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Shi-Ju Ran

Shi-Ju Ran appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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cond-mat.str-elquant-phcond-mat.stat-mechMachine Learningphysics.comp-phComputation and Languagecond-mat.dis-nncond-mat.mtrl-scicond-mat.quant-gashep-latmath.STphysics.app-phphysics.class-phStatistics Theory

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Published work

27 published item(s)

preprint2026arXiv

Confidence Geometry Reveals Trace-Level Correctness in Large Language Model Reasoning

Large language models (LLMs) generate not only reasoning text, but also token-level confidence trajectories that record how uncertainty evolves during inference. Whether these trajectories are relevant to reasoning correctness remains unclear. Here we show that confidence trajectories encode a content-agnostic confidence geometry associated with trace-level final-answer correctness. Using only token-level confidence values, without access to the input question, reasoning text, hidden states, or external verifiers, we find that low-dimensional representations of confidence trajectories separate correct from incorrect reasoning traces. Across GSM8K, MATH, and MMLU, this geometric separation is quantitatively linked to downstream predictability: stronger clustering of correct and incorrect traces, measured by the Davies--Bouldin index, consistently corresponds to higher correctness-discrimination AUC. We further show that correctness-related information is enriched in the tail of reasoning, suggesting that late-stage confidence dynamics carry key correctness signals. We propose NeuralConf, a lightweight estimator that learns from confidence trajectories for correctness evaluation. Under a fixed trace budget, NeuralConf-derived scores improve confidence-weighted answer aggregation over majority voting, tail confidence, and other static baselines. These results reveal that LLMs expose trace-intrinsic statistical signals of correctness through their own confidence dynamics, offering a route to improve inference using information already present within generation.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Accurate simulation and thermal tuning by temperature-adaptive boundary interactions on quantum many-body systems

Constructing quantum Hamiltonians for simulating and controlling the exotic physics of many-body systems belongs to the most important topics of condensed matter physics and quantum technologies. The main challenge that hinders the future investigations is the extremely high complexity for either their numerical simulations or experimental realizations. In this work, we propose the temperature-adaptive entanglement simulator (TAES) that mimics and tunes the thermodynamics of the one-dimensional (1D) many-body system by embedding a small-size model in an entanglement bath. The entanglement bath is described by the interactions located at the boundaries of the small-size model, whose coupling constants are optimized by means of differentiable tensor network at target temperatures. With the benchmark on 1D spin chains, TAES surpasses the state-of-the-art accuracy compared with the existing finite-temperature approaches such as linearized and differential tensor renormalization group algorithms. By tuning the couplings of the entanglement bath with the temperature fixed, the bulk entropy exhibits similar behavior compared to that obtained by tuning the temperature. Our work provides novel opportunities of engineering the distribution of fluctuations and mimicking the non-equilibrium phenomena in a uniform temperature within the canonical ensemble framework using the optimized boundary interactions.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Deep Machine Learning Reconstructing Lattice Topology with Strong Thermal Fluctuations

Applying artificial intelligence to scientific problems (namely AI for science) is currently under hot debate. However, the scientific problems differ much from the conventional ones with images, texts, and etc., where new challenges emerges with the unbalanced scientific data and complicated effects from the physical setups. In this work, we demonstrate the validity of the deep convolutional neural network (CNN) on reconstructing the lattice topology (i.e., spin connectivities) in the presence of strong thermal fluctuations and unbalanced data. Taking the kinetic Ising model with Glauber dynamics as an example, the CNN maps the time-dependent local magnetic momenta (a single-node feature) evolved from a specific initial configuration (dubbed as an evolution instance) to the probabilities of the presences of the possible couplings. Our scheme distinguishes from the previous ones that might require the knowledge on the node dynamics, the responses from perturbations, or the evaluations of statistic quantities such as correlations or transfer entropy from many evolution instances. The fine tuning avoids the "barren plateau" caused by the strong thermal fluctuations at high temperatures. Accurate reconstructions can be made where the thermal fluctuations dominate over the correlations and consequently the statistic methods in general fail. Meanwhile, we unveil the generalization of CNN on dealing with the instances evolved from the unlearnt initial spin configurations and those with the unlearnt lattices. We raise an open question on the learning with unbalanced data in the nearly "double-exponentially" large sample space.

