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arXiv:2504.10919 · Physical review. B./Physical review. B

Symmetry-protected topological order identified via Gutzwiller-guided density matrix renormalization group: SO ( n ) spin chains

Pei-Yuan Cai, Hui-Ke Jin, Yi Zhou

Published Jul 7, 2025• 3 citations
arXivHTMLPDF

Abstract

We present a comprehensive study of topological phases in the SO($n$) spin chains using a combination of analytical parton construction and numerical techniques. For even $n=2l$, we identify a novel SPT$^2$ phase characterized by two distinct topological sectors, exhibiting exact degeneracy at the matrix product state (MPS) exactly solvable point. Through Gutzwiller-projected mean-field theory and density matrix renormalization group (DMRG) calculations, we demonstrate that these sectors remain topologically degenerate in close chains throughout the SPT$^2$ phase, with energy gaps decaying exponentially with system size. For odd $n=2l+1$, we show that the ground state remains unique in close chains. We precisely characterize critical states using entanglement entropy scaling, confirming the central charges predicted by conformal field theories. Our results reveal fundamental differences between even and odd $n$ cases, provide numerical verification of topological protection, and establish reliable methods for studying high-symmetry quantum systems. The Gutzwiller-guided DMRG is demonstrated to be notably efficient in targeting specific topological sectors.

