Mapping quantum phase diagrams through classical simulation is notoriously resource-intensive, as even small systems far from the thermodynamic limit demand prohibitive computational effort. The variational quantum eigensolver (VQE) offers a compelling alternative, exploiting approximate ground states to distinguish phases. An appealing proposal, dubbed as Delta-VQE, determines critical points by contrasting variational energies optimized from reference states of distinct phases. Intriguingly, the diagnostic sharpens as circuit depth decreases, highlighting its promise as a resource-conscious probe of quantum criticality. To probe the broader applicability and underlying mechanisms of this approach, we investigate the one-dimensional transverse-field Ising model with a three-spin cluster interaction, a setting in which the Ising transitions are generally situated beyond the self-dual line. We demonstrate that, whenever dual ansätze are employed, Delta-VQE invariably detects the self-dual points rather than the true criticality. In contrast, when ansätze are carefully tailored to embody the competing phases across the boundary, the genuine Ising critical point can be successfully identified with only minor finite-size effects. Our results establish that, while Delta-VQE provides a resource-efficient probe of quantum criticality without requiring precise ground-state preparation, its diagnostic power is fundamentally contingent upon the judicious selection of physically representative ansätze.

We consider the thermalization hypothesis of pure states in nonintegrable quantum spin chains with conserved charges, including the nonintergable variants of the Ising chain with Z_{2} symmetry, XXZ chain with U(1) symmetry, and XXX chain with SU(2) symmetries. Two kinds of pure states are considered: the energy eigenstates and the typical states evolved unitarily from the random product states for a long enough period. We further group the typical states by their expectation values of the conserved charges and consider the fine-grained thermalization hypothesis. We compare the locally (subsystem) reduced states of typical states/eigenstates with the ones of the corresponding thermal ensemble states. Besides the usual thermal ensembles such as the (micro)canonical ensemble without conserved charges and the generalized Gibbs ensemble (GGE) with all conserved charges included, we also consider the so-called partial-GGEs (p-GGEs), which include only part of the conserved charges in the thermal ensemble. Moreover, in the framework of p-GGE, the Hamiltonian and other conserved charges are on an equal footing. The introduction of p-GGEs extends quantum thermalization to a more general scope. The validity of the subsystem thermalization hypothesis can be quantified by the smallness of the relative entropy of the reduced states obtained from the GGE/p-GGE and the typical states/eigenstates. We examine the validity of the thermalization hypothesis by numerically studying the relative entropy demographics. We show that the thermalization hypothesis holds generically for small enough subsystems for various p-GGEs. Thus, our framework extends the universality of quantum thermalization.

We study three aspects of work statistics in the context of the fluctuation theorem for the quantum spin chains up to $1024$ sites by numerical methods based on matrix-product states (MPS). First, we use our numerical method to evaluate the moments/cumulants of work done by sudden quench process on the Ising or Haldane spin chains and study their behaviors across the quantum phase transitions. Our results show that, up to the fourth cumulant, the work statistics can indicate the quantum phase transition characterized by the local order parameters but barely for purely topological phase transitions. Second, we propose to use the fluctuation theorem, such as Jarzynski's equality, which relates the real-time correlator to the ratio of the thermal partition functions, as a benchmark indicator for the numerical real-time evolving methods. Third, we study the passivity of ground and thermal states of quantum spin chains under some cyclic impulse processes. We show that the passivity of thermal states and ground states under the hermitian actions are ensured by the second laws and variational principles, respectively, and also verify it by numerical calculations. Besides, we also consider the passivity of ground states under non-hermitian actions, for which the variational principle cannot be applied. Despite that, we find no violation of passivity from our numerical results for all the cases considered in the Ising and Haldane chains. {Overall, we demonstrate that the work statistics for the sudden quench and impulse processes can be evaluated precisely by the numerical MPS method to characterize quantum phase transitions and examine the passivity of quantum states. We also propose to exploit the universality of the fluctuation theorem to benchmark the numerical real-time evolutions in an algorithm and model independent way.

We propose a scheme to perform tensor network based finite-size scaling analysis for two-dimensional classical models. In the tensor network representation of the partition function, we use higher-order tensor renormalization group (HOTRG) method to coarse grain the weight tensor. The renormalized tensor is then used to construct the approximated transfer matrix of an infinite strip of finite width. By diagonalizing the transfer matrix we obtain the correlation length, the magnetization, and the energy density which are used in finite-size scaling analysis to determine the critical temperature and the critical exponents. As a benchmark we study the two-dimensional classical Ising model. We show that the critical temperature and the critical exponents can be accurately determined. With HOTRG bond dimension $D=70$, the absolute errors of the critical temperature $T_c$ and the critical exponent $\nu$, $\beta$ are at the order of $10^{-7}, 10^{-5}$, $10^{-4}$ respectively. Furthermore, the results can be systematically improved by increasing the bond dimension of the HOTRG method. Finally, we study the length scale induced by the finite cut-off in bond dimension and elucidate its physical meaning in this context.

