Characterizing criticality in quantum many-body systems of dimension $\ge 2$ is one of the most important challenges of the contemporary physics. In principle, there is no generally valid theoretical method that could solve this problem. In this work, we propose an efficient approach to identify the criticality of quantum systems in higher dimensions. Departing from the analysis of the numerical renormalization group flows, we build a general equivalence between the higher-dimensional ground state and a one-dimensional (1D) quantum state defined in the imaginary time direction in terms of the so-called time matrix product state (tMPS). We show that the criticality of the targeted model can be faithfully identified by the tMPS, using the mature scaling schemes of correlation length and entanglement entropy in 1D quantum theories. We benchmark our proposal with the results obtained for the Heisenberg anti-ferromagnet on honeycomb lattice. We demonstrate critical scaling relation of the tMPS for the gapless case, and a trivial scaling for the gapped case with spatial anisotropy. The critical scaling behaviors are insensitive to the system size, suggesting the criticality can be identified in small systems. Our tMPS scheme for critical scaling shows clearly that the spin-1/2 kagomé Heisenberg antiferromagnet has a gapless ground state. More generally, the present study indicates that the 1D conformal field theories in imaginary time provide a very useful tool to characterize the criticality of higher dimensional quantum systems.
Characterizing criticality in quantum many-body systems of dimension ≥2 is one of the most important challenges of the contemporary physics. In principle, there is no generally valid theoretical method that could solve this problem. In this work, we propose an efficient approach to identify the criticality of quantum systems in higher dimensions. Departing from the analysis of the numerical renormalization group flows, we build a general equivalence between the higher-dimensional ground state and a one-dimensional (1D) quantum state defined in the imaginary time direction in terms of the so-called time matrix product state (tMPS). We show that the criticality of the targeted model can be faithfully identified by the tMPS, using the mature scaling schemes of correlation length and entanglement entropy in 1D quantum theories. We benchmark our proposal with the results obtained for the Heisenberg anti-ferromagnet on honeycomb lattice. We demonstrate critical scaling relation of the tMPS for the gapless case, and a trivial scaling for the gapped case with spatial anisotropy. The critical scaling behaviors are insensitive to the system size, suggesting the criticality can be identified in small systems. Our tMPS scheme for critical scaling shows clearly that the spin-1/2 kagomé Heisenberg antiferromagnet has a gapless ground state. More generally, the present study indicates that the 1D conformal field theories in imaginary time provide a very useful tool to characterize the criticality of higher dimensional quantum systems.
(Please refer to arXiv:1810.08050, which has completely different aims but contains all the main contents of this paper) In this work, we propose to access the information of criticality and excitations of one-dimensional quantum systems by a matrix product state (MPS) defined in the (imaginary) time direction. This state, dubbed as time MPS (tMPS), is a boundary state of tensor network (TN) that represents the ground-state simulation after Trotter-Suzuki decomposition. We show that the tMPS exhibits the structure of the continuous MPS originally proposed for the field theories. The information of excitations, e.g., dynamic correlation length and energy gap, can be accurately calculated from the tMPS. The non-universal renormalization of velocity of the excitations is given by a ratio between the correlations of the ground state and the tMPS. When the system is at the quantum critical point, the tMPS is found to show the logarithmic scaling law, where the scaling coefficient gives the central charge that characterizes the criticality of the theory. Our work also implies that the spectra of the transfer matrices defined by the ground state and tMPS provide information of the low-lying masses of the theory in the continuous limit. This could help to understand the role of the finiteness of the bond dimension as an infra red regulator. Furthermore, we show from the perspective of methodology that, the tMPS emerges from a generalized TN ab-initio optimization principle scheme, which unifies the infinite density matrix renormalization group and the infinite time-evolving block decimation algorithms.
Determination and characterization of criticality in two-dimensional (2D) quantum many-body systems is one of the most important challenges and problems in modern physics. Based on the scaling theory in one-dimensional (1D) systems and tensor networks, we propose an efficient scheme to access the criticality of 2D quantum states. We demonstrate that the boundary state of a critical infinite projected entangled pair state can be described by 1D conformal field theory, which provides a solid indicator that allows one to identify and characterize the criticality of 2D states. Our scheme is verified on the resonating valence bond (RVB) states on kagome and square lattices, where the boundary state of the honeycomb RVB is found to be described by a c=1 conformal field theory. We apply our scheme also to the ground state of the spin-1/2 XXZ model on honeycomb lattice, illustrating the difficulties of standard variational tensor network approaches to study the critical ground states. Our scheme is of high versatility and flexibility, and can be further applied to investigate the quantum criticality in many other phenomena, such as finite-temperature and topological phase transitions.
We demonstrate an efficient method that allows for simultaneous determination of the ground state, low energy excitation properties and excitation gap in quantum many body systems. To this aim we first use the \textit{ab-initio} optimization principle of tensor networks (TN), to show that the infinite density matrix renormalization group (iDMRG) in the real space is associated in a natural manner to the infinite time-evolving block decimation (iTEBD) implemented on a continuous matrix product state (MPS), and defined in imaginary time. We illustrate this association showing that the (imaginary) time matrix product state (MPS) in iTEBD reproduces accurately the properties of the two-dimensional (2D) classical Ising model, verifying in this way that the time MPS corresponds to a well-defined physical state. We apply then our scheme to the one-dimensional (1D) quantum Ising chain, where the time MPS is defined in continuous imaginary time. It is found that the time MPS at or close to the critical point is always maximally entangled, but gives essentially different correlations than those obtained from iDMRG; we argue that these correlations describe the dynamic correlations for the ground state.
