Comments to my papers
On this page, I will write a short comment to remind myself of how I started my research.
[] Numerical study on the theta-dependent spectrum of 2-flavor QED1+1d with DMRG
with Etsuko Itou, Akira Matsumoto
This is the extension of our previous work (2307.16655[hep-lat]) on the particle spectrum of the 2-flavor massive Schwinger model from theta=0 to finite vacuum angles. This is the setup that causes the sign problem in the Monte Carlo simulation with the original field variables, and we have to employ some trick to get around it. There are several options, such as the use of dual variables, bosonizations, etc., and we decided to apply the DMRG with the Hamiltonian formalism as we did in our previous paper. As the mass gap descreases as we approach to the theta=pi point, so we need larger bond dimensions to get reliable results. All the numerical computations were done by Akira, and my contribution was to give him some suggestions on theoretical aspects.
One of the improvements compared with the previous paper is the subtraction of the excited states contribution to determine the particle spectrum for the one-point function. It turns out that the boundary state is contaminated from the excited states than we have thought, and its subtraction gives much better signal for the particle mass. As a result, it turns out that the sigma-to-pi mass ratio obeys the WKB formula Mσ/Mπ = √3 remarkably. We also studied the physics at theta=pi and confirmed its consistency with the SU(2)_1 WZW CFT, and extracting the marginally-relevant JLJR deformation requires more systematic analysis on larger system sizes also with the continuum extrapolations.
with Yui Hayashi
One day in April, after Yui gave her group seminar on our recent work of the eta' periodicity, we chatted about the possibility of the weak-weak continuity between two semiclassical descriptions of color confinement: 3d monopole semiclassics on R3xS1 and 2d center-vortex semiclassics on R2xT2. While drawing several figures and staring blankly at the 3d dual-photon Lagrangian, we suddenly noticed that the fundamental monopoles can be smoothly transmuted to the center vortex.
Over the weekend, Yui checked how this idea works concretely at the level of the partition function, and she identified the classical vacua of the dual photons in the 't Hooft twisted boundary condition serves as the discrete label for the N confinement states in the 2d center-vortex theory. We also solved the classical equation of motion for the monopole in the dual-photon description and found that the magnetic flux gets squeezed into a vortex due to the perturbative mass gap caused by the twisted boundary condition.
I was wondering if there exists a nice connection between these semiclassical descriptions over two years since I developed the center-vortex semiclassics in 2201.06166[hep-th], and I was surprised and quite fascinated that we could solve it in such a neat way. I find our discovery quite exciting as it opens new directions for the semiclassical theories of quark confinement.
[2404.16803[hep-th]] Non-supersymmetric duality cascade of QCD(BF) via semiclassics on ℝ2× T2 with the baryon-'t Hooft flux
with Yui Hayashi and Hiromasa Watanabe
In this paper, we extend our previous paper (2307.13954[hep-th]) and develop the semiclassical techniques for different-rank bifundamental QCD (QCD(BF)) to study their phase diagrams. For the phase diagram of QCD(BF), there is the duality-cascade conjecture by Karasik and Komargodski (1904.09551), which states that SU(N1)xSU(N2) QCD (BF) and SU(N1)xSU(N2-N1) QCD(BF) have the same topological structure of the phase diagram in the (θ1,θ2) space at sufficiently small masses. We carefully examine this conjecture in this paper using our semiclassics: We develop the concrete mapping of parameters to realize the conjectured duality, and numerically confirm it works more nicely than we expected before starting this study.
Theoretical formulations are developed by Yui, including the computation of anomalies and semiclassics. She also gave the proof of duality cascade in certain limits by combining those limits with the semiclassical effective theory and constructed an explicit duality map valid in the large-N limit. Hiromasa developed a numerical code to find the ground states based on these results and showed us the phase diagrams. Comparing the phase diagrams between the parent and daughter theories, they are strikingly similar, and we are quite surprised about the effectiveness of the non-supersymmetric duality cascade. This motivates us to further investigate why the duality is so good, and we argue that its violation is the O(N-4) effect.
with Yoshimasa Hidaka, Arata Yamamoto
Properties of finite-density QCD are still veiled due to the lack of reliable computational tools. Standard Euclidean lattice formalism suffers from the sign problem, and some people expect that the quantum computation with the Hamiltonian formalism might solve this difficulty. Although I am not sure if this is true, it would be interesting to develop a new technology to tacke difficult problems.
When considering the Hamiltonian formalism of field theories, we need to decide how to manage its exponentially large Hilbert space. Especially in quantum gauge theories, the gauge fields have the infinite-dimensional Hilbert space even at finite volumes and the truncation is unavoidable. In the autumn of 2023, Arata, Yoshimasa, and I have a chance to discuss this problem in person at the Yukawa institute, and Arata proposed us that we should develop a nice toy model with the finite gauge theory. In our study, we found that certain finite gauge theories exhibit striking similarities with the anticipated behaviors of finite-density QCD, as evidenced by our sketch of the phase diagram of ℤ3 gauge theories.