We adopt the neural network (NN) flow method to study the Berezinskii-Kosterlitz-Thouless (BKT) phase transitions of the two-dimensional q-state clock model with q≥4. The NN flow consists of a sequence of the same units that proceed with the flow. This unit is a variational autoencoder trained by the data of Monte Carlo configurations in unsupervised learning. To gauge the difference among the ensembles of Monte Carlo configurations at different temperatures and the uniqueness of the ensemble of NN-flow states, we adopt the Jensen-Shannon divergence (JSD) as the information-distance measure "thermometer." This JSD thermometer compares the probability distribution functions of the mean spin value of two ensembles of states. Our results show that the NN flow will flow an arbitrary spin state to some state in a fixed-point ensemble of states. The corresponding JSD of the fixed-point ensemble takes a unique profile with peculiar features, which can help to identify the critical temperatures of BKT phase transitions of the underlying Monte Carlo configurations.

We systematically investigate the ground state phase diagram and the finite temperature phase transitions for a Rydberg-dressed Fermi gas loaded in a bilayer optical lattice. When an effective finite-ranged attraction is induced, our self-consistent mean-field calculation shows that the gapped topological ( p -wave) superfluids in each layer are coupled together by the s -wave pairing in an intermediate inter-layer distance with a spontaneously modulated phases between these two order parameters. The obtained ground state is a gapless topological superfluid with quantized topological charges characterizing the gapless points, leading to a zero energy flat band at the edges. Finally, we calculate the finite temperature phase diagrams of this two-dimensional gapless superfluid and observe two distinct critical temperatures, demonstrating the fruitful many-body effects on a paired topological superfluids.

It is known that a trained restricted Boltzmann machine (RBM) on the binary Monte Carlo Ising spin configurations, generates a series of iterative reconstructed spin configurations which spontaneously flow and stabilize to the critical point of physical system. Here we construct a variety of neural network (NN) flows using the RBM and (variational) autoencoders, to study the q -state Potts and clock models on the square lattice for q = 2, 3, 4. The NN are trained on Monte Carlo spin configurations at various temperatures. We find that the trained NN flow does develop a stable point that coincides with critical point of the q -state spin models. The behavior of the NN flow is nontrivial and generative, since the training is unsupervised and without any prior knowledge about the critical point and the Hamiltonian of the underlying spin model. Moreover, we find that the convergence of the flow is independent of the types of NNs and spin models, hinting a universal behavior. Our results strengthen the potential applicability of the notion of the NN flow in studying various states of matter and offer additional evidence on the connection with the renormalization group flow.

We systematically investigate the topological properties of spin polarized Rydberg-dressed fermionic atoms loaded in a bilayer optical lattice. Through tuning the Rydberg coupling strength and the interlayer tunneling amplitude, we identify different types of topological superfluid states generated from the interlayer pairing and relative pairing phase modulation of the coupled two-dimensional p-wave superfluids. These phases include gapped/gapless with/without time-reversal symmetry. One of the most interesting states is a gapless paired topological superfluid with both the time-reversal symmetry and particle-hole symmetry, which is a realization of a DIII gapless topological superconductor. The flexibility of experimental manipulation in such Rydberg-dressed fermionic systems therefore becomes a promising system for realizing interesting topological superfluids.

We propose a unified scheme to identify phase transitions out of the $\mathbb{Z}_2$ Abelian topological order, including the transition to a non-Abelian chiral spin liquid. Using loop gas and and string gas states [H.-Y. Lee, R. Kaneko, T. Okubo, N. Kawashima, Phys. Rev. Lett. 123, 087203 (2019)] on the star lattice Kitaev model as an example, we compute the overlap of minimally entangled states through transfer matrices. We demonstrate that, similar to the anyon condensation, continuous deformation of a $\mathbb{Z}_2$-injective projected entangled-pair state (PEPS) also allows us to study the transition between Abelian and non-Abelian topological orders. We show that the charge and flux anyons defined in the Abelian phase transmute into the $\sigma$ anyon in the non-Abelian topological order. Furthermore, we show that contrary to the claim in [Phys. Rev. B 101, 035140 (2020)], both the LG and SG states have infinite correlation length in the non-Abelian regime, consistent with the no-go theorem that a chiral PEPS has a gapless parent Hamiltonian.

Using tensor network methods, we perform finite-size scaling analysis to study the parameter-induced phase transitions of two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki states. We use higher-order tensor renormalization group method to evaluate the moments and the correlations. Then, the critical point and critical exponents are determined simultaneously by collapsing the data. Alternatively, the crossing points of the dimensionless ratios are used to determine the critical point, and the scaling at the critical point is used to determine the critical exponents. For the transition between the disordered AKLT phase and the ferromagnetic ordered phase, we demonstrate that both the critical point and the exponents can be determined accurately. Furthermore, the values of the exponents confirm that the AKLT-FM transition belongs to the 2D Ising universality class. We also investigate the Berezinskii-Kosterlitz-Thouless transition from the AKLT phase to the critical XY phase. In this case we show that the critical point can be located by the crossing point of the correlation ratio.

