Xiaojun Tan, Youjin Deng, Jesper Lykke Jacobsen
We study $N$-cluster correlation functions in four- and five-dimensional (4D, 5D) bond percolation by extensive Monte Carlo simulation. We reformulate the transfer Monte Carlo algorithm for percolation [Phys. Rev. E {\bf 72}, 016126 (2005)] using the disjoint-set data structure, and simulate a cylindrical geometry $L^{d-1}\times \infty$, with the linear size up to $L=512$ for 4D and $128$ for 5D. We determine with a high precision all possible $N$-cluster exponents, for $N \! =\!2$ and $3$, and the universal amplitude for a logarithmic correlation function. From the symmetric correlator with $N \! = \!2$, we obtain the correlation-length critical exponent as $1/ν\! =\! 1.4610(12)$ for 4D and $1/ν\! =\! 1.737 (2)$ for 5D, significantly improving over the existing results. Estimates for the other exponents and the universal logarithmic amplitude have not been reported before to our knowledge. Our work demonstrates the validity of logarithmic conformal field theory and adds to the growing knowledge for high-dimensional percolation.
Wei Huang, Pengcheng Hou, Junfeng Wang, Robert M. Ziff, Youjin Deng
We study critical bond percolation on a seven-dimensional (7D) hypercubic lattice with periodic boundary conditions and on the complete graph (CG) of finite volume $V$. We numerically confirm that for both cases, the critical number density $n(s,V)$ of clusters of size $s$ obeys a scaling form $n(s,V) \sim s^{-τ} \tilde{n} (s/V^{d^*_{\rm f}})$ with identical volume fractal dimension $d^*_{\rm f}=2/3$ and exponent $τ= 1+1/d^*_{\rm f}=5/2$. We then classify occupied bonds into {\em bridge} bonds, which includes {\em branch} and {\em junction} bonds, and {\em non-bridge} bonds; a bridge bond is a branch bond if and only if its deletion produces at least one tree. Deleting branch bonds from percolation configurations produces {\em leaf-free} configurations, whereas, deleting all bridge bonds leads to {\em bridge-free} configurations. It is shown that the fraction of non-bridge (bi-connected) bonds vanishes $ρ_{\rm n, CG}$$\rightarrow$0 for large CGs, but converges to a finite value $ ρ_{\rm n, 7D} =0.006 \, 193 \, 1(7)$ for the 7D hypercube. Further, we observe that while the bridge-free dimension $d^*_{\rm bf}=1/3$ holds for both the CG and 7D cases, the volume fractal dimensions of the leaf-free clusters are different: $d^*_{\rm \ell f, 7D} = 0.669 (9) \approx 2/3$ and $d^*_{\rm \ell f, CG} = 0. 333 7 (17) \approx 1/3$. We also study the behavior of the number and the size distribution of leaf-free and bridge-free clusters. For the number of clusters, we numerically find the number of leaf-free and bridge-free clusters on the CG scale as $\sim \ln V$, while for 7D they scale as $\sim V$. Our work demonstrates that the geometric structure of high-dimensional percolation clusters cannot be fully accounted for by their complete-graph counterparts.
Xiaofeng Qian, Youjin Deng, Yuhai Liu, Wenan Guo, Henk W. J. Bloete
We investigate the two-dimensional $q=3$ and 4 Potts models with a variable interaction range by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges as expressed by the number $z$ of equivalent neighbors. For not too large $z$, the transitions fit well in the universality classes of the short-range Potts models. However, at longer ranges the transitions become discontinuous. For $q=3$ we locate a tricritical point separating the continuous and discontinuous transitions near $z=80$, and a critical fixed point between $z=8$ and 12. For $q=4$ the transition becomes discontinuous for $z > 16$. The scaling behavior of the $q=4$ model with $z=16$ approximates that of the $q=4$ merged critical-tricritical fixed point predicted by the renormalization scenario.
