We present exact results for several universal parameters of the tricritical O (n) model in two d... more We present exact results for several universal parameters of the tricritical O (n) model in two dimensions. The results apply to the range −2≤n≤3/2, and include the central charge and three scaling dimensions, associated with temperature, magnetic field and the introduction of an interface. Since these results are based on an extrapolation of known relations between the O (n) and the Potts model, they cannot be considered as rigorous. For this reason, we perform an accurate numerical analysis of the central charge and the critical exponents. This analysis, which is based on transfer-matrix calculations on the honeycomb lattice, is in a full and precise agreement with the theoretical predictions.
The completely packed O(n) loop model on the square lattice This article has been downloaded from... more The completely packed O(n) loop model on the square lattice This article has been downloaded from IOPscience. Please scroll down to see the full text article.
Doping quantum magnets with various impurities can give rise to unusual quantum states and quantu... more Doping quantum magnets with various impurities can give rise to unusual quantum states and quantum phase transitions. A recent example is Sr2CuTeO6, a square-lattice Néel antiferromagnet with superexchange between first-neighbor S = 1/2 Cu spins mediated by plaquette centered Te ions. Substituting Te by W, the affected impurity plaquettes have predominantly second-neighbor interactions, thus causing local magnetic frustration. Here we report a study of Sr2CuTe1−xWxO6 using neutron diffraction and µSR techniques, showing that the Néel order vanishes already at x ≈ 0.03. We explain this extreme order suppression using a two-dimensional Heisenberg spin model, demonstrating that a W-type impurity induces a non-colinear deformation of the order parameter that decays with distance as 1/r 2 at temperature T = 0. Thus, there is a logarithmic singularity and loss of order for any x > 0. Order for small x > 0 and T > 0 in the material is induced by weak inter-plane couplings. In the non-magnetic phase, the µSR relaxation rate exhibits quantum critical scaling with a large dynamic exponent, z ≈ 3, consistent with a random-singlet state.
We use QMC simulations to study effects of disorder on the S = 1/2 Heisenberg model with exchange... more We use QMC simulations to study effects of disorder on the S = 1/2 Heisenberg model with exchange constant J on the square lattice supplemented by multispin interactions Q. It was found recently [L. Lu et al., Phys. Rev. X 8, 041040 (2018)] that the ground state of this J-Q model with random couplings undergoes a quantum phase transition from the Néel state into a randomnessinduced spin-liquid-like state that is a close analogue to the well known random-singlet (RS) state of the random Heisenberg chain. The 2D RS state arises from spinons localized at topological defects. The interacting spinons form a critical state with mean spin-spin correlations decaying with distance r as r −2 , as in the 1D RS state. The dynamic exponent z ≥ 2, varying continuously with the model parameters. We here further investigate the properties of the RS state in the J-Q model with random Q couplings. We study the temperature dependence of the specific heat and various susceptibilities for large enough systems to reach the thermodynamic limit and also analyze the size dependence of the critical magnetic order parameter and its susceptibility in the ground state. For all these quantities, we find consistency with conventional quantum-critical scaling when the condition implied by the r −2 form of the spin correlations is imposed. All quantities can be explained by the same value of the dynamic exponent z at fixed model parameters. We argue that the RS state identified in the J-Q model corresponds to a generic renormalization group fixed point that can be reached in many quantum magnets with random couplings, and may already have been observed experimentally.
We study renormalization group flows in a space of observables computed by Monte Carlo simulation... more 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 Zq 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 cross-over from the U(1) Nambu-Goldstone fixed point to the ultimate Zq 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.
We study the Néel-paramagnetic quantum phase transition in two-dimensional dimerized S = 1/2 Heis... more We study the Néel-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 a new irrelevant field in the staggered model, but, at variance with previous claims, it is not the leading irrelevant field. The new exponent is ω2 ≈ 1.25 and the prefactor of the correction L −ω 2 is large and comes with a different sign from that of the conventional correction with ω1 ≈ 0.78. Our study highlights competing scaling corrections at quantum critical points.