We study critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. The critical point, exponents, and central charge are determined by optimizing them to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling Ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ with mass parameter ${\ensuremath{\mu}}^{2}$ and lattice spacing a, we demonstrate a double data collapse for the correlation length $\ensuremath{\delta}\ensuremath{\xi}(\ensuremath{\mu},\ensuremath{\lambda},D)=\stackrel{\texttildelow{}}{\ensuremath{\xi}}((\ensuremath{\alpha}\ensuremath{-}{\ensuremath{\alpha}}_{c})(\ensuremath{\delta}/a{)}^{\ensuremath{-}1/\ensuremath{\nu}})$ with D the bond dimension, $\ensuremath{\delta}$ the gap between eigenvalues of the transfer matrix, and ${\ensuremath{\alpha}}_{c}={\ensuremath{\mu}}_{R}^{2}/\ensuremath{\lambda}$ the parameter which fixes the critical quantum field theory.
The matrix product state (MPS) is utilized to investigate the ground state properties and quantum phase transitions (QPTs) of the dimerized antiferromagnetic Heisenberg (DAH) model. The ground state MPS wavefunctions determined by the infinite time-evolving block decimation (iTEBD) algorithm are shown to be very efficient descriptions of DAH model. In the thermodynamic limit, the quantum entanglement, the bond energy, and the nearest-neighbor correlations are calculated. It is revealed that the singular behavior of the bipartite entanglement can detect the QPTs directly. The critical point J c 2 = 1.0 is determined evidently, and the quantum phase transition is argued to belong to the second-order category. At the critical point, logarithmic divergent character of the block entanglement is observed, and the system can be described by a free bosonic field theory.
In contrast to classical many-particle systems for which the number of degrees of freedom grows linearly with the system size, it grows exponentially for quantum many-particle systems. On the one hand, the resulting complexity of the quantum systems has a high potential for technological advances if it can be exploited directly. For example, in recent years, the fundamentals for the realization of quantum computers and their possible applications have been investigated intensely. On the other hand, the high complexity, especially in strongly correlated regimes, often makes the analytical or numerical analysis of underlying physical phenomena very difficult. This is why some systems of high technological relevance, like the high-temperature superconductors, are still unsatisfactorily understood. A main topic of this thesis is the development of efficient numerical methods for the simulation of strongly correlated quantum lattice models. For one-dimensional systems, the density-matrix renormalization-group (DMRG) is such a very successful method. The physical states of interest are approximated within a certain class of ansatz states. These ansatz states are designed in a way that the number of degrees of freedom are prevented from growing exponentially. With the DMRG method, ground states of interacting one-dimensional systems can often be determined with very high precision. This observation indicates already that ground states actually do not exhaust the available exponential number of degrees of freedom but can be described efficiently with much fewer parameters. The ansatz states used in DMRG are the so-called matrix product states. A very useful criterion for the construction of corresponding appropriate ansatz states for higher-dimensional systems is the quantum nonlocality as for example quantified by the entanglement entropy. The ansatz states should be able to reproduce the scaling properties of the quantum nonlocality in the physical states of interest with as few parameters as possible. The first part of the thesis, therefore, provides analytical and numerical analysis of the scaling of quantum nonlocality with the system size or time in different, physically relevant scenarios. For example, the scaling of Renyi entropies and their dependence on boundary conditions is derived within the 1+1-dimensional conformal field theory. Conjectures and analytical indications concerning the properties of entanglement entropy in critical fermionic and bosonic systems are confirmed numerically with high precision. For integrable models in the thermodynamic limit, general preconditions are derived under which subsystems converge to steady states. These steady states are non-thermal and retain information about the initial state. It is shown that the entanglement entropy in such steady states is extensive. For short times, the entanglement entropy grows typically linearly with time, causing an exponential increase in computation costs for the DMRG method. Consequently, time-dependent DMRG simulations usually have no direct access to the long time regime. The second part of the thesis focuses on the development and improvement of the above-mentioned numerical techniques. The time-dependent DMRG is complemented with an extrapolation technique for the evaluated observables. In this way, the problem of the entropy increase can be circumvented, allowing for a precise determination of spectral functions. The method is demonstrated using the example of the Heisenberg antiferromagnet and results are compared to Bethe-Ansatz data for T=0 and quantum Monte Carlo data for T>0. For the simulation of higher-dimensional systems, projected-entangled pair-states (PEPS) and the multiscale entanglement renormalization ansatz (MERA) turn out to be appropriate wavefunction classes. Initially, they could only be used for spin systems. Within this thesis, variants of those state classes are being suggested for the fermionic case. It is shown how the known algorithms for spin systems can be translated into corresponding algorithms for the fermionic systems. The occurring computation costs increase only by a marginal overhead. First numerical benchmarking tests proceeded successfully. Thus, new techniques are available which can for example be used to examine the two-dimensional Hubbard model. This model is a candidate for the description of high-temperature superconductivity.