Spectral signatures of fractionalization in the frustrated Heisenberg model on the square lattice Francesco Ferrari1 and Federico Becca2 1SISSA-International School for Advanced Studies, Via Bonomea 265, I-34136 Trieste, Italy 2Democritos National Simulation Center, Istituto Officina dei Materiali del CNR and SISSA-International School for Advanced Studies, Via Bonomea 265, I-34136 Trieste, Italy (Dated: September 3, 2018) We employ a variational Monte Carlo approach to efficiently obtain the dynamical structure factor for the spin-1/2 J1 − J2 Heisenberg model on the square lattice. Upon increasing the frustrating ratio J2=J1, the ground state undergoes a continuous transition from a N´eelantiferromagnet to a Z2 gapless spin liquid. We identify the characteristic spectral features in both phases and highlight the existence of a broad continuum of excitations in the proximity of the spin-liquid phase. The magnon branch, which dominates the spectrum of the unfrustrated Heisenberg model, gradually loses its spectral weight, thus releasing nearly-deconfined spinons, whose signatures are visible even in the magnetically ordered state. Our results provide an important example on how magnons fractionalize into deconfined spinons across a quantum critical point. Introduction. The existence of fractional excitations is Z2 gauge theory [14, 15]. Examples of deconfined excita- a phenomenon due to the strong interaction between the tions have been detected in the Kitaev model [16] or its particles. It refers to the emergence of quasiparticle ex- generalization including Heisenberg terms [17, 18]. Frac- citations having quantum numbers that are non-integer tionalization has been also invoked to describe continu- multiples of those of the constituent particles (such as ous phase transitions, where, in contrast to the conven- electrons) [1]. In two or three spatial dimensions frac- tional Landau-Ginzburg-Wilson paradigm, characterized tionalization is generally associated with emergent gauge by critical fluctuation of the order parameter, the criti- fields and can be described in terms of deconfinement of cal theory contains emergent gauge fields and deconfined quasiparticles that are free to move, rather than forming quasiparticles with fractional quantum numbers [19, 20]. bound states. There are several examples of fractional- In this context, dynamical signatures of the fractionalized ization, ranging from high-energy physics to condensed excitations have been recently considered within a spin matter, where gauge theories show transitions between model with ring-exchange interactions [21]. Furthermore, confining and deconfining phases [2]. One of the most the possibility for a coexistence of nearly-free spinons prominent examples is the fractional quantum Hall ef- and conventional S = 1 magnon excitations has been fect, where the quasiparticles carry fractions of the elec- suggested by recent neutron-scattering experiments on tron charge [3]. Another important case is given by the Cu(DCOO)2 4D2O, which have revealed the presence of spin-charge separation in one-dimensional electronic con- a very broad· spectrum around the wave vector q = (π; 0) ductors: here, the electron splits into a spinon, which [and (0; π)] together with a strong magnon peak around carries spin S = 1=2 and no charge, and a holon, with q = (π; π)[22]. The experimental results were com- S = 0 and unit charge e [4,5]. Also one-dimensional in- bined with a theoretical analysis based upon variational sulators exhibit fractionalization in the spin sector, e.g., wave functions and the unfrustrated Heisenberg model. the Heisenberg model [6], where elementary excitations However, their theoretical description was not fully sat- are S = 1=2 free spinons and not conventional S = 1 spin isfactory, since the magnon branch in the whole Bril- waves [7]. louin zone was recovered by a variational state that in- cludes magnetic order, while a broad continuum of decon- In recent years, motivated by the discovery of high- fined excitations was obtained when using a spin-liquid temperature superconductors, there has been an increas- (i.e., RVB) wave function. Still, an intriguing interpre- ing effort to understand whether spin-charge separation is tation of these results involves the existence of nearly- also possible in two-dimensional Mott insulators. In fact, deconfined spinons [23], while a more conventional one re- in two dimensions, magnetic order is likely to develop in sorts to magnons with a strong attraction at short length the ground state, entailing