Bulletin of the American Physical Society
APS March Meeting 2017
Volume 62, Number 4
Monday–Friday, March 13–17, 2017; New Orleans, Louisiana
Session E8: Quantum Many-Body Systems 3: Tensor Networks and Machine Learning for the Many-Body Problem |
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Sponsoring Units: DCOMP Chair: Julian Rincon, Perimeter Institute Room: 267 |
Tuesday, March 14, 2017 8:00AM - 8:12AM |
E8.00001: Combining integrability and MPS methods to study the two-dimensional Heisenberg model Robert Konik, Andrew James, J.-S. Caux We employ a hybrid of matrix product state methods and exact solvability to study the two dimensional Heisenberg model in a cylindrical geometry. We analyze the two dimensional system by exploiting our ability to solve the one-dimensional subunits (i.e. spin chains) exactly using Bethe ansatz. This provides both the spectrum and the matrix elements of the one-dimensional chain. Having these in hand lessens the numerical burden when it comes to studying the fully two-dimensional system. We argue that at weaker values of the interchain coupling, the dynamics of the two dimensional system is dominated by the two- and four-spinon states of the chains. As the interchain coupling grows however, higher spinon states become more important. To reduce the number of basis states needed in such a case to represent the one-dimensional chains, we explore the use of adaptive bases. [Preview Abstract] |
Tuesday, March 14, 2017 8:12AM - 8:24AM |
E8.00002: Fermionic iPEPS with non-Abelian symmetries Benedikt Bruognolo, Jan von Delft, Andreas Weichselbaum Tensor network (TN) techniques have emerged as one of the most promising tools to resolve many open questions in two-dimensional (2D) lattice models with fermions and frustration. An infinite projected entangled-pair state (iPEPS) represents a particularly versatile TN ansatz working directly in the thermodynamic limit, whose competitiveness has been demonstrated only recently in the context of the 2D $t$--$J$ and Hubbard model [1,2]. Above all, this method makes it possible to study competing low-energy states by constraining specific unit-cell structures or by enforcing that the wavefunction breaks or conserves particular symmetries. In this work, we explore the benefits of conserving global non-Abelian symmetries in the iPEPS ansatz using the QSpace tensor library [3]. To this end, we study an iPEPS with SU(2) spin rotation symmetry for interacting fermionic systems on the square lattice. \\[2ex] [1] P. Corboz, T. M. Rice, and M. Troyer, Phys. Rev. Lett. 113, 046402 (2014). [2] P. Corboz, Phys. Rev. B 93, 045116 (2016). [3] A. Weichselbaum, Ann. Phys. 327, 2972 (2012). [Preview Abstract] |
Tuesday, March 14, 2017 8:24AM - 8:36AM |
E8.00003: Continuous matrix product representations for mixed states Julian Rincon, Martin Ganahl, Guifre Vidal The continuous matrix product state (cMPS) is a powerful variational ansatz for the ground state of strongly interacting quantum field theories in 1+1 spacetime dimensions [F. Verstraete, J.I. Cirac, Phys. Rev. Lett. 104, 190405(2010)]. The cMPS applies to theories that have a finite short-distance behavior under a characteristic length scale. This ultraviolet cut-off may be natural or enforced (e.g. by adding a regulator to the field theory). In this paper we propose a density matrix generalization of the cMPS, the continuous matrix product density operator (cMPDO), and investigate its suitability to represent thermal states and master equation dynamics. We show the existence of such an object by taking the continuum limit of a lattice MPDO and characterize its mathematical properties. For thermal states, we find that the cMPDO offers an accurate description of their corresponding density matrix. We argue that these results can also be extended for the case of master equation dynamics. Finally, we propose and demonstrate an algorithm to find the cMPDO representation of thermal states. [Preview Abstract] |
Tuesday, March 14, 2017 8:36AM - 8:48AM |
E8.00004: Loop optimization for tensor network renormalization Shuo Yang, Zheng-Cheng Gu, Xiao-Gang Wen We introduce a tensor renormalization group scheme for coarse-graining a two-dimensional tensor network, which can be successfully applied to both classical and quantum systems on and off criticality. The key idea of our scheme is to deform a 2D tensor network into small loops and then optimize tensors on each loop. In this way we remove short-range entanglement at each iteration step, and significantly improve the accuracy and stability of the renormalization flow. We demonstrate our algorithm in the classical Ising model and a frustrated 2D quantum model. [Preview Abstract] |
Tuesday, March 14, 2017 8:48AM - 9:00AM |
E8.00005: Tensor Network Monte Carlo for Quantum Lattice Models William Huggins, Edwin Stoudenmire, Norman Tubman, Daniel Freeman, Birgitta Whaley Tensor networks can accurately and efficiently encode the ground states and thermal density matrices of local quantum lattice Hamiltonians. However, extracting expectation values from these representations is known to require the use of approximation schemes; in this work we combine recently developed Monte Carlo techniques for tensor networks with more standard Renormalization Group approaches. We present preliminary results and discuss the current status of our efforts to generalize these methods. [Preview Abstract] |
