Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1102.3562 · Journal of Statistical Mechanics Theory and Experiment

Stiffness in 1D matrix product states with periodic boundary conditions

Davide Rossini, Vittorio Giovannetti, Rosario Fazio

Published May 25, 2011• 15 citations
arXivHTMLPDF

Abstract

We discuss in detail a modified variational matrix product state algorithm for periodic boundary conditions, based on a recent work by Pippan et al (2010 Phys. Rev. B 81 081103(R)), which enables one to study large systems on a ring (composed of N ∼ 10 2 sites). In particular, we introduce a couple of improvements allowing us to enhance the algorithm in terms of stability and reliability. We employ such a method to compute the stiffness of one-dimensional strongly correlated quantum lattice systems. The accuracy of our calculations is tested in the exactly solvable spin-1/2 Heisenberg chain.

DOI
10.1088/1742-5468/2011/05/p05021

Related papers

Efficient matrix-product state method for periodic boundary conditions

Peter Pippan, Steven R. White, Hans Gerd Evertz

We introduce an efficient method to calculate the ground state of one-dimensional lattice models with periodic boundary conditions. The method works in the representation of matrix product states (MPS), related to the density matrix renormalization group (DMRG) method. It improves on a previous approach by Verstraete et al. We introduce a factorization procedure for long products of MPS matrices, which reduces the computational effort from ${m}^{5}$ to ${m}^{3}$, where $m$ is the matrix dimension, and $m\ensuremath{\simeq}100--1000$ in typical cases. We test the method on the $S=\frac{1}{2}$ and $S=1$ Heisenberg chains. It is also applicable to nontranslationally invariant cases. The method makes ground-state calculations with periodic boundary conditions about as efficient as traditional DMRG calculations for systems with open boundaries.

arXiv:0801.1947•Physical Review B•Feb 12, 2010•88 citations

Efficient MPS Algorithm for Periodic Boundary Conditions and Applications Section: Solid matter Original Author's Text: English Abstract: We present the implementation of an efficient algorithm for the calculation of the spectrum of one-dimensional quantum systems with periodic boundary conditions. This algorithm is based on a matrix product representation for quantum states (MPS) and a similar representation for Hamiltonians and other operators (MPO). It is significantly more efficient for systems of about 100 sites and more than for small quantum systems. We apply the formalism to calculate the ground state and the first excited state of a spin-1 Heisenberg ring and deduce the size of a Haldane gap. The results are compared to previous high-precision DMRG calculations. Furthermore, we study the spin-1 systems with a biquadratic nearest-neighbor interaction and show the first results of an application to a mesoscopic Hubbard ring of spinless fermions, which carries a persistent current. Key words: matrix product representation for quantum states (MPS) and Hamiltonians (MPO), spin-1 Heisenberg ring, density matrix renormalization group (DMRG). List Issues Configure

Physikalisch-Technische Bundesanstalt (Bundesallee 100, D-38116 Braunschweig, Germany), Michael Weyrauch, Mykhailo V. Rakov

We present the implementation of an efficient algorithm for the calculation of the spectrum of one-dimensional quantum systems with periodic boundary conditions. This algorithm is based on a matrix product representation for quantum states (MPS) and a similar representation for Hamiltonians and other operators (MPO). It is significantly more efficient for systems of about 100 sites and more than for small quantum systems. We apply the formalism to calculate the ground state and the first excited state of a spin-1 Heisenberg ring and deduce the size of a Haldane gap. The results are compared to previous high-precision DMRG calculations. Furthermore, we study the spin-1 systems with a biquadratic nearest-neighbor interaction and show the first results of an application to a mesoscopic Hubbard ring of spinless fermions, which carries a persistent current.

arXiv:1303.1333•Ukrainian Journal of Physics•Jul 1, 2013•12 citations

Spin-supersolid phase in Heisenberg chains: A characterization via matrix product states with periodic boundary conditions

Davide Rossini, Vittorio Giovannetti, Rosario Fazio

By means of a variational calculation using matrix product states with periodic boundary conditions, we accurately determine the extension of the spin-supersolid phase predicted to exist in the spin-1 anisotropic Heisenberg chain. We compute both the structure factor and the superfluid stiffness and extract the critical exponents of the supersolid-to-solid phase transition.

arXiv:1102.3545•Physical Review B•Apr 25, 2011•14 citations

Matrix product state based algorithm for determining dispersion relations of quantum spin chains with periodic boundary conditions