[2402.04320[hep-th]] Semiclassics for the QCD vacuum structure through T2-compactification with the baryon-'t Hooft flux
with Yui Hayashi
In this paper, we expand on the semiclassics for QCD with fundamental quarks using the anomaly-preserving T2-compactifications (2201.06166[hep-th]). Most parts of this paper were found and carried out by Yui, and, honestly, my contributions are quite tiny. Using semiclassics with center vortices, we derive the 2d effective theory that contains not only massless pions but also eta' meson based on the microscopic computations.
In particular, we have shown that the eta' field has an extended 2πN periodicity instead of the naive 2π periodicity, and the Kobayashi-Maskawa-'t Hooft instanton vertex naturally gets 1/N fractionalization. We carefully examined its implications on the QCD vacuum structures, clarifying the role of the periodicity extension for the global aspects of the phase diagram. Usually, the chiral Lagrangian completely forgets the information of the color number N, but this subtle 2πN extension remembers it, which plays the crucial role to exactly recover the information of 4d global anomaly.
[2307.16655[hep-lat]] Calculating composite-particle spectra in Hamiltonian formalism and demonstration in 2-flavor QED1+1d
with Etuko Itou and Akira Matsumoto
When we started this project, if I remember correctly, Akira was just graduating the PhD from KEK (which means that this took us more than a year). He is interested in the new approach potentially free from the sign problem, and Etsuko, Akira and I started the collaboration on the DMRG approach. This is an extension of the previous study about the 1-flavor Schwinger model (2210.04237[hep-lat]), but the mass spectrum of 2-flavor theory is highly rich compared with the 1-flavor case. Numerical simulations are mostly done by Akira, and I helped the idea for analyses and gave theoretical backgrounds.
We worked on three methods for computing the mass spectrum of gauge theories in the Hamiltonian formalism: (1) correlation-function scheme, (2) one-point function scheme, and (3) dispersion-relation scheme. In all these methods, we find stable composite particles, pions, sigma and eta mesons. Notably, the mass of the sigma meson is lighter than twice the pion mass, making it stable against the decay process σ → ππ, unlike 4d QCD. The numerical findings closely match the analytic prediction based on the WKB approximation, with Mσ/Mπ = √3 pointed out by Coleman long time ago, and it's very curious to me why the semiclassical method work unreasonably so well as parameters seem to be outside the range of its validity.
with Yui Hayashi and Hiromasa Watanabe
This is a joint work with Yui and Hiromasa, postdocs at Yukawa institute. Since they are both interested in the confinement dynamics, I invited them to consider the phase diagram of QCD(BF) using the semiclassical analysis with the anomaly-preserving T2 compactification. Yui then completed the semiclassical computations in a week or so and showed us the phase diagram. What is surprising is that the semiclassical computations cover both the chiral and heavy fermion limits, and we can see how the phase diagram is continuously deformed as a function of the fermion mass.
We considered its physical implications especially in the context of the nonperturbative orbifod equivalence. Our result suggets that the anomaly matching is satisfied by the chiral symmetry breaking, and there is no extra symmetry breaking in the exchange-symmetric point. Under the assumption of the adiabatic continuity, this gives the necessary and sufficient condition for the large-N SYM/QCD(BF) equivalence. We also gave a brief discussion on the domain wall physics, which resolves the confusion in previous literature.
[2307.00939[hep-th]] Solitonic symmetry as non-invertible symmetry: cohomology theories with TQFT coefficients
with Shi Chen
This is a followup work of our previous study (2210.13780[hep-th]) that has uncovered the noninvertible structure in the symmetry of topological solitons for the 4d CP^1 sigma model. One day, Shi found that there is a beutiful mathematical story behind the general structure of the solitonic symmetry, which was very striking and unexpected for me.
We develop a new approach called solitonic cohomology or cohomology with TQFT coefficients to study noninvertible solitonic symmetries, and this perspective allows for the identification of invertible solitonic subsymmetries and sheds light on the topological origin of non-invertibility in solitonic symmetry. To treat it systematically, we need resort to higher category theories, with which I was completely unfamiliar before doing this study and Shi helped me a lot with detailed explanations.
[2306.02485[hep-th]] Study of gapped phases of 4d gauge theories using temporal gauging of the ℤN 1-form symmetry
with Mendel Nguyen and Mithat Ünsal
This research began during my two-week visit to North Carolina State University in May 2023, when Mendel, Mitat, and I were chatting about random topics related to non-Abelian gauge theory. One of the topics we were discussing was whether the Wilson-'t Hooft criterion for the 4d gap phase really makes sense in the modern perspectives of quantum field theory (QFT). The Wilson-'t Hooft criterion differentiates the gapped phases by the set of the deconfined dyonic line operators. This cannot be simply understood as the spontaneous breakdown of 1-form symmetry, because the set of dyonic lines does not have the mutual locality.
We noticed that gauging the 1-form symmetry only in the temporal direction gives the framework that justifies the Wilson-'t Hooft criterion, and we call this operation the temporal gauging. The temporal gauging produces the effective 3d QFT with dyonic ℤN x ℤN 1-form symmetry from the 4d QFT with ℤN 1-form symmetry. The 4d Lorentz invariance gives a strong constraint, and the dyonic 1-form symmetry must be spontaneously broken to an order-N subgroup in gapped phases, and this unbroken order-N subgroup has the 1-to-1 correspondence with the 4d ℤN/n topological orders enriched with ℤn SPT states. This result suggests that the temporal gauging provides a suitable Kennedy-Tasaki transformation for 4d QFT with ℤN 1-form symmetry.
with Motokazu Abe, Okuto Morikawa, Soma Onoda, and Hiroshi Suzuki
When I visited Kyushu University in the last December, we also started considering about the properties of magnetic monopoles on the lattice. In the previous study by Motokazu, Okuto and Hiroshi, they defined the topological theta angle for the 4D Maxwell theory. To this end, the admissibility condition plays the important role, but it rules out the magnetic objects such as the 't Hooft loop, and we should develop some new techniques to observe the interesting phenomena, such as Witten effect, on the lattice field theories with well-defined topological properties.