Tuesday, March 14, 2017 9:00AM - 9:12AM |
E8.00006: Series-Expansion Thermal Tensor Network Approach for Quantum Lattice Models Wei Li, Bin-Bin Chen, yun-jing liu, ziyu chen In this work we propose a series-expansion thermal tensor network (SETTN) approach for efficient simulations of quantum manybody systems. This continuous-time SETTN method is based on the numerically exact Taylor series expansion of equilibrium density operator $e^{-\beta H}$ (with $H$ the total Hamiltonian and $\beta$ the imaginary time), and is thus Trotter-error free. We discover, through simulating the XXZ spin chain and square-lattice quantum Ising models, that not only the Hamiltonian $H$, but also its powers $H^n$, can be efficiently expressed as matrix product operators, which enables us to calculate with high precision the equilibrium and dynamical properties of quantum lattice models at finite temperatures. Our SETTN method provides an alternative to conventional Trotter-Suzuki renormalization group (RG) approaches, and achieves an unprecedented standard of thermal RG simulations in terms of accuracy and flexibility. [Preview Abstract] |
Tuesday, March 14, 2017 9:12AM - 9:24AM |
E8.00007: Effective theory and emergent $SU(2)$ symmetry in the flat bands of attractive Hubbard models Murad Tovmasyan, Sebastiano Peotta, P\"aivi T\"orm\"a, Sebastian Huber We study fermions interacting via attractive Hubbard interaction on a lattice with a flat Bloch band separated from the other bands by a finite energy gap. First, we project the Hamiltonian into the flat band Wannier functions. Then, we do a further approximation which leads to an effective ferromagnetic spin chain with an emergent $SU(2)$ symmetry. As a specific example, we consider a one-dimensional ladder with two perfectly flat Bloch bands. We show that as a manifestation of the emergent $SU(2)$ symmetry the Bardeen-Cooper-Schrieffer (BCS) wavefunction is the exact ground state of the projected Hamiltonian, and that the compressibility is diverging. We extend the projected model by using the Schrieffer-Wolf transformation and show that the $SU(2)$ symmetry is broken by second order interband transitions also resulting in a finite compressibility, which we calculate analytically and compare to the result obtained via quasi-exact DMRG simulations. Our predictions can be tested via transport measurements in cold atom experiments. [Preview Abstract] |
Tuesday, March 14, 2017 9:24AM - 9:36AM |
E8.00008: Time evolution of two holes in $t-J$ chains with anisotropic couplings Salvatore R. Manmana, Holger Thyen, Thomas K\"ohler, Stephan C. Kramer Using time-dependent Matrix Product State (MPS) methods we study the real-time evolution of hole-excitations in t-J chains close to filling $n=1$. The dynamics in 'standard' $t-J$ chains with SU(2) invariant spin couplings is compared to the one when introducing anisotropic, XXZ-type spin interactions as realizable, e.g., by ultracold polar molecules on optical lattices. The simulations are performed with MPS implementations based on the usual singular value decompositions (SVD) as well as ones using the adaptive cross approximation (ACA) instead. The ACA can be seen as an iterative approach to SVD which is often used, e.g., in the context of finite-element-methods, leading to a substantial speedup. A comparison of the performance of both algorithms in the MPS context is discussed. [Preview Abstract] |
Tuesday, March 14, 2017 9:36AM - 9:48AM |
E8.00009: Luttinger Liquids coupled to Quantum Impurities; Exact Solutions via Bethe Ansatz Colin Rylands, Natan Andrei The Bethe ansatz method has been hugely successful in finding exact solutions of quantum impurity systems, the Kondo model and Anderson model to name but two. Up till now such solutions have been restricted to models where the bulk is non-interacting or the impurity is located on the boundary. It is known, however that placing impurities in a Luttinger liquid can have different and remarkable consequences. A backscattering impurity (Kane-Fisher model) will cause the system to be cut in two at low temperature for repulsive interactions or completely healed if the interactions are attractive. Coupling instead a dynamic impurity like a resonance level modifies this picture so that even attractive interactions can result in the system being split. Such systems can be realised experimentally, in edge states of quantum Hall systems or quantum dots attached to one dimensional leads. In this talk I will describe how to solve models which couple a Luttinger liquid to an impurity in the bulk via Bethe ansatz. Along the way I will exhibit the solutions of the Kane-Fisher model as well as a Luttinger liquid coupled to a resonance level. [Preview Abstract] |
Tuesday, March 14, 2017 9:48AM - 10:00AM |