B. Pirvu, Jutho Haegeman, Frank Verstraete

We study a matrix product state algorithm to approximate excited states of translationally invariant quantum spin systems with periodic boundary conditions. By means of a momentum eigenstate ansatz generalizing the one of \"Ostlund and Rommer [see S. \"Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995);S. Rommer and S. \"Ostlund, Phys. Rev. B 55, 2164 (1997)], we separate the Hilbert space of the system into subspaces with different momentum. This gives rise to a direct sum of effective Hamiltonians, each one corresponding to a different momentum, and we determine their spectrum by solving a generalized eigenvalue equation. Surprisingly, many branches of the dispersion relation are approximated to a very good precision. We benchmark the accuracy of the algorithm by comparison with the exact solutions and previous numerical results for the quantum Ising, the antiferromagnetic Heisenberg spin-$1/2$, and the bilinear-biquadratic spin-1 models.

arXiv:1103.2735•Physical Review B•Jan 31, 2012•42 citations

Exploiting translational invariance in matrix product state simulations of spin chains with periodic boundary conditions

B. Pirvu, Frank Verstraete, Guifré Vidal

We present a matrix product state (MPS) algorithm to approximate ground states of translationally invariant systems with periodic boundary conditions. For a fixed value of the bond dimension $D$ of the MPS, we discuss how to minimize the computational cost to obtain a seemingly optimal MPS approximation to the ground state. In a chain with $N$ sites and correlation length $\ensuremath{\xi}$, the computational cost formally scales as $g(D,\ensuremath{\xi}/N){D}^{3}$, where $g(D,\ensuremath{\xi}/N)$ is a nontrivial function. For $\ensuremath{\xi}\ensuremath{\ll}N$, this scaling reduces to ${D}^{3}$, independent of the system size $N$, making our method $N$ times faster than previous proposals. We apply the algorithm to obtain MPS approximations for the ground states of the critical quantum Ising and Heisenberg spin-$1/2$ models as well as for the noncritical Heisenberg spin-$1$ model. In the critical case, for any chain length $N$, we find a model-dependent bond dimension $D(N)$ above which the polynomial decay of correlations is faithfully reproduced throughout the entire system.

arXiv:1005.5195•Physical Review B•Mar 17, 2011•52 citations

An efficient density matrix renormalization group algorithm for chains with periodic boundary condition

Dayasindhu Dey, Debasmita Maiti, Manoranjan Kumar

The Density Matrix Renormalization Group (DMRG) is a state-of-the-art numerical technique for a one dimensional quantum many-body system; but calculating accurate results for a system with Periodic Boundary Condition (PBC) from the conventional DMRG has been a challenging job from the inception of DMRG. The recent development of the Matrix Product State (MPS) algorithm gives a new approach to find accurate results for the one dimensional PBC system. The most efficient implementation of the MPS algorithm can scale as O($p \times m^3$), where $p$ can vary from 4 to $m^2$. In this paper, we propose a new DMRG algorithm, which is very similar to the conventional DMRG and gives comparable accuracy to that of MPS. The computation effort of the new algorithm goes as O($m^3$) and the conventional DMRG code can be easily modified for the new algorithm.Received: 2 August 2016, Accepted: 12 October 2016; Edited by: K. Hallberg; DOI: http://dx.doi.org/10.4279/PIP.080006Cite as: D Dey, D Maiti, M Kumar, Papers in Physics 8, 080006 (2016)This paper, by D Dey, D Maiti, M Kumar, is licensed under the Creative Commons Attribution License 3.0.