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preprint2022arXiv

Efficient Simulation of Quantum Many-body Thermodynamics by Tailoring Zero-temperature Tensor Network

Numerical annealing and renormalization group have conceived various successful approaches to study the thermodynamics of strongly-correlated systems where perturbation or expansion theories fail to work. As the process of lowering the temperatures is usually involved in different manners, these approaches in general become much less efficient or accurate at the low temperatures. In this work, we propose to access the finite-temperature properties from the tensor network (TN) representing the zero-temperature partition function. We propose to "scissor" a finite part from such an infinite-size TN, and "stitch" it to possess the periodic boundary condition along the imaginary-time direction. We dub this approach as TN tailoring. Exceptional accuracy is achieved with a fine-tune process, surpassing the previous methods including the linearized tensor renormalization group [Phys. Rev. Lett. 106, 127202 (2011)], continuous matrix product operator [Phys. Rev. Lett. 125, 170604 (2020)], and etc. High efficiency is demonstrated, where the time cost is nearly independent of the target temperature including the extremely-low temperatures. The proposed idea can be extended to higher-dimensional systems of bosons and fermions.

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preprint2022arXiv

Experimental realization of a quantum image classifier via tensor-network-based machine learning

Quantum machine learning aspires to overcome intractability that currently limits its applicability to practical problems. However, quantum machine learning itself is limited by low effective dimensions achievable in state-of-the-art experiments. Here we demonstrate highly successful classifications of real-life images using photonic qubits, combining a quantum tensor-network representation of hand-written digits and entanglement-based optimization. Specifically, we focus on binary classification for hand-written zeroes and ones, whose features are cast into the tensor-network representation, further reduced by optimization based on entanglement entropy and encoded into two-qubit photonic states. We then demonstrate image classification with a high success rate exceeding 98%, through successive gate operations and projective measurements. Although we work with photons, our approach is amenable to other physical realizations such as nitrogen-vacancy centers, nuclear spins and trapped ions, and our scheme can be scaled to efficient multi-qubit encodings of features in the tensor-product representation, thereby setting the stage for quantum-enhanced multi-class classification.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Functional Tensor Network Solving Many-body Schrödinger Equation

Schrödinger equation belongs to the most fundamental differential equations in quantum physics. However, the exact solutions are extremely rare, and many analytical methods are applicable only to the cases with small perturbations or weak correlations. Solving the many-body Schrödinger equation in the continuous spaces with the presence of strong correlations is an extremely important and challenging issue. In this work, we propose the functional tensor network (FTN) approach to solve the many-body Schrödinger equation. Provided the orthonormal functional bases, we represent the coefficients of the many-body wave-function as tensor network. The observables, such as energy, can be calculated simply by tensor contractions. Simulating the ground state becomes solving a minimization problem defined by the tensor network. An efficient gradient-decent algorithm based on the automatically differentiable tensors is proposed. We here take matrix product state (MPS) as an example, whose complexity scales only linearly with the system size. We apply our approach to solve the ground state of coupled harmonic oscillators, and achieve high accuracy by comparing with the exact solutions. Reliable results are also given with the presence of three-body interactions, where the system cannot be decoupled to isolated oscillators. Our approach is simple and with well-controlled error, superior to the highly-nonlinear neural-network solvers. Our work extends the applications of tensor network from quantum lattice models to the systems in the continuous space. FTN can be used as a general solver of the differential equations with many variables. The MPS exemplified here can be generalized to, e.g., the fermionic tensor networks, to solve the electronic Schrödinger equation.