E8.00010: Single-particle plus local reduced density matrix functional theory for Fermionic lattice models Zhengqian Cheng, Chris Marianetti We formulate a functional of the reduced density matrix (RDM) for interacting Fermionic lattice models, which depends on all local elements of the RDM (ie. all local N-body contributions); while the only nonlocal contributions are confined to the single-particle density matrix. We propose an ansatz for the unknown kinetic energy functional and evaluate it as compared to numerous exact results. In the one dimensional Hubbard model, the insulating state is properly obtained at infinitesimal $U$; in addition to an accurate prediction of the ground state energy over a broad range of $t/U$. In the infinite dimension, single band Hubbard model, we properly find a finite-$U$ metal-insulator transition with reasonable quantitative accuracy; in addition to the ground state energy. While our approach does not address frequency dependent observables, it has a negligible computational cost as compared to dynamical mean field theory and could be highly applicable in the context total energies of strongly correlated materials and molecules. [Preview Abstract] |
Tuesday, March 14, 2017 10:00AM - 10:12AM |
E8.00011: Markovian marginals Isaac Kim We introduce the notion of so called Markovian marginals, which is a natural framework for constructing solutions to the quantum marginal problem. A set of reduced density matrices obeying a certain set of local constraints necessarily has a global state that is compatible with all the given reduced density matrices. This leads to an algorithm to study interacting quantum many-body systems in two spatial dimensions. [Preview Abstract] |
Tuesday, March 14, 2017 10:12AM - 10:24AM |
E8.00012: Machine Learning for the Many-Body Problem Roger Melko The machine learning (ML) community has developed computational techniques with remarkable abilities to recognize, classify, and characterize complex sets of data. In this talk, we review recent applications of ML to classical and quantum many-body problems in condensed matter physics. Using architectures for supervised learning, such as fully-connected, deep, and convolutional neural networks, we demonstrate how ML can be used to identify phases and phase transitions in a variety of condensed matter Hamiltonians, including those with topological order. We also discuss how unsupervised learning, implemented through a restricted Boltzmann machine, is capable of faithfully modeling thermodynamic observables in a variety of systems. The combination of modern ML methods coupled with conventional Monte Carlo simulations promises to lead to new generations of hybrid computational techniques, with broad consequences for our theoretical understanding of condensed matter. [Preview Abstract] |
Tuesday, March 14, 2017 10:24AM - 10:36AM |
E8.00013: Exact Machine Learning Topological States Dong-Ling Deng, Xiaopeng Li Artificial neural networks play a prominent role in the rapidly growing field of machine learning and are recently introduced to quantum many-body systems to tackle complex problems. Here, we show that even topological states with long-range quantum entanglement can be represented with classical artificial neural networks. This is demonstrated by using two concrete spin systems, the one-dimensional (1D) symmetry-protected topological cluster state and the 2D toric code state with an intrinsic topological order. For both cases we show rigorously that the topological ground states can be represented by short-range neural networks in an {\it exact} fashion. This neural network representation, in addition to being exact, is surprisingly {\it efficient} as the required number of hidden neurons is as small as the number of physical spins. Our results demonstrate explicitly the exceptional power of neural networks in describing exotic quantum states, and at the same time provides valuable guidances to supervise machine learning topological quantum orders in generic lattice models. [Preview Abstract] |
Tuesday, March 14, 2017 10:36AM - 10:48AM |
E8.00014: Numerical implementation of ``rigorous renormalization group'' for ground states and low-energy excited states of 1d Hamiltonians Brenden Roberts, Thomas Vidick, Olexei Motrunich The ``rigorous renormalization group'' (RRG) is a recently-developed algorithm (Landau, Vazirani, Vidick, Nat.~Phys.~11, 2015) for obtaining MPS ansatz approximations to the ground spaces and low-lying excited spectra of local Hamiltonians. The technique is related to constructions used recently to tighten the bound in the proof of the area law in one dimension (Arad, Kitaev, Landau, Vazirani, Proc.~4th ITCS, 2013). The RRG algorithm does not rely on the iterated optimization of local degrees of freedom; rather, it operates in a tree-like manner on subspaces of Hilbert space to increase overlap with the low-energy eigenspace of a Hamiltonian. Because of this, the algorithm handles in a natural way both ground state degeneracy and approximation of low-energy excited spectra. We describe the background and implementation of the algorithm and exhibit results comparing our implementation with DMRG for various systems of interest. [Preview Abstract] |
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