I pointed out that these problems should exist in a simpler model, i.e. 2D compact scalar theories, and we should try to solve them in this setup. By proposing the excision method, we find that many observations in the continuum descriptions are nicely derived at the finite lattice spacing. Moreover, we consider a model with two compact scalars, which accepts the continuous theta angle, and we show that the Witten effect can be reproduced on the lattice in the ultra-local way in this 2D lattice model.
with Motokazu Abe, Okuto Morikawa, Soma Onoda, and Hiroshi Suzuki
I visited Kyushu university in the last December and we started the discussion on topological aspects of lattice gauge theories. Motokazu, Okuto, and Hiroshi worked on the lattice U(1) gauge theories coupled with 2-form gauge fields, and I heard from them that it was successful to identify the topological structure of lattice U(1) gauge theories but its extension to SU(N) seemed to be quite nontrivial. There is an important work by Lüscher that shows the presence of topological sectors for SU(N) lattice gauge fields, so we try to extend it to the case when it couples with Z_N two-form gauge fields.
We established the topological sectors for SU(N) lattice gauge theories when they couple to Z_N two-form gauge fields. We extend Luscher's construction of topological charge to our setup, and the manifest 1-form gauge invariance plays the important role for our extension. The local topological charge density is defined in the SU(N) and Z_N 1-form gauge invariant way. As a byproduct, we show that the lattice SU(N) Yang-Mills theory with the admissibility condition enjoys the mixed 't Hooft anomaly between its Z_N 1-form symmetry and the theta-angle periodicity.
[2210.13780[hep-th]] Solitonic symmetry beyond homotopy: invertibility from bordism and non-invertibility from topological quantum field theory
with Shi Chen
Solitons are topologically stable objects in QFTs, especially in non-linear sigma models, and their classification is believed to obey the homotopy group. We uncover that this is not always the case, and demonstrate that a more sophisticated algebraic structure appears as the solitonic symmetry of 4d CP^1 sigma model.
4d CP^1 model has the particle-like soliton, called hopfion or Hopf soliton, which name comes from the Hopf fibration. There are also string-like solitons, and their charged are characterized by the U(1) 1-form symmetry. However, we cannot find the local conserved current for the hopfion number. We have shown that the only good symmetry is just Z_2 instead of U(1).
What is more surprising is that the full structure of the hopfion number symmetry is characterized by the non-invertible symmetry. Because of this, we can prove the stronger selection rule, and the hopfion number is conserved as integers as long as the vortex-line operator is not inserted. Once the vortex operator is inserted, the hopfion number is conserved only in mod 2, and this subtlety is nicely explained by the non-invertibility of the correct hopfion number symmetry.
Originally, this project started from the study on baryons of SO(N_c) QCD, but we unexpectedly achieved the interesting applications of non-invertible symmetries.
with Masazumi Honda and Etsuko Itou
This is the follow-up work of the previous numerical study (2110.14105) of charge-q Schwinger model. In the continuum analysis, the charge-q Schwinger model in the chiral limit has an 't Hooft anomaly between the 1-form symmetry and the discrete chiral symmetry, and thus the chiral symmetry has to be spontaneously broken to satisfy the anomaly matching condition. We would like to confirm if the lattice Hamiltonian formulation reproduces the presence of the 't Hooft anomaly.
As we use the staggered lattice fermion, most of the chiral symmetry is explicitly broken by the artifact of lattice regularization, so we should carefully take the continuum limit to obtain the chiral symmetry. This is done using the density-matrix renormalization group (DMRG) with taking into account the recent proposal by Dempsey, Klebanov, Pufu, and Zan (2206.05308) about the discrepancy between the continuum fermion mass and the lattice staggered fermion mass. We confirm that the phase of the chiral condensate rotates by 2π/q across the Wilson loop as suggested by the 't Hooft anomaly.
with Mendel Nguyen and Mithat Ünsal
During my visit to NCSU, Mendel, Mithat, and I discussed what happens if we take the chirally twisted boundary condition to obtain the weak-coupling setup for the asymptotically free QFTs. We noticed that it is related to the case where the theta parameter has the nonzero winding number as a result of the ABJ anomaly, and we decided to work on it first. Moreover, I had been personally interested in this setup from another perspective. By shifting the theta parameter by 2pi, the vacuum is replaced by the different SPT states, and thus the nonzero winding number of theta requires the interface between different SPT states. Since we do not have an anti-interface, it means that the total Hilbert space has 't Hooft anomalous symmetry.