We discuss the concept of typicality of quantum states at quantum-critical points, using projecto... more We discuss the concept of typicality of quantum states at quantum-critical points, using projector Monte Carlo simulations of an S = 1 2 bilayer Heisenberg antiferromagnet as an illustration. With the projection (imaginary) time τ scaled as τ = aL z , L being the system length and z the dynamic critical exponent (which takes the value z = 1 in the bilayer model studied here), a critical point can be identified which asymptotically flows to the correct location and universality class with increasing L, independently of the prefactor a and the initial state. Varying the proportionality factor a and the initial state only changes the cross-over behavior into the asymptotic large-L behavior. In some cases, choosing an optimal factor a may also lead to the vanishing of the leading finite-size corrections. The observation of typicality can be used to speed up simulations of quantum criticality, not only within the Monte Carlo approach but also with other numerical methods where imaginarytime evolution is employed, e.g., tensor network states, as it is not necessary to evolve fully to the ground state but only for sufficiently long times to reach the typicality regime.
We study the mechanism of decay of a topological (winding-number) excitation due to finitesize ef... more We study the mechanism of decay of a topological (winding-number) excitation due to finitesize effects in a two-dimensional valence-bond solid state, realized in an S = 1/2 spin model (J-Q model) with six-spin interactions and studied using projector Monte Carlo simulations in the valence bond basis. A topological excitation with winding number |W | > 0 contains domain walls, which are unstable due to the emergence of long valence bonds in the wave function, unlike in effective descriptions with the quantum dimer model (which by construction includes only short bonds). We find that the life time of the winding number in imaginary time, which is directly accessible in the simulations, diverges as a power of the system length L. The energy can be computed within this time (i.e., it converges toward a "quasi-eigenvalue" before the winding number decays) and agrees for large L with the domain-wall energy computed in an open lattice with boundary modifications enforcing a domain wall. Constructing a simplified two-state model which can be solved in real and imaginary time, and using the imaginary-time behavior from the simulations as input, we find that the real-time decay rate out of the initial winding sector is exponentially small in L. Thus, the winding number rapidly becomes a well-defined conserved quantum number for large systems, supporting the conclusions reached by computing the energy quasi-eigenvalues. Including Heisenberg exchange interactions which brings the system to a quantum-critical point separating the valencebond solid from an antiferromagnetic ground state (the putative "deconfined" quantum-critical point), we can also converge the domain wall energy here and find that it decays as a power-law of the system size. Thus, the winding number is an emergent quantum number also at the critical point, with all winding number sectors becoming degenerate in the thermodynamic limit. This supports the description of the critical point in terms of a U(1) gauge-field theory.
Journal of Physics A: Mathematical and Theoretical, 2012
We present an investigation of the completely packed O(n) loop model on the square lattice by mea... more We present an investigation of the completely packed O(n) loop model on the square lattice by means of the transfer-matrix method and finite-size scaling. We investigate the model for a number of n values covering a wide range. This model is known to be equivalent with the q-state Potts model with q = n 2 , but here we also investigate the range n < 0, including rather large negative numbers. In the critical range |n| < 2, we find an energy-like scaling dimension X = 4, which is the leading one for n < 1 and the second leading one for 1 < n < 2. The point n = −2 is special, with a conformal anomaly c = −∞. For n < −2, the model is no longer critical, as evidenced e.g. by the exponentially fast convergence of the finite-size estimates of the free energy density to the infinite-system value. For |n| > 2, the system is in an ordered phase, where the majority of the loops cover part of the elementary faces of the lattice in one of two checkerboard patterns that are in phase coexistence. Furthermore, we find that the numerical results for the free energy density are in agreement with the expressions obtained from the exact analysis of the equivalent six-vertex model.
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 2002
We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a f... more We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a finite-sizescaling analysis. Using a minimal set of assumptions we find that the spectrum of correction-to-scaling exponents is consistent with that of the exactly solved Ising model, and that the critical exponents and correlationlength amplitudes closely follow the relation predicted by conformal invariance. Assuming that these spectra are exactly identical, and conformal invariance, we determine the critical point, the conformal anomaly, and the temperature and magnetic exponents with numerical margins of 10 Ϫ11 or less. These results are in a perfect agreement with the exactly known Ising universal parameters in two dimensions. In order to obtain this degree of precision, we included system sizes as large as feasible, and used extended-precision floating-point arithmetic. The latter resource provided a substantial improvement of the analysis, despite the fact that it restricted the transfer-matrix calculations to finite sizes of at most 34 lattice units.
Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, b... more Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, because distinct surface universality classes can be realized at the same bulk critical point by tuning the surface interactions. The exploration of surface critical behavior provides a window looking into higher-dimensional boundary conformal field theories. In this work, we study the surface critical behavior of a two-dimensional (2D) quantum critical Heisenberg model by tuning the surface coupling strength, and discover a direct special transition on the surface from the ordinary phase into an extraordinary phase. The extraordinary phase has a long-range antiferromagnetic order on the surface, in sharp contrast to the logarithmic decaying spin correlations in the 3D classical O(3) model. The special transition point has a new set of critical exponents, y_{s}=0.86(4)ys=0.86(4) and \eta_{\parallel}=-0.33(1)η∥=−0.33(1), which are distinct from the special transition of the classical O(3) ...
We study the surface behavior of the two-dimensional columnar dimerized quantum antiferromagnetic... more We study the surface behavior of the two-dimensional columnar dimerized quantum antiferromagnetic XXZ model with easy-plane anisotropy, with particular emphasis on the surface critical behaviors of the (2+1)-dimensional quantum critical points of the model that belong to the classical three-dimensional O(2) universality class, for both S = 1/2 and S = 1 spins using quantum Monte Carlo simulations. We find completely different surface behaviors on two different surfaces of geometrical settings: the dangling-ladder surface, which is exposed by cutting a row of weak bonds, and the dangling-chain surface, which is formed by cutting a row of strong bonds along the direction perpendicular to the strong bonds of a periodic system. Similar to the Heisenberg limit, we find an ordinary transition on the dangling-ladder surface for both S = 1 and S = 1/2 spin systems. However, the dangling-chain surface shows much richer surface behaviors than in the Heisenberg limit. For the S = 1/2 easy-plane model, at the bulk critical point, we provide evidence supporting an extraordinary surface transition with a long-range order established by effective long-range interactions due to bulk critical fluctuations. The possibility that the state is an extraordinary-log state seems unlikely. For the S = 1 system, we find surface behaviors similar to that of the three-dimensional classical XY model with sufficiently enhanced surface coupling, suggesting an extraordinary-log state at the bulk critical point.
We study phase transitions of the Potts model on the centered-triangular lattice with two types o... more We study phase transitions of the Potts model on the centered-triangular lattice with two types of couplings, namely K between neighboring triangular sites, and J between the centered and the triangular sites. Results are obtained by means of a finite-size analysis based on numerical transfermatrix calculations and Monte Carlo simulations. Our investigation covers the whole (K, J) phase diagram, but we find that most of the interesting physics applies to the antiferromagnetic case K < 0, where the model is geometrically frustrated. In particular, we find that there are, for all finite J, two transitions when K is varied. Their critical properties are explored. In the limits J → ±∞ we find algebraic phases with infinite-order transitions to the ferromagnetic phase.
We investigate the O(n) nonintersecting loop model on the square lattice under the constraint tha... more We investigate the O(n) nonintersecting loop model on the square lattice under the constraint that the loops consist of ninety-degree bends only. The model is governed by the loop weight n, a weight x for each vertex of the lattice visited once by a loop, and a weight z for each vertex visited twice by a loop. We explore the (x, z) phase diagram for some values of n. For 0 < n < 1, the diagram has the same topology as the generic O(n) phase diagram with n < 2, with a first-order line when z starts to dominate, and an O(n)-like transition when x starts to dominate. Both lines meet in an exactly solved higher critical point. For n > 1, the O(n)-like transition line appears to be absent. Thus, for z = 0, the (n, x) phase diagram displays a line of phase transitions for n ≤ 1. The line ends at n = 1 in an infinite-order transition. We determine the conformal anomaly and the critical exponents along this line. These results agree accurately with a recent proposal for the universal classification of this type of model, at least in most of the range −1 ≤ n ≤ 1. We also determine the exponent describing crossover to the generic O(n) universality class, by introducing topological defects associated with the introduction of 'straight' vertices violating the ninety-degree-bend rule. These results are obtained by means of transfer-matrix calculations and finite-size scaling.
