F. Gliozzi, M. A. Rajabpour
We calculated numerically the fractal dimension of the boundaries of the Fortuin-Kasteleyn clusters of the $q$-state Potts model for integer and non-integer values of $q$ on the square lattice. In addition we calculated with high accuracy the fractal dimension of the boundary points of the same clusters on the square domain. Our calculation confirms that this curves can be described by SLE$_κ$.
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Boundary Criticality of Complex Conformal Field Theory: A Case Study in the Non-Hermitian 5-State Potts Model
本文研究了非厄米特5态Potts模型中边界临界现象，识别了不同共形边界条件并揭示了共形塔结构，为复杂CFTs的理解提供了新视角。
We study the famous example of weakly first order phase transitions in the 1+1D quantum Q-state Potts model at Q>4. We numerically show that these weakly first order transitions have approximately conformal invariance. Specifically, we find entanglement entropy on considerably large system sizes fits perfectly with the universal scaling law of this quantity in the conformal field theories (CFTs). This supports that the weakly first order transitions is proximate to complex fixed points, which are described by recent conjectured complex CFTs. Moreover, the central charge extracted from this fit…
We perform a detailed investigation of the scaling Potts field theory using the truncated conformal space approach.
Based on the spectrum identified in our earlier work [arXiv:1809.02191], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the $Q$-state Potts model. Crucial in our approach is the existence of "interchiral conformal blocks", which arise from the degeneracy of fields with conformal weight $h_{r,1}$, with $r\in\mathbb{N}^*$, and are related to the underlying presence of the "interchiral algebra" introduced in [arXiv:1207.6334]. We also find evidence for the existence of "renormalized" recursions, replacing those that follow fro…
We investigate the operator content of the Q-state Potts model in arbitrary dimension, using the representation theory of the symmetric group. In particular we construct all possible tensors acting on N spins, corresponding to given symmetries under $S_Q$ and $S_N$, in terms of representations involving any Young diagram. These operators transform non-trivially under the group of spatial rotations, with a definite conformal spin. The two-point correlation functions are then computed, and their physical interpretation is given in terms of Fortuin-Kasteleyn clusters propagating between two neigh…
Fortuin-Kastelyn clusters in the critical $Q$-state Potts model are conformally invariant fractals. We obtain simulation results for the fractal dimension of the complete and external (accessible) hulls for Q=1, 2, 3, and 4, on clusters that wrap around a cylindrical system. We find excellent agreement between these results and theoretical predictions. We also obtain the probability distributions of the hull lengths and maximal heights of the clusters in this geometry and provide a conjecture for their form.
Reclaiming the Lost Conformality in a non-Hermitian Quantum 5-state Potts Model
本文通过在非厄密itian量子5态Potts模型中引入额外相互作用，成功恢复了因固定点碰撞而失去的共形性，并揭示了复CFTs的径向量化及新算符谱。
A universal amplitude ratio for the q<4 Potts model from a solvable lattice model
利用稀A模型在区域1中的结果确定了二维（$q\leqslant 4$）态Potts模型的普适振幅比$R_ξ$。讨论了与$ϕ_{2,1}$扰动共形场论有关的Potts模型和稀A模型之间的关系的本质。
In this paper, we present a systematical account of the descending procedure from six-vertex model to the $N$-state chiral Potts model through fusion relations of $τ^{(j)}$-operators, following the works of Bazhanov-Stroganov and Baxter-Bazhanov-Perk. A careful analysis of the descending process leads to appearance of the high genus curve as rapidities' constraint for the chiral Potts models. Full symmetries of the rapidity curve are identified, so is its symmetry group structure. By normalized transfer matrices of the chiral Potts model, the $τ^{(2)}T$ relation can be reduced to functional eq…
