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arXiv:2306.06021

Coexistence of symmetry-protected topological order and Neel order in the spin-1/2 ladder antiferromagnet C9H18N2CuBr4

Tao Hong, Imam Makhfudz, Xianglin Ke, May, Andrew F., Andrey A. Podlesnyak, Daniel M. Pajerowski, Barry Winn, Deumal, Merce, Takano, Yasumasa, Mark M. Turnbull

Published Jun 9, 2023• 0 citations

Abstract

Topological phases of matter are beyond the paradigm of Landau's symmetry breaking and have challenged our understanding of condensed matter systems. Here we report a new type of symmetry-protected topological phase of matter in the spin-1/2 coupled two-leg ladder antiferromagnet C9H18N2CuBr4, DLCB for short. In this two-sublattice antiferromagnet with a weak easy-axis anisotropy, we find no evidence of a conventional spin-flop transition in the magnetization with the magnetic field applied parallel to the easy axis at T=0.4 K, well below TN=2.0 K. Moreover, the temperature dependence of the gapped transverse excitations across TN indicates that they are not the conventional S=1 magnons associated with explicit symmetry breaking. Instead, the thermal renormalization of the gap energy shows a remarkable agreement with a calculation for the three-dimensional O(3) nonlinear sigma model. Accordingly, the spin gap in DLCB is not due to the spin anisotropy but to the separation between a spin singlet state and a triplet excited state. Since an antiferromagnetic spin-1/2 ladder systems can be mapped onto the spin-1 chain, the notion of the Haldane gap is proposed to explain the opening of the spin gap in DLCB. Therefore, the ground state of DLCB is best described as a quantum superposition of a Haldane phase and a Neel-ordered phase, which resembles the quantum state of a qubit in quantum computing. Our results indicate the presence of a symmetry-protected topological order coexisting with an antiferromagnetic order in this material.

DOI: 10.48550/arxiv.2306.06021

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Topological phases of matter are beyond the paradigm of Landau's symmetry breaking and have challenged our understanding of condensed matter systems. Here we report a new type of symmetry-protected topological phase of matter in the spin-1/2 coupled two-leg ladder antiferromagnet C9H18N2CuBr4, DLCB for short. In this two-sublattice antiferromagnet with a weak easy-axis anisotropy, we find no evidence of a conventional spin-flop transition in the magnetization with the magnetic field applied parallel to the easy axis at T=0.4 K, well below TN=2.0 K. Moreover, the temperature dependence of the gapped transverse excitations across TN indicates that they are not the conventional S=1 magnons associated with explicit symmetry breaking. Instead, the thermal renormalization of the gap energy shows a remarkable agreement with a calculation for the three-dimensional O(3) nonlinear sigma model. Accordingly, the spin gap in DLCB is not due to the spin anisotropy but to the separation between a spin singlet state and a triplet excited state. Since an antiferromagnetic spin-1/2 ladder systems can be mapped onto the spin-1 chain, the notion of the Haldane gap is proposed to explain the opening of the spin gap in DLCB. Therefore, the ground state of DLCB is best described as a quantum superposition of a Haldane phase and a Neel-ordered phase, which resembles the quantum state of a qubit in quantum computing. Our results indicate the presence of a symmetry-protected topological order coexisting with an antiferromagnetic order in this material.

It is well known that the Haldane phase of a one-dimensional spin-1 chain is a symmetry-protected topological (SPT) phase, which is described by a nonlinear sigma model (NLSM) with a $\ensuremath{\Theta}$ term at $\ensuremath{\Theta}=2\ensuremath{\pi}$. In this work we study a three-dimensional (3d) SPT phase of a SU(2

N

) antiferromagnetic spin system with a self-conjugate representation on every site. The spin-ordered N\'eel phase of this system has a ground state manifold $\mathcal{M}=\frac{\mathrm{U}(2N)}{\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)}$, and this system is described by a NLSM defined with manifold

\mathcal{M}

. Since the homotopy group ${\ensuremath{\pi}}_{4}[\mathcal{M}]=\mathbb{Z}$ for

N>1

, this NLSM can naturally have a $\ensuremath{\Theta}$ term. We will argue that when $\ensuremath{\Theta}=2\ensuremath{\pi}$ this NLSM describes a SPT phase. This SPT phase is protected by the SU(2

N

) spin symmetry, or its subgroup SU(

N

)$\ifmmode\times\else\texttimes\fi{}$SU$(N)\ensuremath{\rtimes}{Z}_{2}$, without assuming any other discrete symmetry. We will also construct a trial SU(2

N

) spin state on a 3d lattice; we argue that the long-wavelength physics of this state is precisely described by the aforementioned NLSM with $\ensuremath{\Theta}=2\ensuremath{\pi}$.