We investigate the topological properties of spin polarized fermionic polar molecules loaded in a multi-layer structure with the electric dipole moment polarized to the normal direction. When polar molecules are paired by attractive inter-layer interaction, unpaired Majorana fermions can be macroscopically generated in the top and bottom layers in dilute density regime. We show that the resulting topological state is effectively composed by a bundle of 1D Kitaev ladders labeled by in-plane momenta k and -k, and hence belongs to BDI class characterized by the winding number Z, protected by the time reversal symmetry. The Majorana surface modes exhibit a flatband at zero energy, fully gapped from Bogoliubov excitations in the bulk, and hence becomes an idea system to investigate the interaction effects on quantum degenerate Majorana fermions. We further show that additional interference fringes can be identified as a signature of such 2D Majorana surface modes in the time-of-flight experiment.

We study spin-2 deformed-AKLT models on the square lattice, specifically a two-parameter family of $O(2)$-symmetric ground-state wavefunctions as defined by Niggemann, Klumper, and Zittartz, who found previously that the phase diagram consists of a Neel-ordered phase and a disordered phase which contains the AKLT point. Using tensor-network methods, we not only confirm the Neel phase but also find an XY phase with quasi-long-range order and a region adjacent to it, within the AKLT phase, with very large correlation length, and investigate the consequences of a perfectly factorizable point at the corner of that phase.

Recent progress in the characterization of gapped quantum phases has also triggered the search for a universal resource for quantum computation in symmetric gapped phases. Prior works in one dimension suggest that it is a feature more common than previously thought, in that nontrivial one-dimensional symmetry-protected topological (SPT) phases provide quantum computational power characterized by the algebraic structure defining these phases. Progress in two and higher dimensions so far has been limited to special fixed points. Here we provide two families of two-dimensional Z(2) symmetric wave functions such that there exists a finite region of the parameter in the SPT phases that supports universal quantum computation. The quantum computational power appears to lose its universality at the boundary between the SPT and the symmetry-breaking phases.

In numerical simulations of classical and quantum lattice systems, 2D corner transfer matrices (CTMs) and 3D corner tensors (CTs) are a useful tool to compute approximate contractions of infinite-size tensor networks. In this paper we show how the numerical CTMs and CTs can be used, additionally, to extract universal information from their spectra. We provide examples of this for classical and quantum systems, in 1D, 2D, and 3D. Our results provide, in particular, practical evidence for a wide variety of models of the correspondence between d-dimensional quantum and (d + 1)-dimensional classical spin systems. We show also how corner properties can be used to pinpoint quantum phase transitions, topological or not, without the need for observables. Moreover, for a chiral topological PEPS we show by examples that corner tensors can be used to extract the entanglement spectrum of half a system, with the expected symmetries of the SU(2)(k) Wess-Zumino-Witten model describing its gapless edge for k = 1,2. We also review the theory behind the quantum-classical correspondence for spin systems and provide a numerical scheme for quantum state renormalization in 2D using CTs. Our results show that bulk information of a lattice system is encoded holographically in efficiently-computable properties of its corners.

We investigate the topological properties of spin-polarized fermionic polar molecules loaded in a multi-layer structure with the electric dipole moment polarized to the normal direction by an external electric field. When polar molecules are paired by attractive inter-layer interaction, unpaired Majorana fermions can be macroscopically generated in the 2D surface layers, if their in-plane kinetic energy satisfies a certain condition. Protected by the time reversal symmetry, the resulting topological state is composed by 1D effective Kitaev ladders labeled by in-plane momentum ${bfk_|}$ and ${-bfk_|}$, and belongs to BDI class characterized by the winding number $mathbb{Z}$. We further calculate the entanglement entropy, entanglement spectrum, and the critical temperature for realizing such Majorana fermions, and then discuss the signature for observing such 2D quantum degenerate Majorana fermions in the time-of-flight experiments.

Symmetry-protected topological (SPT) phases exhibit nontrivial order if symmetry is respected but are adiabatically connected to the trivial product phase if symmetry is not respected. However, unlike the symmetry-breaking phase, there is no local order parameter for SPT phases. Here we employ a tensor-network method to compute the topological invariants characterized by the simulated modular $S$ and $T$ matrices to study transitions in a few families of two-dimensional (2D) wave functions which are ${\mathbb{Z}}_{N}$ $(N=2\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}3)$ symmetric. We find that in addition to the topologically ordered phases, the modular matrices can be used to identify nontrivial SPT phases and detect transitions between different SPT phases as well as between symmetric and symmetry-breaking phases. Therefore modular matrices can be used to characterize various types of gapped phases in a unifying way.