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arXiv:1301.3207 · Physical Review Letters

Confinement and Deconfinement of Spinons in Two Dimensions

Ying Tang, Anders W. Sandvik

Published May 24, 2013• 36 citations
arXivHTMLPDF

Abstract

We use Monte Carlo methods to study spinons in two-dimensional quantum spin systems, characterizing their intrinsic size $\ensuremath{\lambda}$ and confinement length $\ensuremath{\Lambda}$. We confirm that spinons are deconfined, $\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$ and $\ensuremath{\lambda}$ finite, in a resonating valence-bond spin-liquid state. In a valence-bond solid, we find finite $\ensuremath{\lambda}$ and $\ensuremath{\Lambda}$, with $\ensuremath{\lambda}$ of a single spinon significantly larger than the bound state---the spinon is soft and shrinks as the bound state is formed. Both $\ensuremath{\lambda}$ and $\ensuremath{\Lambda}$ diverge upon approaching the critical point separating valence-bond solid and N\'eel ground states. We conclude that the spinon deconfinement is marginal in the lowest-energy state in the spin-1 sector, due to weak attractive spinon interactions. Deconfinement in the vicinity of the critical point should occur at higher energies.

DOI
10.1103/physrevlett.110.217213

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We use Monte Carlo methods to study spinons in two-dimensional quantum spin systems, characterizing their intrinsic size \ensuremathλ\ensuremath{\lambda}\ensuremathλ and confinement length \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ. We confirm that spinons are deconfined, \ensuremathΛ\ensuremath→\ensuremath∞\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}\ensuremathΛ\ensuremath→\ensuremath∞ and \ensuremathλ\ensuremath{\lambda}\ensuremathλ finite, in a resonating valence-bond spin-liquid state. In a valence-bond solid, we find finite \ensuremathλ\ensuremath{\lambda}\ensuremathλ and \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ, with \ensuremathλ\ensuremath{\lambda}\ensuremathλ of a single spinon significantly larger than the bound state---the spinon is soft and shrinks as the bound state is formed. Both \ensuremathλ\ensuremath{\lambda}\ensuremathλ and \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ diverge upon approaching the critical point separating valence-bond solid and N'eel ground states. We conclude that the spinon deconfinement is marginal in the lowest-energy state in the spin-1 sector, due to weak attractive spinon interactions. Deconfinement in the vicinity of the critical point should occur at higher energies.
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We study the level structure of excitations at the ``deconfined'' critical point separating antiferromagnetic and valence-bond-solid phases in two-dimensional quantum spin systems using the $J\text{\ensuremath{-}}Q$ model as an example. Energy gaps in different spin (S)(S)(S) and momentum (k)(\mathbf{k})(k) sectors are extracted from imaginary-time correlation functions obtained in quantum Monte Carlo simulations. We find strong quantitative evidence for deconfined linearly dispersing spinons with gapless points at $\mathbf{k}=(0,0),(\ensuremath{\pi},0),(0,\ensuremath{\pi})$, and $(\ensuremath{\pi},\ensuremath{\pi})$, as inferred from two-spinon excitations (S=0S=0S=0 and S=1S=1S=1 states) around these points. We also observe a duality between singlet and triplet excitations at the critical point and inside the ordered phases, in support of an enhanced symmetry, possibly SO(5).
Strongly correlated low-dimensional quantum spin models provide a well-established frame- \nwork to study magnetic properties of insulators, and are of great theoretical interest and experimental relevance in condensed-matter physics. In this thesis, I use quantum Monte Carlo methods to numerically study quantum critical behavior in low-dimensional quantum spin models and wavefunctions. \n \nFirst, I study spinons &ndash emergent spin-1/2 bosonic excitations &ndash at certain one- and two-dimensional quantum phase transitions (QPTs) in spin models, by characterizing their size and confinement length quantitatively. In particular, I focus on the QPT from an antiferromagnetic (AFM) phase into a valence-bond solid (VBS) phase, which is an example of a violation of the standard Landau-Ginzburg-Wilson paradigm for phase transitions. This transition in two dimensions (2D) is instead likely described by a novel theory called "deconfined quantum criticality" (DQC). According to the theory, spinons should be deconfined. The degree of deconfinement is quantified in my calculations. \n \nSecond, I present a comprehensive study of so-called short-bond resonating-valence-bond (RVB) spin liquids in 2D, which have been suggested as a good starting point for understanding the spin physics of high-temperature cuprates. I find that these RVB states can also be classified as quantum-critical VBS states, which indicates that RVB is less disordered than expected. This work suggests a possible mapping from the quantum RVB states to classical dimer models via a classical continuum field theory--the height model. This map explicitly bridges well-established classical results to future quantum studies. \n \nThird, I consider 1D amplitude product (AP) states, which are generalized versions of RVB states, with different wavefunction weightings of bonds according to their lengths. AP states constitute a good ansatz for certain Hamiltonians and are of broad interest in quantum magnetism. I study phase transitions from AFM-VBS phases in AP states by tuning their amplitudes, and obtain continuously varying critical exponents. In addition, I classify the 1D AP states through entanglement entropy calculations of the central charge in (1+1)D conformal field theory. This new classification could serve as guide for AP states as trial wavefunctions to search for ground states of corresponding quantum spin models.
The transition between the N\'eel antiferromagnet and the valence-bond solid state in two dimensions has become a paradigmatic example of deconfined quantum criticality, a non-Landau transition characterized by fractionalized excitations (spinons). We consider an extension of this scenario whereby the deconfined spinons are subject to a magnetic field. The primary purpose is to identify the exotic scenario of a Bose-Einstein condensate of spinons. We employ quantum Monte Carlo simulations of the $J\ensuremath{-}Q$ model with a magnetic field, and we perform a quantum field theoretic analysis of the magnetic field and temperature dependence of thermodynamic quantities. The combined analysis provides evidence for Bose-Einstein condensation of spinons and also demonstrates an extended temperature regime in which the system is best described as a gas of spinons interacting with an emergent gauge field.
We consider the quantum phase transition between a Néel antiferromagnet and a valence-bond solid (VBS) in a two-dimensional system of S = 1/2 spins. Assuming that the excitations of the critical ground state are linearly dispersing deconfined spinons obeying Bose statistics, we derive expressions for the specific heat C and the magnetic susceptibility χ at low temperature T in terms of a correlation length ξ(T). Comparing with quantum Monte Carlo results for the J-Q model, which is a candidate for a deconfined Néel-VBS transition, we obtain an almost perfect consistency between C, χ, and ξ. The corresponding expressions for magnon (triplet) excitations are not internally consistent, however, lending strong support for spinon excitations in the J-Q model.
Using quantum Monte Carlo and numerical analytic continuation methods, we study the dynamic spin structure factor and the single-hole spectral function of a two-dimensional quantum magnet (JJJ-QQQ model) at its quantum phase transition separating Néel antiferromagnetic and spontaneously dimerized ground states. At this putative deconfined quantum-critical point, we find a broad continuum of spinon excitations that can be accounted for by the fermionic πππ-flux state; a known mean-field model for deconfined quantum criticality. We find that the best description of the two-spinon continuum is with a version of the model with a 2×22\times 22×2 unit cell, reflecting non-trivial mutual statistics of spinons and anti-spinons. The single-hole spectral function can be described by the same spinon dispersion relation and an independently propagating holon. Thus, the system exhibits spin-charge separation and will likely evolve into an extended holon metal phase at finite doping.