We introduce a matrix product state (MPS) with an incommensurate periodicity by applying the spin-rotation operator of each site to a uniform MPS in the thermodynamic limit. The relation between the improvement of the variational method due to the spin-rotation operators and the spontaneous symmetry breaking of the MPS is studied. The optimized pitch of the rotational operator reflects the commensurate/incommensurate properties of spin-spin correlation functions in the
S=1/2 Heisenberg chain and the
S=1/2 ferromagnetic-antiferromagnetic zigzag chain.
We investigate the use of matrix product states (MPS) to approximate ground states of critical quantum spin chains with periodic boundary conditions (PBC). We identify two regimes in the
(N,D) parameter plane, where
N is the size of the spin chain and
D is the dimension of the MPS matrices. In the first regime, MPS can be used to perform finite size scaling (FSS). In the complementary regime, the MPS simulations show instead the clear signature of finite entanglement scaling (FES). In the thermodynamic limit (or large
N limit), only MPS in the FSS regime maintain a finite overlap with the exact ground state. This observation has implications on how to correctly perform FSS with MPS as well as on the performance of recent MPS algorithms for systems with PBC. It also gives clear evidence that critical models can actually be simulated very well with MPS by using the right scaling relations; in the appendixes, we give an alternative derivation of the result of Pollmann et al. [Phys. Rev. Lett. 102, 255701 (2009)] relating the bond dimension of the MPS to an effective correlation length.
A variational ansatz for momentum eigenstates of translation-invariant quantum spin chains is formulated. The matrix product state ansatz works directly in the thermodynamic limit and allows for an efficient implementation (cubic scaling in the bond dimension) of the variational principle. Unlike previous approaches, the ansatz includes topologically nontrivial states (kinks, domain walls) for systems with symmetry breaking. The method is benchmarked using the spin-\textonehalf{}
XXZ antiferromagnet and the spin-1 Heisenberg antiferromagnet, and we obtain surprisingly accurate results.
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
D3, 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.
Matrix product states (MPS) are quintessential examples of frustration-free gapped ground states of local interactions called parent Hamiltonians. In this work, we investigate parent Hamiltonians for a class of ergodic matrix product states (EMPS), which are MPS defined by site-dependent random tensors
{Xj[k]}j=1D which are homogeneously distributed at every site
k in the spin chain. Here, the EMPS are not translation-invariant but rather statistically translation-invariant. Under a mild injectivity assumption, we show the thermodynamic limit of an EMPS is the unique frustration-free ground state of a parent Hamiltonian on the whole spin chain, which, depending on the statistical properties of the EMPS, may or may not be finite-range. In contrast to the translation-invariant regime, these Hamiltonians need not be gapped. Nevertheless, applying the martingale method while keeping track of local statistics gives conditions for a gap, in addition to pointing towards why there need not be a gap in general. We include examples of EMPS both with and without spectral gaps to illustrate our results.
We report on a systematic implementation of su (2) invariance for matrix product states (MPS) with concrete computations cast in a diagrammatic language. As an application we present a variational MPS study of a spin-1/2 quantum chain. For efficient computations, we make systematic use of the su (2) symmetry at all steps of the calculations: (i) the matrix space is set up as a direct sum of irreducible representations, (ii) the local matrices with state-valued entries are set up as superposition of su (2) singlet operators, (iii) products of operators are evaluated algebraically by making use of identities for 3 j and 6 j symbols. The remaining numerical computations like the diagonalization of the associated transfer matrix and the minimization of the energy expectation value are done in spaces free of symmetry degeneracies. The energy expectation value is a strict upper bound of the true ground-state energy and yields definite conclusions about the accuracy of density-matrix renormalization group results reported in the literature. Furthermore, we present explicit results with accuracy better than 10 −4 for nearest- and next-nearest-neighbour spin correlators and for general dimer–dimer correlators in the thermodynamical limit of the spin- Heisenberg chain with frustration.