arXiv subjects
Publications and source records attributed to Ken Kikuchi.
We present a compact formula for the exact partition function of the $d$-dimensional arithmetic Ising model (AIM). For a $2\times2$ system, we express it analytically using the $q$-Hurwitz-Lerch zeta function and derive explicit forms for the free energy and entropy. Additionally, we find that the entropy increases at high temperatures, supporting the presence of entropic order.
Renormalization group flows are constrained by symmetries. Traditionally, we have made the most of 't Hooft anomalies associated to the symmetries. The anomaly is mathematically part of the data for the monoidal structure on symmetry categories. The symmetry categories sometimes admit additional structures such as braiding. It was found that the additional structures give further constraints on renormalization group flows. One of these constraints is the half-integer condition. The condition claims the following. Braidings are characterized by conformal dimensions. A symmetry object $c$ in a braided symmetry category surviving all along the flow thus has two conformal dimensions, one in ultraviolet $h_c^\text{UV}$ and the other in infrared $h_c^\text{IR}$. In a renormalization group flow with a renormalization group defect, they add up to a half-integer $h_c^\text{UV}+h_c^\text{IR}\in\frac12\mathbb Z$. We find a necessary condition for the sum to be half-integer. We solve some flows with the refined half-integer condition.
We prove conformal and global dimensions monotonically decrease under the infinitely many Tanaka-Nakayama renormalization group flows between Virasoro minimal models. The flows also satisfy the half-integer condition.
Consider a renormalization group flow preserving a pre-modular fusion category $\mathcal S_1$. If it flows to a rational conformal field theory, the surviving symmetry $\mathcal S_1$ flows to a pre-modular fusion category $\mathcal S_2$ with monoidal functor $F:\mathcal S_1\to\mathcal S_2$. By clarifying mathematical (especially category theoretical) meaning of renormalization group domain wall/interface or boundary condition, we find the hidden extended vertex operator (super)algebra gives a unique (up to braided equivalence) completely $(\mathcal S_1\boxtimes\mathcal S_2)'$-anisotropic representative of the Witt equivalence class $[\mathcal S_1\boxtimes\mathcal S_2]$. The mathematical conjecture is supported physically, and passes various tests in concrete examples including non/unitary minimal models, and Wess-Zumino-Witten models. In particular, the conjecture holds beyond diagonal cosets. The picture also establishes the conjectured half-integer condition, which fixes infrared conformal dimensions mod $\frac12$. It further leads to the double braiding relation, namely braiding structures jump at conformal fixed points. As an application, we solve the flow from the $E$-type minimal model $(A_{10},E_6)\to M(4,3)$.
We discuss anyon condensation in mixed-state topological order. The phases were recently conjectured to be classified by pre-modular fusion categories. Just like anyon condensation in pure-state topological order, a bootstrap analysis shows condensable anyons are given by connected \'etale algebras. We explain how to perform generic anyon condensation including non-invertible anyons and successive condensations. Interestingly, some condensations lead to pure-state topological orders. We clarify when this happens. We also compute topological invariants of equivalence classes.
We classify connected \'etale algebras $A$'s in multiplicity-free modular fusion categories $\mathcal B$'s with $\text{rank}(\mathcal B)\le9$. We also identify categories $\mathcal B_A$'s of right $A$-modules. The results have physical applications in constraining renormalization group flows. As demonstration, we study massive renormalization group flows from non-unitary minimal models to predict ground state degeneracies and prove spontaneous $\mathcal B$-symmetry breaking.
We classify connected \'etale algebras $A$'s in multiplicity-free modular fusion categories (MFCs) $\mathcal{B}$'s at rank six, namely $\text{rank}(\mathcal{B})=6$. There are eight MFCs in total and the result indicates that only $so(5)_2$ has nontrivial connected \'etale algebra. We briefly mention anyon condensation as it is used to determine the category of right $A$-modules in $so(5)_2$. Finally, we discuss physical applications, specifically proving spontaneous $\mathcal{B}$-symmetry breaking (SSB) of these MFCs. The discussion also includes predicting ground state degeneracies and SSB in massive renormalization group flows from two non-unitary minimal models.
We classify connected \'etale algebras in (possibly non-unitary) modular fusion categories $\mathcal B$'s with $\text{rank}(\mathcal B)\le5$. We also comment on Lagrangian algebra, anyon condensation, and physical applications. Concretely, we prove certain spontaneous $\mathcal B$-symmetry breaking and predict ground state degeneracies in massive renormalization group flows from non-unitary minimal models.
We classify connected \'etale algebras $A$'s in pre-modular fusion categories $\mathcal B$ with $\text{rank}(\mathcal B)\le3$ including degenerate and non-(pseudo-)unitary ones. We comment on Lagrangian algebras and physical applications to ground state degeneracy and proof of spontaneous $\mathcal B$-symmetry breaking.
