Finite-size scaling of string order parameters characterizing the Haldane phase

Ueda, Hiroshi; Nakano, Hiroki; Kusakabe, Koichi

doi:10.1103/physrevb.78.224402

Cited by 37 publications

(63 citation statements)

References 29 publications

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“…Applying finite-size-scaling analysis [41] allows an accurate location of the corresponding phase-transition lines depicted in Fig. 3 of the main text.…”

Section: Appendix B: Numerical Analysismentioning

confidence: 99%

Ultracold bosons in zig-zag optical lattices

2013

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Ultracold bosons in zig-zag optical lattices present a rich physics due to the interplay between frustration induced by lattice geometry, two-body interactions, and a three-body constraint. Unconstrained bosons may develop chiral superfluidity and become a Mott insulator even at vanishingly small interactions. Bosons with a three-body constraint allow for a Haldane-insulator phase in nonpolar gases, as well as pair superfluidity and density-wave phases for attractive interactions. These phases may be created and detected within the current state-of-the-art techniques.

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“…Applying finite-size-scaling analysis [41] allows an accurate location of the corresponding phase-transition lines depicted in Fig. 3 of the main text.…”

Section: Appendix B: Numerical Analysismentioning

confidence: 99%

Ultracold bosons in zig-zag optical lattices

2013

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“…This prediction is supported by conformal field theory arguments 69 and a level spectroscopy method based on a renormalization group analysis and the SU(2)/Z 2 symmetry of the BKT transition 61,[70][71][72] . Numerically, λ 1 ≈ 0 has been verified via finite-size scaling 64,73,74 and DMRG 65 . Finally, at J z = λ 0 = −1, a first-order phase transition from the XY phase to a ferromagnetic Ising phase takes place 61,66,75 .…”

Section: Model Hamiltonian and Phase Diagramsmentioning

confidence: 99%

Kaleidoscope of quantum phases in a long-range interacting spin-1 chain

Gong

Maghrebi

et al. 2016

Phys. Rev. B

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Motivated by recent trapped-ion quantum simulation experiments, we carry out a comprehensive study of the phase diagram of a spin-1 chain with XXZ-type interactions that decay as $1/r^{\alpha}$, using a combination of finite and infinite-size DMRG calculations, spin-wave analysis, and field theory. In the absence of long-range interactions, varying the spin-coupling anisotropy leads to four distinct phases: a ferromagnetic Ising phase, a disordered XY phase, a topological Haldane phase, and an antiferromagnetic Ising phase. If long-range interactions are antiferromagnetic and thus frustrated, we find primarily a quantitative change of the phase boundaries. On the other hand, ferromagnetic (non-frustrated) long-range interactions qualitatively impact the entire phase diagram. Importantly, for $\alpha\lesssim3$, long-range interactions destroy the Haldane phase, break the conformal symmetry of the XY phase, give rise to a new phase that spontaneously breaks a $U(1)$ continuous symmetry, and introduce an exotic tricritical point with no direct parallel in short-range interacting spin chains. We show that the main signatures of all five phases found could be observed experimentally in the near future.Comment: 11 pages, 6 figure

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“…This large systematic error could be reduced using more advanced finite size-scaling methods [35,37], which is beyond the scope of this manuscript. In the inset of Fig.…”

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Observation of Correlated Particle-Hole Pairs and String Order in Low-Dimensional Mott Insulators

Endres

Cheneau

Fukuhara

et al. 2011

Science

290

341

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Quantum phases of matter are characterized by the underlying correlations of the many-body system. Although this is typically captured by a local order parameter, it has been shown that a broad class of many-body systems possesses a hidden non-local order. In the case of bosonic Mott insulators, the ground state properties are governed by quantum fluctuations in the form of correlated particle-hole pairs that lead to the emergence of a non-local string order in one dimension. Using high-resolution imaging of low-dimensional quantum gases in an optical lattice, we directly detect these pairs with single-site and single-particle sensitivity and observe string order in the one-dimensional case.The realization of strongly correlated quantum manybody systems using ultracold atoms has enabled the direct observation and control of fundamental quantum effects [1][2][3]. A prominent example is the transition from a superfluid (SF) to a Mott insulator (MI), occurring when interactions between bosonic particles on a lattice dominate over their kinetic energy [4][5][6][7][8]. At zero temperature, and in the limit where the ratio of kinetic energy over interaction energy vanishes, particle fluctuations are completely suppressed and the lattice sites are occupied by an integer number of particles. However, at a finite tunnel coupling, but still in the Mott insulating regime, quantum fluctuations create correlated particlehole pairs on top of this fixed-density background, which can be understood as virtual excitations. These particlehole pairs fundamentally determine the properties of the Mott insulator such as its residual phase coherence [9] and lie at the heart of superexchange-mediated spin interactions that form the basis of quantum magnetism in multi-component quantum gas mixtures [10][11][12].In a one-dimensional system, the appearance of correlated particle-hole pairs at the transition point from a superfluid to a Mott insulator is intimately connected to the emergence of a hidden string-order parameter O P [13,14]:Here δn j =n j −n denotes the deviation in occupation of the jth lattice site from the average background density, and k is an arbitrary position along the chain. In the simplest case of a Mott insulator with unity filling * Electronic address: manuel.endres@mpq.mpg.de (n = 1), relevant to our experiments, each factor in the product of operators in Eq. 1 yields −1 instead of +1, when a single-particle fluctuation from the unit background density is encountered. In the superfluid, particle and hole fluctuations occur independently and are uncorrelated, such that O P = 0. However, in the Mott insulating phase, density fluctuations always occur as correlated particle-hole pairs, resulting in O P = 0. For a homogeneous system, O P is expected to follow a scaling of Berezinskii-Kosterlitz-Thouless (BKT) type [15]. Non-local correlation functions, like the string-order parameter defined above, have been introduced in the context of low-dimensional quantum systems. They classify many-body quantum phases that are n...

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Finite-size scaling of string order parameters characterizing the Haldane phase

Cited by 37 publications

References 29 publications

Ultracold bosons in zig-zag optical lattices

Ultracold bosons in zig-zag optical lattices

Kaleidoscope of quantum phases in a long-range interacting spin-1 chain

Observation of Correlated Particle-Hole Pairs and String Order in Low-Dimensional Mott Insulators

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