Spin models in complex magnetic fields: A hard sign problem

Coupling spin models to complex external fields can give rise to interesting phenomena like zeroes of the partition function (Lee-Yang zeroes, edge singularities) or oscillating propagators. Unfortunately, it usually also leads to a severe sign problem that can be overcome only in special cases; if the partition function has zeroes, the sign problem is even representation-independent at these points. In this study, we couple the N-state Potts model in different ways to a complex external magnetic field and discuss the above mentioned phenomena and their relations based on analytic calculations (1D) and results obtained using a modified cluster algorithm (general D) that in many cases either cures or at least drastically reduces the sign-problem induced by the complex external field.Coupling spin models to complex external fields can give rise to interesting phenomena like zeroes of the partition function (Lee-Yang zeroes, edge singularities) or oscillating propagators. Unfortunately, it usually also leads to a severe sign problem that can be overcome only in special cases; if the partition function has zeroes, the sign problem is even representation-independent at these points. In this study, we couple the N-state Potts model in different ways to a complex external magnetic field and discuss the above mentioned phenomena and their relations based on analytic calculations (1D) and results obtained using a modified cluster algorithm (general D) that in many cases either cures or at least drastically reduces the sign-problem induced by the complex external field.

arXiv:1711.00042EPJ Web of ConferencesJan 1, 20185 citations

Yang-Lee zeros and the critical behavior of the infinite-range two- and three-state Potts models

Zvonko Glumac, Katarina Uzelac

The phase diagram of the two- and three-state Potts model with infinite-range interactions in the external field is analyzed by studying the partition function zeros in the complex field plane. The tricritical point of the three-state model is observed as the approach of the zeros to the real axis at the nonzero field value. Different regimes, involving several first- and second-order transitions of the complicated phase diagram of the three-state model, are identified from the scaling properties of the zeros closest to the real axis. The critical exponents related to the tricritical point and the Yang-Lee edge singularity are well reproduced. Calculations are extended to the negative fields, where the exact implicit expression for the transition line is derived.The phase diagram of the two- and three-state Potts model with infinite-range interactions in the external field is analyzed by studying the partition function zeros in the complex field plane. The tricritical point of the three-state model is observed as the approach of the zeros to the real axis at the nonzero field value. Different regimes, involving several first- and second-order transitions of the complicated phase diagram of the three-state model, are identified from the scaling properties of the zeros closest to the real axis. The critical exponents related to the tricritical point and the Yang-Lee edge singularity are well reproduced. Calculations are extended to the negative fields, where the exact implicit expression for the transition line is derived.

arXiv:1302.7197Physical Review EFeb 25, 20137 citations

Yang-Lee Edge Singularity of the Ising Model on a Honeycomb Lattice in an External Magnetic Field

Seung‐Yeon Kim

The Ising model in an external magnetic field is one of the most outstanding and intriguing unsolved problems. One of the important properties of the Ising model in an external magnetic field is the Yang-Lee edge singularity. The zeros in the complex magnetic-field plane of the grand partition function of the Ising model on a honeycomb lattice in an external magnetic field are exactly calculated by using the exact integer values for the density of states on L × 2L honeycomb lattices (up to L = 14). The critical line of the Yang-Lee edge singularity of the Ising model on a honeycomb lattice in an external magnetic field is accurately investigated from its Yang-Lee zeros.The Ising model in an external magnetic field is one of the most outstanding and intriguing unsolved problems. One of the important properties of the Ising model in an external magnetic field is the Yang-Lee edge singularity. The zeros in the complex magnetic-field plane of the grand partition function of the Ising model on a honeycomb lattice in an external magnetic field are exactly calculated by using the exact integer values for the density of states on L × 2L honeycomb lattices (up to L = 14). The critical line of the Yang-Lee edge singularity of the Ising model on a honeycomb lattice in an external magnetic field is accurately investigated from its Yang-Lee zeros.

Journal of the Korean Physical SocietyAug 1, 20204 citations

Generalized partition function zeros of 1D spin models and their critical behavior at edge singularities

Denis Dalmazi, Fernanda Lopes Sá

Here we study the partition function zeros of the one-dimensional Blume–Emery–Griffiths model close to their edge singularities. The model contains four couplings ( H , J , Δ, K ) including the magnetic field H and the Ising coupling J . We assume that only one of the three couplings ( J , Δ, K ) is complex and the magnetic field is real. The generalized zeros z i tend to form continuous curves on the complex z -plane in the thermodynamic limit. The linear density at the edges z E diverges usually with ρ( z ) ∼ | z − z E | σ and σ = −1/2. However, as in the case of complex magnetic fields (Yang–Lee edge singularity), if we have a triple degeneracy of the transfer matrix eigenvalues a new critical behavior with σ = −2/3 can appear as we prove here explicitly for the cases where either Δ or K is complex. Our proof applies for a general three-state spin model with short-range interactions. The Fisher zeros (complex J ) are more involved; in practice, we have not been able to find an explicit example with σ = −2/3 as far as the other couplings ( H , Δ, K ) are kept as real numbers. Our results are supported by numerical computations of zeros. We show that it is absolutely necessary to have a non-vanishing magnetic field for a new critical behavior. The appearance of σ = −2/3 at the edge closest to the positive real axis indicates its possible relevance for tricritical phenomena in higher-dimensional spin models.Here we study the partition function zeros of the one-dimensional Blume–Emery–Griffiths model close to their edge singularities. The model contains four couplings ( H , J , Δ, K ) including the magnetic field H and the Ising coupling J . We assume that only one of the three couplings ( J , Δ, K ) is complex and the magnetic field is real. The generalized zeros z i tend to form continuous curves on the complex z -plane in the thermodynamic limit. The linear density at the edges z E diverges usually with ρ( z ) ∼ | z − z E | σ and σ = −1/2. However, as in the case of complex magnetic fields (Yang–Lee edge singularity), if we have a triple degeneracy of the transfer matrix eigenvalues a new critical behavior with σ = −2/3 can appear as we prove here explicitly for the cases where either Δ or K is complex. Our proof applies for a general three-state spin model with short-range interactions. The Fisher zeros (complex J ) are more involved; in practice, we have not been able to find an explicit example with σ = −2/3 as far as the other couplings ( H , Δ, K ) are kept as real numbers. Our results are supported by numerical computations of zeros. We show that it is absolutely necessary to have a non-vanishing magnetic field for a new critical behavior. The appearance of σ = −2/3 at the edge closest to the positive real axis indicates its possible relevance for tricritical phenomena in higher-dimensional spin models.