Youjin Deng, Xiaofeng Qian, Henk W. J. Blote
We formulate a single-cluster Monte Carlo algorithm for the simulation of the random-cluster model. This algorithm is a generalization of the Wolff single-cluster method for the $q$-state Potts model to non-integer values $q>1$. Its results for static quantities are in a satisfactory agreement with those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which involves a full cluster decomposition of random-cluster configurations. We explore the critical dynamics of this algorithm for several two-dimensional Potts and random-cluster models. For integer $q$, the single-cluster algorithm can be reduced to the Wolff algorithm, for which case we find that the autocorrelation functions decay almost purely exponentially, with dynamic exponents $z_{\rm exp} =0.07 (1), 0.521 (7)$, and $1.007 (9)$ for $q=2, 3$, and 4 respectively. For non-integer $q$, the dynamical behavior of the single-cluster algorithm appears to be very dissimilar to that of the SWCM algorithm. For large critical systems, the autocorrelation function displays a range of power-law behavior as a function of time. The dynamic exponents are relatively large. We provide an explanation for this peculiar dynamic behavior.
Jin-Yi Zhang, Si-Cong Ji, Zhu Chen, Long Zhang, Zhi-Dong Du, Bo Yan, Ge-Sheng Pan, Bo Zhao, Youjin Deng, Hui Zhai, Shuai Chen, Jian-Wei Pan
We present an experimental study of the collective dipole oscillation of a spin-orbit coupled Bose-Einstein condensate in a harmonic trap. Dynamics of the center-of-mass dipole oscillation is studied in a broad parameter region, as a function of spin-orbit coupling parameters as well as oscillation amplitude. Anharmonic properties beyond effective-mass approximation are revealed, such as amplitude-dependent frequency and finite oscillation frequency at place with divergent effective mass. These anharmonic behaviors agree quantitatively with variational wave-function calculations. Moreover, we experimentally demonstrate a unique feature of spin-orbit coupled system predicted by a sum-rule approach, stating that spin polarization susceptibility--a static physical quantity--can be measured via dynamics of dipole oscillation. The divergence of polarization susceptibility is observed at the quantum phase transition that separates magnetic nonzero-momentum condensate from nonmagnetic zero-momentum phase. The good agreement between the experimental and theoretical results provides a bench mark for recently developed theoretical approaches.
Long Zhang, Jin-Yi Zhang, Si-Cong Ji, Zhi-Dong Du, Hui Zhai, Youjin Deng, Shuai Chen, Peng Zhang, Jian-Wei Pan
We study the decay behaviors of ultracold atoms in metastable states with spin-orbit coupling (SOC), and demonstrate that there are two SOC-induced decay mechanisms. One arises from the trapping potential and the other is due to interatomic collision. We present general schemes for calculating decay rates from these two mechanisms, and illustrate how the decay rates can be controlled by experimental parameters.We experimentally measure the decay rates over a broad parameter region, and the results agree well with theoretical calculations. This work provides an insight for both quantum simulation involving metastable dressed states and studies on few-body problems with SO coupling.
Youjin Deng, Henk W. J. Blote
We investigate the critical scaling behavior of finite systems in the canonical ensemble. The essential difference with the grand canonical ensemble. i.e., the constraint on the number of particles, is already known to lead to the Fisher renormalization phenomenon that modifies the thermal critical singularities. We show that, in observables that are not Fisher renormalized, it also leads to a finite-size effect governed by an exponent $y_1$ that depends on the temperature exponent $y_t$ and the dimensionality $d$ as $y_1=-|2y_t-d|$. We verify this prediction by a Monte Carlo analysis of several two-dimensional lattice models in the percolation, the Ising and the 3-state Potts universality classes.