Journal of Physics A: Mathematical and Theoretical, 2017
We study a self-dual generalization of the Baxter-Wu model, employing results obtained by transfe... more We study a self-dual generalization of the Baxter-Wu model, employing results obtained by transfer matrix calculations of the magnetic scaling dimension and the free energy. While the pure critical Baxter-Wu model displays the critical behavior of the four-state Potts fixed point in two dimensions, in the sense that logarithmic corrections are absent, the introduction of different couplings in the upand down triangles moves the model away from this fixed point, so that logarithmic corrections appear. Real couplings move the model into the first-order range, away from the behavior displayed by the nearest-neighbor, four-state Potts model. We also use complex couplings, which bring the model in the opposite direction characterized by the same type of logarithmic corrections as present in the four-state Potts model. Our finite-size analysis confirms in detail the existing renormalization theory describing the immediate vicinity of the four-state Potts fixed point.
We test the performance of the Monte Carlo renormalization method using the Ising model on the tr... more We test the performance of the Monte Carlo renormalization method using the Ising model on the triangular lattice. We apply block-spin transformations which allow for adjustable parameters so that the transformation can be optimized. This optimization takes into account the relation between corrections to scaling and the location of the fixed point. To this purpose we determine corrections to scaling of the triangular Ising model with nearest-and next-nearestneighbor interactions, by means of transfer matrix calculations and finite-size scaling. We find that the leading correction to scaling just vanishes for the nearest-neighbor model. However, the fixed point of the commonly used majority-rule block-spin transformation lies far away from the nearest-neighbour critical point. This raises the question whether the majority rule is suitable as a renormalization transformation, because corrections to scaling are supposed to be absent at the fixed point. We define a modified block-spin transformation which shifts the fixed point back to the vicinity of the nearest-neighbour critical Hamiltonian. This modified transformation leads to results for the Ising critical exponents that converge faster, and are more accurate than those obtained with the majority rule.
We present exact results for several universal parameters of the tricritical O (n) model in two d... more We present exact results for several universal parameters of the tricritical O (n) model in two dimensions. The results apply to the range −2≤n≤3/2, and include the central charge and three scaling dimensions, associated with temperature, magnetic field and the introduction of an interface. Since these results are based on an extrapolation of known relations between the O (n) and the Potts model, they cannot be considered as rigorous. For this reason, we perform an accurate numerical analysis of the central charge and the critical exponents. This analysis, which is based on transfer-matrix calculations on the honeycomb lattice, is in a full and precise agreement with the theoretical predictions.
The completely packed O(n) loop model on the square lattice This article has been downloaded from... more The completely packed O(n) loop model on the square lattice This article has been downloaded from IOPscience. Please scroll down to see the full text article.
Doping quantum magnets with various impurities can give rise to unusual quantum states and quantu... more Doping quantum magnets with various impurities can give rise to unusual quantum states and quantum phase transitions. A recent example is Sr2CuTeO6, a square-lattice Néel antiferromagnet with superexchange between first-neighbor S = 1/2 Cu spins mediated by plaquette centered Te ions. Substituting Te by W, the affected impurity plaquettes have predominantly second-neighbor interactions, thus causing local magnetic frustration. Here we report a study of Sr2CuTe1−xWxO6 using neutron diffraction and µSR techniques, showing that the Néel order vanishes already at x ≈ 0.03. We explain this extreme order suppression using a two-dimensional Heisenberg spin model, demonstrating that a W-type impurity induces a non-colinear deformation of the order parameter that decays with distance as 1/r 2 at temperature T = 0. Thus, there is a logarithmic singularity and loss of order for any x > 0. Order for small x > 0 and T > 0 in the material is induced by weak inter-plane couplings. In the non-magnetic phase, the µSR relaxation rate exhibits quantum critical scaling with a large dynamic exponent, z ≈ 3, consistent with a random-singlet state.
