Exact blocking formulas for spin and gauge models

Liu, Yuzhi; Meurice, Yannick; Qin, Mingpu; Unmuth-Yockey, Judah; Xiang, Tao; Xie, Z. Y.; Yu, Ji-Feng; Zou, Haiyuan

doi:10.1103/physrevd.88.056005

Cited by 114 publications

(168 citation statements)

References 37 publications

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“…Although our method can be formally extended to 2D lattice QCD by adopting a tensor network formulation of SUðNÞ gauge theory proposed by Liu et al [8], it is necessary to check how much computational cost is actually required for numerical calculations. The biggest difficulty is to develop a practical method to calculate 4D systems.…”

Section: Discussionmentioning

confidence: 99%

“…β is the inverse coupling constant squared. Liu et al have shown that a finite-dimensional tensor network representation of pure lattice gauge theory is derived by the character expansion (CE) with truncation [8], and its numerical accuracy with the TRG is verified for the XY model [9][10][11]. Using the character expansion, the Boltzmann weight per plaquette is decomposed as…”

Section: B Grassmann Valued Tensor Networkmentioning

confidence: 99%

“…It has been applied to a couple of models consisting of continuous bosonic variables [7][8][9][10][11]. Gu et al generalized the TRG to a Grassmann valued tensor network in order to investigate fermionic systems [12,13].…”

Section: Introductionmentioning

confidence: 99%

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Grassmann tensor renormalization group approach to one-flavor lattice Schwinger model

2014

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We apply the Grassmann tensor renormalization group to the lattice regularized Schwinger model with one-flavor of the Wilson fermion. We study the phase diagram in the ðβ; κÞ plane performing a detailed analysis of the scaling behavior of the Lee-Yang zeros and the peak height of the chiral susceptibility. Our results strongly indicate that the whole range of the phase transition line starting from ðβ; κÞ ¼ ð0.0; 0.380665ð59ÞÞ and ending at ð∞; 0.25Þ belongs to the two-dimensional Ising universality class similar to the free fermion case.

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Section: Discussionmentioning

confidence: 99%

Section: B Grassmann Valued Tensor Networkmentioning

confidence: 99%

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Grassmann tensor renormalization group approach to one-flavor lattice Schwinger model

2014

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“…In Ref. [1], we showed that TRG methods can be applied to models studied by lattice gauge theorists, namely spin models (O(N) and principal chiral models) and pure gauge models (Ising, U(1) and SU (2)). The case of actions quadratic in Grassmann variables is briefly discussed in [6].…”

Section: Introductionmentioning

confidence: 99%

Blocking versus Sampling

Meurice¹,

Liu²,

Unmuth-Yockey³

et al. 2015

Proceedings of the 32nd International Symposium on Lattice Field Theory — PoS(LATTICE2014)

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The idea of blocking in configuration space has played an important role in the development of the RG ideas. However, despite being half a century old and having had a huge intellectual impact, generic numerical methods to perform blocking for lattice models have progressed more slowly than sampling methods. Blocking may be essential to deal with near conformal situations. Typically, blocking methods have smaller statistical errors but larger systematic errors than sampling methods. This situation is evolving with recent developments based on the Tensor RG (TRG) method. We report recent results for spin and gauge lattice models obtained with this new method regarding searches for fixed points, calculations of critical exponents and resolutions of sign problems. An interesting model for comparison is the 2-dimensional O(2) model with a chemical potential which has a sign problem with conventional Monte Carlo but allows sampling with the worm algorithm and blocking with various TRG formulations. We compare the efficiency and accuracy of these two methods.

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“…The von Neumann entanglement for the isotropic model has been calculated numerically [29] using tensor Renormalization Group (TRG) methods [28,30]. In order to connect the O(2) model with quantum simulators, it is possible to take a highly anisotropic limit of the transfer matrix where the time becomes continuous and we can identify a quantum Hamiltonian [28,[31][32][33][34] :…”

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confidence: 99%

Probing the conformal Calabrese-Cardy scaling with cold atoms

et al. 2017

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We demonstrate that current experiments using cold bosonic atoms trapped in one-dimensional optical lattices and designed to measure the second-order Rényi entanglement entropy S2, can be used to verify detailed predictions of conformal field theory (CFT) and estimate the central charge c. We discuss the adiabatic preparation of the ground state at half-filling where we expect a CFT with c = 1. This can be accomplished with a very small hopping parameter J, in contrast to existing studies with density one where a much larger J is needed. We provide two complementary methods to estimate and subtract the classical entropy generated by the experimental preparation and imaging processes. We compare numerical calculations for the classical O(2) model with a chemical potential on a 1+1 dimensional lattice, and the quantum Bose-Hubbard Hamiltonian implemented in the experiments. S2 is very similar for the two models and follows closely the Calabrese-Cardy scaling, (c/8) ln(Ns), for Ns sites with open boundary conditions, provided that the large subleading corrections are taken into account.PACS numbers: 05.10. Cc, 11.15.Ha, 11.25.Hf, 37.10.Jk, 67.85.Hj, 75.10.Hk The concept of universality provides a unified approach to the critical behavior of lattice models studied in condensed matter, lattice gauge theory (LGT) and experimentally accessible systems of cold atoms trapped in optical lattices. Conformal field theory (CFT) [1, 2] offers many interesting examples of universal behavior that can be observed for lattice models in two [3][4][5], three [6], and four [7,8] dimensions. Practical simulations for these models unavoidably involve a finite volume that breaks explicitly the conformal invariance. However, this symmetry breaking follows definite patterns dictated by the restoration of the symmetry at infinite volume and allows us to identify the universality class. In view of the rich collection of interesting CFTs, it would be highly desirable to study their universality classes using quantum simulations. In order to start this ambitious program, one needs a simple concrete example to demonstrate the feasibility of the idea.In this Letter, we propose to use the setup of ongoing cold atom experiments to quantum simulate the O(2) model with a chemical potential and check the predictions of CFT for the growth of the entanglement entropy with the size of the system corresponding to the universality class of the superfluid (SF) phase. The O(2) model is an extension of the Ising model where the spin is allowed to move on a circle, making an angle θ with respect to a direction of reference. This model can be used to describe easy plane ferromagnetism and the compactness of θ leads to topological configurations called vortices. Their unbinding provides a prime example of a Berezinski-Kosterlitz-Thouless transition [9,10] in a way that has also been advocated to apply for gauge theories near the boundary of the conformal window [11]. When space and Euclidean time are treated isotropically, this model has important common ...

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Exact blocking formulas for spin and gauge models

Cited by 114 publications

References 37 publications

Grassmann tensor renormalization group approach to one-flavor lattice Schwinger model

Grassmann tensor renormalization group approach to one-flavor lattice Schwinger model

Blocking versus Sampling

Probing the conformal Calabrese-Cardy scaling with cold atoms

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