Youjin Deng, Timothy M. Garoni, Wenan Guo, Henk W. J. Blote, Alan D. Sokal
We develop cluster algorithms for a broad class of loop models on two-dimensional lattices, including several standard O(n) loop models at n \ge 1. We show that our algorithm has little or no critical slowing-down when 1 \le n \le 2. We use this algorithm to investigate the honeycomb-lattice O(n) loop model, for which we determine several new critical exponents, and a square-lattice O(n) loop model, for which we obtain new information on the phase diagram.
Wei Zhang, Youjin Deng
We formulate a Swendsen-Wang-like version of the geometric cluster algorithm. As an application,we study the hard-core lattice gas on the triangular lattice with the first- and the second-neighbor exclusions. The data are analyzed by finite-size scaling, but the possible existence of logarithmic corrections is not considered due to the limited data. We determine the critical chemical potential as $μ_c=1.75682 (2)$ and the critical particle density as $ρ_c=0.180(4)$. The thermal and magnetic exponents $y_t=1.51(1) \approx 3/2$ and $y_h=1.8748 (8) \approx 15/8$, estimated from Binder ratio $Q$ and susceptibility $χ$, strongly support the general belief that the model is in the 4-state Potts universality class. On the other hand, the analyses of energy-like quantities yield the thermal exponent $y_t$ ranging from $1.440(5)$ to $1.470(5)$. These values differ significantly from the expected value 3/2, and thus imply the existence of logarithmic corrections.
Youjin Deng, Evgeny Kozik, Nikolay V. Prokof'ev, Boris V. Svistunov
A significant part of the phase diagram of the two-dimensional fermionic Hubbard model for moderate interactions and filling factors ($U < 4, \, n<0.7$) is governed by effective Fermi liquid physics with weak BCS-type instabilities. We access this regime in a controlled way by a combination of the bold-line diagrammatic Monte Carlo method with an additional ladder-diagram summation trick and semi-analytic treatment of the weak instability in the Cooper channel. We obtain the corresponding ground-state phase diagram in the $(n,U)$ plane describing the competition between the $p-$ and $d-$wave superfluid states. We also claim the values of the dimensionless BCS coupling constants controlling the superfluid $T_c$ at the phase boundaries, which prove to be very small up to $U=4, n = 0.6$.
Youjin Deng, Jesper Lykke Jacobsen, Xuan-Wen Liu
We introduce a simple lattice model in which percolation is constructed on top of critical percolation clusters, and show that it can be repeated recursively any number $n$ of generations. In two dimensions, we determine the percolation thresholds up to $n=5$. The corresponding critical clusters become more and more compact as $n$ increases, and define universal scaling functions of the standard two-dimensional form and critical exponents that are distinct for any $n$. This family of exponents differs from any previously known universality class, and cannot be accommodated by existing analytical methods. We confirm that recursive percolation is well defined also in three dimensions.
Xiaomei Feng, Youjin Deng, Henk W. J. Blote
We investigate bond- and site-percolation models on several two-dimensional lattices numerically, by means of transfer-matrix calculations and Monte Carlo simulations. The lattices include the square, triangular, honeycomb kagome and diced lattices with nearest-neighbor bonds, and the square lattice with nearest- and next-nearest-neighbor bonds. Results are presented for the bond-percolation thresholds of the kagome and diced lattices, and the site-percolation thresholds of the square, honeycomb and diced lattices. We also include the bond- and site-percolation thresholds for the square lattice with nearest- and next-nearest-neighbor bonds. We find that corrections to scaling behave according to the second temperature dimension $X_{t2}=4$ predicted by the Coulomb gas theory and the theory of conformal invariance. In several cases there is evidence for an additional term with the same exponent, but modified by a logarithmic factor. Only for the site-percolation problem on the triangular lattice such a logarithmic term appears to be small or absent. The amplitude of the power-law correction associated with $X_{t2}=4$ is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.