Our main goal of this paper is to understand the semiclassical realization of the above observation. We pick up the 2d CP^{N-1} sigma model as an example, and then the interface should be in the projective representation of PSU(N) symmetry, and there have to be N-fold degenerate ground states. Although this is the immediate consequence of the anomaly matching condition, it is quite nontrivial from the concrete analysis of semiclassical analysis. To make the dilute gas approximation reliable, we take the flavor twisted boundary condition, and then there are N classical vacua related by (fractional) instantons. Usually, instantons lift the classical degeneracy, but the anomaly tells us that instantons should not do the job at all.
We have such a situation when fermion exists due to the spurious axial symmetry, but how we can eliminate the instanton effects at all in purely bosonic systems? This is the question we answered in this paper, and the moduli parameter of instanton plays an essential role here.
with Mithat Ünsal
In our previous paper (2201.06166[hep-th]), Mithat and I developed the novel semiclassical theory to describe the nonperturbative aspects of 4d gauge theories by considering the T2 compactification with magnetic flux. In this paper, we applied this technique to study QCD(Sym/ASym), in which the quark field is in the two-index (anti-)symmetric representation of the SU(N) gauge group.
These theories have been expected to have an interesting relationship with the N=1 super YM theory, called the large-N orientifold equivalence. This is a conjecture that claims that non-supersymmetric gauge theories are equivalent to supersymmetric ones for correlation functions of certain observables, and it is theoretically quite appealing. For this claim to be true at the nonperturbative level, we must show that a certain symmetry (charge conjugation C) is not spontaneously broken, and this is usually a tough question as it requires the knowledge of strong dynamics.
This paper shows that, at least for sufficiently small R2×T2, the discrete chiral symmetry is spontaneously broken for QCD(Sym/Asym), while the C-symmetry is kept intact in those vacua. Assuming that this compactified regime is adiabatically connected to the strong dynamics on R4, our analysis gives strong support for the validity of large-N orientifold equivalence.
This paper was completed during the workshop "Continuous Advances in QCD 2022" at the University of Minnesota and during my stay at North Carolina State University. This was my first visit to the US since the pandemic started, and I really enjoyed both the on-site workshop and my stay at NCSU.
[2204.07440[hep-th]] Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly
with Yui Hayashi
Recently, people have just started to understand non-invertible self-duality symmetry in 4d gauge theories (arXiv:2109.05992, arXiv:2111.01139, and arXiv:2111.01141). Since the electromagnetic duality exchanges the electric and magnetic charges, we cannot naively realize it if there is an imbalance between the electric and magnetic 1-form symmetries. These recent developments overcome this difficulty, by noticing that the duality defects can be constructed by using the half-space gauging procedure.
This progress reminded me that I could not fully understand the electromagnetic duality of 4d U(1) gauge theory when studying the Cardy-Rabinovici model with Masazumi (arXiv:2009.10183). We could construct the self-duality for the partially gauged CR model, which constrains the low-energy dynamics thanks to the mixed-gravitational anomaly. However, it was not clear if there exists such constraints on the original model. So I thought that this would be a good opportunity to revisit this problem. Yui Hayashi, a Ph.D. student at Chiba Univ., just visited me for discussions so I talked to her about this problem and started this collaboration.
We immediately noticed that we can have the ST^p duality by gauging the Z_N 1-form symmetry with the level-p discrete topological term. At the self-dual point, this gives the non-invertible symmetry as the fusion rule becomes non-group-like. Moreover, we find that there exists a mixed gravitational anomaly that rules out the trivially gapped phase.
[2202.00375[hep-th]] Exploring the θ-vacuum structure in the functional renormalization group approach
with Kenji Fukushima and Takuya Shimazaki
In this work, we test the applicability of the functional renormalization group (FRG) to the quantum phase transition that cannot be characterized by local order parameters. As the simplest example, we consider the quantum mechanics on the circle with the θ angle and discuss how to deal with the level-crossing phenomena in FRG. Kenji came up with this problem inspired by Mithat's question at a conference, and he invited me to join this project.
Since the FRG cannot naively treat the topological term because of its differential nature, we embed the S1 quantum mechanics to the R2 quantum mechanics with the double-well potential and take the infinite depth limit after the calculations to recover the original system. Kenji and Takuya found that the simplest approximation (local potential approximation, LPA) works well until the level crossing point, but the FRG flow turns out to diverge for larger values of θ. We did some analytic calculations to demonstrate that the quantum effective action becomes nonlocal at the level-crossing point. This shows that we must have a controllable nonlocal ansatz to study the level crossing with the functional methods.
As another remark, we find that the quantum effective action defined by the Legendre transformation may not have the desired property when the path integral suffers from the sign problem, because of the loss of convexity. The S1 quantum mechanics with the θ angle turns out to deserve the useful test ground in these perspectives.
[2201.06166[hep-th]] Center vortex and confinement in Yang-Mills theory and QCD with anomaly-preserving compactifications
with Mithat Ünsal
In this paper, we constructed a novel semiclassical theory of color confinement for 4d gauge theories by considering the T2 compactification with 't Hooft flux. This is the project that took us more than 2 years, as I remember that Mithat and I chatted about this idea when I was doing a postdoc in North Carolina. Interestingly, we already reached the possibility of adding the baryon-number monopole flux to apply the T2 compactification to QCD with fundamental quarks at that time, but we could not be so confident if such a naive idea works to study the strong-coupling nature of QCD. Then, time flew, and this project was stopped for a while.