We initiate a study of the boundary version of the square-lattice $Q$-state Potts antiferromagnet, with $Q \in [0,4]$ real, motivated by the fact that the continuum limit of the corresponding bulk model is a non-compact CFT, closely related with the $SL(2,\mathbb{R})_k/U(1)$ Euclidian black-hole coset model. While various types of conformal boundary conditions (discrete and continuous branes) have been formally identified for the the $SL(2,\mathbb{R})_k/U(1)$ coset CFT, we are only able in this work to identify conformal boundary conditions (CBC) leading to a discrete boundary spectrum. The co…
Higher Conformal Multifractality
从共形不变性和量子引力出发，导出了二维共形不变分形附近调和测度（或静电势，或扩散场）的多重分形谱f（α，c），对应于中心荷c的共形场论.它给出了边界点集的豪斯多夫维数，其中势随到分形边界的距离r而变化，作为r^{alpha}。第一个例子是布朗前沿，自避免行走，或渗流集群。Potts，O（N）模型，和所谓的SLE过程也被考虑。可以导出更高的多重分形函数，如泛函数f_2（alpha，alpha '），它给出了势随距离r共同变化的点的Hausdorff维数，在随机曲线的一侧为r^{alpha}，在另一侧为r^{alpha'}。我们提出了一个对偶Potts集团和O（N）圈在其临界点，在以前的工作，以及相应的对偶SLE_{kappa}过程中的kappa kappa '= 16的外部周长。
For the two-dimensional Q-state Potts model at criticality, we consider Fortuin-Kasteleyn and spin clusters and study the average number N_Gamma of clusters that intersect a given contour Gamma. To leading order, N_Gamma is proportional to the length of the curve. Additionally, however, there occur logarithmic contributions related to the corners of Gamma. These are found to be universal and their size can be calculated employing techniques from conformal field theory. For the Fortuin-Kasteleyn clusters relevant to the thermal phase transition we find agreement with these predictions from larg…
We address the geometrical critical behavior of the two-dimensional Q-state Potts model in terms of the spin clusters (i.e., connected domains where the spin takes a constant value). These clusters are different from the usual Fortuin-Kasteleyn clusters, and are separated by domain walls that can cross and branch. We develop a transfer matrix technique enabling the formulation and numerical study of spin clusters even when Q is not an integer. We further identify geometrically the crossing events which give rise to conformal correlation functions. This leads to an infinite series of fundamenta…
We study the phase diagram of the triangular-lattice $Q$-state Potts model in the real $(Q,v)$-plane, where $v=e^J-1$ is the temperature variable. Our first goal is to provide an obviously missing feature of this diagram: the position of the antiferromagnetic critical curve. This curve turns out to possess a bifurcation point with two branches emerging from it, entailing important consequences for the global phase diagram. We have obtained accurate numerical estimates for the position of this curve by combining the transfer-matrix approach for strip graphs with toroidal boundary conditions and…
Free boson formulation of boundary states in W_3 minimal models and the critical Potts model
我们发展了具有$W$对称性的边界共形场论的库仑气体形式，并使用三态Potts模型说明了它的操作。我们发现六个$W$守恒边界态存在自由场表示，它们产生固定的和混合的物理边界条件，两个$W$破坏边界态产生自由的和新的边界条件.其他$W$违反边界状态可以构造，但他们从理论的其余部分解耦。这样我们就有了三态Potts模型已知边界态的一个完整的自由场实现。然后，我们使用的形式来计算在各种情况下的边界关联函数。我们发现在自由边界条件和新边界条件下计算$ϕ_{2,3}$的两点函数时所产生的共形块确实是由奇异向量生成的六阶微分方程的最后两个解。
"Magic" is the degree to which a state cannot be approximated by Clifford gates. We study mana, a measure of magic, in the ground state of the $\mathbb Z_3$ Potts model, and argue that it is a broadly useful diagnostic for many-body physics. In particular we find that the $q = 3$ ground state has large mana at the model's critical point, and that this mana resides in the system's correlations. We explain the form of the mana by a simple tensor-counting calculation based on a MERA representation of the state. Because mana is present at all length scales, we conclude that the conformal field the…
We study numerically the fractal dimensions and the bulk three-point connectivity for the spin clusters of the Q-state Potts model in two dimensions with $1\leq Q\leq 4$. We check that the usually invoked correspondence between FK clusters and spin clusters works at the level of fractal dimensions. However, the fine structure of the conformal field theories describing critical clusters first manifests at the level of the three-point connectivities. Contrary to what recently found for FK clusters, no obvious relation emerges for generic Q between the spin cluster connectivity and the structure …