We use QMC simulations to study effects of disorder on the S = 1/2 Heisenberg model with exchange... more We use QMC simulations to study effects of disorder on the S = 1/2 Heisenberg model with exchange constant J on the square lattice supplemented by multispin interactions Q. It was found recently [L. Lu et al., Phys. Rev. X 8, 041040 (2018)] that the ground state of this J-Q model with random couplings undergoes a quantum phase transition from the Néel state into a randomnessinduced spin-liquid-like state that is a close analogue to the well known random-singlet (RS) state of the random Heisenberg chain. The 2D RS state arises from spinons localized at topological defects. The interacting spinons form a critical state with mean spin-spin correlations decaying with distance r as r −2 , as in the 1D RS state. The dynamic exponent z ≥ 2, varying continuously with the model parameters. We here further investigate the properties of the RS state in the J-Q model with random Q couplings. We study the temperature dependence of the specific heat and various susceptibilities for large enough systems to reach the thermodynamic limit and also analyze the size dependence of the critical magnetic order parameter and its susceptibility in the ground state. For all these quantities, we find consistency with conventional quantum-critical scaling when the condition implied by the r −2 form of the spin correlations is imposed. All quantities can be explained by the same value of the dynamic exponent z at fixed model parameters. We argue that the RS state identified in the J-Q model corresponds to a generic renormalization group fixed point that can be reached in many quantum magnets with random couplings, and may already have been observed experimentally.
We study renormalization group flows in a space of observables computed by Monte Carlo simulation... more 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 Zq 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 cross-over from the U(1) Nambu-Goldstone fixed point to the ultimate Zq 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.
We study the Néel-paramagnetic quantum phase transition in two-dimensional dimerized S = 1/2 Heis... more We study the Néel-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 a new irrelevant field in the staggered model, but, at variance with previous claims, it is not the leading irrelevant field. The new exponent is ω2 ≈ 1.25 and the prefactor of the correction L −ω 2 is large and comes with a different sign from that of the conventional correction with ω1 ≈ 0.78. Our study highlights competing scaling corrections at quantum critical points.
We discuss the concept of typicality of quantum states at quantum-critical points, using projecto... more We discuss the concept of typicality of quantum states at quantum-critical points, using projector Monte Carlo simulations of an S = 1 2 bilayer Heisenberg antiferromagnet as an illustration. With the projection (imaginary) time τ scaled as τ = aL z , L being the system length and z the dynamic critical exponent (which takes the value z = 1 in the bilayer model studied here), a critical point can be identified which asymptotically flows to the correct location and universality class with increasing L, independently of the prefactor a and the initial state. Varying the proportionality factor a and the initial state only changes the cross-over behavior into the asymptotic large-L behavior. In some cases, choosing an optimal factor a may also lead to the vanishing of the leading finite-size corrections. The observation of typicality can be used to speed up simulations of quantum criticality, not only within the Monte Carlo approach but also with other numerical methods where imaginarytime evolution is employed, e.g., tensor network states, as it is not necessary to evolve fully to the ground state but only for sufficiently long times to reach the typicality regime.
We study the mechanism of decay of a topological (winding-number) excitation due to finitesize ef... more We study the mechanism of decay of a topological (winding-number) excitation due to finitesize effects in a two-dimensional valence-bond solid state, realized in an S = 1/2 spin model (J-Q model) with six-spin interactions and studied using projector Monte Carlo simulations in the valence bond basis. A topological excitation with winding number |W | > 0 contains domain walls, which are unstable due to the emergence of long valence bonds in the wave function, unlike in effective descriptions with the quantum dimer model (which by construction includes only short bonds). We find that the life time of the winding number in imaginary time, which is directly accessible in the simulations, diverges as a power of the system length L. The energy can be computed within this time (i.e., it converges toward a "quasi-eigenvalue" before the winding number decays) and agrees for large L with the domain-wall energy computed in an open lattice with boundary modifications enforcing a domain wall. Constructing a simplified two-state model which can be solved in real and imaginary time, and using the imaginary-time behavior from the simulations as input, we find that the real-time decay rate out of the initial winding sector is exponentially small in L. Thus, the winding number rapidly becomes a well-defined conserved quantum number for large systems, supporting the conclusions reached by computing the energy quasi-eigenvalues. Including Heisenberg exchange interactions which brings the system to a quantum-critical point separating the valencebond solid from an antiferromagnetic ground state (the putative "deconfined" quantum-critical point), we can also converge the domain wall energy here and find that it decays as a power-law of the system size. Thus, the winding number is an emergent quantum number also at the critical point, with all winding number sectors becoming degenerate in the thermodynamic limit. This supports the description of the critical point in terms of a U(1) gauge-field theory.