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preprint2021arXiv

Cold atoms meet lattice gauge theory

The central idea of this review is to consider quantum field theory models relevant for particle physics and replace the fermionic matter in these models by a bosonic one. This is mostly motivated by the fact that bosons are more ``accessible'' and easier to manipulate for experimentalists, but this ``substitution'' also leads to new physics and novel phenomena. It allows us to gain new information about among other things confinement and the dynamics of the deconfinement transition. We will thus consider bosons in dynamical lattices corresponding to the bosonic Schwinger or Z$_2$ Bose-Hubbard models. Another central idea of this review concerns atomic simulators of paradigmatic models of particle physics theory such as the Creutz-Hubbard ladder, or Gross-Neveu-Wilson and Wilson-Hubbard models. Finally, we will briefly describe our efforts to design experimentally friendly simulators of these and other models relevant for particle physics.

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preprint2021arXiv

Non-parametric Semi-Supervised Learning in Many-body Hilbert Space with Rescaled Logarithmic Fidelity

In quantum and quantum-inspired machine learning, the very first step is to embed the data in quantum space known as Hilbert space. Developing quantum kernel function (QKF), which defines the distances among the samples in the Hilbert space, belongs to the fundamental topics for machine learning. In this work, we propose the rescaled logarithmic fidelity (RLF) and non-parametric semi-supervised learning in the quantum space, which we name as RLF-NSSL. The rescaling takes advantage of the non-linearity of the kernel to tune the mutual distances of samples in the Hilbert space, and meanwhile avoids the exponentially-small fidelities between quantum many-qubit states. Being non-parametric excludes the possible effects from the variational parameters, and evidently demonstrates the advantages from the space itself. We compare RLF-NSSL with several well-known non-parametric algorithms including naive Bayes classifiers, k-nearest neighbors, and spectral clustering. Our method exhibits better accuracy particularly for the unsupervised case with no labeled samples and the few-shot cases with small numbers of labeled samples. With the visualizations by t-stochastic neighbor embedding, our results imply that the machine learning in the Hilbert space complies with the principles of maximal coding rate reduction, where the low-dimensional data exhibit within-class compressibility, between-class discrimination, and overall diversity. Our proposals can be applied to other quantum and quantum-inspired machine learning, including the methods using the parametric models such as tensor networks, quantum circuits, and quantum neural networks.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Bayesian Tensor Network with Polynomial Complexity for Probabilistic Machine Learning

It is known that describing or calculating the conditional probabilities of multiple events is exponentially expensive. In this work, Bayesian tensor network (BTN) is proposed to efficiently capture the conditional probabilities of multiple sets of events with polynomial complexity. BTN is a directed acyclic graphical model that forms a subset of TN. To testify its validity for exponentially many events, BTN is implemented to the image recognition, where the classification is mapped to capturing the conditional probabilities in an exponentially large sample space. Competitive performance is achieved by the BTN with simple tree network structures. Analogous to the tensor network simulations of quantum systems, the validity of the simple-tree BTN implies an ``area law'' of fluctuations in image recognition problems.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Encoding of Matrix Product States into Quantum Circuits of One- and Two-Qubit Gates

The matrix product state (MPS) belongs to the most important mathematical models in, for example, condensed matter physics and quantum information sciences. However, to realize an $N$-qubit MPS with large $N$ and large entanglement on a quantum platform is extremely challenging, since it requires high-level qudits or multi-body gates of two-level qubits to carry the entanglement. In this work, an efficient method that accurately encodes a given MPS into a quantum circuit with only one- and two-qubit gates is proposed. The idea is to construct the unitary matrix product operators that optimally disentangle the MPS to a product state. These matrix product operators form the quantum circuit that evolves a product state to the targeted MPS with a high fidelity. Our benchmark on the ground-state MPS's of the strongly-correlated spin models show that the constructed quantum circuits can encode the MPS's with much fewer qubits than the sizes of the MPS's themselves. This method paves a feasible and efficient path to realizing quantum many-body states and other MPS-based models as quantum circuits on the near-term quantum platforms.