The situation has changed since I got an invitation to talk at the conference "Paths to QFT", and I started rethinking this problem by taking this opportunity. I become confident about the vortex-induced confinement by concretely solving 2d Abelian Higgs models in a semiclassical way and the result is consistent with anomaly matching. We understood the 2d theory well, so the next problem became to apply this idea to 4d gauge theories with compactification. When discussing 4d Yang-Mills theory, what annoyed us was the absence of analytic solutions for the center vortex under compactified setup with 't Hooft flux. Still, there was numerical evidence for its existence and important properties were also numerically justified in previous studies, so we decided to use those center vortices in our semiclassical description. The results are remarkably successful, and the weak-coupling analysis with center vortices describes many important features of 4d gauge theories including QCD, which turn out to be consistent with 4d strong dynamics.
Understanding color confinement is one of my biggest goals since I became a graduate student, so I am very pleased that I could write this paper achieving a small step toward it.
[2110.14105[hep-th]] Negative string tension of higher-charge Schwinger model via digital quantum simulation
with Masazumi Honda, Etsuko Itou, and Yuta Kikuchi
We realize the multi-charge Schwinger model on the open interval by a certain spin chain using the Jordan-Winger transformation and check the appearance of the negative string tension, which is one of the most exotic features of (1+1)-dim QFTs with 1-form symmetry. Even though this has been already suggested in the path integral formalism, we constructed the state of the Hilbert space explicitly using the adiabatic state preparation to see this property. Since the Hilbert space is the most natural language of quantum computations, we believe that it becomes important to understand or reinterpret various aspects of QFTs in the Hamiltonian formalism, and this is the first step in that direction.
About the technical aspects of quantum computation, our study is based on the previous work (arxiv:2105.03276) by other collaborators of this paper except me. Since Masazumi is my office mate at Yukawa institute, we chatted about their paper when they were trying to finish it (possibly in March), and I told him that the 1-form symmetry of their system allows the negative string tension. Since it is a quite unusual feature from the physical intuition, I told them that direct observation should be exciting. Then, this collaboration started, but I think that, honestly, telling this story was my only basic contribution to this project.
[2109.10051[cond-mat.str-el]] Sigma-model analysis of SU(3) antiferromagnetic spins on the triangular lattice
with Itsuki Takahashi
This is a paper with my student, Itsuki Takahashi, studying the Neel-VBS transition for the (2+1)d SU(3) spin systems on the triangular lattice. I was interested in extending the sigma model analysis for the SU(3) antiferromagnetic spin chain to the 2d quantum antiferromagnets since I did the related previous work (1805.11423) with Tin Sulejmanpasic. I suggested Itsuki study this system using the nonlinear sigma model, and we derived the 3d SU(3)/U(1)^2 nonlinear sigma model with the emergent relativistic invariance.
To apply this to the Neel-VBS phase transition, it is crucial to study the monopole effective theory when all the electric particles are massive. The possible monopole perturbations are restricted by the destructive interference determined by the Berry phase. We have derived the graphical rule to compute the Berry phase for various tunneling processes, and we show that there is a nontrivial magnetic symmetry even with dynamical monopoles. This leads to the 't Hooft anomaly matching condition, which supports a direct phase transition between Neel and VBS orders.
with Mendel Nguyen and Mithat Ünsal
Motivated by the previous work with Mendel and Mithat (2101.02227[hep-th]), we studied the classification of confining strings in the 2d Yang-Mills theory. That model is exactly solvable, and the string tensions are known to obey the Casimir scaling, i.e. the confining force is proportional to the second Casimir element of the test quark.
This means that, although the center symmetry is finite, there are infinitely many string tensions. In 2d, this is not surprising because the gluons in 2d do not propagate due to the gauge invariance. Still, it is very curious for us if it's merely a dynamical consequence or not: that is, we are wondering whether this can be understood from the symmetry consideration. We have strongly believed that there should be kinematical reasoning but needed to come up with a nice observable and struggled over a month.
Then, one day, Mendel suggested a nice defect operator, and demonstrated that it fits our every criterion! It turns out to be a 2d version of the Gukov-Witten surface operator in 4d gauge theories, and we can now explain the symmetry reasoning behind the exact Casimir scaling for 2d YM theories.
with Ryohei Kobayashi, Yasunori Lee, Ken Shiozaki
Recently, it has been suggested that possible topological terms in QFTs should be regarded as the partition function of SPT phases. According to this proposal, topological terms can be classified by the bordism group with an appropriate structure for the spacetime manifolds. As I have worked on the 2d flag-manifold sigma model, I got interested in its extension to 3d version and wondered what would be the possible topological terms for Lagrangians of such models.
The last December, Yasunori visited the CREST workshop at YITP organized by Ken, and, after some conversations, Ken and I decided to invite Yasunori and Ryohei for more discussions this January. During discussions with them, I learned a lot about spectral sequences, and we finally decided to work on the above problem using those techniques. With the help of these abstract topology techniques, we determined the explicit form of possible effective Lagrangians and discussed the physical consequences of those topological terms with concrete computations.
Above all, I enjoyed working on this project a lot as we could discuss in person getting together at YITP.