We revisit in this paper the problem of connectivity correlations in the Fortuin-Kasteleyn cluster representation of the two-dimensional $Q$-state Potts model conformal field theory. In a recent work [M. Picco, S. Ribault and R. Santachiara, SciPost Phys. 1, 009 (2016); arXiv:1607.07224], results for the four-point functions were obtained, based on the bootstrap approach, combined with simple conjectures for the spectra in the different fusion channels. In this paper, we test these conjectures using lattice algebraic considerations combined with extensive numerical studies of correlations on i…
Reexamining algebraic curves found in the eight-vertex model, we propose an asymptotic form of the correlation functions for off-critical systems possessing rotational and mirror symmetries of the square lattice, i.e., the $C_{4v}$ symmetry. In comparison with the use of the Ornstein-Zernike form, it is efficient to investigate the correlation length with its directional dependence (or anisotropy). We investigate the $Q$-state Potts model on the square lattice. Monte Carlo (MC) simulations are performed using the infinite-size algorithm by Evertz and von der Linden. Fitting the MC data with th…
Magnetic critical behavior of two-dimensional random-bond Potts ferromagnets in confined geometries
我们对二维随机键Potts铁磁体进行了数值研究。该模型的研究低于和高于临界值$Q_c=4$，区分在纯系统中的二阶和一阶过渡。两种几何形状被认为是，即圆柱体和正方形系统，和临界行为进行了研究，通过共形不变性技术，最近被证明是有效的，即使在随机诱导的二阶相变制度Q>4。在圆柱体的几何形状，连接传递矩阵计算提供了一个简单的测试，以找到无序振幅的范围，这是无序固定点的特性。然后，标度维度遵循沿着条带的相关性的指数衰减。另一方面，自旋系统的蒙特卡罗模拟通常是在正方形晶格上的矩形系统上进行的，但是数据会受到强表面效应的干扰。一个正方形内的半无限系统的保角映射，使我们能够考虑边界效应明确，并导致准确的确定的缩放尺寸。这些技术被应用到范围3-64中的不同Q值。
By computing the low-lying energy excitation spectra with the density matrix renormalization group algorithm we show that boundaries polarized in the direction of the transverse field lead to scale-invariant conformal towers of states at the critical point of the quantum four-state Potts model - a special symmetric case of the Ashkin-Teller model. Furthermore, by direct comparison of the excitation spectra we phenomenologically establish the duality between the transverse-polarized and three-state-mixed boundary conditions at the four-state Potts critical point. Finally, for completeness, we v…
Harmonic Measure and Winding of Conformally Invariant Curves
在二维空间中导出了任意共形不变标度曲线附近静电势线标度和缠绕的精确联合多重分形分布。它的谱f（alpha，lambda）给出了点的Hausdorff维数，其中势随距离$r$缩放为$H \sim r^α$，而曲线以旋转角phi=lambda ln r螺旋。它服从标度律f（α，λ）=（1+λ^2）f（\bar α）-bλ^2，其中\bar α=α/（1+λ^2），b=（25-c）/{12}$，其中f（α）\equiv f（α，0）是纯调和测度谱，c是共形中心荷.结果适用于O（N）和Potts模型，以及{ \rm SLE}_κ。
Conformally Invariant Fractals and Potential Theory
在二维空间中求解了任意共形不变分形边界附近静电势的多重分形分布，如临界O（N）圈或$Q$态Potts团簇.具有局部楔角$θ$的边界集的维数$\hat f(θ)$是$\hat f(θ)=\fracπθ -\frac{25-c}{12} \frac{(π-θ)^2}{θ(2π-θ)}$，其中$c$是模型的中心荷。作为推论，Potts团簇的外周长的维数$D_{\rm EP} =sup_θ\hat f(θ)$和壳的维数$D_{\rm H}$服从对偶方程$(D_{\rm EP}-1)(D_{\rm H}-1)={1/4}$。一个相关的协变MF频谱获得锚定在集群边界的自避免行走。
We consider the two dimensional $Q-$ random-cluster Potts model on the torus and at the critical point. We study the probability for two points to be connected by a cluster for general values of $Q\in [1,4]$. Using a Conformal Field Theory (CFT) approach, we provide the leading topological corrections to the plane limit of this probability. These corrections have universal nature and include, as a special case, the universality class of two-dimensional critical percolation. We compare our predictions to Monte Carlo measurements. Finally, we take Monte Carlo measurements of the torus energy one…
We propose exact expressions for the conformal anomaly and for three critical exponents of the tricritical O(n) loop model as a function of n in the range $-2 \leq n \leq 3/2$. These findings are based on an analogy with known relations between Potts and O(n) models, and on an exact solution of a 'tri-tricritical' Potts model described in the literature. We verify the exact expressions for the tricritical O(n) model by means of a finite-size scaling analysis based on numerical transfer-matrix calculations.