Journal of Physics A: Mathematical and Theoretical, 2012
We present an investigation of the completely packed O(n) loop model on the square lattice by mea... more We present an investigation of the completely packed O(n) loop model on the square lattice by means of the transfer-matrix method and finite-size scaling. We investigate the model for a number of n values covering a wide range. This model is known to be equivalent with the q-state Potts model with q = n 2 , but here we also investigate the range n < 0, including rather large negative numbers. In the critical range |n| < 2, we find an energy-like scaling dimension X = 4, which is the leading one for n < 1 and the second leading one for 1 < n < 2. The point n = −2 is special, with a conformal anomaly c = −∞. For n < −2, the model is no longer critical, as evidenced e.g. by the exponentially fast convergence of the finite-size estimates of the free energy density to the infinite-system value. For |n| > 2, the system is in an ordered phase, where the majority of the loops cover part of the elementary faces of the lattice in one of two checkerboard patterns that are in phase coexistence. Furthermore, we find that the numerical results for the free energy density are in agreement with the expressions obtained from the exact analysis of the equivalent six-vertex model.
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 2002
We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a f... more We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a finite-sizescaling analysis. Using a minimal set of assumptions we find that the spectrum of correction-to-scaling exponents is consistent with that of the exactly solved Ising model, and that the critical exponents and correlationlength amplitudes closely follow the relation predicted by conformal invariance. Assuming that these spectra are exactly identical, and conformal invariance, we determine the critical point, the conformal anomaly, and the temperature and magnetic exponents with numerical margins of 10 Ϫ11 or less. These results are in a perfect agreement with the exactly known Ising universal parameters in two dimensions. In order to obtain this degree of precision, we included system sizes as large as feasible, and used extended-precision floating-point arithmetic. The latter resource provided a substantial improvement of the analysis, despite the fact that it restricted the transfer-matrix calculations to finite sizes of at most 34 lattice units.
Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, b... more Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, because distinct surface universality classes can be realized at the same bulk critical point by tuning the surface interactions. The exploration of surface critical behavior provides a window looking into higher-dimensional boundary conformal field theories. In this work, we study the surface critical behavior of a two-dimensional (2D) quantum critical Heisenberg model by tuning the surface coupling strength, and discover a direct special transition on the surface from the ordinary phase into an extraordinary phase. The extraordinary phase has a long-range antiferromagnetic order on the surface, in sharp contrast to the logarithmic decaying spin correlations in the 3D classical O(3) model. The special transition point has a new set of critical exponents, y_{s}=0.86(4)ys=0.86(4) and \eta_{\parallel}=-0.33(1)η∥=−0.33(1), which are distinct from the special transition of the classical O(3) ...
We study the surface behavior of the two-dimensional columnar dimerized quantum antiferromagnetic... more We study the surface behavior of the two-dimensional columnar dimerized quantum antiferromagnetic XXZ model with easy-plane anisotropy, with particular emphasis on the surface critical behaviors of the (2+1)-dimensional quantum critical points of the model that belong to the classical three-dimensional O(2) universality class, for both S = 1/2 and S = 1 spins using quantum Monte Carlo simulations. We find completely different surface behaviors on two different surfaces of geometrical settings: the dangling-ladder surface, which is exposed by cutting a row of weak bonds, and the dangling-chain surface, which is formed by cutting a row of strong bonds along the direction perpendicular to the strong bonds of a periodic system. Similar to the Heisenberg limit, we find an ordinary transition on the dangling-ladder surface for both S = 1 and S = 1/2 spin systems. However, the dangling-chain surface shows much richer surface behaviors than in the Heisenberg limit. For the S = 1/2 easy-plane model, at the bulk critical point, we provide evidence supporting an extraordinary surface transition with a long-range order established by effective long-range interactions due to bulk critical fluctuations. The possibility that the state is an extraordinary-log state seems unlikely. For the S = 1 system, we find surface behaviors similar to that of the three-dimensional classical XY model with sufficiently enhanced surface coupling, suggesting an extraordinary-log state at the bulk critical point.