[2101.02227[hep-th]] Semi-Abelian gauge theories, non-invertible symmetry, and string tensions beyond N-ality
with Mendel Nguyen, Mithat Ünsal
In this paper, we consider a 3d Abelian lattice gauge theory, whose gauge group is corresponding to the maximal Abelian subgroup of SU(N). The model has the S_N permutation symmetry, corresponding to the Weyl reflections, and thus all the monopoles in the root system equally contribute to the semiclassical analysis, and this has the sharp contrast compared with other Abelianized confining theories such as the Polyakov model. We can explicitly compute the mass gap and string tensions of this model, and we can also perform the gauging of S_N, which makes the gauge group U(1)N-1⋊ SN closer to SU(N).
The original idea of the model and computation of the mass gap generation were already discussed by Mendel and Mithat when I joined the project. When calculating the gauging of S_N, I noticed that the center symmetry is not large enough to explain the spectral properties of string tensions. The center symmetry is given by ℤN , and thus it only explains the N-ality part of sring tensions. However, since it is originated from the pure Abelian theory, there are infinitely many string tensions, which carry detailed information beyond N-ality. We find that non-invertible symmetry is the key to having a natural explanation.
with Masazumi Honda
Since I started working on recent anomaly matching, I was interested in understanding the oblique confinement phase proposed by 't Hooft, which is a very exotic but interesting scenario for the SU(2) Yang-Mills theory at θ=π. In this paper, with Masazumi, we revisit the model proposed by Cardy and Rabinovici from the recent topological aspects and anomaly matching.
The Cardy-Rabinovici model is expected to show the rich phase diagram thanks to the presence of both electric and magnetic particles. We show that this model has the anomaly and/or global inconsistency involving the one-form and CP symmetries, and it is the same anomaly with the SU(N) YM theory. As we can understand the microscopic details of this model, we got very excited that this simple model provides various concrete and nontrivial ways to match the 't Hooft anomaly and the global inconsistency conditions. This study not only justifies the conjectured phase diagram but also uncovers the topological aspect of each phase, including the oblique confinement phase.
It has been thought that the Cardy-Rabinovici model enjoys the SL(2, ℤ) self-duality, which is almost true for the local dynamics. However, we noticed that it cannot be correct for the global nature of the theory if we pay attention that whether the low-energy theories are topological or not. To rescue the original idea, we constructed the SL(2, ℤ) self-dual model by gauging a part of the one-form symmetry of the Cardy-Rabinovici model. We discuss various anomalies, including the mixed anomaly of SL(2, ℤ) and gravity, to constrain the phase diagram of gauged Cardy-Rabinovici model.
[2006.01487[hep-th]] Deconfinement and CP-breaking at θ=π in Yang-Mills theories and a novel phase for SU(2)
with Shi Chen, Kenji Fukushima, Hiromichi Nishimura
In the summer of 2017, when I was a postdoc at RIKEN BNL, Kenji visited BNL for a while. During his stay, Rob Pisarski told him about a recent discovery about the θ angle physics by anomaly matching (Gaiotto et al. 1703.00501). Since I started working on related subjects, I explained about their paper, and Kenji asked if we can understand it by Polyakov-loop models.
An important point of this development was the discovery of a new anomaly, so we should work on the model which has the same property. We have reached the work on the semiclassical confinement-deconfinement transition for super Yang-Mills theories (Poppitz et al. 1212.1238), and Kenji and I started reading their work inviting Hiromichi. The anomaly tells Tdec≤ TCP , but we wanted to know if the deconfinement and CP restoration occurs at the same time or not by understanding microscopic dynamics. It took me a while to comprehend Poppitz et al., but it is suggesting that Tdec =TCP , and I told my collaborators that nothing exotic seems to happen.
After a while, Kenji gave this problem to his student, Shi Chen, and, very surprisingly, he told that he found an exotic phase diagram for SU(2) super Yang-Mills theory. Indeed, the previous study has shown that the deconfinement transition is of the 1st order for all the simple gauge groups except SU(2), and in that case, the exotic thing does not happen. But, this was not the whole story. For SU(2), the deconfinement transition is of the 2nd order, and the first order CP-broken line goes across this 2nd-order deconfinement curve. So, there is a finite window of the temperatures for the novel deconfined, CP-broken phase.
[2005.13822[hep-th]] Finite-density massless two-color QCD at isospin Roberge-Weiss point and 't Hooft anomaly
with Takuya Furusawa, Etsuko Itou
We studied how the phase diagram of finite-density two-color QCD can be constrained by 't Hooft anomaly. This is along the line of some of my previous studies (1710.08923 and 1711.10487), and it was pointed out by Itou-san that two-color QCD has a chance to show very different behaviors compared with the usual QCD. Moreover, two-color QCD can be tested by numerical lattice simulation even at finite densities, so it is worthwhile to think about the anomaly constraint on the two-color QCD phase diagram in detail.
Takuya immediately computed the discrete anomaly with full generality for this purpose, and we discussed its physical meanings. It gives a meaningful constraint on the phase diagram with imaginary isospin chemical potentials, but I am surprised as the way to match the anomaly turns to be very different from that of 3-flavor 3-color QCD. What is more interesting to me personally is that the anomaly suggests the similarity between two-color QCD and (2+1)d anti-ferromagnetic systems. It's very curious if this similarity on symmetry and anomaly extends to that of dynamics, too, or not.
