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Register a new userZero-site DMRG and the optimal low-rank correction
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Added by
Yuriel N\\'u\\~nez-Fern\\'andez
Publication date
2019
fields
Physics
and research's language is
English
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A zero-site density matrix renormalization algorithm (DMRG0) is proposed to minimize the energy of matrix product states (MPS). Instead of the site tensors themselves, we propose to optimize sequentially the message tensors between neighbor sites, which contain the singular values of the bipartition. This leads to a local minimization step that is independent of the physical dimension of the site. Conceptually, it separates the optimization and decimation steps in DMRG. Furthermore, we introduce two new global perturbations based on the optimal low-rank correction to the current state, which are used to avoid local minima. They are determined variationally as the MPS closest to the one-step correction of the Lanczos or Jacobi-Davidson eigensolver, respectively. These perturbations mainly decrease the energy and are free of hand-tuned parameters. Compared to existing single-site enrichment proposals, our approach gives similar convergence ratios per sweep while the computations are cheaper by construction. Our methods may be useful in systems with many physical degrees of freedom per lattice site. We test our approach on the periodic Heisenberg spin chain for various spins, and on free electrons on the lattice.
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We analyze the antiferromagnetic $text{SU}(3)$ Heisenberg chain by means of the Density Matrix Renormalization Group (DMRG). The results confirm that the model is critical and the computation of its central charge and the scaling dimensions of the first excited states show that the underlying low energy conformal field theory is the $text{SU}(3)_1$ Wess-Zumino-Novikov-Witten model.
We describe the occurrence and physical role of zero-energy modes in the Dirac equation with a topologically non-trivial background.
The infinite Density Matrix Renormalisation Group (iDMRG) algorithm is a highly successful numerical algorithm for the study of low-dimensional quantum systems, and is also frequently used to initialise the more popular finite DMRG algorithm. Implementations of both finite and infinite DMRG frequently incorporate support for the protection and exploitation of symmetries of the Hamiltonian. In common with other variational tensor network algorithms, convergence of iDMRG to the ground state is not guaranteed, with the risk that the algorithm may become stuck in a local minimum. In this paper I demonstrate the existence of a particularly harmful class of physically irrelevant local minima affecting both iDMRG and to a lesser extent also infinite Time-Evolving Block Decimation (iTEBD), for which the ground state is compatible with the protected symmetries of the Hamiltonian but cannot be reached using the conventional iDMRG or iTEBD algorithms. I describe a modified iDMRG algorithm which evades these local minima, and which also admits a natural interpretation on topologically ordered systems with a boundary.
We describe the phase diagram of a 2+1 dimensional SU(2) gauge theory of fluctuating incommensurate spin density waves for the hole-doped cuprates. Our primary assumption is that all low energy fermionic excitations are gauge neutral and electron-like, while the spin density wave order is fractionalized into Higgs fields transforming as adjoints of the gauge SU(2). The confining phase of the gauge theory is a conventional Fermi liquid with a large Fermi surface (and its associated $d$-wave superconductor). There is a quantum phase transition to a Higgs phase describing the `pseudogap at lower doping. Depending upon the quartic terms in the Higgs potential, the Higgs phase exhibits one or more of charge density wave, Ising-nematic, time-reversal odd scalar spin chirality, and $mathbb{Z}_2$ topological orders. It is notable that the emergent broken symmetries in our theory of fluctuating spin density waves co-incide with those observed in diverse experiments. For the electron-doped cuprates, the spin density wave fluctuations are at wavevector $(pi,pi)$, and then the corresponding SU(2) gauge theory only has a crossover between the confining and Higgs regimes, with an exponentially large confinement scale deep in the Higgs regime. On the Higgs side, for both the electron- and hole-doped cases, and at scales shorter than the confinement scale (which can be infinite when $mathbb{Z}_2$ topological order is present), the electron spectral function has a `fractionalized Fermi liquid (FL*) form with small Fermi surfaces. We also describe the deconfined quantum criticality of the Higgs transition in the limit of a large number of Higgs flavors, and perturbatively discuss its coupling to fermionic excitations.
Zero modes arising from a planar Majorana equation in the presence of $N$ vortices require an $mathcal{N}$-dimensional state-space, where $mathcal{N} = 2^{N/2}$ for $N$ even and $mathcal{N} = 2^{(N + 1)/2}$ for $N$ odd. The mode operators form a restricted $mathcal{N}$-dimensional Clifford algebra.
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