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preprint2020arXiv

Tangent-Space Gradient Optimization of Tensor Network for Machine Learning

The gradient-based optimization method for deep machine learning models suffers from gradient vanishing and exploding problems, particularly when the computational graph becomes deep. In this work, we propose the tangent-space gradient optimization (TSGO) for the probabilistic models to keep the gradients from vanishing or exploding. The central idea is to guarantee the orthogonality between the variational parameters and the gradients. The optimization is then implemented by rotating parameter vector towards the direction of gradient. We explain and testify TSGO in tensor network (TN) machine learning, where the TN describes the joint probability distribution as a normalized state $\left| ψ\right\rangle $ in Hilbert space. We show that the gradient can be restricted in the tangent space of $\left\langle ψ\right.\left| ψ\right\rangle = 1$ hyper-sphere. Instead of additional adaptive methods to control the learning rate in deep learning, the learning rate of TSGO is naturally determined by the angle $θ$ as $η= \tan θ$. Our numerical results reveal better convergence of TSGO in comparison to the off-the-shelf Adam.

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preprint2019arXiv

Generative Tensor Network Classification Model for Supervised Machine Learning

Tensor network (TN) has recently triggered extensive interests in developing machine-learning models in quantum many-body Hilbert space. Here we purpose a generative TN classification (GTNC) approach for supervised learning. The strategy is to train the generative TN for each class of the samples to construct the classifiers. The classification is implemented by comparing the distance in the many-body Hilbert space. The numerical experiments by GTNC show impressive performance on the MNIST and Fashion-MNIST dataset. The testing accuracy is competitive to the state-of-the-art convolutional neural network while higher than the naive Bayes classifier (a generative classifier) and support vector machine. Moreover, GTNC is more efficient than the existing TN models that are in general discriminative. By investigating the distances in the many-body Hilbert space, we find that (a) the samples are naturally clustering in such a space; and (b) bounding the bond dimensions of the TN's to finite values corresponds to removing redundant information in the image recognition. These two characters make GTNC an adaptive and universal model of excellent performance.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Lecture Notes of Tensor Network Contractions

Tensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarse-graining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multi-linear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches.

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preprint2019arXiv

Quantum Compressed Sensing with Unsupervised Tensor-Network Machine Learning

We propose tensor-network compressed sensing (TNCS) by combining the ideas of compressed sensing, tensor network (TN), and machine learning, which permits novel and efficient quantum communications of realistic data. The strategy is to use the unsupervised TN machine learning algorithm to obtain the entangled state $|Ψ\rangle$ that describes the probability distribution of a huge amount of classical information considered to be communicated. To transfer a specific piece of information with $|Ψ\rangle$, our proposal is to encode such information in the separable state with the minimal distance to the measured state $|Φ\rangle$ that is obtained by partially measuring on $|Ψ\rangle$ in a designed way. To this end, a measuring protocol analogous to the compressed sensing with neural-network machine learning is suggested, where the measurements are designed to minimize uncertainty of information from the probability distribution given by $|Φ\rangle$. In this way, those who have $|Φ\rangle$ can reliably access the information by simply measuring on $|Φ\rangle$. We propose q-sparsity to characterize the sparsity of quantum states and the efficiency of the quantum communications by TNCS. The high q-sparsity is essentially due to the fact that the TN states describing nicely the probability distribution obey the area law of entanglement entropy. Testing on realistic datasets (hand-written digits and fashion images), TNCS is shown to possess high efficiency and accuracy, where the security of communications is guaranteed by the fundamental quantum principles.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Reentrance of Topological Phase in Spin-1 Frustrated Heisenberg Chain

For the Haldane phase, the magnetic field usually tends to break the symmetry and drives the system into a topologically trivial phase. Here, we report a novel reentrance of the Haldane phase at zero temperature in the spin-1 antiferromagnetic Heisenberg model on sawtooth chain. A partial Haldane phase is induced by the magnetic field, which is the combination of the Haldane state in one sublattice and a ferromagnetically ordered state in the other sublattice. Such a partial topological order is a result of the zero-temperature entropy due to quantum fluctuations caused by geometrical frustration.

