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arXiv:1607.05110 · Physical review. B./Physical review. B

Level spectroscopy in a two-dimensional quantum magnet: Linearly dispersing spinons at the deconfined quantum critical point

Hidemaro Suwa, Arnab Sen, Anders W. Sandvik

Published Oct 14, 2016• 40 citations
arXivHTMLPDF

Abstract

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 $(S)$ and momentum $(\mathbf{k})$ 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 ($S=0$ and $S=1$ 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).

DOI
10.1103/physrevb.94.144416

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•Jan 1, 2014•0 citations
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\ensuremath-QJ\text{\ensuremath{-}}QJ\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 k=(0,0),(\ensuremathπ,0),(0,\ensuremathπ)\mathbf{k}=(0,0),(\ensuremath{\pi},0),(0,\ensuremath{\pi})k=(0,0),(\ensuremathπ,0),(0,\ensuremathπ), and (\ensuremathπ,\ensuremathπ)(\ensuremath{\pi},\ensuremath{\pi})(\ensuremathπ,\ensuremathπ), 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).
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.
We resolve the nature of the quantum phase transition between antiferromagnetic and valence-bond solid ground states in square-lattice spin-1/2 magnets - the putative deconfined quantum-critical point. Using quantum Monte Carlo simulations of J-Q models, in which Heisenberg exchange J competes with multi-spin interactions Qn formed by n singlet projectors, we demonstrate first-order discontinuities increasing with n. For n=2 and n=3, the first-order signatures are very weak and we extract universal scaling dimensions of a proximate multicritical point. Their values coincide closely with known bounds for an SO(5) symmetric conformal field theory (CFT) containing a relevant SO(5) singlet operator that destabilizes the generic deconfined critical point. We characterize the emergent SO(5) symmetry by its scaling dimensions of the leading irrelevant perturbations. Using the J-Q2-Q6 model, we tune the transition from weak to stronger first-order and find scaling of the coexisting order parameters with a large exponent beta ~ 0.85, in agreement with the relevant scaling dimensions of the multicritical point. The exponent that had previously been identified with the correlation length does not have a corresponding CFT level, and we explain it as pseudo-critical length scale of bubbles of one phase inside the other that are compressed in size by an SO(5)-violating irrelevant repulsive interaction between the two order parameters. The bubble scale reflects a hierarchy of excitation gaps generated by the SO(5) perturbation in the neighborhood of the first-order transition, akin to a Higgs mechanism. We argue that the multicritical point is the tip of a gapless spin liquid phase in frustrated Heisenberg models into which the J-Q models can be deformed, but the spin liquid, and perhaps the multicritical point itself, exist outside the regime amenable to sign-free simulations.
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.
We consider spin-half quantum antiferromagnets in two spatial dimensions in the quantum limit, where the spins are in a valence bond solid (VBS) phase. The transition between two such VBS phases is studied. In some cases, an interesting second-order transition controlled by a fixed line with varying critical exponents is found. A specific example is provided by an antiferromagnetically coupled bilayer system on the honeycomb lattice where a continuous quantum phase transition can generically exist between two VBS phases. Furthermore, these critical points are deconfined, in the sense that gapped spin-1∕21∕21∕2 spinon excitations emerge right at the transition. The low-energy physics of this critical point (up to marginally irrelevant interactions) contains just a free quadratically dispersing ``photon.'' The phase structure on one side of this continuous transition is very intricate, consisting of a series of infinitely closely spaced further transitions in a ``devil's staircase'' form. Analogies with previous examples of deconfined quantum criticality are emphasized. Closely related transitions in single layer systems are explored. These are second order only at some multicritical points. The solvable Rokshar-Kivelson point of quantum dimer models of single layer systems is found to correspond to a nongeneric multicritical point.
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.