Si-Cong Ji, Long Zhang, Xiao-Tian Xu, Zhan Wu, Youjin Deng, Shuai Chen, Jian-Wei Pan
Roton-type excitations usually emerge from strong correlations or long-range interactions, as in superfluid helium or dipolar ultracold atoms. However, in weakly short-range interacting quantum gas, the recently synthesized spin-orbit (SO) coupling can lead to various unconventional phases of superfluidity, and give rise to an excitation spectrum of roton-maxon character. Using Bragg spectroscopy we study a SO coupled Bose-Einstein condensate of $^{87}$Rb atoms, and show that the excitation spectrum in a "magnetized" phase clearly possesses a two-branch and roton-maxon structure. As Raman coupling strength $Ω$ is decreased, a roton-mode softening is observed, as a precursor of the phase transition to a stripe phase that spontaneously breaks spatially translational symmetry. The measured roton gaps agree well with theoretical calculations. Further, we determine sound velocities both in the magnetized and the non-magnetized phase, and a phonon-mode softening is observed around the phase transition in between. The validity of the $f$-sum rule is examined.
Youjin Deng, Wenan Guo, Henk W. J. Blote
We study the O(N) loop model on the Honeycomb lattice with real value $N \geq 1$ by means of a cluster algorithm. The formulation of the algorithm is based on the equivalence of the O(N) loop model and the low-temperature graphical representation of a $N$-color Ashkin-Teller model on the triangular lattice. The latter model with integer $N$ can be simulated by means of an embedding Swendsen-Wang-type cluster method. By taking into account the symmetry among loops of different colors, we develop another version of the Swendsen-Wang-type method. This version allows the number of colors $N$ to take any real value $N \geq 1$. As an application, we investigate the $N=1.25, 1.50, 1.75$, and 2 loop model at criticality. The determined values of various critical exponents are in excellent agreement with theoretical predictions. In particular, from quantities associated with half of the loops, we determine some critical exponents that corresponds to those for the tricritical $q=N^2$ Potts model but have not been observed yet. Dynamic scaling behavior of the algorithm is also analyzed. The numerical data strongly suggest that our cluster algorithm {\it hardly} suffers from critical slowing down.
Xiaofeng Qian, Youjin Deng, Lev N. Shchur, Henk W. J. Blöte
We perform a Monte Carlo analysis of the Ising model on many three-dimensional lattices. By means of finite-size scaling we obtain the critical points and determine the scaling dimensions. As expected, the critical exponents agree with the three-dimensional Ising universality class for all models. The irrelevant field, as revealed by the correction-to-scaling amplitudes, appears to be relatively large. Combining the Monte Carlo results for the hydrogen peroxide lattice with those for five other three-dimensional lattices, we obtain a set of data covering a wide range of the irrelevant temperature field. This is helpful in the determination of the parameters describing the corrections to scaling. As a consequence, new results are obtained for the universal parameters describing Ising criticality in three dimensions, with reduced error margins in comparison with earlier Monte Carlo analyses. The critical exponents describing the thermodynamic singularities are determined by the temperature renormalization exponent $y_t = 1.58693 (9)$ and the magnetic renormalization exponent $y_h = 2.48178 (5)$. The corrections to scaling are governed by the irrelevant exponent $y_1 = -0.821 (5)$.
Wenan Guo, Youjin Deng, Henk W. J. Blote
We derive the scaling dimension associated with crossing bonds in the random-cluster representation of the two-dimensional Potts model, by means of a mapping on the Coulomb gas. The scaling field associated with crossing bonds appears to be irrelevant, on the critical as well as on the tricritical branch. The latter result stands in a remarkable contrast with the existing result for the tricritical O(n) model that crossing bonds are relevant. In order to obtain independent confirmation of the Coulomb gas result for the crossing-bond exponent, we perform a finite-size-scaling analysis based on numerical transfer-matrix calculations.
