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Akiyama S, Meurice Y, Sakai R. Tensor renormalization group for fermions. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2024; 36:343002. [PMID: 38701830 DOI: 10.1088/1361-648x/ad4760] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/06/2024] [Accepted: 05/03/2024] [Indexed: 05/05/2024]
Abstract
We review the basic ideas of the tensor renormalization group method and show how they can be applied for lattice field theory models involving relativistic fermions and Grassmann variables in arbitrary dimensions. We discuss recent progress for entanglement filtering, loop optimization, bond-weighting techniques and matrix product decompositions for Grassmann tensor networks. The new methods are tested with two-dimensional Wilson-Majorana fermions and multi-flavor Gross-Neveu models. We show that the methods can also be applied to the fermionic Hubbard model in 1+1 and 2+1 dimensions.
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Affiliation(s)
- Shinichiro Akiyama
- Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan
- Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Yannick Meurice
- Department of Physics and Astronomy, The University of Iowa, Iowa City, IA 52242, United States of America
| | - Ryo Sakai
- Jij Inc., Bunkyo-ku, Tokyo 113-0031, Japan
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Huang RZ, Zhang L, Läuchli AM, Haegeman J, Verstraete F, Vanderstraeten L. Emergent Conformal Boundaries from Finite-Entanglement Scaling in Matrix Product States. PHYSICAL REVIEW LETTERS 2024; 132:086503. [PMID: 38457723 DOI: 10.1103/physrevlett.132.086503] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/03/2023] [Accepted: 01/23/2024] [Indexed: 03/10/2024]
Abstract
The use of finite entanglement scaling with matrix product states (MPS) has become a crucial tool for studying one-dimensional critical lattice theories, especially those with emergent conformal symmetry. We argue that finite entanglement introduces a relevant deformation in the critical theory. As a result, the bipartite entanglement Hamiltonian defined from the MPS can be understood as a boundary conformal field theory with a physical and an entanglement boundary. We are able to exploit the symmetry properties of the MPS to engineer the physical conformal boundary condition. The entanglement boundary, on the other hand, is related to the concrete lattice model and remains invariant under this relevant perturbation. Using critical lattice models described by the Ising, Potts, and free compact boson conformal field theories, we illustrate the influence of the symmetry and the relevant deformation on the conformal boundaries in the entanglement spectrum.
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Affiliation(s)
- Rui-Zhen Huang
- Department of Physics and Astronomy, University of Ghent, 9000 Ghent, Belgium
| | - Long Zhang
- Kavli Institute for Theoretical Sciences and CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
| | - Andreas M Läuchli
- Laboratory for Theoretical and Computational Physics, Paul Scherrer Institute, 5232 Villigen, Switzerland
- Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
| | - Jutho Haegeman
- Department of Physics and Astronomy, University of Ghent, 9000 Ghent, Belgium
| | - Frank Verstraete
- Department of Physics and Astronomy, University of Ghent, 9000 Ghent, Belgium
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom
| | - Laurens Vanderstraeten
- Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Brussels, Belgium
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Mortier Q, Li MH, Haegeman J, Bultinck N. Finite-Entanglement Scaling of 2D Metals. PHYSICAL REVIEW LETTERS 2023; 131:266202. [PMID: 38215387 DOI: 10.1103/physrevlett.131.266202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/17/2023] [Revised: 09/27/2023] [Accepted: 11/21/2023] [Indexed: 01/14/2024]
Abstract
We extend the study of finite-entanglement scaling from one-dimensional gapless models to two-dimensional systems with a Fermi surface. In particular, we show that the entanglement entropy of a contractible spatial region with linear size L scales as S∼Llog[ξf(L/ξ)] in the optimal tensor network, and hence area-law entangled, state approximation to a metallic state, where f(x) is a scaling function which depends on the shape of the Fermi surface and ξ is a finite correlation length induced by the restricted entanglement. Crucially, the scaling regime can be realized with numerically tractable bond dimensions. We also discuss the implications of the Lieb-Schultz-Mattis theorem at fractional filling for tensor network state approximations of metallic states.
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Affiliation(s)
- Quinten Mortier
- Department of Physics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
| | - Ming-Hao Li
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
| | - Jutho Haegeman
- Department of Physics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
| | - Nick Bultinck
- Department of Physics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
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Eberharter AA, Vanderstraeten L, Verstraete F, Läuchli AM. Extracting the Speed of Light from Matrix Product States. PHYSICAL REVIEW LETTERS 2023; 131:226502. [PMID: 38101351 DOI: 10.1103/physrevlett.131.226502] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2023] [Revised: 10/09/2023] [Accepted: 10/18/2023] [Indexed: 12/17/2023]
Abstract
We provide evidence that the spectrum of the local effective Hamiltonian and the transfer operator in infinite-system matrix product state simulations are identical up to a global rescaling factor, i.e., the speed of light of the system, when the underlying system is described by a 1+1 dimensional CFT. We provide arguments for this correspondence based on a path integral point of view. This observation turns out to yield very precise estimates for the speed of light in practice, confirming exact results to high precision where available, but also allowing us to finally determine the speed of light of the non-integrable, critical SU(2) Heisenberg chains with half-integer spin S>1/2 with unprecedented accuracy. We also show that the same technology applied to doped Hubbard ladders provides highly accurate velocities for a range of dopings. Combined with measurements of compressibilities we present new results for the Luttinger liquid parameter in the Luther-Emery regime of doped Hubbard ladders, outperforming earlier approaches based on the fitting of real-space correlation functions.