[2004.10328[hep-th]] Universality between vector-like and chiral quiver gauge theories: Anomalies and domain walls
with Tin Sulejmanpasic, Mithat Ünsal
Just a few days before my leave from North Carolina, Mithat asked me if we can understand the dynamics of chiral quiver gauge theories. Chiral quiver theory is a cousin of QCD coupled to flavor gauge fields (QCD with bi-fundamental quarks called QCD(BF)). In the large-N limit, there is an interesting perturbative equivalence to the super Yang-Mills theory. It is also known that this perturbative planar equivalence does not extend nonperturbatively, so it's an interesting problem to identify the unknown ground-state structures.
Because of the chiral nature, we do not have many techniques to understand these theories. What we tried first is to identify the global symmetry correctly including the discrete factors, because this is important to understand recent 't Hooft anomalies for these systems: Understanding such constraints by symmetry and anomaly, we proposed a reasonable guess on the possible dynamics.
Based on these analyses, we argue that chiral symmetry should be spontaneously broken. We, however, wanted to obtain further information beyond these kinematical constraints. For this purpose, putting the theory on a small cylinder after some deformations, we confirmed these conjectural dynamics by an explicit computation and find a consistent picture. Interestingly, their dynamics turn out to be analogous to those of vector-like gauge theories: For odd-site quiver theories, symmetry realizations are the same with super Yang-Mills, and for even-site quiver theories, they are the same with QCD(BF). We also conjecture that these ground-state structures are continuously connected, in other words, these chiral quiver models provide technicolor-like realizations for vector-like theories, super YM, and QCD(BF), under a suitable setup.
with Mithat Ünsal
One day in November, I asked Mithat if the instanton sectors are really necessary for resolving the U(1)_A problem. We have indeed worked on a related issue in the 2d case, higher-charge 2d U(1) gauge theory (1905.05781), so motivated by those studies, Mithat and I were wondering if a similar thing occurs in 4d gauge theory. In order to study it, we restrict the possible instanton sectors to multiples of a certain integer p for SU(N) Yang-Mills theory and quantum chromodynamics (QCD).
To make such restrictions consistent with locality, we have to introduce the topological degrees of freedom to the theory. As a consequence, the theory acquires the 3-form symmetry, although the local dynamics are completely identical with the usual theory. Quite often, this 3-form symmetry has a mixed anomaly, and it leads to the new selection rule prohibiting the domain-wall excitation or the false-vacuum decay. We are enjoying a lot with this stronger superselection rule.
with Tatsuhiro Misumi
Around the time I started the research on anomaly matching, Tatsu sent me an email suggesting many new possibilities of its applications. The phase structure of lattice Wilson fermion was one of the topics, and we decided to come back to this problem almost after two years!
2d lattice Gross-Neveu model with Wilson fermion is claimed to have the Aoki phase depending on its Wilson parameter, which breaks parity spontaneously by pseudo-scalar condensate. In the mean-field approximation, there is the central cusp inside the Aoki phase, at which there are always two vacua for any four-fermion GN couplings. Wilson fermion with this choice of Wilson parameter is named the central-branch Wilson fermion, and it has been shown that it has extra onsite U(1) symmetry by Tatsu and his collaborators, 1101.4239 and 1111.0402.
In this paper, we find (1) the central-branch Wilson fermion has no sign problem, so MC numerical simulation is possible, and (2) the extra U(1) symmetry has the mixed anomaly with lattice translation and 90-degree lattice rotation. This triple mixed anomaly is the same as the LSM anomaly of the XXZ spin chain, which has a close connection with the Haldane conjecture. This also explains why there is the central cusp in the Aoki phase.
with Aleksey Cherman, Theodore Jacobson, Mithat Ünsal
After finishing the paper about the Schwinger model, I got more interested in other 2d gauge theories. Aleksey, Theo, and Mithat were already working on 2d adjoint QCD at that time, and I started to join the collaboration. This theory has a history of almost 25 years, and many claims about it sound being mysterious (at least to us).
We decided to look at what symmetry tells us. This is a system of 1+1d interacting Majorana fermions, which can belong to an interesting nontrivial class of SPT phases if we turn on a negative mass. I adapt the mod 2 index theorem in a suitable form for our purpose, and it enables us to find many mixed 't Hooft anomalies. These anomalies, fortunately, tell us many new things about the confinement and chiral symmetry breaking of this model. Moreover, semiclassical analysis on small cylinders gives a consistent result to those findings.
Unfortunately, however, these new things are inconsistent with many previous studies, so we had to check carefully how these inconsistencies appear. After those surveys, the new technique, mod 2 index theorem, turns out to play an important role to understand the dynamics of 2d adjoint QCD. In our understandings, the previous studies without it failed to capture the fully non-Abelian dynamics.
Addendum: About a year later, an interesting paper appears, by Komargodski, Ohmori, Roumpedakis, Seifnashri (2008.07567). There, it has been reported that, at the superrenormalizable points, the 2d adjoint QCD enjoys many non-invertible topological lines. As a result, there are exponentially many vacua for this theory. Therefore, when we include generic four-fermion couplings which do not break any 0-form symmetries, our suggestion is correct. But, when those four-fermion couplings satisfy certain constraints, all the Wilson loops are deconfined as expected in the '90s but with completely different reasonings, and there are much more deconfined line operators as a result of non-invertible symmetry.