Backbone three-point correlation function in the two-dimensional Potts model
研究二维Q态Potts模型中背骨的三点相关函数，利用Fortuin-Kasteleyn表示法，通过高效集群算法计算背骨和FK簇的通用三点幅度比，验证数值方法的可靠性，并发现临界区背骨比大于FK簇比，三临界分支中两者一致，表明两者具有相同的几何普适性。
Exploring Replica-Potts CFTs in Two Dimensions
本文对具有 $S_n \ltimes (S_Q)^n$ 全局对称性的二维 CFTs 进行了非微扰共形 bootstrap 研究，重点分析了三个耦合的 3态 Potts 模型，所得算符标度维数界限与先前结果一致，为未来精确研究奠定基础。
We consider the random-bond Potts model in the large-$Q$ limit and calculate the excess entropy, $S_Γ$, of a contour, $Γ$, which is given by the mean number of Fortuin-Kasteleyn clusters which are crossed by $Γ$. In two dimensions $S_Γ$ is proportional to the length of $Γ$, to which - at the critical point - there are universal logarithmic corrections due to corners. These are calculated by applying techniques of conformal field theory and compared with the results of large scale numerical calculations. The central charge of the model is obtained from the corner contributions to the excess ent…
Boundary critical behaviour of two-dimensional random Potts models
我们考虑随机q态Potts模型$3\le q \le 8$的正方形晶格上的铁磁耦合采取两个值$J_1>J_2$具有相等的概率。对于任何q，该模型表现出连续的相变，无论是在本体和在边界。用Monte Carlo方法研究了临界温度附近的表面磁化和体磁化，确定了临界指数$β_1$和$β$。在条形几何中，利用自由-固定自旋边界条件研究了临界磁化强度分布，并根据共形场论计算了特征标度维数$β_1/ν$.临界指数和标度维数随q单调增加。估计了相关标度场的反常维数，并分析了临界时的多重分形行为。
We consider the 3-state Potts model in $d\geq2$ dimensions. For $d$ less than the upper critical dimension $d_\text{crit}$, the model has a critical and a tricritical fixed point. In $d=2$, these fixed points are described by minimal models, and so are exactly solvable. For $d>2$, however, strong coupling makes them difficult to study and there is no consensus on the value of $d_\text{crit}$. We use the numerical conformal bootstrap to compute critical exponents of both the critical and tricritical fixed points for general $d$. In $d=2$ our results match the expected values, and as we increase…
We conjecture an exact form for an universal ratio of four-point cluster connectivities in the critical two-dimensional $Q$-color Potts model. We also provide analogous results for the limit $Q\rightarrow 1$ that corresponds to percolation where the observable has a logarithmic singularity. Our conjectures are tested against Monte Carlo simulations showing excellent agreement for $Q=1,2,3$.
We have studied the iso-height lines on the $\mathrm{WO_3}$ surface as a physical candidate for conformally invariant curves. We have shown that these lines are conformally invariant with the same statistics of domain walls in the critical Ising model. They belong to the family of conformal invariant curves called Schramm-Loewner evolution (or $SLE_κ$), with diffusivity of $κ\sim 3$. This can be regarded as the first experimental observation of SLE curves. We have also argued that Ballistic Deposition (BD) can serve as a growth model giving rise to contours with similar statistics at large sca…
Lattice Realizations of Topological Defects in the critical (1+1)-d Three-State Potts Model
本文在三态Potts共形场论中提出自旋链正则化方案，构造所有基本拓扑缺陷的晶格实现，并通过精确对角化、密度矩阵重整化群及纠缠熵计算验证其与共形场论预期一致。
The $q-$state Potts model from the Nonperturbative Renormalization Group
本研究利用非微扰重整化群的局域势近似，研究任意q和d的q-态Potts模型，确定了2.8<d≤4下的相变分界曲线q_c(d)，验证了三维三态Potts模型的一级相变性质。
Conformal Field Theories (CFTs) have been used extensively to understand the physics of critical lattice models at equilibrium. However, the applicability of CFT calculations to the behavior of the lattice systems in the out-of-equilibrium setting is not entirely understood. In this work, we compare the CFT results of the evolution of the entanglement spectrum after a quantum quench with numerical calculations of the entanglement spectrum of the three-state Potts model using matrix product state simulations. Our results lead us to conjecture that CFT does not describe the entanglement spectrum…
We provide a representation for the scaling limit of the d=2 critical Ising magnetization field as a (conformal) random field using SLE (Schramm-Loewner Evolution) clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. Extensions to off-critical scaling limits, to d=3 and to Potts models are also considered.
Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models II. Extended Results for Square-Lattice Chromatic Polynomial
本文研究了m个\times n个正方形格子条的色多项式，其宽度为9 <= m <= 13（具有周期边界条件），长度为n（具有自由边界条件）.我们已经使用了一个传递矩阵的方法，使我们也可以提取的极限曲线时，n \to \infty。在此极限下，我们还得到了这些正方形格条的孤立极限点，并检验了与Beraha数有关的一些定理。
Monte Carlo (MC) simulations and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the ferromagnetic 4-state Potts model on the square lattice are analyzed in a vicinity of the critical point in order to estimate universal combinations of critical amplitudes. The quality of the fits is improved using predictions of the renormalization group (RG) approach and of conformal invariance, and restricting the data within an appropriate temperature window. The RG predictions on the cancelation of the logarithmic corrections in the universal amplitude ratios …
A crossing probability for the critical four-state Potts model on an $L\times M$ rectangle on a square lattice is numerically studied. The crossing probability here denotes the probability that spin clusters cross from one side of the boundary to the other. First, by employing a Monte Carlo method, we calculate the fractal dimension of a spin cluster interface with a fluctuating boundary condition. By comparison of the fractal dimension with that of the Schramm-Loewner evolution (SLE), we numerically confirm that the interface can be described by the SLE with $κ=4$, as predicted in the scaling…
We study four-point functions of critical percolation in two dimensions, and more generally of the Potts model. We propose an exact ansatz for the spectrum: an infinite, discrete and non-diagonal combination of representations of the Virasoro algebra. Based on this ansatz, we compute four-point functions using a numerical conformal bootstrap approach. The results agree with Monte-Carlo computations of connectivities of random clusters.
We discuss walking behavior in gauge theories and weak first-order phase transitions in statistical physics. Despite appearing in very different systems (QCD below the conformal window, the Potts model, deconfined criticality) these two phenomena both imply approximate scale invariance in a range of energies and have the same RG interpretation: a flow passing between pairs of fixed point at complex coupling. We discuss what distinguishes a real theory from a complex theory and call these fixed points complex CFTs. By using conformal perturbation theory we show how observables of the walking th…
Dynamic critical behavior of cluster algorithms for 2D Ashkin-Teller and Potts models
We study the dynamic critical behavior of two algorithms: the Swendsen-Wang algorithm for the two-dimensional Potts model with q=2,3,4 and a Swendsen-Wang-type algorithm for the two-dimensional symmetric Ashkin-Teller model on the self-dual curve. We find that the Li--Sokal bound on the autocorrelation time τ_{{ \rm int},{ \cal E}} \geq const \times C_H is almost, but not quite sharp. The ratio τ_{{ \rm int},{ \cal E}}/C_H appears to tend to infinity either as a logarithm or as a small power (0.05 \ltapprox p \ltapprox 0.12). We also show that the exponential autocorrelation time τ_{{ \rm exp}…
Tensor Renormalization Group Calculations of Partition-Function Ratios
本文利用张量重整化组计算了不同通用类别的二维模型的分区函数比值，验证了其与共形场理论预测的一致性，并发现了四态Potts模型中的对数修正。
We point out that the probability law of a single domain wall separating clusters in ADE lattice models in a simply connected domain is identical to that of corresponding chordal curves in the lattice O(n) and Q-state Potts models, for suitable n or Q. They are conjectured to be described in the scaling limit by chordal SLE(kappa) with kappa rational and >2. However in a multiply-connected domain the laws can differ from those for the corresponding O(n) or Potts model. The correspondence also sheds light on the scaling limit of multiple curves.