0 citations0 reviewsNo code linkedOpen access
preprint2018arXiv

Entanglement-guided architectures of machine learning by quantum tensor network

It is a fundamental, but still elusive question whether the schemes based on quantum mechanics, in particular on quantum entanglement, can be used for classical information processing and machine learning. Even partial answer to this question would bring important insights to both fields of machine learning and quantum mechanics. In this work, we implement simple numerical experiments, related to pattern/images classification, in which we represent the classifiers by many-qubit quantum states written in the matrix product states (MPS). Classical machine learning algorithm is applied to these quantum states to learn the classical data. We explicitly show how quantum entanglement (i.e., single-site and bipartite entanglement) can emerge in such represented images. Entanglement characterizes here the importance of data, and such information are practically used to guide the architecture of MPS, and improve the efficiency. The number of needed qubits can be reduced to less than 1/10 of the original number, which is within the access of the state-of-the-art quantum computers. We expect such numerical experiments could open new paths in charactering classical machine learning algorithms, and at the same time shed lights on the generic quantum simulations/computations of machine learning tasks.

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preprint2015arXiv

Identification of Degeneracy and Criticality of Two-Dimensional Statistical and Quantum Systems by the Boundary States of Tensor Networks

We propose a systematic scheme to reach the properties of two-dimensional (2D) statistical and quantum systems by studying the effective (1+1)-dimensional theory that is constructed from the tensor network representation. On on hand, we discover that the degeneracy of the 2D system can be determined by the purity of the boundary thermal state, which is the density operator of the effective theory at zero (effective) temperature. On the other hand, we find that the gapped (or critical) 2D system leads to a gapped (or critical) effective (1+1)-dimensional theory whose criticality can be accessed by the entanglement entropy $S$ of its ground state dubbed as boundary pure state. We also uncover that for the critical systems, $S$ obeys the same logarithmic law as that found in the critical 1D quantum chains, which reads $S = (κc/6)\log_2 D + const.$, with $c$ the central charge and $κ$ a constant related to the scaling property of the correlation length $ξ$ as $ξ\sim D^κ$. Such a scaling law presents an efficient way to characterize the critical universality class of the original 2D systems. An important implication of our work is that many well-established theories for 1D quantum chains become available for studying 2D systems with the help of the proposed lower dimensional correspondence.

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preprint2015arXiv

Phase diagram and exotic spin-spin correlations of anisotropic Ising model on the Sierpiński gasket

The anisotropic antiferromagnetic Ising model on the fractal Sierpiński gasket is intensively studied, and a number of exotic properties are disclosed. The ground state phase diagram in the plane of magnetic field-interaction of the system is obtained. The thermodynamic properties of the three plateau phases are probed by exploring the temperature-dependence of magnetization, specific heat, susceptibility and spin-spin correlations. No phase transitions are observed in this model. In the absence of a magnetic field, the unusual temperature dependence of the spin correlation length is obtained with $0 \leq$J$_b/$J$_a<1$, and an interesting crossover behavior between different phases at J$_b/$J$_a=1$ is unveiled, whose dynamics can be described by the J$_b/$J$_a$-dependence of the specific heat, susceptibility and spin correlation functions. The exotic spin-spin correlation patterns that share the same special rotational symmetry as that of the Sierpiński gasket are obtained in both the $1/3$ plateau disordered phase and the $5/9$ plateau partially ordered ferrimagnetic phase. Moreover, a quantum scheme is formulated to study the thermodynamics of the fractal Sierpiński gasket with Heisenberg interactions. We find that the unusual temperature dependence of the correlation length remains intact in a small quantum fluctuation.

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preprint2014arXiv

Featureless quantum spin liquid, 1/3-magnetization plateau state and exotic thermodynamic properties of spin-1/2 frustrated Heisenberg antiferromagnet on an infinite Husimi lattice

By utilizing tensor-network-based methods, we investigate the zero- and finite-temperature properties of the spin-1/2 Heisenberg antiferromagnetic (HAF) model on an infinite Husimi lattice that contains 3/2 sites per triangle. The ground state of this model is found to possess vanishing local magnetization and is featureless; the spin-spin and dimer-dimer correlation functions are verified to decay exponentially; and its ground-state energy per site is determined to be $e_0=-0.4343(1)$, which is very close to that [$e_0=-0.4386(5)$] of the intriguing kagome HAF model. The magnetization curve shows the absence of a zero-magnetization plateau, implying a gapless excitation. A $1/3$-magnetization plateau with spin up-up-down state is observed, which is selected and stabilized by quantum fluctuations. A ground state phase diagram under magnetic fields is presented. Moreover, both magnetic susceptibility and the specific heat are studied, whose low-temperature behaviors reinforce the conclusion that the HAF model on the infinite Husimi lattice owns a gapless and featureless spin liquid ground state.