DOI
10.1103/tdgz-ky5v

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We present a comprehensive study of topological phases in the SO(nnn) spin chains using a combination of analytical parton construction and numerical techniques. For even n=2ln=2ln=2l, we identify a novel SPT2^22 phase characterized by two distinct topological sectors, exhibiting exact degeneracy at the matrix product state (MPS) exactly solvable point. Through Gutzwiller-projected mean-field theory and density matrix renormalization group (DMRG) calculations, we demonstrate that these sectors remain topologically degenerate in close chains throughout the SPT2^22 phase, with energy gaps decaying exponentially with system size. For odd n=2l+1n=2l+1n=2l+1, we show that the ground state remains unique in close chains. We precisely characterize critical states using entanglement entropy scaling, confirming the central charges predicted by conformal field theories. Our results reveal fundamental differences between even and odd nnn cases, provide numerical verification of topological protection, and establish reliable methods for studying high-symmetry quantum systems. The Gutzwiller-guided DMRG is demonstrated to be notably efficient in targeting specific topological sectors.
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Effective field theory is a very useful technique for understanding quantum many body systems. We use this approach to study a certain class of critical quantum spin-1 chains and symmetry protected topological phases of matter in two spatial dimensions. In the first context we consider the SU(3) symmetric spin-1 chain with additional interactions. We use bosonization to demonstrate that the effective field theories describing the critical behaviour of these spin chains can be mapped to free compact boson conformal field theories (CFTs) with central charge c=2. We also describe how some predictions from field theory can be verified in numerical calculations using exact diagonalization (ED) and the density matrix renormalization group (DMRG) algorithm. In particular, the bosonization method gives a formula for the evolution of four Tomonaga-Luttinger liquid (TLL) parameters as a function of the lattice parameters. Using the analytic formulae for the scaling dimensions in terms of the TLL parameters and matching of lowest scaling dimensions, we numerically calculate these field-theoretic parameters and confirm that their evolution agrees with the prediction using bosonization. We also tackle aspects of the challenging problem of understanding interacting topological phases. In this context, we aim to understand the effects if interactions in certain classes of symmetry protected topological (SPT) phases of matter. We consider non-chiral SPT phases in two spatial dimensions protected by a discrete symmetry such as Z_K or Z_K x Z_K symmetry. We argue that modular invariance/noninvariance of the partition function of the one-dimensional edge theory can be used to diagnose whether, by adding a suitable potential, the edge theory can be gapped or not without breaking the symmetry. By taking bosonic phases described by Chern-Simons K-matrix theories and fermionic phases relevant to topological superconductors as examples, we demonstrate explicitly that when modular invariance is achieved, we can construct an interaction potential that is consistent with the symmetry and can completely gap out the edge. We also briefly discuss preliminary results of a numerical approach for simulating the 3 dimensional Landau Level problem. This includes a brief description of a potential application of variational Monte Carlo to spin-orbit coupled ab initio systems.
This thesis studies a pair of symmetry protected topological (SPT) phases which arise when considering one-dimensional quantum spin systems possessing a natural orthogonal group symmetry. Particular attention is given to a family of exactly solvable models whose ground states admit a matrix product state description and generalize the AKLT chain. We call these models ``SO(n)SO(n)SO(n) AKLT chains'' and the phase they occupy the ``SO(n)SO(n)SO(n) Haldane phase''. We present new results describing their ground state structure and, when nnn is even, their peculiar O(n)O(n)O(n)-to-SO(n)SO(n)SO(n) symmetry breaking. We also prove that these states have arbitrarily large correlation and injectivity length by increasing nnn, but all have a 2-local parent Hamiltonian, in contrast to the natural expectation that the interaction range of a parent Hamiltonian should diverge as these quantities diverge. We extend Ogata's definition of an SPT index for a split state for a finite symmetry group GGG to an SPT index for a compact Lie group GGG. We then compute this index, which takes values in the second Borel group cohomology H2(SO(n),U(1))H^2(SO(n),U(1))H2(SO(n),U(1)), at a single point in each of the SPT phases. The two points have different indices, confirming the two SPT phases are indeed distinct. Chapter 1 contains an introduction with a detailed overview of the contents of this thesis, which includes several chapters of background information before presenting new results in Chapter 7 and Chapter 8.
We reexamine the one-dimensional spin-1 XXZXXZXXZ model with on-site uniaxial single-ion anisotropy as to the appearance and characterization of the symmetry-protected topological Haldane phase. By means of large-scale density-matrix renormalization group (DMRG) calculations the central charge can be determined numerically via the von Neumann entropy, from which the ground-state phase diagram of the model can be derived with high precision. The nontrivial gapped Haldane phase shows up in between the trivial gapped even Haldane and N\'eel phases, appearing at large single-ion and spin-exchange interaction anisotropies, respectively. We furthermore carve out a characteristic degeneracy of the lowest entanglement level in the topological Haldane phase, which is determined using a conventional finite-system DMRG technique with both periodic and open boundary conditions. Defining the spin and neutral gaps in analogy to the single-particle and neutral gaps in the intimately connected extended Bose-Hubbard model, we show that the excitation gaps in the spin model qualitatively behave just as for the bosonic system. We finally compute the dynamical spin structure factor in the three different gapped phases and find significant differences in the intensity maximum which might be used to distinguish these phases experimentally.
Symmetry-protected trivial (SPt) phases of matter are the product-state analog of symmetry-protected topological (SPT) phases. This means, SPt phases can be adiabatically connected to a product state by some path that preserves the protecting symmetry. Moreover, SPt and SPT phases can be adiabatically connected to each other when interaction terms that break the symmetries protecting the SPT order are added in the Hamiltonian. It is also known that spin-1 SPT phases in quantum spin chains can emerge as effective intermediate phases of spin-2 Hamiltonians. In this paper we show that a similar scenario is also valid for SPt phases. More precisely, we show that for a given spin-2 quantum chain, effective intermediate spin-1 SPt phases emerge in some regions of the phase diagram, these also being adiabatically connected to nontrivial intermediate SPT phases. We characterize the phase diagram of our model by studying quantities such as the entanglement entropy, symmetry-related order parameters, and 1-site fidelities. Our numerical analysis uses matrix product states and the infinite time evolving block decimation method to approximate ground states of the system in the thermodynamic limit. Moreover, we provide a field theory description of the possible quantum phase transitions between the SPt phases. Together with the numerical results, such a description shows that the transitions may be described by conformal field theories with central charge c=1c=1c=1. Our results are in agreement with, and further generalize, those of Y. Fuji, F. Pollmann, and M. Oshikawa [Phys. Rev. Lett. 114, 177204 (2015)].
Symmetry protected topological (SPT) phases are characterized by robust boundary features, which do not disappear unless passing through a phase transition. These boundary features can be quantified by a topological invariant which, in some cases, is related to a physical quantity, such as the spin conductivity for the quantum spin Hall insulators. In other cases, the boundary features give rise to new physics, such as the Majorana fermion. In all cases the boundary features can be analyzed with the help of an entanglement spectrum and their robustness make them promising candidates for storing quantum information. The topological invariant characterizing SPT phases is strictly only invariant under deformations which respect a certain \nsymmetry. For example, the boundary currents of the quantum spin Hall insulator are only robust against non-magnetic, i.e. time-reversal invariant, impurities. In this thesis we study the SPT phases of spin chains. \nAs a result of our work we find a topological invariant for SPT phases of spin chains which are protected by continuous symmetries. By means of a non-local order parameter we find a way to extract this invariant from the ground state wave function of the system. Using density-matrix-renormalization-group techniques we verify that this invariant is a tool to detect transitions between different topological phases. We find a non-local transformation that maps SPT phases to conventional phases characterized by a local order parameter. This transformation suggests an analogy between topological phases and conventional phases and thus give a deeper understanding of the role of topology in spin systems.