conventional spin-wave exci- scales [24]. tations (i.e., S = 1 magnons)[8]. In this regard, frus- trated systems are ideal candidates, since the competi- In this paper, we report variational calculations for tion among different super-exchange couplings may pre- the frustrated J1 J2 spin-1/2 Heisenberg model on the vent the system from developing long-range order [9, 10]. square lattice. Different− numerical approaches suggested There have been several attempts to obtain fractional the existence of a quantum critical point at J2=J1 0:5, excitations in two-dimensional antiferromagnets, ranging separating an ordered N´eelantiferromagnet and a≈ non- from the earliest approaches based upon the resonating- magnetic phase [25{31], whose precise nature is still un- valence bond (RVB) theory [11{13] to alternative theo- der debate. By using a recently developed variational retical frameworks based on boson-vortex dualities and technique that can deal with low-energy states with given 2 6 6 5 5 100 100 4 4 1 1 ω 3 10− ω 3 10− 2 2 2 2 10− 10− 1 1 3 3 0 10− 0 10− π π π π π π π π ( 2 , 2 ) (π, 0) (π, π) ( 2 , 2 ) (0, 0) (0, π) ( 2 , 2 ) (π, 0) (π, π) ( 2 , 2 ) (0, 0) (0, π) q q 6 6 5 5 100 100 4 4 1 1 ω 3 10− ω 3 10− 2 2 2 2 10− 10− 1 1 3 3 0 10− 0 10− π π π π π π π π ( 2 , 2 ) (π, 0) (π, π) ( 2 , 2 ) (0, 0) (0, π) ( 2 , 2 ) (π, 0) (π, π) ( 2 , 2 ) (0, 0) (0, π) q q Figure 1: Dynamical spin structure factor of the spatially anisotropic Heisenberg model with J2 = 0. Different values of the ? k inter-chain couplings are reported: J1 =J1 = 0:1 (upper-left), 0:3 (upper-right), 0:5 (lower-left), and 0:7 (lower-right). The square cluster contains N = 22 × 22 sites. Spectral functions have been convoluted with normalized Gaussians with standard k deviation 0:1J1 . momentum q [32, 33], we report the evolution of the dy- model on the square lattice with N = L L sites: × namical structure factor from the N´eelto a gapless 2 Z k X ? X = J Si Si+y + J Si Si+x spin-liquid phase, which is obtained within our approach. H 1 · 1 · The gradual softening and broadening of the spectral i i X signal around q = (π; 0) and (0; π) when approaching +J2 (Si Si+x+y + Si Si+x−y) ; (1) · · the critical point represents an important hallmark of i the spin model. A gapless continuum of free spinons in x y z the spin-liquid phase is ascribed to four Dirac points at where Si = (Si ;Si ;Si ) is the spin-1/2 operator on the q = ( π=2; π=2). Moreover, our results suggest the site i; the Hamiltonian contains both nearest-neighbor ± ± ? k possibility that nearly-deconfined spinons already exist terms along x and y spatial directions (J1 and J1 , re- within the N´eelphase close to the critical point. This spectively) and next-nearest-neighbor ones along x + y ? k work provides the evidence that the dynamical signa- and x y (J2). Here, we consider J1 = J1 only when tures of fractionalization observed for the sign-free model − ? k 6 J2 = 0, while we take J1 = J1 = J1 when J2 > 0. of Ref. [21] is a generic feature of continuous quantum The central aim of this work is to assess the dynamical phase transitions in frustrated spin models. structure factor at zero temperature, defined by: a X q a 2 q S (q; !) = Υα Sq Υ0 δ(! Eα + E0); (2) α jh j j ij − Model and method. We consider the J1 J2 Heisenberg where Υ0 is the ground state of the system with energy − j i 3 q E0, Υα are the excited states with momentum q (rel- fj ig q a 0.35 ative to the ground state) and energy Eα, and Sq is the a Fourier transform of the spin operator Si . 0.30 For generic values of the frustrating ratio J2=J1, there 0.25 are no exact methods that allow us to evaluate the dy- namical structure factor. Therefore, we resort to consid- 0.20 ering a suitable approximation, by employing Gutzwiller 0.15 projected fermionic wave functions to construct accurate 0.10 variational states to describe both the ground state and magnetization low-energy excitations. In particular, we consider an aux- 0.05 iliary superconducting (BCS) Hamiltonian: 0.00 0.00 0.10 0.20 0.30 0.40 0.48 X y X y y J /J 0 = ti;jc c + ∆i;jc c + h:c: 2 1 H i,σ j,σ i;" j;# i;j,σ i;j X iQRi y y + ∆AF e ci;"ci;# + ci;#ci;" ; (3) Figure 2: Variational results for the ground-state magne- i tization of the J1 − J2 Heisenberg model. The results are obtained in the thermodynamic limit, by extrapolating the y where, ci,σ (ci,σ) creates (destroys) an electron with spin isotropic spin-spin correlations at the maximum distance in σ = 1=2 on the site i; ti;j is a complex nearest-neighbor the L × L clusters with L ranging from 10 to 22.