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arXiv:1905.03719 · Physical review. B./Physical review. B

Critical properties of the Néel–algebraic-spin-liquid transition

Nikolai Zerf, Rufus Boyack, Peter Marquard, J. A. Gracey, Joseph Maciejko

Published Dec 19, 2019• 18 citations
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Abstract

The algebraic spin liquid is a long-sought-after phase of matter characterized by the absence of quasiparticle excitations, a low-energy description in terms of emergent Dirac fermions and gauge fields interacting according to $(2+1)$-dimensional quantum electrodynamics $({\mathrm{QED}}_{3})$, and power-law correlations with universal exponents. The prototypical algebraic spin liquid is the Affleck-Marston $\ensuremath{\pi}$-flux phase, originally proposed as a possible ground state of the spin-$\frac{1}{2}$ Heisenberg model on the two-dimensional square lattice. While the latter model is now known to order antiferromagnetically at zero temperature, recent sign-problem-free quantum Monte Carlo simulations of spin-$\frac{1}{2}$ fermions coupled to a compact $\text{U}(1)$ gauge field on the square lattice have shown that quantum fluctuations can destroy N\'eel order and drive a direct quantum phase transition to the $\ensuremath{\pi}$-flux phase. We show this transition is in the universality class of the chiral Heisenberg ${\mathrm{QED}}_{3}$-- Gross-Neveu-Yukawa model with a single $\text{SU}(2)$ doublet of four-component Dirac fermions (i.e., ${N}_{f}=1)$, pointing out important differences with the corresponding putative transition on the kagome lattice. Using an $\ensuremath{\epsilon}$ expansion below four space-time dimensions to four-loop order, and a large-${N}_{f}$ expansion up to second order, we show the transition is continuous and compute various thermodynamic and susceptibility critical exponents at this transition, setting the stage for future numerical determinations of these quantities. As a by-product of our analysis, we also obtain charge-density-wave and valence-bond-solid susceptibility exponents at the semimetal-N\'eel transition for interacting fermions on the honeycomb lattice.

