Progress on stochastic analytic continuation of quantum Monte Carlo data

We report multipronged progress on the stochastic averaging approach to numerical analytic continuation of quantum Monte Carlo data. With the sampled spectrum parametrized with delta-functions in continuous frequency space, a calculation of the configurational entropy lends support to a simple goodness-of-fit criterion for the optimal sampling temperature. To further investigate entropic effects, we compare spectra sampled in continuous frequency with results of amplitudes sampled on a fixed frequency grid. We demonstrate equivalences between sampling and optimizing spectral functions with the maximum-entropy approach with different forms of the entropy. These insights revise prevailing notions of the maximum-entropy method and its relationship to stochastic analytic continuation. We further explore various adjustable (optimized) constraints that allow sharp spectral features to be resolved, in particular at the lower frequency edge. The constraints, e.g., the location of the edge or the spectral weight of a quasi-particle peak, are optimized using a statistical criterion. We show that this method can correctly reproduce both narrow and broad quasi-particle peaks. We next introduce a parametrization for more intricate spectral functions with sharp edges, e.g., power-law singularities. Tests with synthetic data as well as with real simulation data for the spin-1/2 Heisenberg chain demonstrate that constrained sampling methods can reproduce spectral functions with sharp edge features at unprecedented fidelity. We present new results for S=1/2 Heisenberg 2-leg and 3-leg ladders to illustrate the ability of the methods to resolve spectral features arising from both elementary and composite excitations. Finally, we also propose how the methods developed here could be used as "pre processors" for analytic continuation by machine learning.Comment: 87 pages, 55 figures. v2: expanded discussion of quasi-particle peaks + other minor change

Random-Singlet Phase in Disordered Two-Dimensional Quantum Magnets

We study effects of disorder (randomness) in a 2D square-lattice $S=1/2$ quantum spin system, the $J$-$Q$ model with a 6-spin interaction $Q$ supplementing the Heisenberg exchange $J$. In the absence of disorder the system hosts antiferromagnetic (AFM) and columnar valence-bond-solid (VBS) ground states. The VBS breaks $Z_4$ symmetry, and in the presence of arbitrarily weak disorder it forms domains. Using QMC simulations, we demonstrate two kinds of such disordered VBS states. Upon dilution, a removed site leaves a localized spin in the opposite sublattice. These spins form AFM order. For random interactions, we find a different state, with no order but algebraically decaying mean correlations. We identify localized spinons at the nexus of domain walls between different VBS patterns. These spinons form correlated groups with the same number of spinons and antispinons. Within such a group, there is a strong tendency to singlet formation, because of spinon-spinon interactions mediated by the domain walls. Thus, no long-range AFM order forms. We propose that this state is a 2D analog of the well-known 1D random singlet (RS) state, though the dynamic exponent $z$ in 2D is finite. By studying the T-dependent magnetic susceptibility, we find that $z$ varies, from $z=2$ at the AFM--RS phase boundary and larger in the RS phase The RS state discovered here in a system without geometric frustration should correspond to the same fixed point as the RS state recently proposed for frustrated systems, and the ability to study it without Monte Carlo sign problems opens up opportunities for further detailed characterization of its static and dynamic properties. We also discuss experimental evidence of the RS phase in the quasi-two-dimensional square-lattice random-exchange quantum magnets Sr$_2$CuTe$_{1-x}$W$_x$O$_6$.Comment: 31 pages, 29 figures; substantial additions in v2; additional analysis in v

Random-singlet phase in disordered two-dimensional quantum magnets

We study effects of disorder (randomness) in a 2D square-lattice S=1/2 quantum spin system, the J-Q model with a 6-spin interaction Q supplementing the Heisenberg exchange J. In the absence of disorder the system hosts antiferromagnetic (AFM) and columnar valence-bond-solid (VBS) ground states. The VBS breaks Z4 symmetry, and in the presence of arbitrarily weak disorder it forms domains. Using QMC simulations, we demonstrate two kinds of such disordered VBS states. Upon dilution, a removed site leaves a localized spin in the opposite sublattice. These spins form AFM order. For random interactions, we find a different state, with no order but algebraically decaying mean correlations. We identify localized spinons at the nexus of domain walls between different VBS patterns. These spinons form correlated groups with the same number of spinons and antispinons. Within such a group, there is a strong tendency to singlet formation, because of spinon-spinon interactions mediated by the domain walls. Thus, no long-range AFM order forms. We propose that this state is a 2D analog of the well-known 1D random singlet (RS) state, though the dynamic exponent z in 2D is finite. By studying the T-dependent magnetic susceptibility, we find that z varies, from z=2 at the AFM--RS phase boundary and larger in the RS phase The RS state discovered here in a system without geometric frustration should correspond to the same fixed point as the RS state recently proposed for frustrated systems, and the ability to study it without Monte Carlo sign problems opens up opportunities for further detailed characterization of its static and dynamic properties. We also discuss experimental evidence of the RS phase in the quasi-two-dimensional square-lattice random-exchange quantum magnets Sr2CuTe1−xWxO6.Accepted manuscrip