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preprint2013arXiv

Honeycomb Heisenberg Spin Ladder: Unusual Ground State and Thermodynamic Properties

The unusual ground state and thermodynamic properties of spin-1/2 two-leg honeycomb (HC) spin ladder are systematically studied by jointly utilizing various analytical and numerical methods. The HC spin ladder is found to exhibit behaviors dramatically different from those of the conventional square spin ladder. A strong relevant term nJ and a half saturation magnetization plateau that can be attributed to the formation of diluted dimer states are observed in the HC ladder, both of which are absent in the square ladder. The ground state phase diagram of the HC spin ladder is identified, and the thermodynamic properties of the specific heat and susceptibility for different couplings are thoroughly explored, where two kinds of excitations are unveiled. The distinct Wilson ratios for both spin ladders at the lower critical fields are also obtained. Our calculated result is well fitted to the experimental data of the two-leg HC spin ladder compound [Cu2L1(N3)4]n.

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preprint2013arXiv

Kosterlitz-Thouless phase transition and reentrance in an anisotropic 3-state Potts model on the generalized Kagome lattice

The unusual reentrant phenomenon is observed in the anisotropic 3-state Potts model on a gen- eralized Kagome lattice. By employing the linearized tensor renormalization group method, we find that the reentrance can appear in the region not only under a partial ordered phase as commonly known but also a phase without a local order parameter, which is uncovered to fall into the uni- versality of the Kosterlitz-Thouless (KT) type. The region of the reentrance depends strongly on the ratios of the next nearest couplings α = J2 /|J1 | and β = J3 /|J1 |. The phase diagrams in the plane of temperature versus β for different α are obtained. Through massive calculations, it is also revealed that the quasi-entanglement entropy can be used to accurately detect the KT transition temperature.

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preprint2013arXiv

Theory of Network Contractor Dynamics for Exploring Thermodynamic Properties of Two-dimensional Quantum Lattice Models

Based on the tensor network state representation, we develop a nonlinear dynamic theory coined as network contractor dynamics (NCD) to explore the thermodynamic properties of two-dimensional quantum lattice models. By invoking the rank-$1$ decomposition in the multi-linear algebra, the NCD scheme makes the contraction of the tensor network of the partition function be realized through a contraction of a local tensor cluster with vectors on its boundary. An imaginary-time-sweep algorithm for implementation of the NCD method is proposed for practical numerical simulations. We benchmark the NCD scheme on the square Ising model, which shows a great accuracy. Besides, the results on the spin-1/2 Heisenberg antiferromagnet on honeycomb lattice are disclosed in good agreement with the quantum Monte Carlo calculations. The quasi-entanglement entropy $S$, Lyapunov exponent $I^{lya}$ and loop character $I^{loop}$ are introduced within the dynamic scheme, which are found to display the ``nonlocality" near the critical point, and can be applied to determine the thermodynamic phase transitions of both classical and quantum systems.

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preprint2012arXiv

Optimized Decimation of Tensor Networks with Super-orthogonalization for Two-Dimensional Quantum Lattice Models

A novel algorithm based on the optimized decimation of tensor networks with super-orthogonalization (ODTNS) that can be applied to simulate efficiently and accurately not only the thermodynamic but also the ground state properties of two-dimensional (2D) quantum lattice models is proposed. By transforming the 2D quantum model into a three-dimensional (3D) closed tensor network (TN) comprised of the tensor product density operator and a 3D brick-wall TN, the free energy of the system can be calculated with the imaginary time evolution, in which the network Tucker decomposition is suggested for the first time to obtain the optimal lower-dimensional approximation on the bond space by transforming the TN into a super-orthogonal form. The efficiency and accuracy of this algorithm are testified, which are fairly comparable with the quantum Monte Carlo calculations. Besides, the present ODTNS scheme can also be applicable to the 2D frustrated quantum spin models with nice efficiency.

