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Optimal Matrix Product States for the Heisenberg Spin Chain
Other Condensed Matter · Submitted 9 Dec 2008
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This allows to show that results of standard simulations, e.g. density matrix renormalization group and infinite time evolving block decimation, do correspond to the result obtained by this minimization strategy and, thus, both methods deliver optimal MPS with the same energy but, otherwise, different properties.
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We present some exact results for the optimal Matrix Product State (MPS) approximation to the ground state of the infinite isotropic Heisenberg spin-1/2 chain. Our approach is based on the systematic use of Schmidt decompositions to reduce the problem of approximating for the ground state of a spin chain to an analytical minimization. This allows to show that results of standard simulations, e.g. density matrix renormalization group and infinite time evolving block decimation, do correspond to the result obtained by this minimization strategy and, thus, both methods deliver optimal MPS with the same energy but, otherwise, different properties. We also find that translational and rotational symmetries cannot be maintained simultaneously by the MPS ansatz of minimum energy and present explicit constructions for each case. Furthermore, we analyze symmetry restoration and quantify it to uncover new scaling relations. The method we propose can be extended to any translational invariant Hamiltonian.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present some exact results for the optimal Matrix Product State (MPS) approximation to the ground state of the infinite isotropic Heisenberg spin-1/2 chain. | contribution 0.99 | 0.95 | 0.27 | asserted 1.00 |
| s1 | Our approach is based on the systematic use of Schmidt decompositions to reduce the problem of approximating for the ground state of a spin chain to an analytical minimization. | method 1.00 | 0.88 | 0.08 | asserted 1.00 |
| s2 | This allows to show that results of standard simulations, e.g. density matrix renormalization group and infinite time evolving block decimation, do correspond to the result obtained by this minimization strategy and, thus, both methods deliver optimal MPS with the same energy but, otherwise, different properties. | result 0.81 | 0.76 | 0.39 | asserted 1.00 |
| s3 | We also find that translational and rotational symmetries cannot be maintained simultaneously by the MPS ansatz of minimum energy and present explicit constructions for each case. | result 0.98 | 0.96 | 0.18 | asserted 1.00 |
| s4 | Furthermore, we analyze symmetry restoration and quantify it to uncover new scaling relations. | contribution 0.72 | 0.95 | 0.09 | asserted 1.00 |
| s5 | The method we propose can be extended to any translational invariant Hamiltonian. | future work 0.72 | 0.53 | 0.13 | asserted 0.52 |
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Original abstract
We present some exact results for the optimal Matrix Product State (MPS) approximation to the ground state of the infinite isotropic Heisenberg spin-1/2 chain. Our approach is based on the systematic use of Schmidt decompositions to reduce the problem of approximating for the ground state of a spin chain to an analytical minimization. This allows to show that results of standard simulations, e.g. density matrix renormalization group and infinite time evolving block decimation, do correspond to the result obtained by this minimization strategy and, thus, both methods deliver optimal MPS with the same energy but, otherwise, different properties. We also find that translational and rotational symmetries cannot be maintained simultaneously by the MPS ansatz of minimum energy and present explicit constructions for each case. Furthermore, we analyze symmetry restoration and quantify it to uncover new scaling relations. The method we propose can be extended to any translational invariant Hamiltonian.
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A number of results can be found in the literature [ref] (and references therein) concerning exact MPS in particular cases, exploiting rotational or translational invariance.
Matrix product operator representations · Introduction · 0.92
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