Topological phases, like the Haldane phase in spin-1 chains, defy characterization through local order parameters. Instead, nonlocal string order parameters can be employed to reveal their hidden order. Similar diluted magnetic correlations appear in doped one-dimensional lattice systems owing to the phenomenon of spin-charge separation. Here we report on the direct observation of such hidden magnetic correlations via quantum gas microscopy of hole-doped ultracold Fermi-Hubbard chains. The measurement of nonlocal spin-density correlation functions reveals a hidden finite-range antiferromagnetic order, a direct consequence of spin-charge separation. Our technique, which measures nonlocal order directly, can be readily extended to higher dimensions to study the complex interplay between magnetic order and density fluctuations.Topological phases, like the Haldane phase in spin-1 chains, defy characterization through local order parameters. Instead, nonlocal string order parameters can be employed to reveal their hidden order. Similar diluted magnetic correlations appear in doped one-dimensional lattice systems owing to the phenomenon of spin-charge separation. Here we report on the direct observation of such hidden magnetic correlations via quantum gas microscopy of hole-doped ultracold Fermi-Hubbard chains. The measurement of nonlocal spin-density correlation functions reveals a hidden finite-range antiferromagnetic order, a direct consequence of spin-charge separation. Our technique, which measures nonlocal order directly, can be readily extended to higher dimensions to study the complex interplay between magnetic order and density fluctuations.

arXiv:1702.00642ScienceAug 3, 2017215 citations

Constraints on Order and Disorder Parameters in Quantum Spin Chains

Michael Levin

arXiv:1903.09028Communications in Mathematical PhysicsJul 16, 202032 citations

Magnetization plateaus, Haldane-like gap, string order, and hidden symmetry in a spin- 12 tetrameric Heisenberg antiferromagnetic chain

Shou-Shu Gong, Gang Su

The ground-state properties of a spin $S=1/2$ tetrameric Heisenberg antiferromagnetic chain with alternating couplings ${\text{AF}}_{1}{\text{-AF}}_{2}{\text{-AF}}_{1}\text{-F}$ (AF and F denote antiferromagnetic and ferromagnetic couplings, respectively) are studied by means of the density-matrix renormalization-group method. Two plateaus of magnetization $m$ are found at $m=0$ and 1/4. It is shown that in such a spin-1/2 AF system, there is a gap from the singlet ground state to the triplet excited states in the absence of a magnetic field. The spin-spin correlation function decays exponentially, and the gapped states have a nonvanishing string order, which measures a hidden symmetry in the system. By a dual transformation, the string order is transformed into a ferromagnetic order and the hidden symmetry is unveiled to be a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ discrete symmetry, which is fully broken in the gapped states. This half-integer spin Heisenberg AF chain is in a Haldane-like phase suggesting that the present findings extend the substance of Haldane's scenario. A valence-bond-solid state picture is also proposed for the gapped states.The ground-state properties of a spin $S=1/2$ tetrameric Heisenberg antiferromagnetic chain with alternating couplings ${\text{AF}}_{1}{\text{-AF}}_{2}{\text{-AF}}_{1}\text{-F}$ (AF and F denote antiferromagnetic and ferromagnetic couplings, respectively) are studied by means of the density-matrix renormalization-group method. Two plateaus of magnetization $m$ are found at $m=0$ and 1/4. It is shown that in such a spin-1/2 AF system, there is a gap from the singlet ground state to the triplet excited states in the absence of a magnetic field. The spin-spin correlation function decays exponentially, and the gapped states have a nonvanishing string order, which measures a hidden symmetry in the system. By a dual transformation, the string order is transformed into a ferromagnetic order and the hidden symmetry is unveiled to be a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ discrete symmetry, which is fully broken in the gapped states. This half-integer spin Heisenberg AF chain is in a Haldane-like phase suggesting that the present findings extend the substance of Haldane's scenario. A valence-bond-solid state picture is also proposed for the gapped states.

arXiv:0809.0093Physical Review BSep 18, 200822 citations

Magnetized States of Quantum Spin Chains

C. Broholm, G. Aeppli, Youzhe Chen +15 more

Lecture notes in physicsJan 1, 200210 citations

The ground-state properties of a spin

S=1/2

tetrameric Heisenberg antiferromagnetic chain with alternating couplings

{\text{AF}}_{1}{\text{-AF}}_{2}{\text{-AF}}_{1}\text{-F}

(AF and F denote antiferromagnetic and ferromagnetic couplings, respectively) are studied by means of the density-matrix renormalization-group method. Two plateaus of magnetization

m

are found at

m=0

and 1/4. It is shown that in such a spin-1/2 AF system, there is a gap from the singlet ground state to the triplet excited states in the absence of a magnetic field. The spin-spin correlation function decays exponentially, and the gapped states have a nonvanishing string order, which measures a hidden symmetry in the system. By a dual transformation, the string order is transformed into a ferromagnetic order and the hidden symmetry is unveiled to be a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ discrete symmetry, which is fully broken in the gapped states. This half-integer spin Heisenberg AF chain is in a Haldane-like phase suggesting that the present findings extend the substance of Haldane's scenario. A valence-bond-solid state picture is also proposed for the gapped states.