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arXiv:1603.07769 · Journal of Statistical Mechanics Theory and Experiment

Matrix product states for su (2) invariant quantum spin chains

Rubina Zadourian, Andreas Fledderjohann, Andreas Klümper

Published Aug 1, 2016• 3 citations
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Abstract

Abstract A systematic and compact treatment of arbitrary su (2) invariant spin- s quantum chains with nearest-neighbour interactions is presented. The ground-state is derived in terms of matrix product states (MPS). The fundamental MPS calculations consist of taking products of basic tensors of rank 3 and contractions thereof. The algebraic su (2) calculations are carried out completely by making use of Wigner calculus. As an example of application, the spin-1 bilinear-biquadratic quantum chain is investigated. Various physical quantities are calculated with high numerical accuracy of up to 8 digits. We obtain explicit results for the ground-state energy, entanglement entropy, singlet operator correlations and the string order parameter. We find an interesting crossover phenomenon in the correlation lengths.

DOI
10.1088/1742-5468/2016/08/083101

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Abstract A systematic and compact treatment of arbitrary su (2) invariant spin- s quantum chains with nearest-neighbour interactions is presented. The ground-state is derived in terms of matrix product states (MPS). The fundamental MPS calculations consist of taking products of basic tensors of rank 3 and contractions thereof. The algebraic su (2) calculations are carried out completely by making use of Wigner calculus. As an example of application, the spin-1 bilinear-biquadratic quantum chain is investigated. Various physical quantities are calculated with high numerical accuracy of up to 8 digits. We obtain explicit results for the ground-state energy, entanglement entropy, singlet operator correlations and the string order parameter. We find an interesting crossover phenomenon in the correlation lengths.
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.
We investigate chains of ddd-dimensional quantum spins (qudits) on a line with generic nearest-neighbor interactions without translational invariance. We find the conditions under which these systems are not frustrated, that is, when the ground states are also the common ground states of all the local terms in the Hamiltonians. The states of a quantum spin chain are naturally represented in the matrix product states (MPS) framework. Using imaginary time evolution in the MPS ansatz, we numerically investigate the range of parameters in which we expect the ground states to be highly entangled and find them hard to approximate using our MPS method.
We present a new general and much simpler scheme to construct various quantum phase transitions (QPTs) in spin chain systems with matrix product ground states. By use of the scheme we take into account one kind of matrix product state (MPS) QPT and provide a concrete model. We also study the properties of the concrete example and show that a kind of QPT appears, accompanied by the appearance of the discontinuity of the parity absent block physical observable, diverging correlation length only for the parity absent block operator, and other properties which are that the fixed point of the transition point is an isolated intermediate-coupling fixed point of renormalization flow and the entanglement entropy of a half-infinite chain is discontinuous.
I first give an overview of the thesis and Matrix Product States (MPS) representation of quantum spin chains with an improvement on the conventional notation. The rest of this thesis is divided into two parts. The first part is devoted to eigenvalues of quantum many-body systems (QMBS). I introduce Isotropic Entanglement, which draws from various tools in random matrix theory and free probability theory (FPT) to accurately approximate the eigenvalue distribution of QMBS on a line with generic interactions. Next, I discuss the energy distribution of one particle hopping random Schrödinger operator in 1D from FPT in context of the Anderson model. The second part is devoted to ground states and gap of QMBS. I first give the necessary background on frustration free (FF) Hamiltonians, real and imaginary time evolution within MPS representation and a numerical implementation. I then prove the degeneracy and FF condition for quantum spin chains with generic local interactions, including corrections to our earlier assertions. I then summarize my efforts in proving lower bounds for the entanglement of the ground states, which includes some new results, with the hope that they inspire future work resulting in solving the conjecture given therein. Next I discuss two interesting measure zero examples where FF Hamiltonians are carefully constructed to give unique ground states with high entanglement. One of the examples (i.e., d=4d=4d=4) has not appeared elsewhere. In particular, we calculate the Schmidt numbers exactly, entanglement entropies and introduce a novel technique for calculating the gap which may be of independent interest. The last chapter elaborates on one of the measure zero examples (i.e., d=3d=3d=3) which is the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits signatures of a critical behavior.
The matrix product state (MPS) is utilized to study the ground state properties and quantum phase transitions (QPTs) of the one-dimensional quantum compass model (QCM). The MPS wavefunctions are argued to be very efficient descriptions of QCM ground states, and are numerically determined by imaginary time projections. The ground state energy, correlations, quantum entanglement and its spectrum, local and nonlocal order parameters, etc., are calculated and studied in details. It is revealed that the bipartite and block entanglement entropies, as well as the nearest neighbor correlation functions can be used to detect the second-order QPTs, but not the first-order ones, while fidelity detections can recognize both. The entanglement spectrum is extracted from the MPS wavefunction, and found to be doubly degenerate in disordered phases of QCM, where non-local string order parameters exist. Moreover, with linearized tensor renormalization group method, the specific heat curves are evaluated and their low temperature behaviors are investigated.
For the past twenty years, Matrix Product States (MPS) have been widely used in solid state physics to approximate the ground state of one-dimensional spin chains. In this paper, we study homogeneous MPS (hMPS), or MPS constructed via site-independent tensors and a boundary condition. Exploiting a connection with the theory of matrix algebras, we derive two structural properties shared by all hMPS, namely: a) there exist local operators which annihilate all hMPS of a given bond dimension; and b) there exist local operators which, when applied over any hMPS of a given bond dimension, decouple (cut) the particles where they act from the spin chain while at the same time join (glue) the two loose ends back again into a hMPS. Armed with these tools, we show how to systematically derive `bond dimension witnesses', or 2-local operators whose expectation value allows us to lower bound the bond dimension of the underlying hMPS. We extend some of these results to the ansatz of Projected Entangled Pairs States (PEPS). As a bonus, we use our insight on the structure of hMPS to: a) derive some theoretical limitations on the use of hMPS and hPEPS for ground state energy computations; b) show how to decrease the complexity and boost the speed of convergence of the semidefinite programming hierarchies described in [Phys. Rev. Lett. 115, 020501 (2015)] for the characterization of finite-dimensional quantum correlations.