We study the q-state Potts model with nearest-neighbor coupling v=e^{βJ}-1 in the limit q,v \to 0 with the ratio w = v/q held fixed. Combinatorially, this limit gives rise to the generating polynomial of spanning forests; physically, it provides information about the Potts-model phase diagram in the neighborhood of (q,v) = (0,0). We have studied this model on the square and triangular lattices, using a transfer-matrix approach at both real and complex values of w. For both lattices, we have computed the symbolic transfer matrices for cylindrical strips of widths 2 \le L \le 10, as well as the …
We consider two models with disorder dominated critical points and study the distribution of clusters which are confined in strips and touch one or both boundaries. For the classical random bond Potts model in the large-q limit we study optimal Fortuin-Kasteleyn clusters by combinatorial optimization algorithm. For the random transverse-field Ising chain clusters are defined and calculated through the strong disorder renormalization group method. The numerically calculated density profiles close to the boundaries are shown to follow scaling predictions. For the random bond Potts model we have …
We numerically obtain the conformal spectrum of several classical spin models on a two-dimensional lattice with open boundaries, for every boundary fixed point obtained by the Cardy's derivation [J. L. Cardy, Nucl. Phys. B 324, 581 (1989)]. In order to extract accurate conformal data, we implement the tensor network renormalization algorithm [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)] extended so as to be applicable to a square lattice with open boundaries. We successfully compute the boundary conformal spectrum consistent with the underlying boundary conformal field theorie…
We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q^{-47} (resp. q^{-46}). Finally, we compute chromatic roots for strips of wi…
Multi-point functions of a full-plane two-state fuzzy $4$-Potts model
通过研究全平面两态模糊 $4$-Potts 模型，证明了所有偶数多点自旋关联函数在适当归一化后收敛到显式的共形协变库仑气体型中性电荷和，并由此得到重整化磁化场的分布收敛及其矩。
We obtain long series expansions for the bulk, surface and corner free energies for several two-dimensional statistical models, by combining Enting's finite lattice method (FLM) with exact transfer matrix enumerations. The models encompass all integrable curves of the Q-state Potts model on the square and triangular lattices, including the antiferromagnetic transition curves and the Ising model (Q=2) at temperature T, as well as a fully-packed O(n) type loop model on the square lattice. The expansions are around the trivial fixed points at infinite Q, n or 1/T. By using a carefully chosen expa…
We study, using transfer-matrix methods, the partition-function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature (= square-lattice chromatic polynomial) for the special boundary conditions that are obtained from an m x n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column. We provide numerical evidence that the partition-function zeros are becoming dense everywhere in the complex q-plane outside the limiting curve B_\infty(sq) for th…
Fractal geometry of random curves appearing in the scaling limit of critical two-dimensional statistical systems is characterized by their harmonic measure and winding angle. The former is the measure of the jaggedness of the curves while the latter quantifies their tendency to form logarithmic spirals. We show how these characteristics are related to local operators of conformal field theory and how they can be computed using conformal invariance of critical systems with central charge $c \leqslant 1$.
Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-Lattice Chromatic Polynomial
本文研究了正方形晶格的m个\times n矩形子集的色多项式P_G（q），其中m \le为8（自由或周期性横向边界条件），n为任意（自由纵向边界条件），使用Fortuin-Kasteleyn表示中的转移矩阵.特别地，我们提取了当n个\to \infty时，由主特征值模的交叉引起的配分函数零点的极限曲线（Beraha-Kahane-Weiss定理）。证明了当带宽m为\ge 7时，Beraha数B_2，B_3，B_4，B_5是n个\to \infty的配分函数零点的极限点.或\ge 8（游离横b.c.）。一路上，我们证明了一个非整数Beraha数（也许除了B_{10}）不能是任何图的色根。
Emergent conformal boundaries from finite-entanglement scaling in matrix product states
该研究针对1+1维临界格点理论的有限纠缠标度问题，提出有限纠缠会引入相关形变，证明MPS的双体纠缠哈密顿量对应带物理与纠缠边界的边界共形场论，可通过MPS对称性设计物理边界，并借助三类CFT模型验证了相关影响。
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