We develop a systematic method to classify connected \'etale algebras $A$'s in (possibly degenerate) pre-modular category $\mathcal B$. In particular, we find the category of $A$-modules, $\mathcal B_A$, have ranks bounded from above by $\lfloor\text{FPdim}(\mathcal B)\rfloor$. For demonstration, we classify connected \'etale algebras in some $\mathcal B$'s, which appear in physics. Physically, the results constrain (or fix) ground state degeneracies of (certain) $\mathcal B$-symmetric gapped phases. We study massive deformations of rational conformal field theories such as minimal models and Wess-Zumino-Witten models. In most of our examples, the classification suggests the symmetries $\mathcal B$'s are spontaneously broken.
We constrain renormalization group flows from $ABCDE$ type Wess-Zumino-Witten models triggered by adjoint primaries. We propose positive Lagrangian coupling leads to massless flow and negative to massive. In the conformal phase, we prove an interface with the half-integral condition obeys the double braiding relations. Distinguishing simple and non-simple flows, we conjecture the former satisfies the half-integral condition. If the conjecture is true, some previously allowed massless flows are ruled out. For $A$ type, known mixed anomalies fix the ambiguity in identifications of Verlinde lines; an object is identified with its charge conjugate. In the massive phase, we compute ground state degeneracies.
We investigate the fate of the non-supersymmetric Gross-Neveu-Yukawa fixed point found by Fei et al in $4-\epsilon$ dimensions with a two-component Majorana fermion continued to two dimensions. Assuming that it is a fermionic minimal model which possesses a chiral $\mathbb{Z}_2$ symmetry (in addition to fermion number parity) and just two relevant singlet operators, we can zero in on four candidates. Assuming further that the least relevant deformation leads to the supersymmetric Gross-Neveu-Yukawa fixed point (i.e. fermionic tricritical Ising model), we can rule out two of them by matching the spin contents of the preserved topological defect lines. The final candidates are the fermionic $(11,4)$ minimal model if it is non-unitary, and the fermionic $(E_{6}, A_{10})$ minimal model if it is unitary. If we further use a constraint from the double braiding relation proposed by one of the authors, the former scenario is preferable.
We axiomatize rational massless renormalization group flow as Kan extension.
Just as spontaneous symmetry breaking can be understood in terms of energy, emergent symmetry (more precisely, its `size' and structure) can also be explained by free energy. In particular, in renormalization group flow to rational conformal field theory, we find infrared symmetry category is realized by consistent modular tensor category with minimal free energy. For demonstration, we study non-unitary theories.
We explain when and why symmetries enhance in fermionic rational conformal field theories. In order to achieve the goal, we first clarify invariants under renormalization group flows. In particular, we find the Ocneanu rigidity is not enough to protect some quantities. Concretely, while (double) braidings are subject to the rigidity, they jump at conformal fixed points. The jump happens in a specific way, so the double braiding relation further constrains renormalization group flows. The new constraints enable us three things; 1) to predict infrared conformal dimensions in massless flow, 2) to reveal some structures of the theory space, and 3) to obtain a necessary condition for a flow to be massless. We also find scaling dimensions ``monotonically'' decrease along massless flows. Combining the discovery with predictions, sometimes, we can uniquely fix infrared conformal dimensions.
We propose a renormalization group flow with emergent supersymmetry in two dimensions from a non-Lagrangian theory. The ultraviolet theory does not have supersymmetry while the infrared theory does. We constrain the flow both analytically and numerically (truncated conformal space approach). Analytic constraints include a new spin constraint.
We propose when and why symmetry enhancements happen in massless renormalization group (RG) flows to two-dimensional rational conformal field theories (RCFTs). We test our proposal against known RG flows from unitary minimal models. We also suggest which sign of the relevant coupling triggers the massless RG flow. The other sign triggers massive RG flows to topological quantum field theories (TQFTs). We comment on their ground state degeneracies.
We show that three-dimensional trace anomalies lead to new universal anomalous transport effects on a conformally-flat spacetime with background scalar fields. In contrast to conventional anomalous transports in quantum chromodynamics (QCD) or quantum electrodynamics (QED), our current is independent of background gauge fields. Therefore, our anomalous transport survives even in the absence of vector-like external sources. By manipulating background fields, we suggest a setup to detect our anomalous transport. If one turns on scalar couplings in a finite interval and considers a conformal factor depending just on (conformal) time, we find anomalous transport localized at the interfaces of the interval flows perpendicularly to the interval. The magnitude of the currents is the same on the two interfaces but with opposite directions. Without the assumption on scalar couplings, and only assuming the conformal factor depending solely on (conformal) time as usually done in cosmology, one also finds the three-dimensional Hubble parameter naturally appears in our current.