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中国科学通报：英文版

Symmetry protected topological Luttinger liquids and the phase transition between them

Hong‐Chen Jiang, Zixiang Li, Alexander Seidel, D. H. Lee

Published May 17, 2018• 63 citations
Publisher

Abstract

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.

DOI
10.1016/j.scib.2018.05.010

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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 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 dxyd_{xy}dxy​ in the non-trivial phase and triplet Sz=0S_z=0Sz​=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.
In the present work, we provide a comprehensive numerical investigation of the magnetic properties and phase spectra of three types of spin-1/2 branched chains consisting of one, two and three side spins per unit block with intra-chain interaction and a uniform inter-chain interaction in the presence of an external magnetic field. In a specific magnetic field interval, the low-temperature magnetization of these chains shows a step-like behavior with a pronounced plateau depending on the strength and the type of intra-chain interaction being ferromagnetic or antiferromagnetic. We demonstrate that when inter-chain interaction J1 is antiferromagnetic and intra-chain interaction J2 is ferromagnetic, the magnetization of the models manifests a smooth increase without a plateau, which is evidence of the existence of a Luttinger-like spin liquid phase before reaching its saturation value. On the other hand, when J1 is ferromagnetic and J2 is antiferromagnetic, the low-temperature magnetization of the chain with two branches shows an intermediate plateau at one-half of the saturation magnetization that breaks a quantum spin liquid phase into two regions. The magnetization of the chain with three branches exhibits two intermediate plateaus and two regions of a quantum spin liquid. We demonstrate that the chains with more than one side spin illustrate in their ground-state phase diagram a Kosterlitz-Thouless transition from a gapful phase to a gapless spin liquid phase.
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.
In this paper, we propose strongly correlated gapless states (or critical states) of spin-1/2 electrons in 1+11+11+1 dimensions, such as the doped ferromagnetic and antiferromagnetic spin-1/2 Ising chains. We find that the metallic phases in the doped ferromagnetic and antiferromagnetic Ising chain are different strongly correlated gapless phases, despite the fact that the two phases have the same symmetry. The doped antiferromagnetic Ising chain has a finite energy gap for all charge-1 fermionic excitations even without pairing caused by the attractive interaction, resembling the pseudogap phase of underdoped high Tc{T}_{c}Tc​ superconductors. Applying a transverse field to the ferromagnetic and antiferromagnetic metallic phase can restore the Z2{Z}_{2}Z2​ symmetry, which gives rise to two distinct critical points despite the fact that the two transitions have exactly the same symmetry-breaking pattern. We also propose chiral metallic states. Some of these gapless states are strongly correlated in the sense that they do not belong to the usual Tomonaga-Luttinger phase of fermions, i.e., they cannot be smoothly deformed into noninteracting fermion systems with the same symmetry. Our nonperturbative results are obtained by noting that gapless quantum systems have emergent categorical symmetries (i.e., noninvertible gravitational anomalies), which are described by multicomponent partition functions that are modular covariant. This allows us to calculate the scaling dimensions and quantum numbers of all the low-energy operators for those strongly correlated gapless states. This demonstrates an application of emergent categorical symmetries in determining low-energy properties of strongly correlated gapless states, which are hard to obtain otherwise.
We study topology in gapless phases of an interacting spinful model with spin-charge separation. We focus on the gapless boundaries between Z2\mathbb{Z}_2Z2​ symmetry-breaking phases. We find two topologically non-trivial gapless states that occur at the boundary between a non-trivial and a trivial insulator. They correspond to topological Luther-Emery liquid and topological Mott insulator. The Luther-Emery liquid is characterized by gapless charge excitations and features topological edge modes that carry fractional spin, while the topological Mott insulator has gapless spin sector and features edge states that carry fractional charge. Surprisingly, even though there is no mean-field description of the interacting gapless phases, as there is no local order parameter, we show that they can be adiabatically connected to a non-interacting topological metal. This non-interacting state is a phase boundary between decoupled Su-Schrieffer-Heeger chains with the winding number ν=2ν=2ν=2 and chains with ν=1ν=1ν=1.
We study a three-leg spin-1/2 ladder with geometrically frustrated interleg interactions. We call this model an anisotropic triangular-strip (ATS) model. We numerically and field-theoretically show that its ground state belongs to a gapless symmetry-protected topological (SPT) phase. The numerical approach is based on density-matrix renormalization group analyses of the entanglement entropy and the entanglement spectrum. Whereas the entanglement entropy exhibits a critical behavior, the entanglement spectrum is nontrivially degenerate. These entanglement properties imply that the ground state is a gapless topological phase. We investigate the ATS model using a quantum field theory to support the numerical findings. When the frustrated interchain interaction is deemed a perturbation acting on the three spin chains, the frustrated interchain interaction almost isolates the second chain from the other two chains. However, at the same time, the second chain mediates a ferromagnetic interaction between the first and third chains. Therefore, the ground state of the ATS model is a gapless Tomonaga-Luttinger liquid weakly coupled to a spin-1 Haldane chain with irrelevant interactions. Last but not least, we show that the gapless SPT phase of the ATS model is a symmetry-protected critical phase. We point out that the symmetry protection of criticality is essential in characterization of the gapless SPT phase.