Unconstrained tree tensor network: An adaptive gauge picture for enhanced performance

Gerster, Matthias; Silvi, Pietro; Rizzi, Matteo; Fazio, Rosario; Calarco, Tommaso; Montangero, Simone

doi:10.1103/physrevb.90.125154

Cited by 62 publications

(86 citation statements)

References 88 publications

(106 reference statements)

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“…Moreover, we double-check our results via another method for the determination of the MI-SF transition, namely the value of the Luttinger parameter K obtained from the hopping correlation functions [18] In the clean case the criterion for locating the BKT transition is given by K 1 2 c = . Indeed, as critical exponents can be obtained with high numerical precision in TTN simulations [42], we expect this method to work well in the present scenario. Moreover, the PBC setting allows for an easy treatment of finite size effects in equation (4), simply by replacing the distance r with the effective distance r r crd…”

Section: Correlation Functionsmentioning

confidence: 68%

“…The absence of loops in the network is crucial for the applicability of efficient energy minimization techniques needed for the ground state search. For algorithm details we refer the reader to [42], however it is important to emphasize that with our TTN approach we will be able to compute the superfluid density for large system sizes (up to N s =256 in the absence of disorder) and therefore perform a reliable finite-size scaling analysis. As we will show, at equilibrium in 1D this approach provides matching results to Monte Carlo calculations.…”

Section: The Disordered Bose-hubbard Modelmentioning

confidence: 99%

“…In the present work we apply a tree tensor network (TTN) method [37][38][39][40][41] (specifically our recently introduced gauge-adaptive TTN technique [42]), a natural ansatz for the simulation of one-dimensional quantum many-body systems with PBC, to the disordered Bose-Hubbard model. We compute the superfluid stiffness and correlation functions, avoiding the detrimental boundary effects arising in simulations with open boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Superfluid density and quasi-long-range order in the one-dimensional disordered Bose–Hubbard model

et al. 2016

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We study the equilibrium properties of the one-dimensional disordered Bose-Hubbard model by means of a gauge-adaptive tree tensor network variational method suitable for systems with periodic boundary conditions. We compute the superfluid stiffness and superfluid correlations close to the superfluid to glass transition line, obtaining accurate locations of the critical points. By studying the statistics of the exponent of the power-law decay of the correlation, we determine the boundary between the superfluid region and the Bose glass phase in the regime of strong disorder and in the weakly interacting region, not explored numerically before. In the former case our simulations are in agreement with previous Monte Carlo calculations.

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Section: Correlation Functionsmentioning

confidence: 68%

Section: The Disordered Bose-hubbard Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Superfluid density and quasi-long-range order in the one-dimensional disordered Bose–Hubbard model

et al. 2016

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“…We emphasize that these distances therefore correspond to quite different measures than those studied in the various sciences mentioned before. We also note that in tensor networks the leaf-to-leaf "a.goldsborough@warwick.ac.uk; www.warwick.ac.uk/andrewgoIdsborough ' s.a.rautu@warwick.ac.uk *r.roemer@ Warwick.ac.uk; www.warwick.ac.uk/rudoroemer distance is referred to as the path length [13], but in graph theory this term usually refers to the sum of the levels of each of the vertices in the tree [1], In the present work, we shall concentrate on full and complete trees that have the same structure as regular tree tensor networks [14,15], We derive the average leaf-to-leaf distances for varying leaf separation with leaves ordered in a one-dimensional line as shown, e.g., in Fig. 1(a) for a binary tree [16], The method is then generalized to m-ary trees and the moments of the leaf-to-leaf distances.…”

Section: Introductionmentioning

confidence: 99%

Leaf-to-leaf distances and their moments in finite and infinite ordered m -ary tree graphs

2015

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We study the leaf-to-leaf distances on one-dimensionally ordered, full and complete m-ary tree graphs using a recursive approach. In our formulation, unlike in traditional graph theory approaches, leaves are ordered along a line emulating a one-dimensional lattice. We find explicit analytical formulas for the sum of all paths for arbitrary leaf separation r as well as the average distances and the moments thereof. We show that the resulting explicit expressions can be recast in terms of Hurwitz-Lerch transcendants. Results for periodic trees are also given. For incomplete random binary trees, we provide first results by numerical techniques; we find a rapid drop of leaf-to-leaf distances for large r.

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“…The density-matrix renormalisation method [16] can indeed be viewed as a variational principle over matrix-product states [11,13,[17][18][19]. Generalising these ideas, a number of exciting methods have been proposed [20][21][22][23][24][25][26], some of which also allow to study open quantum systems. In most cases matrixproduct operators (MPO) are at the heart of these methods.…”

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confidence: 99%

Positive Tensor Network Approach for Simulating Open Quantum Many-Body Systems

et al. 2016

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Open many-body quantum systems play an important role in quantum optics and condensed-matter physics, and capture phenomena like transport, interplay between Hamiltonian and incoherent dynamics, and topological order generated by dissipation. We introduce a versatile and practical method to numerically simulate one-dimensional open quantum many-body dynamics using tensor networks. It is based on representing mixed quantum states in a locally purified form, which guarantees that positivity is preserved at all times. Moreover, the approximation error is controlled with respect to the trace norm. Hence, this scheme overcomes various obstacles of the known numerical open-system evolution schemes. To exemplify the functioning of the approach, we study both stationary states and transient dissipative behaviour, for various open quantum systems ranging from few to many bodies.Open quantum systems are ubiquitous in physics. To some extent any quantum system is coupled to an environment, and in many instances this interaction significantly alters the system's dynamics. Traditionally such decoherence processes are seen as an enemy to coherent state manipulation. However, suitably engineered dissipation can also have beneficial effects and can be exploited for state preparation [1][2][3][4][5][6], even of states containing strong entanglement or featuring topological order [7,8]. In condensed matter physics, many concepts such as transport are often studied within the closed systems paradigm, but, it is becoming increasingly clear that some familiar concepts may have to be revisited in the open system setting [9], where the interplay between coherent quantum many-body and open systems dynamics, i.e. the competition between Hamiltonian interactions and dissipation leads to interesting physical effects. Since few analytical methods are available for such systems, the design of novel numerical tools for the simulation of dissipative quantum many-body systems is of the utmost importance. In this work, we present a new algorithm which captures the open many-body dynamics in one spatial dimension -for both transient and steady regimes -based on a locally purified tensor network ansatzclass. It comprises a new approach in that the positivity of the operators is maintained during the whole simulation. Importantly, the approximation errors can be controlled in a way that yields a trace-norm certificate. Hence, the algorithm provides not only a conceptually new approach to the problem, but also combines several desired features of the existing schemes and overcomes previous limitations.Tensor-network ansatz-classes have proven to be widely successful to capture the physics of many-body states [10][11][12][13][14][15]. They rely on the idea that relevant quantum states lie in the very small sub-manifold with local correlations, which in turn can be efficiently captured in terms of tensor networks. The density-matrix renormalisation method [16] can indeed be viewed as a variational principle over matrix-product states [11,13,[17][18][...

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Unconstrained tree tensor network: An adaptive gauge picture for enhanced performance

Cited by 62 publications

References 88 publications

Superfluid density and quasi-long-range order in the one-dimensional disordered Bose–Hubbard model

Superfluid density and quasi-long-range order in the one-dimensional disordered Bose–Hubbard model

Leaf-to-leaf distances and their moments in finite and infinite ordered m -ary tree graphs

Positive Tensor Network Approach for Simulating Open Quantum Many-Body Systems

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