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Affiliation(s)
| | - Laurens Vanderstraeten
- Department of Physics and Astronomy, University of Ghent, Ghent, Belgium
- Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Brussels, Belgium
| | - Frank Verstraete
- Department of Physics and Astronomy, University of Ghent, Ghent, Belgium
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom
| | - Andreas M Läuchli
- Laboratory for Theoretical and Computational Physics, Paul Scherrer Institute, 5232 Villigen, Switzerland
- Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
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Burgelman L, Devos L, Vanhecke B, Verstraete F, Vanderstraeten L. Contrasting pseudocriticality in the classical two-dimensional Heisenberg and RP^{2} models: Zero-temperature phase transition versus finite-temperature crossover. Phys Rev E 2023; 107:014117. [PMID: 36797854 DOI: 10.1103/physreve.107.014117] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2022] [Accepted: 12/09/2022] [Indexed: 01/15/2023]
Abstract
Tensor-network methods are used to perform a comparative study of the two-dimensional classical Heisenberg and RP^{2} models. We demonstrate that uniform matrix product states (MPSs) with explicit SO(3) symmetry can probe correlation lengths up to O(10^{3}) sites accurately, and we study the scaling of entanglement entropy and universal features of MPS entanglement spectra. For the Heisenberg model, we find no signs of a finite-temperature phase transition, supporting the scenario of asymptotic freedom. For the RP^{2} model we observe an abrupt onset of scaling behavior, consistent with hints of a finite-temperature phase transition reported in previous studies. A careful analysis of the softening of the correlation length divergence, the scaling of the entanglement entropy, and the MPS entanglement spectra shows that our results are inconsistent with true criticality, but are rather in agreement with the scenario of a crossover to a pseudocritical region which exhibits strong signatures of nematic quasi-long-range order at length scales below the true correlation length. Our results reveal a fundamental difference in scaling behavior between the Heisenberg and RP^{2} models: Whereas the emergence of scaling in the former shifts to zero temperature if the bond dimension is increased, it occurs at a finite bond-dimension independent crossover temperature in the latter.
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Affiliation(s)
- Lander Burgelman
- Department of Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
| | - Lukas Devos
- Department of Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
| | - Bram Vanhecke
- Department of Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium.,Faculty of Physics, University of Vienna, Boltzmanngasse 9, 1090 Vienna, Austria
| | - Frank Verstraete
- Department of Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
| | - Laurens Vanderstraeten
- Department of Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
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Vanhecke B, Hasik J, Verstraete F, Vanderstraeten L. Scaling Hypothesis for Projected Entangled-Pair States. PHYSICAL REVIEW LETTERS 2022; 129:200601. [PMID: 36461996 DOI: 10.1103/physrevlett.129.200601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/09/2021] [Accepted: 10/17/2022] [Indexed: 06/17/2023]
Abstract
We introduce a new paradigm for scaling simulations with projected entangled-pair states (PEPS) for critical strongly correlated systems, allowing for reliable extrapolations of PEPS data with relatively small bond dimensions D. The key ingredient consists of using the effective correlation length ξ for inducing a collapse of data points, f(D,χ)=f(ξ(D,χ)), for arbitrary values of D and the environment bond dimension χ. As such we circumvent the need for extrapolations in χ and can use many distinct data points for a fixed value of D. Here, we need that the PEPSs have been optimized using a fixed-χ gradient method, which can be achieved using a novel tensor-network algorithm for finding fixed points of 2D transfer matrices, or by using the formalism of backwards differentiation. We test our hypothesis on the critical 3D dimer model, the 3D classical Ising model, and the 2D quantum Heisenberg model.
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Affiliation(s)
- Bram Vanhecke
- Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium
| | - Juraj Hasik
- Laboratoire de Physique Théorique UMR5152, C.N.R.S. and Université de Toulouse, 118 Route de Narbonne, 31062 Toulouse, France
| | - Frank Verstraete
- Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium
| | - Laurens Vanderstraeten
- Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium
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Mortier Q, Schuch N, Verstraete F, Haegeman J. Tensor Networks Can Resolve Fermi Surfaces. PHYSICAL REVIEW LETTERS 2022; 129:206401. [PMID: 36461985 DOI: 10.1103/physrevlett.129.206401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2022] [Accepted: 10/12/2022] [Indexed: 06/17/2023]
Abstract
We demonstrate that projected entangled-pair states are able to represent ground states of critical, fermionic systems exhibiting both 1d and 0d Fermi surfaces on a 2D lattice with an efficient scaling of the bond dimension. Extrapolating finite size results for the Gaussian restriction of fermionic projected entangled-pair states to the thermodynamic limit, the energy precision as a function of the bond dimension is found to improve as a power law, illustrating that an arbitrary precision can be obtained by increasing the bond dimension in a controlled manner. In this process, boundary conditions and system sizes have to be chosen carefully so that nonanalyticities of the Ansatz, rooted in its nontrivial topology, are avoided.
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Affiliation(s)
- Quinten Mortier
- Department of Physics and Astronomy, Ghent University, Krijgslaan 281, S9, 9000 Gent, Belgium
| | - Norbert Schuch
- Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Straße 1, 85748 Garching, Germany and Munich Center for Quantum Science and Technology, Schellingstraße 4, 80799 München, Germany
- University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Wien, Austria and University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
| | - Frank Verstraete
- Department of Physics and Astronomy, Ghent University, Krijgslaan 281, S9, 9000 Gent, Belgium
| | - Jutho Haegeman
- Department of Physics and Astronomy, Ghent University, Krijgslaan 281, S9, 9000 Gent, Belgium
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