We study phase transitions of the Potts model on the centered-triangular lattice with two types o... more We study phase transitions of the Potts model on the centered-triangular lattice with two types of couplings, namely K between neighboring triangular sites, and J between the centered and the triangular sites. Results are obtained by means of a finite-size analysis based on numerical transfermatrix calculations and Monte Carlo simulations. Our investigation covers the whole (K, J) phase diagram, but we find that most of the interesting physics applies to the antiferromagnetic case K < 0, where the model is geometrically frustrated. In particular, we find that there are, for all finite J, two transitions when K is varied. Their critical properties are explored. In the limits J → ±∞ we find algebraic phases with infinite-order transitions to the ferromagnetic phase.
We investigate the O(n) nonintersecting loop model on the square lattice under the constraint tha... more We investigate the O(n) nonintersecting loop model on the square lattice under the constraint that the loops consist of ninety-degree bends only. The model is governed by the loop weight n, a weight x for each vertex of the lattice visited once by a loop, and a weight z for each vertex visited twice by a loop. We explore the (x, z) phase diagram for some values of n. For 0 < n < 1, the diagram has the same topology as the generic O(n) phase diagram with n < 2, with a first-order line when z starts to dominate, and an O(n)-like transition when x starts to dominate. Both lines meet in an exactly solved higher critical point. For n > 1, the O(n)-like transition line appears to be absent. Thus, for z = 0, the (n, x) phase diagram displays a line of phase transitions for n ≤ 1. The line ends at n = 1 in an infinite-order transition. We determine the conformal anomaly and the critical exponents along this line. These results agree accurately with a recent proposal for the universal classification of this type of model, at least in most of the range −1 ≤ n ≤ 1. We also determine the exponent describing crossover to the generic O(n) universality class, by introducing topological defects associated with the introduction of 'straight' vertices violating the ninety-degree-bend rule. These results are obtained by means of transfer-matrix calculations and finite-size scaling.
Journal of Physics A: Mathematical and Theoretical, 2017
We study a self-dual generalization of the Baxter-Wu model, employing results obtained by transfe... more We study a self-dual generalization of the Baxter-Wu model, employing results obtained by transfer matrix calculations of the magnetic scaling dimension and the free energy. While the pure critical Baxter-Wu model displays the critical behavior of the four-state Potts fixed point in two dimensions, in the sense that logarithmic corrections are absent, the introduction of different couplings in the upand down triangles moves the model away from this fixed point, so that logarithmic corrections appear. Real couplings move the model into the first-order range, away from the behavior displayed by the nearest-neighbor, four-state Potts model. We also use complex couplings, which bring the model in the opposite direction characterized by the same type of logarithmic corrections as present in the four-state Potts model. Our finite-size analysis confirms in detail the existing renormalization theory describing the immediate vicinity of the four-state Potts fixed point.
We test the performance of the Monte Carlo renormalization method using the Ising model on the tr... more We test the performance of the Monte Carlo renormalization method using the Ising model on the triangular lattice. We apply block-spin transformations which allow for adjustable parameters so that the transformation can be optimized. This optimization takes into account the relation between corrections to scaling and the location of the fixed point. To this purpose we determine corrections to scaling of the triangular Ising model with nearest-and next-nearestneighbor interactions, by means of transfer matrix calculations and finite-size scaling. We find that the leading correction to scaling just vanishes for the nearest-neighbor model. However, the fixed point of the commonly used majority-rule block-spin transformation lies far away from the nearest-neighbour critical point. This raises the question whether the majority rule is suitable as a renormalization transformation, because corrections to scaling are supposed to be absent at the fixed point. We define a modified block-spin transformation which shifts the fixed point back to the vicinity of the nearest-neighbour critical Hamiltonian. This modified transformation leads to results for the Ising critical exponents that converge faster, and are more accurate than those obtained with the majority rule.
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