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preprint2012arXiv

Phase Diagrams, Distinct Conformal Anomalies and Thermodynamics of Spin-1 Bond-Alternating Heisenberg Antiferromagnetic Chain in Magnetic Fields

The ground state phase diagrams and thermodynamic properties of a spin-1 bond-alternating Heisenberg antiferromagnetic chain with a single-ion anisotropy in longitudinal and transverse magnetic fields are investigated jointly by means of the infinite time evolving block decimation, the linearized tensor renormalization group and the density matrix renormalization group methods. It is found that in the magnetic field-bond alternating ratio plane six phases such as singlet dimer, Haldane, two Tomonaga-Luttinger liquid, 1/2 magnetization plateau and spin polarized phases are identified in a longitudinal field, while in a transverse field there are fives phases including singlet dimer, Haldane, Z$_2$ symmetry breaking, quasi-1/2 magnetization plateau and quasi spin polarized phases. A reentrant behavior of the staggered magnetization in a transverse field is observed. The quantum critical behaviors in longitudinal and transverse magnetic fields are disclosed to fall into different universality classes with corresponding conformal field central charge $c=1$ and 1/2, respectively. The experimental data of the compound NTENP under both longitudinal and transverse magnetic fields are nicely fitted, and the Luttinger liquid behavior of low temperature specific heat experimentally observed is also confirmed.

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preprint2011arXiv

Linearized Tensor Renormalization Group Algorithm for Thermodynamics of Quantum Lattice Models

A linearized tensor renormalization group (LTRG) algorithm is proposed to calculate the thermodynamic properties of one-dimensional quantum lattice models, that is incorporated with the infinite time-evolving block decimation technique, and allows for treating directly the two-dimensional transfer-matrix tensor network. To illustrate its feasibility, the thermodynamic quantities of the quantum XY spin chain are calculated accurately by the LTRG, and the precision is shown to be comparable with (even better than) the transfer matrix renormalization group (TMRG) method. Unlike the TMRG scheme that can only deal with the infinite chains, the present LTRG algorithm could treat both finite and infinite systems, and may be readily extended to boson and fermion quantum lattice models.

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preprint2011arXiv

Modified Tucker Decomposition for Tensor Network and Fast Linearized Tensor Renormalization Group Algorithm for Two-Dimensional Quantum Spin Lattice Systems

We propose a novel algorithm with a modified Tucker decomposition for tensor network that allows for efficiently and precisely calculating the ground state and thermodynamic properties of two-dimensional (2D) quantum spin lattice systems, and is coined as the fast linearized tensor renormalization group (fLTRG). Its amazing efficiency and precision are examined by studying the spin-1/2 anisotropic Heisenberg antiferromagnet on a honeycomb lattice, and the results are found to be fairly in agreement with the quantum Monte Carlo calculations. It is also successfully applied to tackle a quasi-2D spin-1/2 frustrated bilayer honeycomb Heisenberg model, where a quantum phase transition from an ordered antiferromagnetic state to a gapless quantum spin liquid phase is found. The thermodynamic behaviors of this frustrated spin system are also explored. The present fLTRG algorithm could be readily extended to other quantum lattice systems.

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preprint2010arXiv

Phase transitions and thermodynamics of the two-dimensional Ising model on a distorted Kagomé lattice

The two-dimensional Ising model on a distorted Kagomé lattice is studied by means of exact solutions and the tensor renormalisation group (TRG) method. The zero-field phase diagrams are obtained, where three phases such as ferromagnetic, ferrimagnetic and paramagnetic phases, along with the second-order phase transitions, have been identified. The TRG results are quite accurate and reliable in comparison to the exact solutions. In a magnetic field, the magnetization ($m$), susceptibility and specific heat are studied by the TRG algorithm, where the $m=1/3$ plateaux are observed in the magnetization curves for some couplings. The experimental data of susceptibility for the complex Co(N$_3$)$_2$(bpg)$\cdot$ DMF$_{4/3}$ are fitted with the TRG results, giving the couplings of the complex $J=22K$ and $J'=33K$.

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