We study the quantum phase diagram of the spin-1/2 Heisenberg model on the kagomé lattice with first-, second-, and third-neighbor interactions J1, J2, and J3 by means of density matrix renormalization group. For small J2 and J3, this model sustains a time-reversal invariant quantum spin liquid phase. With increasing J2 and J3, we find in addition a q = (0, 0) Néel phase, a chiral spin liquid phase, an apparent valence-bond crystal phase, and a complex non-coplanar magnetically ordered state with spins forming the vertices of a cuboctahedron known as a cuboc1 phase. Both the chiral spin liquid and cuboc1 phase break time reversal symmetry in the sense of spontaneous scalar spin chirality. We show that the chiralities in the chiral spin liquid and cuboc1 are distinct, and that these two states are separated by a strong first order phase transition. The transitions from the chiral spin liquid to both the q = (0, 0) phase and to time-reversal symmetric spin liquid, however, are consistent with continuous quantum phase transitions.

Low dimensional quantum magnets are interesting because of the emerging collective behavior arising from strong quantum fluctuations. The one-dimensional (1D) S = 1/2 Heisenberg antiferromagnet is a paradigmatic example, whose low-energy excitations, known as spinons, carry fractional spin S = 1/2. These fractional modes can be reconfined by the application of a staggered magnetic field. Even though considerable progress has been made in the theoretical understanding of such magnets, experimental realizations of this low-dimensional physics are relatively rare. This is particularly true for rare-earth-based magnets because of the large effective spin anisotropy induced by the combination of strong spin–orbit coupling and crystal field splitting. Here, we demonstrate that the rare-earth perovskite YbAlO3 provides a realization of a quantum spin S = 1/2 chain material exhibiting both quantum critical Tomonaga–Luttinger liquid behavior and spinon confinement–deconfinement transitions in different regions of magnetic field–temperature phase diagram.

We study the spin liquid candidate of the spin-1/2 J1-J2 Heisenberg antiferromagnet on the triangular lattice by means of density matrix renormalization group (DMRG) simulations. By applying an external Aharonov-Bohm flux insertion in an infinitely long cylinder, we find unambiguous evidence for gapless U (1) Dirac spin liquid behavior. The flux insertion overcomes the finite size restriction for energy gaps and clearly shows gapless behavior at the expected wave-vectors. Using the DMRG transfer matrix, the low-lying excitation spectrum can be extracted, which shows characteristic Dirac cone structures of both spinon-bilinear and monopole excitations. Finally, we confirm that the entanglement entropy follows the predicted universal response under the flux insertion.

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We study the spin-1/2 Heisenberg model on the triangular lattice with the nearest-neighbor J1 > 0, the nextnearest-neighobr J2 > 0 Heisenberg interactions, and the additional scalar chiral interaction Jχ( Si × Sj) · S k for the three spins in all the triangles using large-scale density matrix renormalization group calculation on cylinder geometry. With increasing J2 (J2/J1 ≤ 0.3) and Jχ (Jχ/J1 ≤ 1.0) interactions, we establish a quantum phase diagram with the magnetically ordered 120 • phase, stripe phase, and non-coplanar tetrahedral phase. In between these magnetic order phases, we find a chiral spin liquid (CSL) phase, which is identified as a ν = 1/2 bosonic fractional quantum Hall state with possible spontaneous rotational symmetry breaking. By switching on the chiral interaction, we find that the previously identified spin liquid in the J1 − J2 triangular model (0.08 J2/J1 0.15) shows a phase transition to the CSL phase at very small Jχ. We also compute spin triplet gap in both spin liquid phases, and our finite-size results suggest large gap in the odd topological sector but small or vanishing gap in the even sector. We discuss the implications of our results to the nature of the spin liquid phases.

We study the effects of local projective measurements on the quantum quench dynamics. As a concrete example, one-dimensional Bose-Hubbard model is simulated by using matrix product state and time evolving block decimation. We map out a global phase diagram in terms of the measurement rate in spatial space and time domain, which demonstrates a volume-to-area law entanglement phase transition. When the measurement rates reach the critical values, we observe a logarithmic growth of entanglement entropy as the sub-system size or evolved time increases. This is akin to the character in the many-body localization, implying a general picture of the dynamical transitions separating quantum systems with different entanglement features. Moreover, we find that the probability distribution of the single-site entanglement entropy distinguishes the volume and area law phases just as the case of disorder-induced many-body localization. This suggests that the different type entanglement phase transitions may be understood as a nonlocalized-to-localized transition. We also investigate the scaling behavior of entanglement entropy and mutual information between two separated intervals, which is indicative of a single universality class and thus suggests a possible unified description of this transition.

The topological order is equivalent to the pattern of long-range quantum entanglements, which cannot be measured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice boson models. We identify multiple independent minimal entangled states (MESs) in the groundstate manifold on a torus. The extracted modular S matrix from MESs faithfully demonstrates the Majorana quasiparticle or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings support that MESs manifest the eigenstates of quasiparticles for the non-Abelian topological states and encode the full information of the topological order. Introduction.-One of the most striking phenomena in the fractional quantum Hall (FQH) system is the emergent fractionalized quasiparticles obeying Abelian [1] or non-Abelian [2-4] braiding statistics. Interchange of two Abelian quasiparticles leads to a nontrivial phase acquired by their wavefunction, whereas interchange of two non-Abelian quasiparticles results in an operation of matrix to the degenerating groundstate space and the final state will depend on the order of operations being carried out. The non-Abelian quasiparticles and its braiding statistics are fundamentally important for understanding the topological order and also have potential application in topologically fault-tolerant quantum computation [5][6][7]. So far, such quasiparticles have not been definitely identified in nature. However, it is generally believed that they exist in the FQH systems at filling factor ν = 5/2 [8] and 12/5 [9], described by the Moore-Read (MR) [2,10,11] and Read-Rezayi (RR) states [4,12], respectively. The topological band model for optical lattices with bosonic particles is another promising platform to realize the non-Abelian topological states [13][14][15][16][17][18][19][20][21][22][23]. In the MR and RR states, the quasiparticles satisfy the following characteristic fusion rules that specify how the quasiparticles combine and fuse into more than one type of quasiparticles [7]:

Multicomponent quantum Hall systems with internal degrees of freedom provide a fertile ground for the emergence of exotic quantum liquids. Here we investigate the possibility of non-Abelian topological order in the half-filled fractional quantum Hall (FQH) bilayer system driven by the tunneling effect between two layers. By means of the state-of-the-art density-matrix renormalization group, we unveil "finger print" evidence of the non-Abelian Moore-Read Pfaffian state emerging in the intermediate-tunneling regime, including the ground-state degeneracy on the torus geometry and the topological entanglement spectroscopy (entanglement spectrum and topological entanglement entropy) on the spherical geometry, respectively. Remarkably, the phase transition from the previously identified Abelian (331) Halperin state to the non-Abelian Moore-Read Pfaffian state is determined to be continuous, which is signaled by the continuous evolution of the universal part of the entanglement spectrum, and discontinuities in the excitation gap and the derivative of the ground-state energy. Our results not only provide a "proof-of-principle" demonstration of realizing a non-Abelian state through coupling different degrees of freedom, but also open up a possibility in FQH bilayer systems for detecting different chiral p−wave pairing states.