix, we find that the asymptotic correlations are very well preserved under time evolution, including the critical nature of the singlet correlations, as expected from the Lieb-Robinson (LR) bound theorem. Consistently, the system (asymptotic) boundary is found to be described by the same Conformal Field Theory of central charge c = 1 c = 1 during time evolution. We also compute the time-evolution of the short distance spin-spin correlations and estimate the LR velocity.
A one-parameter family of long-range resonating-valence-bond (RVB) state on the square lattice was previously proposed to describe a critical spin liquid (SL) phase of the spin-1/2 frustrated Heisenberg model. We provide evidence that this RVB state in fact also realizes a topological (long-range entangled)
Z2 SL, limited by two transitions to critical SL phases. The topological phase is naturally connected to the
Z2 gauge symmetry of the local tensor. This Rapid Communication shows that, on one hand, spin-1/2 topological SL with
C4v point-group symmetry and
SU(2) spin rotation symmetry exists on the square lattice and, on the other hand, criticality and nonbipartiteness are compatible. We also point out that strong similarities between our phase diagram and the ones of classical interacting dimer models suggest both can be described by similar Kosterlitz-Thouless transitions. This scenario is further supported by the analysis of the one-dimensional boundary state. Forms of parent Hamiltonians hosting the
Z2 SL are suggested.
We generalize the construction of the spin-1/2 SU(2) resonating valence bond (RVB) state to the case of the self-conjugate
6 representation of SU(4). As for the case of SU(2) [J.-Y. Chen and D. Poilblanc, Phys. Rev. B 97, 161107(R) (2018)], we use the projected entangled pair state (PEPS) formalism to derive a simple (two-dimensional) family of generalized SU(4) RVB states on the square lattice. We show that, when longer-range SU(4)-singlet bonds are included, a local gauge symmetry is broken down from U(1) to
Z2, leading to the emergence of a short-range spin liquid. Evidence for the topological nature of this spin liquid is provided by the investigation of the Renyi entanglement entropy of infinitely long cylinders and of the modular matrices. Relevance to microscopic models and experiments of ultracold atoms is discussed.