Abstract We present several features of a thermal pure quantum matrix product state (TPQ-MPS). TPQ-MPS can simulate finite temperature quantum many-body systems with a numerical cost comparable to the matrix product algorithm for the ground state in one-dimensional systems. Starting from a random matrix product state with auxiliary sites attached to the edges of the system, we successively operate the Hamiltonian with a shift-constant and obtain a series of TPQ-MPS that represents the microcanonical pure state at the highest temperature to the one at the lowest available temperature. How the parameters of the TPQ-MPS protocol influence the effective bond dimension χ of the matrix and the temperature distribution of these microcanonical states is examined.
While most of the features of the Kitaev honeycomb model are already clarified from the combined
Z2 plaquette and Majorana fermion picture, we show that retrieving the exact quantum many-body form gives a clue to fix the remaining issues on its spin liquid property. We provide an exact two-dimensional tensor network description of a full set of solutions from a random product state, by sequential quantum-number projection and Gutzwiller projection, both expressed as local matrix-product operators of dimension
D=2. The ground and excited states share the same bond dimension yet exhibit distinct Schmidt spectra, distinguishing their area-law and volume-law entanglement. We demonstrate that the topological entanglement entropy emerges {\it for all eigenstates only after} the Gutzwiller projection onto the physical spin Hilbert space, where the gauge constraint removes exactly half of the Schmidt states. This establishes that excited states are likewise topologically ordered. We further show that there exists a projection-induced collapse of the wave function into a single gauge sector in the Parton mean-field states that is known to give a good description of the spin liquids. The demonstrations on the
S=1/2 kagome and triangular Heisenberg models suggest that the present perspective is applied to a wider class of models hosting
Z2 spin liquids. Our finding visualizes a bulk-boundary correspondence with the Wilson-loop characterization of topological order.
We propose a way to construct a thermal pure quantum matrix product state (TPQ-MPS) that can simulate finite-temperature quantum many-body systems with a minimal numerical cost comparable to the matrix product algorithm for the ground state.The MPS was originally designed for the wave function with area law entanglement.However, by attaching the auxiliary sites to the edges of the random matrix product state, we find that the degree of entanglement is automatically tuned so as to recover the volume law of the entanglement entropy that characterizes the TPQ state.The finite-temperature physical quantities of the transverse Ising and the spin-1/2 Heisenberg chains evaluated by a TPQ-MPS show excellent agreement even for bond dimension ∼10-20 with those of the exact results.
The long-sought quantum spin liquid is a quantum-entangled magnetic state leading to the fractionalization of spin degrees of freedom. Quasiparticles emergent from the fractionalization affect not only the ground-state properties but also thermodynamic behavior in a peculiar manner. Here we investigate how the spin dynamics evolves from the high-temperature paramagnet to the quantum spin-liquid ground state, for the Kitaev spin model describing the fractionalization into itinerant matter fermions and localized
Z2 fluxes. Beyond the previous studies [J. Yoshitake, J. Nasu, and Y. Motome, Phys. Rev. Lett. 117, 157203 (2016); J. Yoshitake, J. Nasu, Y. Kato, and Y. Motome, Phys. Rev. B 96, 024438 (2017)], in which the mean-field nature of the cluster dynamical mean-field theory prevented us from studying low-temperature properties, we develop a numerical technique by applying the continuous-time quantum Monte Carlo (CTQMC) method to statistical samples generated by the quantum Monte Carlo (QMC) method in a Majorana fermion representation. This QMC+CTQMC method is fully unbiased and enables us to investigate the low-temperature spin dynamics dominated by thermally excited
Z2 fluxes, including the unconventional phase transition caused by
Z2 flux loops in three dimensions, which was unreachable by the previous methods. We apply this technique to the Kitaev model in both two and three dimensions. Our results clearly distinguish two cases: while the dynamics changes smoothly through the crossover in the two-dimensional honeycomb case, it exhibits singular behaviors at the phase transition in the three-dimensional hyperhoneycomb case. We show that the low-temperature spin dynamics is a sensitive probe for thermally fluctuating
Z2 fluxes that behave very differently between two and three dimensions.
We study the 3D Kitaev and Kitaev-Heisenberg models, respectively, on the hyperhoneycomb and hyperoctagon lattices, both at zero and finite-temperature, in the thermodynamic limit. Our analysis relies on advanced tensor network (TN) simulations based on graph projected entangled-pair states (gPEPS). We map out the TN phase diagrams of the models and characterize their underlying gapped and gapless phases both at zero and finite temperature. In particular, we demonstrate how cooling down the hyperhoneycomb system from high temperature leads to fractionalization of spins to Majorana fermions and gauge fields that occurs in two separate temperature regimes, leaving their fingerprint on specific heat as a double-peak feature as well as on other quantities such as the thermal entropy, spin-spin correlations, and bond entropy. Using the Majorana representation of the Kitaev model, we further show that the low-temperature thermal transition to the Kitaev quantum spin liquid (QSL) phase is associated with the nontrivial Majorana band topology and the presence of Weyl nodes, which manifests itself via nonvanishing Chern number and finite thermal Hall conductivity. Beyond the pure Kitaev limit, we study the 3D Kitaev-Heisenberg (KH) model on the hyperoctagon lattice and extract the full phase diagram for different Heisenberg couplings. We further explore the thermodynamic properties of the magnetically-ordered regions in the KH model and show that, in contrast to the QSL phase, here the thermal phase transition follows the standard Landau symmetry-breaking theory.
We propose a framework to design concurrently a frustration-free quantum many-body Hamiltonian and its numerically exact ground states on a sufficiently large finite-size cluster in one and two dimensions using an elementary matrix product state (MPS) representation. Our approach strategically chooses a local cluster Hamiltonian, which is arranged to overlap with neighboring clusters on a designed lattice. The frustration-free Hamiltonian is given as the sum of the cluster Hamiltonians by ensuring that there exists a state that has its local submanifolds as the lowest-energy eigenstate of every cluster. The key to find such a solution is a systematic protocol, which projects out excited states on every cluster using MPS and effectively entangles the cluster states. This algorithm not only offers an exact solution to general frustration-free Hamiltonian, but even provide the excited eigenstates by setting a proper target. It has several advantages, including the ability to achieve the calculation of states at nearly equal cost in one and two dimensions including those belonging to gapless or long-range entangled classes, flexibility in designing Hamiltonians unbiasedly across various forms of models, and numerically feasible validation of exactness through energy calculations. It enables the exploration of exact phase boundaries and the analysis of even a spatially nonuniform random system, providing platforms for quantum simulations and benchmarks.