arXiv:1605.09301•Papers in Physics•Nov 18, 2016•18 citations
We discuss in detail a modified variational matrix product state algorithm for periodic boundary conditions, based on a recent work by Pippan et al (2010 Phys. Rev. B 81 081103(R)), which enables one to study large systems on a ring (composed of N ∼ 10 2 sites). In particular, we introduce a couple of improvements allowing us to enhance the algorithm in terms of stability and reliability. We employ such a method to compute the stiffness of one-dimensional strongly correlated quantum lattice systems. The accuracy of our calculations is tested in the exactly solvable spin-1/2 Heisenberg chain.
We introduce an efficient method to calculate the ground state of one-dimensional lattice models with periodic boundary conditions. The method works in the representation of matrix product states (MPS), related to the density matrix renormalization group (DMRG) method. It improves on a previous approach by Verstraete et al. We introduce a factorization procedure for long products of MPS matrices, which reduces the computational effort from m5{m}^{5}m5 to m3{m}^{3}m3, where mmm is the matrix dimension, and $m\ensuremath{\simeq}100--1000$ in typical cases. We test the method on the S=12S=\frac{1}{2}S=21​ and S=1S=1S=1 Heisenberg chains. It is also applicable to nontranslationally invariant cases. The method makes ground-state calculations with periodic boundary conditions about as efficient as traditional DMRG calculations for systems with open boundaries.
Efficient MPS Algorithm for Periodic Boundary Conditions and Applications Section: Solid matter Original Author's Text: English Abstract: We present the implementation of an efficient algorithm for the calculation of the spectrum of one-dimensional quantum systems with periodic boundary conditions. This algorithm is based on a matrix product representation for quantum states (MPS) and a similar representation for Hamiltonians and other operators (MPO). It is significantly more efficient for systems of about 100 sites and more than for small quantum systems. We apply the formalism to calculate the ground state and the first excited state of a spin-1 Heisenberg ring and deduce the size of a Haldane gap. The results are compared to previous high-precision DMRG calculations. Furthermore, we study the spin-1 systems with a biquadratic nearest-neighbor interaction and show the first results of an application to a mesoscopic Hubbard ring of spinless fermions, which carries a persistent current. Key words: matrix product representation for quantum states (MPS) and Hamiltonians (MPO), spin-1 Heisenberg ring, density matrix renormalization group (DMRG). List Issues Configure
We present the implementation of an efficient algorithm for the calculation of the spectrum of one-dimensional quantum systems with periodic boundary conditions. This algorithm is based on a matrix product representation for quantum states (MPS) and a similar representation for Hamiltonians and other operators (MPO). It is significantly more efficient for systems of about 100 sites and more than for small quantum systems. We apply the formalism to calculate the ground state and the first excited state of a spin-1 Heisenberg ring and deduce the size of a Haldane gap. The results are compared to previous high-precision DMRG calculations. Furthermore, we study the spin-1 systems with a biquadratic nearest-neighbor interaction and show the first results of an application to a mesoscopic Hubbard ring of spinless fermions, which carries a persistent current.
We study a matrix product state algorithm to approximate excited states of translationally invariant quantum spin systems with periodic boundary conditions. By means of a momentum eigenstate ansatz generalizing the one of \"Ostlund and Rommer [see S. \"Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995);S. Rommer and S. \"Ostlund, Phys. Rev. B 55, 2164 (1997)], we separate the Hilbert space of the system into subspaces with different momentum. This gives rise to a direct sum of effective Hamiltonians, each one corresponding to a different momentum, and we determine their spectrum by solving a generalized eigenvalue equation. Surprisingly, many branches of the dispersion relation are approximated to a very good precision. We benchmark the accuracy of the algorithm by comparison with the exact solutions and previous numerical results for the quantum Ising, the antiferromagnetic Heisenberg spin-1/21/21/2, and the bilinear-biquadratic spin-1 models.
We present a matrix product state (MPS) algorithm to approximate ground states of translationally invariant systems with periodic boundary conditions. For a fixed value of the bond dimension DDD of the MPS, we discuss how to minimize the computational cost to obtain a seemingly optimal MPS approximation to the ground state. In a chain with NNN sites and correlation length $\ensuremath{\xi}$, the computational cost formally scales as $g(D,\ensuremath{\xi}/N){D}^{3}$, where $g(D,\ensuremath{\xi}/N)$ is a nontrivial function. For $\ensuremath{\xi}\ensuremath{\ll}N$, this scaling reduces to D3{D}^{3}D3, independent of the system size NNN, making our method NNN times faster than previous proposals. We apply the algorithm to obtain MPS approximations for the ground states of the critical quantum Ising and Heisenberg spin-1/21/21/2 models as well as for the noncritical Heisenberg spin-111 model. In the critical case, for any chain length NNN, we find a model-dependent bond dimension D(N)D(N)D(N) above which the polynomial decay of correlations is faithfully reproduced throughout the entire system.
The Density Matrix Renormalization Group (DMRG) is a state-of-the-art numerical technique for a one dimensional quantum many-body system; but calculating accurate results for a system with Periodic Boundary Condition (PBC) from the conventional DMRG has been a challenging job from the inception of DMRG. The recent development of the Matrix Product State (MPS) algorithm gives a new approach to find accurate results for the one dimensional PBC system. The most efficient implementation of the MPS algorithm can scale as O(p×m3p \times m^3p×m3), where ppp can vary from 4 to m2m^2m2. In this paper, we propose a new DMRG algorithm, which is very similar to the conventional DMRG and gives comparable accuracy to that of MPS. The computation effort of the new algorithm goes as O(m3m^3m3) and the conventional DMRG code can be easily modified for the new algorithm.Received: 2 August 2016, Accepted: 12 October 2016; Edited by: K. Hallberg; DOI: http://dx.doi.org/10.4279/PIP.080006Cite as: D Dey, D Maiti, M Kumar, Papers in Physics 8, 080006 (2016)This paper, by D Dey, D Maiti, M Kumar, is licensed under the Creative Commons Attribution License 3.0.