Anomalous quantum-critical scaling corrections in two-dimensional antiferromagnets

We study the N\'eel-paramagnetic quantum phase transition in two-dimensional dimerized $S=1/2$ Heisenberg antiferromagnets using finite-size scaling of quantum Monte Carlo data. We resolve the long standing issue of the role of cubic interactions arising in the bond-operator representation when the dimer pattern lacks a certain symmetry. We find non-monotonic (monotonic) size dependence in the staggered (columnar) dimerized model, where cubic interactions are (are not) present. We conclude that there is an irrelevant field in the staggered model that is not present in the columnar case, but, at variance with previous claims, it is not the leading irrelevant field. The new exponent is $\omega_2 \approx 1.25$ and the prefactor of the correction $L^{-\omega_2}$ is large and comes with a different sign from that of the formally leading conventional correction with exponent $\omega_1 \approx 0.78$. Our study highlights the possibility of competing scaling corrections at quantum critical points.Comment: 6 pages, 6 figure

Nearly deconfined spinon excitations in the square-lattice spin-1/2 Heisenberg antiferromagnet

We study the spin-excitation spectrum (dynamic structure factor) of the spin-1/2 square-lattice Heisenberg antiferromagnet and an extended model (the J−Q model) including four-spin interactions Q in addition to the Heisenberg exchange J. Using an improved method for stochastic analytic continuation of imaginary-time correlation functions computed with quantum Monte Carlo simulations, we can treat the sharp (δ-function) contribution to the structure factor expected from spin-wave (magnon) excitations, in addition to resolving a continuum above the magnon energy. Spectra for the Heisenberg model are in excellent agreement with recent neutron-scattering experiments on Cu(DCOO)2⋅4D2O, where a broad spectral-weight continuum at wave vector q=(π,0) was interpreted as deconfined spinons, i.e., fractional excitations carrying half of the spin of a magnon. Our results at (π,0) show a similar reduction of the magnon weight and a large continuum, while the continuum is much smaller at q=(π/2,π/2) (as also seen experimentally). We further investigate the reasons for the small magnon weight at (π,0) and the nature of the corresponding excitation by studying the evolution of the spectral functions in the J−Q model. Upon turning on the Q interaction, we observe a rapid reduction of the magnon weight to zero, well before the system undergoes a deconfined quantum phase transition into a nonmagnetic spontaneously dimerized state. Based on these results, we reinterpret the picture of deconfined spinons at (π,0) in the experiments as nearly deconfined spinons—a precursor to deconfined quantum criticality. To further elucidate the picture of a fragile (π,0)-magnon pole in the Heisenberg model and its depletion in the J−Q model, we introduce an effective model of the excitations in which a magnon can split into two spinons that do not separate but fluctuate in and out of the magnon space (in analogy to the resonance between a photon and a particle-hole pair in the exciton-polariton problem). The model can reproduce the reduction of magnon weight and lowered excitation energy at (π,0) in the Heisenberg model, as well as the energy maximum and smaller continuum at (π/2,π/2). It can also account for the rapid loss of the (π,0) magnon with increasing Q and the remarkable persistence of a large magnon pole at q=(π/2,π/2) even at the deconfined critical point. The fragility of the magnons close to (π,0) in the Heisenberg model suggests that various interactions that likely are important in many materials—e.g., longer-range pair exchange, ring exchange, and spin-phonon interactions—may also destroy these magnons and lead to even stronger spinon signatures than in Cu(DCOO)2⋅4D2O.We thank Wenan Guo, Akiko Masaki-Kato, Andrey Mishchenko, Martin Mourigal, Henrik Ronnow, Kai Schmidt, Cenke Xu, and Seiji Yunoki for useful discussions. Experimental data from Ref. [33] were kindly provided by N. B. Christensen and H. M. Ronnow. H. S. was supported by the China Postdoctoral Science Foundation under Grants No. 2016M600034 and No. 2017T100031. St.C. was funded by the NSFC under Grants No. 11574025 and No. U1530401. Y. Q. Q. and Z. Y. M. acknowledge funding from the Ministry of Science and Technology of China through National Key Research and Development Program under Grant No. 2016YFA0300502, from the key research program of the Chinese Academy of Sciences under Grant No. XDPB0803, and from the NSFC under Grants No. 11421092, No. 11574359, and No. 11674370, as well as the National Thousand-Young Talents Program of China. A. W. S. was funded by the NSF under Grants No. DMR-1410126 and No. DMR-1710170, and by the Simons Foundation. In addition H. S., Y. Q. Q., and Sy. C. thank Boston University's Condensed Matter Theory Visitors program for support, and A. W. S. thanks the Beijing Computational Science Research Center and the Institute of Physics, Chinese Academy of Sciences for visitor support. We thank the Center for Quantum Simulation Sciences at the Institute of Physics, Chinese Academy of Sciences, the Tianhe-1A platform at the National Supercomputer Center in Tianjin, Boston University's Shared Computing Cluster, and CALMIP (Toulouse) for their technical support and generous allocation of CPU time. (2016M600034 - China Postdoctoral Science Foundation; 2017T100031 - China Postdoctoral Science Foundation; 11574025 - NSFC; U1530401 - NSFC; 11421092 - NSFC; 11574359 - NSFC; 11674370 - NSFC; 2016YFA0300502 - Ministry of Science and Technology of China; XDPB0803 - Chinese Academy of Sciences; National Thousand-Young Talents Program of China; DMR-1410126 - NSF; DMR-1710170 - NSF; Simons Foundation; Boston University's Condensed Matter Theory Visitors program)Accepted manuscript and published version