We show that a doped spin-1/2 ladder with antiferromagnetic intra-chain and ferromagnetic inter-chain coupling is a symmetry protected topologically non-trivial Luttinger liquid. Turning on a large easy-plane spin anisotropy drives the system to a topologically-trivial Luttinger liquid. Both phases have full spin gaps and exhibit power-law superconducting pair correlation. The Cooper pair symmetry is singlet

d_{xy}

in the non-trivial phase and triplet

S_z=0

in the trivial phase. The topologically non-trivial Luttinger liquid exhibits gapless spin excitations in the presence of a boundary, and it has no non-interacting or mean-field theory analog even when the fluctuating phase in the charge sector is pinned. As a function of the strength of spin anisotropy there is a topological phase transition upon which the spin gap closes. We speculate these Luttinger liquids are relevant to the superconductivity in metalized integer spin ladders or chains.

We study a spin-

\frac{1}{2}

XXZ model with a four-spin interaction on a two-leg ladder. By means of effective field theory and matrix product state calculations, we obtain rich ground-state phase diagrams that consist of eight distinct gapped phases. Four of them exhibit spontaneous symmetry breaking with either a magnetic or valence-bond-solid (VBS) long-range order. The other four are featureless, i.e., the bulk ground state is unique and does not break any symmetry. The featureless phases include the rung singlet (RS) and Haldane phases as well as their variants, the RS* and Haldane* phases, in which twisted singlet pairs $(|\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\rangle}+|\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\rangle})/\sqrt{2}$ are formed between the two legs. We argue and demonstrate that Gaussian transitions with the central charge

c=1

occur between the featureless phases and between the ordered phases while Ising transitions with

c=1/2

occur between the featureless and ordered phases. The two types of transition lines cross at the

\mathrm{SU}(2)

-symmetric point, where the criticality is described by the

\mathrm{SU}{(2)}_{2}

Wess-Zumino-Witten theory with

c=3/2

. The RS-Haldane* and RS*-Haldane transitions give examples of topological phase transitions. Interestingly, the RS* and Haldane* phases, which have highly anisotropic nature, appear even in the vicinity of the isotropic case. We demonstrate that all the four featureless phases are distinguished by topological indices in the presence of certain symmetries.

We show that a doped spin-1/2 ladder with antiferromagnetic intra-chain and ferromagnetic inter-chain coupling is a symmetry protected topologically non-trivial Luttinger liquid. Turning on a large easy-plane spin anisotropy drives the system to a topologically-trivial Luttinger liquid. Both phases have full spin gaps and exhibit power-law superconducting pair correlation. The Cooper pair symmetry is singlet dxy in the non-trivial phase and triplet Sz=0 in the trivial phase. The topologically non-trivial Luttinger liquid exhibits gapless spin excitations in the presence of a boundary, and it has no non-interacting or mean-field theory analog even when the fluctuating phase in the charge sector is pinned. As a function of the strength of spin anisotropy there is a topological phase transition upon which the spin gap closes. We speculate these Luttinger liquids are relevant to the superconductivity in metalized integer spin ladders or chains.

We report the experimental realization of a spin-1/2 extended diamond chain in a verdazyl-Cu complex, where competing interactions and lattice distortions give rise to exotic quantum phases. The magnetic properties exhibit a zero-field energy gap and 1/2 magnetization plateau, which is explained by a dimer-monomer model. Considering the effective interactions between the monomers, three types of dimer-dimer phases are expected to appear as the ground state, depending on the magnitude of the lattice distortions. By mapping to the nonlinear sigma model, three phases are distinguished topologically, and a symmetry-protected topological phase equivalent to the Haldane phase is identified. Furthermore, a nontrivial magnetization is observed above the 1/2 plateau region, suggesting a gapped dimer phase accompanied by a spontaneous breaking of translational symmetry. The discovery of this rare quantum state has broad implications for strongly correlated systems, topological matter, and quantum information science, where symmetry and topology play crucial roles.

Abstract Topology in quantum many-body systems has profoundly changed our understanding of quantum phases of matter. The model that has played an instrumental role in elucidating these effects is the antiferromagnetic spin-1 Haldane chain 1,2 . Its ground state is a disordered state, with symmetry-protected fourfold-degenerate edge states due to fractional spin excitations. In the bulk, it is characterized by vanishing two-point spin correlations, gapped excitations and a characteristic non-local order parameter 3,4 . More recently it has been understood that the Haldane chain forms a specific example of a more general classification scheme of symmetry-protected topological phases of matter, which is based on ideas connected to quantum information and entanglement 5–7 . Here, we realize a finite-temperature version of such a topological Haldane phase with Fermi–Hubbard ladders in an ultracold-atom quantum simulator. We directly reveal both edge and bulk properties of the system through the use of single-site and particle-resolved measurements, as well as non-local correlation functions. Continuously changing the Hubbard interaction strength of the system enables us to investigate the robustness of the phase to charge (density) fluctuations far from the regime of the Heisenberg model, using a novel correlator.