Journal of Physics A Mathematical and TheoreticalMay 28, 20105 citations

Meron-Cluster Simulation of Quantum Spin Ladders in a Magnetic Field

Shailesh Chandrasekharan A, B. Scarlet, U.‐J. Wiese

Numerical simulations of numerous quantum systems suffer from the notorious sign problem. Meron-cluster algorithms lead to an efficient solution of sign problems for both fermionic and bosonic models. Here we apply the meron concept to quantum spin systems in an arbitrary external magnetic field, in which case standard cluster algorithms fail. As an example, we simulate antiferromagnetic quantum spin ladders in a uniform external magnetic field that competes with the spin-spin interaction. The numerical results are in agreement with analytic predictions for the magnetization as a function of the external field.Numerical simulations of numerous quantum systems suffer from the notorious sign problem. Meron-cluster algorithms lead to an efficient solution of sign problems for both fermionic and bosonic models. Here we apply the meron concept to quantum spin systems in an arbitrary external magnetic field, in which case standard cluster algorithms fail. As an example, we simulate antiferromagnetic quantum spin ladders in a uniform external magnetic field that competes with the spin-spin interaction. The numerical results are in agreement with analytic predictions for the magnetization as a function of the external field.

arXiv:cond-mat/9909451Sep 30, 19992 citations

On properties of the Ising model for complex energy/temperature and magnetic field

V. Matveev, Robert E. Shrock

We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable $u=e^{-4K}$ , where $K=J/(k_BT)$ , for nonzero external magnetic field, $H$ . Exact results are given for the phase diagram in the $u$ plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real $h=H/(k_BT)$ , these results provide new insights into features of our earlier study of this case. We also consider complex $h=H/(k_BT)$ and $μ=e^{-2h}$ . Calculations of complex- $u$ zeros of the partition function on sections of the square lattice are presented. For the case of imaginary $h$ , i.e., $μ=e^{iθ}$ , we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the $u$ plane. We find that in this case, the phase boundary ${\cal B}_u$ contains a real line segment extending through part of the physical ferromagnetic interval $0 \le u \le 1$ , with a right-hand endpoint $u_{rhe}$ at the temperature for which the Yang-Lee edge singularity occurs at $μ=e^{\pm iθ}$ . Conformal field theory arguments are used to relate the singularities at $u_{rhe}$ and the Yang-Lee edge.We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable $u=e^{-4K}$, where $K=J/(k_BT)$, for nonzero external magnetic field, $H$. Exact results are given for the phase diagram in the $u$ plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real $h=H/(k_BT)$, these results provide new insights into features of our earlier study of this case. We also consider complex $h=H/(k_BT)$ and $μ=e^{-2h}$. Calculations of complex-$u$ zeros of the partition function on sections of the square lattice are presented. For the case of imaginary $h$, i.e., $μ=e^{iθ}$, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the $u$ plane. We find that in this case, the phase boundary ${\cal B}_u$ contains a real line segment extending through part of the physical ferromagnetic interval $0 \le u \le 1$, with a right-hand endpoint $u_{rhe}$ at the temperature for which the Yang-Lee edge singularity occurs at $μ=e^{\pm iθ}$. Conformal field theory arguments are used to relate the singularities at $u_{rhe}$ and the Yang-Lee edge.

arXiv:0711.4639Journal of Physics A Mathematical and TheoreticalMar 17, 200823 citations

We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable

u=e^{-4K}

, where

K=J/(k_BT)

, for nonzero external magnetic field,

H

. Exact results are given for the phase diagram in the

u

plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real

h=H/(k_BT)

, these results provide new insights into features of our earlier study of this case. We also consider complex

h=H/(k_BT)

and

μ=e^{-2h}

. Calculations of complex-

u

zeros of the partition function on sections of the square lattice are presented. For the case of imaginary

h

, i.e.,

μ=e^{iθ}

, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the

u

plane. We find that in this case, the phase boundary

{\cal B}_u

contains a real line segment extending through part of the physical ferromagnetic interval

0 \le u \le 1

, with a right-hand endpoint

u_{rhe}

at the temperature for which the Yang-Lee edge singularity occurs at

μ=e^{\pm iθ}

. Conformal field theory arguments are used to relate the singularities at

u_{rhe}

and the Yang-Lee edge.