Multiscale entanglement renormalization ansatz for spin chains with continuously varying criticality

Bridgeman, Jacob C.; O’Brien, Aroon; Bartlett, Stephen D.; Doherty, Andrew C.

doi:10.1103/physrevb.91.165129

Cited by 27 publications

(39 citation statements)

References 78 publications

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“…The calculated exponent is exact at the Ising decoupling point (the point in parameter space where the AT model reduces to two uncoupled Ising models) and the relative error in this calculated exponent is comparable to the numerically-intensive MERA simulations performed in Ref. [13]. Specifically, the SRS RG gives a relative error in the prediction of < 10% in the range λ ∈ [−0.4, 0.9], and much small error for |λ| 1, compared with < 5% error in Ref.…”

Section: Introductionsupporting

confidence: 60%

“…Specifically, the SRS RG gives a relative error in the prediction of < 10% in the range λ ∈ [−0.4, 0.9], and much small error for |λ| 1, compared with < 5% error in Ref. [13]. The accuracy decreases significantly towards the endpoints of the critical line.…”

Section: Introductionmentioning

confidence: 90%

“…We can compare the calculated critical exponents from SRS RG with those obtained from the six-vertex model [12] as well as the known behavior of the CFT. A recent study of the critical behavior of the AT model and a detailed comparison with the c = 1 orbifold boson CFT predictions was performed using MERA [13] (a much more sophisticated and numerically-intensive real-space RG method), and this provides a useful benchmark.…”

Section: Introductionmentioning

confidence: 99%

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Symmetry-respecting real-space renormalization for the quantum Ashkin-Teller model

et al. 2015

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We use a simple real-space renormalization group approach to investigate the critical behavior of the quantum Ashkin-Teller model, a one-dimensional quantum spin chain possessing a line of criticality along which critical exponents vary continuously. This approach, which is based on exploiting the on-site symmetry of the model, has been shown to be surprisingly accurate for predicting some aspects of the critical behavior of the quantum transverse-field Ising model. Our investigation explores this approach in more generality, in a model where the critical behavior has a richer structure but which reduces to the simpler Ising case at a special point. We demonstrate that the correlation length critical exponent as predicted from this real-space renormalization group approach is in broad agreement with the corresponding results from conformal field theory along the line of criticality. Near the Ising special point, the error in the estimated critical exponent from this simple method is comparable to that of numerically-intensive simulations based on much more sophisticated methods, although the accuracy decreases away from the decoupled Ising model point.

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Section: Introductionsupporting

confidence: 60%

Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

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Symmetry-respecting real-space renormalization for the quantum Ashkin-Teller model

et al. 2015

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“…Thus the low-energy description in terms of N fermions with a linear dispersion naively suggests that the corresponding critical points are described by a product theory of N Ising CFTs with c = N/2. Indeed, central charges of c = 1 and c = 3/2 have been obtained for the 3-cluster model at (J, h) = (0, ±1) [30] and (J, h) = (±1, 0) [23], respectively.…”

mentioning

confidence: 99%

“…While in 1D these models are universal resources only for a single qubit, they have proven accessible settings to study the entanglement [21][22][23][25][26][27][28][29][30] and the computational power [31][32][33] of symmetry protected states, as well as the robustness of edge states in a many-body localized phase [34]. Perturbing the pure stabilizer models with Ising and Zeeman terms, we define the generalized cluster models with periodic boundary conditions by the Hamiltonians (the original 1D cluster model corresponds to A = 3)…”

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Realizing Allso(N)1Quantum Criticalities in Symmetry Protected Cluster Models

Lahtinen

Ardonne

2015

Phys. Rev. Lett.

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We show that all so(N )1 universality class quantum criticalities emerge when one-dimensional generalized cluster models are perturbed with Ising or Zeeman terms. Each critical point is described by a low-energy theory of N linearly dispersing fermions, whose spectrum we show to precisely match the prediction by so(N )1 conformal field theory. Furthermore, by an explicit construction we show that all the cluster models are dual to non-locally coupled transverse field Ising chains, with the universality of the so(N )1 criticality manifesting itself as N of these chains becoming critical. This duality also reveals that the symmetry protection of cluster models arises from the underlying Ising symmetries and it enables the identification of local representations for the primary fields of the so(N )1 conformal field theories. For the simplest and experimentally most realistic case that corresponds to the original one-dimensional cluster model with local three-spin interactions, our results show that the su(2)2 so(3)1 Wess-Zumino-Witten model can emerge in a local, translationally invariant and Jordan-Wigner solvable spin-1/2 model.A striking property of quantum phase transitions is that of universality. Near a critical regime separating distinct quantum states of matter, the system specific microscopic details are lost and the system acquires universal behavior that is characterized by scaling exponents of distinct observables near the transition [1]. A crucial step in putting order to the zoo of distinct universalities was made by noting that the diverging correlation length at the critical point implies conformal invariance, and thus a description by a conformal field theory (CFT) [2,3]. This approach is particularly powerful in one spatial dimension (1D), where the relevant CFT fully describes (the scaling of) the correlation functions.1D quantum models with critical points are thus the natural playgrounds for quantum criticality. The simplest and the most celebrated ones are the transverse field Ising (TFI) chain [4], that at criticality is described by the Ising CFT, and the XY chain [5] where the anisotropic transition is described by the so called so(2) 1 CFT. So ubiquitous are these universality classes that criticality beyond them is often filed under "exotic". Still, considering this ubiquitous nature these simple models have played in various fields of physics, it was only rather recently when the details of the criticality of the TFI chain were experimentally probed [6]. The challenge of probing quantum criticality beyond these simple models owes to the lack of tractable models that admit accessible experimental realization. To go beyond them, often higher spins [7][8][9][10][11][12], infinite range couplings [13,14], broken translational invariance [15], strongly interacting systems [16][17][18], or combinations there of [19,20], are required. Here we show that a hierarchy of local, exactly solvable and translationally invariant spin-1/2 models provides the simplest setting to generalize the Ising unive...

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Tensor Network Contractions

Ran

Tirrito

Peng

et al. 2020

Lecture Notes in Physics

112

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PrefaceTensor 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, coarsegraining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multilinear 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. v AcknowledgementsWe are indebted to

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Multiscale entanglement renormalization ansatz for spin chains with continuously varying criticality

Cited by 27 publications

References 78 publications

Symmetry-respecting real-space renormalization for the quantum Ashkin-Teller model

Symmetry-respecting real-space renormalization for the quantum Ashkin-Teller model

Realizing Allso(N)1Quantum Criticalities in Symmetry Protected Cluster Models

Tensor Network Contractions

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