DOI
10.1103/physrevb.100.235130

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We study the universal critical properties of the QED3{\mathrm{QED}}_{3}QED3​-Gross-Neveu-Yukawa model with NNN flavors of four-component Dirac fermions coupled to a real scalar order parameter at four-loop order in the $\ensuremath{\epsilon}$ expansion below four dimensions. For N=1N=1N=1, the model is conjectured to be infrared dual to the SU(2)\text{SU}(2)SU(2)-symmetric noncompact CP1\mathbb{C}{\mathrm{P}}^{1}CP1 model, which describes the deconfined quantum critical point of the N\'eel-valence-bond-solid transition of spin-1/2 quantum antiferromagnets on the two-dimensional square lattice. For N=2N=2N=2, the model describes a quantum phase transition between an algebraic spin liquid and a chiral spin liquid in the spin-1/2 kagome antiferromagnet. For general NNN we determine the order-parameter anomalous dimension, the correlation length exponent, the stability critical exponent, as well as the scaling dimensions of SU(N)\text{SU}(N)SU(N) singlet and adjoint fermion mass bilinears at the critical point. We use Pad\'e approximants to obtain estimates of critical properties in 2+1 dimensions.
Quantum spin liquids are exotic phases of quantum matter especially pertinent to many modern condensed matter systems. Dirac spin liquids (DSLs) are a class of gapless quantum spin liquids that do not have a quasi-particle description and are potentially realized in a wide variety of spin 1/2 1 / 2 magnetic systems on 2d 2 d lattices. In particular, the DSL in square lattice spin- 1/2 1 / 2 magnets is described at low energies by (2+1)d ( 2 + 1 ) d quantum electrodynamics with N_f=4 N f = 4 flavors of massless Dirac fermions minimally coupled to an emergent U(1) U ( 1 ) gauge field. The existence of a relevant, symmetry-allowed monopole perturbation renders the DSL on the square lattice intrinsically unstable. We argue that the DSL describes a stable continuous phase transition within the familiar Neel phase (or within the Valence Bond Solid (VBS) phase). In other words, the DSL is an ``unnecessary’’ quantum critical point within a single phase of matter. Our result offers a novel view of the square lattice DSL in that the critical spin liquid can exist within either the Neel or VBS state itself, and does not require leaving these conventional states.
Algebraic spin liquids are quantum disordered phases of insulating magnets that exhibit fractionalized gapless excitations and power-law correlations. Quantum spin liquids in this category include the experimentally established one-dimensional (1D) Luttinger liquid, as well as the U(1) Dirac spin liquid (DSL) that has been a focus of recent candidate materials searches. Most notably, several exchange-frustrated Heisenberg materials on the triangular lattice have shown evidence of the U(1) DSL. In this work, we measure the algebraic correlations of spin-singlet excitations in the J 1 − J 2 antiferromagnetic Heisenberg model on the triangular lattice, prompting a detailed investigation of this model’s stability under spin-lattice coupling with dynamical phonons using variational Monte Carlo. As seen before in 1D spin chains, we observe a low-temperature transition from a U(1) DSL to valence bond order and predict the parameter regime where the model realizes a stable DSL ground state. To achieve this, we employ a series of finite-size scaling Ansätze inspired by the low-energy DSL’s conformal description in terms of quantum electrodynamics and show that emergent monopole operators drive the instability. We compare the physics of this transition to the 1D Luttinger liquid throughout our analysis. We derive the regime of stability against spin-Peierls ordering and argue that the DSL ground state might still be achievable in candidate materials, despite its tendency to valence bond solid ordering.
The interplay between lattice gauge theories and fermionic matter accounts for fundamental physical phenomena ranging from the deconfinement of quarks in particle physics to quantum spin liquid with fractionalized anyons and emergent gauge structures in condensed matter physics.However, except for certain limits (for instance, a large number of flavors of matter fields), analytical methods can provide few concrete results.Here we show that the problem of compact Uð1Þ lattice gauge theory coupled to fermionic matter in ð2 þ 1ÞD is possible to access via sign-problem-free quantum Monte Carlo simulations.One can hence map out the phase diagram as a function of fermion flavors and the strength of gauge fluctuations.By increasing the coupling constant of the gauge field, gauge confinement in the form of various spontaneous-symmetrybreaking phases such as the valence-bond solid (VBS) and Néel antiferromagnet emerge.Deconfined phases with algebraic spin and VBS correlation functions are also observed.Such deconfined phases are incarnations of exotic states of matter, i.e., the algebraic spin liquid, which is generally viewed as the parent state of various quantum phases.The phase transitions between the deconfined and confined phases, as well as that between the different confined phases provide various manifestations of deconfined quantum criticality.In particular, for four flavors N f ¼ 4, our data suggest a continuous quantum phase transition between the VBS and Néel order.We also provide preliminary theoretical analysis for these quantum phase transitions.
The theory for the vanishing of N\'eel order in the spin S=1/2S=1/2S=1/2 square lattice antiferromagnet has been the focus of attention for many decades. A consensus appears to have emerged in recent numerical studies on the antiferromagnet with first and second neighbor exchange interactions (the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model): A gapless spin liquid is present for a narrow window of parameters between the vanishing of the N\'eel order and the onset of a gapped valence bond solid state. We propose a deconfined critical SU(2) gauge theory for a transition into a stable Z2{\mathbb{Z}}_{2}Z2​ spin liquid with massless Dirac spinon excitations; on the other side of the critical point, the SU(2) spin liquid (the `$\ensuremath{\pi}$-flux' phase) is presumed to be unstable to confinement to the N\'eel phase. We identify a dangerously irrelevant coupling in the critical SU(2) gauge theory, which contributes a logarithm-squared renormalization. This critical theory is also not Lorentz invariant and weakly breaks the SO(5) symmetry which rotates between the N\'eel and valence bond solid order parameters. We also propose a distinct deconfined critical U(1) gauge theory for a transition into the same gapless Z2{\mathbb{Z}}_{2}Z2​ spin liquid; on the other side of the critical point, the U(1) spin liquid (the `staggered flux' phase) is presumed to be unstable to confinement to the valence bond solid. This critical theory has no dangerously irrelevant coupling, dynamic critical exponent $z\ensuremath{\ne}1$, and no SO(5) symmetry. All of these phases and critical points are unified in a SU(2) gauge theory with Higgs fields and fermionic spinons which can naturally realize the observed sequence of phases with increasing J2/J1{J}_{2}/{J}_{1}J2​/J1​: N\'eel, gapless Z2{\mathbb{Z}}_{2}Z2​ spin liquid, and valence bond solid.
At sufficiently low temperatures, interacting electron systems tend to develop orders. Exceptions are quantum critical point (QCP) and quantum spin liquid (QSL), where fluctuations prevent the highly entangled quantum matter to an ordered state down to the lowest temperature. While the ramification of these states may have appeared in high-temperature superconductors, ultracold atoms, frustrated magnets, and quantum moir\'e materials, their unbiased presence remains elusive in microscopic two-dimensional lattice models. Here, we show, by means of large-scale quantum Monte Carlo simulations of correlated electrons on the $\ensuremath{\pi}$-flux square lattice subjected to plaquette Hubbard interaction, that a Gross-Neveu QCP separating massless Dirac fermions and a columnar valence bond solid at finite interaction and a possible Dirac QSL at the infinite yet tractable interaction limit emerge in a coherent sequence. These unexpected quantum states reside in this simple-looking model, unifying ingredients including emergent symmetry, deconfined fractionalization, and the dynamic coupling between emergent matter and gauge fields and will have profound implications both in quantum many-body theory and understanding of the aforementioned experimental systems.