Unconventional U(1) to Zq crossover in quantum and classical q -state clock models

We consider two-dimensional q-state quantum clock models with quantum fluctuations connecting states with all-to-all clock transitions with different choices for the matrix elements. We study the quantum phase transitions in these models using quantum Monte Carlo simulations and finite-size scaling, with the aim of characterizing the crossover from emergent U(1) symmetry at the transition (for q≥4) to Zq symmetry of the ordered state. We also study classical three-dimensional clock models with spatial anisotropy corresponding to the space-time anisotropy of the quantum systems. The U(1) to Zq symmetry crossover in all these systems is governed by a so-called dangerously irrelevant operator. We specifically study q=5 and q=6 models with different forms of the quantum fluctuations and different anisotropies in the classical models. In all cases, we find the expected classical XY critical exponents and scaling dimensions yq of the clock fields. However, the initial weak violation of the U(1) symmetry in the ordered phase, characterized by a Zq symmetric order parameter ϕq, scales in an unexpected way. As a function of the system size (length) L, close to the critical temperature ϕq∝Lp, where the known value of the exponent is p=2 in the classical isotropic clock model. In contrast, for strongly anisotropic classical models and the quantum models, we find p=3. For weakly anisotropic classical models, we observe a crossover from p=2 to p=3 scaling. The exponent p directly impacts the exponent ν′ governing the divergence of the U(1) to Zq crossover length scale ξ′ in the thermodynamic limit, according to the relationship ν′=ν(1+|yq|/p), where ν is the conventional correlation length exponent. We present a phenomenological argument for p=3 based on an anomalous renormalization of the clock field in the presence of anisotropy, possibly as a consequence of topological (vortex) line defects. Thus, our study points to an intriguing interplay between conventional and dangerously irrelevant perturbations, which may also affect other quantum systems with emergent symmetries.First author draf

Monte Carlo renormalization flows in the space of relevant and irrelevant operators: application to three-dimensional clock models

We study renormalization group flows in a space of observables computed by Monte Carlo simulations. As an example, we consider three-dimensional clock models, i.e., the XY spin model perturbed by a Z_{q} symmetric anisotropy field. For q=4, 5, 6, a scaling function with two relevant arguments describes all stages of the complex renormalization flow at the critical point and in the ordered phase, including the crossover from the U(1) Nambu-Goldstone fixed point to the ultimate Z_{q} symmetry-breaking fixed point. We expect our method to be useful in the context of quantum-critical points with inherent dangerously irrelevant operators that cannot be tuned away microscopically but whose renormalization flows can be analyzed as we do here for the clock models.Accepted manuscrip