Kun Chen, Yuan Huang, Youjin Deng, Boris Svistunov
The halon is a special critical state of an impurity in a quantum-critical environment. The hallmark of the halon physics is that a well-defined integer charge gets fractionalized into two parts: a microscopic core with half-integer charge and a critically large halo carrying a complementary charge of $\pm 1/2$. The halon phenomenon emerges when the impurity--environment interaction is fine-tuned to the vicinity of a boundary quantum critical point (BQCP), at which the energies of two quasiparticle states with adjacent integer charges approach each other. The universality class of such BQCP is captured by a model of pseudo-spin-$1/2$ impurity coupled to the quantum-critical environment, in such a way that the rotational symmetry in the pseudo-spin $xy$-plane is respected, with a small local "magnetic" field along the pseudo-spin $z$-axis playing the role of control parameter driving the system away from the BQCP. On the approach to BQCP, the half-integer projection of the pseudo-spin on its $z$-axis gets delocalized into a halo of critically divergent radius, capturing the essence of the phenomenon of charge fractionalization. With large-scale Monte Carlo simulations, we confirm the existence of halons---and quantify their universal features---in O(2) and O(3) quantum critical systems.
Fedor Šimkovic, J. P. F. LeBlanc, Aaram J. Kim, Youjin Deng, N. V. Prokof'ev, B. V. Svistunov, Evgeny Kozik
The ground state of the Hubbard model with nearest-neighbor hopping on the square lattice at half filling is known to be that of an antiferromagnetic (AFM) band insulator for any on-site repulsion. At finite temperature, the absence of long-range order makes the question of how the interaction-driven insulator is realized nontrivial. We address this problem with controlled accuracy in the thermodynamic limit using self-energy diagrammatic determinant Monte Carlo and dynamical cluster approximation methods and show that development of long-range AFM correlations drives an extended crossover from Fermi liquid to insulating behavior in the parameter regime that precludes a metal-to-insulator transition. The intermediate crossover state is best described as a non-Fermi liquid with a partially gapped Fermi surface.
Xiaojun Tan, Romain Couvreur, Youjin Deng, Jesper Lykke Jacobsen
Percolation, a paradigmatic geometric system in various branches of physical sciences, is known to possess logarithmic factors in its correlators. Starting from its definition, as the $Q\rightarrow1$ limit of the $Q$-state Potts model with $S_Q$ symmetry, in terms of geometrical clusters, its operator content as $N$-cluster observables has been classified. We extensively simulate critical bond percolation in two and three dimensions and determine with high precision the $N$-cluster exponents and non-scalar features up to $N \! =\! 4$ (2D) and $N \! =\! 3$ (3D). The results are in excellent agreement with the predicted exact values in 2D, while such families of critical exponents have not been reported in 3D, to our knowledge. Finally, we demonstrate the validity of predictions about the logarithmic structure between the energy and two-cluster operators in 3D.
Fedor Šimkovic, Youjin Deng, Evgeny Kozik
In nonperturbative regimes, the superfluid instability in the two-dimensional Hubbard model can be described by an emergent BCS theory with small effective pairing constants. We compute the effective couplings using a controlled bold-line diagrammatic Monte Carlo approach, which stochastically sums all skeleton Feynman diagrams dressed in the one- and two-particle channels to high expansion orders, and map out the resulting superfluid ground-state phase diagram in a range of next-nearest-neighbor hopping $0 \leq t^{\prime} \leq 0.3t$, interaction strength $0 \leq U \leq 3t$, and lattice filling $0 \leq n \leq 2$. The phase diagram is dramatically transformed in the hole-doped region and becomes particularly rich at larger doping and $t'$. At $t'=0.3$, the weak-coupling picture with the dominant triplet pairing sharply peaked at $n \approx 0.73$ due to the Van Hove singularity is replaced by a plateau of the singlet $d_{x^2-y^2}$ paring, while for $U \gtrsim 3t$ the effective couplings are consistent with the $d_{x^2-y^2}$ high-temperature superconductivity in the hole-doped region near cuprates' optimal doping.