Infinite boundary conditions for response functions and limit cycles within the infinite-system density matrix renormalization group approach demonstrated for bilinear-biquadratic spin-1 chains

Abstract

Response functions Ax(t)By(0) for one-dimensional strongly correlated quantum many-body systems can be computed with matrix product state (MPS) techniques. Especially, when one is interested in spectral functions or dynamic structure factors of translation-invariant systems, the response for some range |x − y| < is needed. We demonstrate how the number of required time-evolution runs can be reduced substantially: (a) If finite-system simulations are employed, the number of time-evolution runs can be reduced from to 2 √ . (b) To go beyond, one can employ infinite MPS (iMPS) such that two evolution runs suffice. To this purpose, iMPS that are heterogeneous only around the causal cone of the perturbation are evolved in time, i.e., the simulation is done with infinite boundary conditions. Computing overlaps of these states, spatially shifted relative to each other, yields the response functions for all distances |x−y|. As a specific application, we compute the dynamic structure factor for ground states of bilinear-biquadratic spin-1 chains with very high resolution and explain the underlying low-energy physics. To determine the initial uniform iMPS for such simulations, infinite-system density-matrix renormalization group (iDMRG) can be employed. We discuss that, depending on the system and chosen bond dimension, iDMRG with a cell size nc may converge to a non-trivial limit cycle of length m. This then corresponds to an iMPS with an enlarged unit cell of size mnc. arXiv:1804.09163v2 [cond-mat.str-el]

Efficient tensor network representation for Gutzwiller projected states of paired fermions

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Low-energy physics of isotropic spin-1 chains in the critical and Haldane phases

“…We have explored the low-energy physics of isotropic spin-1 chains. Using an MPS algorithm [26], we were able to compute precise dynamic structure factors, even in the highly entangled critical phase with c = 2. We have found that the NLσM and the Majorana field theory fail to capture the influence of the biquadratic term and provide only a rather unsatisfactory description for the Haldane phase.…”

“…While the groundstate phase diagram has been studied extensively, much less is known about the low-energy dynamics. We use a recently introduced algorithm [26] based on the density matrix renormalization group (DMRG) [27][28][29] and the time evolution of matrix product states (MPS) [30][31][32] with infinite boundary conditions [33,34] to compute dynamic structure factors S(k, ω) = x e −ikx dt e iωt ψ| Âx (t) B0 (0)|ψ ,…”

“…They can also be measured in neutron-scattering or ARPES experiments. Starting from high-resolution dynamic structure factors computed with the MPS algorithm [26], we study the relevant excitations of the model to explain the observed features. To this end, we compare the numerical results to Bethe ansatz and fieldtheoretical treatments.…”

Loop-free tensor networks for high-energy physics

“…They originated from the understanding that the density matrix renormalization group (DMRG) technique could be recast in a variational formulation by means of matrix product states (MPS) [6][7][8]. This stimulated the further development of such a framework in the last decade, extending the TN paradigm to encompass higher dimensionality [9][10][11][12][13][14], peculiar geometries [15,16], directly addressing the thermodynamical limit [17][18][19], as well as the continuum [20][21][22][23].…”