[1905.05781[hep-th]] Fractional θ angle, 't Hooft anomaly, and quantum instanton in charge-q multi-flavor Schwinger model
with Tatsuhiro Misumi and Mithat Ünsal
The Schwinger model is 2-dim QED with Dirac fermion. This quantum gauge theory can be solved explicitly, while it has many interesting features similar to those of 4-dim QCD. Therefore, it has been used as a toy model to understand complicated phenomena in 4-dim non-Abelian gauge theories.
In my previous paper (1812.02259), the flag sigma model was studied on the cylinder with a twisted boundary condition, and this is believed to have a connection with SU(3) WZW model with the same twist. After this paper appears, Mithat suggested to Tatsu and me that we should study the WZW model itself and its fermionized theory. By current algebra, it was known that the N-flavor Schwinger model shows SU(N) level-1 WZW conformal behavior at low energies, and thus we started to collaborate on the charge-q N-flavor Schwinger model.
This model turns out to have interesting features: Depending on the boundary conditions, the theory on the cylinder has different 't Hooft anomalies. We confirm this explicitly by semiclassical analysis both in operator and path-integral formalisms. Especially, we cannot explain the behavior of chiral condensate with the usual instanton, and we have to introduce quantum instanton. This is a funny object that saturates the BPS bound between classical Lagrangian and quantum-induced potential.
with Yuji Hirono
This is the follow-up paper of our previous paper (1811.10608[hep-th]). In the previous paper, we found it useful to consider the generalized BF theory with non-square charge matrix K when discussing the possibility of topological order under the existence of superfluidity.
In this paper, Yuji and I analyzed the generalized BF theory in details, and discussed its structure in view of higher-form symmetry. We generally identify the generators of discrete and continuous higher-form symmetries, and give the criteria when the symmetry is spontaneously broken. We revisit the analysis of CFL phase in high-density QCD in this framework and no topological order appears there. We also discuss an example of superfluidity with topological order within our framework.
with Hiromichi Nishimura
In this paper, we discussed how the high-temperature phase of QCD can be nontrivial as symmetry-protected topological (SPT) phases.
QCD with imaginary chemical potential has the Roberge-Weiss (RW) periodicity as a consequence of gauge invariance, and there exists the RW phase transition in the high-temperature phase to be consistent with the RW periodicity. In 1706.06104, it was found that this first-order phase transition is a consequence of parity-anomaly matching of 3d field theory when quark masses are set zero, and the RW parity must be spontaneously broken when chiral symmetry is respected.
In this setup, we can consider the domain wall connecting those two pure states. We find that the domain wall supports 2d U(N-1) gauge theory with massless Dirac fermions, and this theory has an 't Hooft anomaly that cancels the anomaly inflow from the bulk. This allows us to interpret the RW phase transition as a quantum phase transition between two different SPT phases protected by chiral symmetry. We also find that the same observation applies for high-temperature phases of Z(N)-QCD.
with Masaru Hongo, Tatsuhiro Misumi
Previously, in 1805.11423, I worked on the two-dimensional flag sigma model to generalize the Haldane conjecture to SU(N) spin chains with Tin Sulejmanpasic. I wanted to check the conjectured phase diagram obtained by anomaly and global inconsistency. Then, Masaru, Tatsuhiro and I worked together on this project by using reliable semiclassical analysis with twisted boundary condition.
Interestingly, the knowledge of fractional instanton on CP^N sigma model becomes quite useful to construct the BPS solutions of the SU(3)/U(1)^2 flag sigma model. By employing the symmetry-twisted boundary condition, the dilute instanton gas approximation (DIGA) gives the reliable answer without suffering from the infrared divergences. We found that the interesting phase structure conjectured by anomaly and global inconsistency can be concretely understood thanks to DIGA.
We also gave some speculations about what happens beyond DIGA, by estimating the imaginary ambiguity about the complex bion amplitude and assuming the structure of resurgence theory. The size of imaginary ambiguity shows the difference from the one expected from two-dimensional renormalon diagrams. This difference appears for 4d adjoint QCD but does not occur for 2d CP^N model. In this respect, our model might be more similar to adjoint QCD compared with CP^N model.
with Yuji Hirono
In this November, I had a chance to visit APCTP in Korea for one week invited by Yuji Hirono. We decided to work very intensely on topological nature of color-flavor locking (CFL), one of cold dense QCD matters, during this visit.
The work is motivated by a recent work by Cherman, Sen, and Yaffe, which finds the anyonic statistics between test quarks and superfluid CFL vortex. Conventional wisdom in QCD says that hadronic superfluid and quark superfluid can be connected by smooth crossover, according to Landau's classification of phases of matter. This new finding questions this continuity when looking at quantum nature of QCD, and poses a new interesting puzzle in QCD physics. Also, I had a chance to talk with Aleksey one-week before the visit, so I got a nice motivation to work on this puzzle.
In this work, we established the consistency of quark-hadron continuity as quantum matters and solved this puzzle, so let us call it quark-hadron continuity beyond Ginzburg-Landau paradigm. We have derived the effective field theory that can compute superfluid phonon, vortex, and above anyonic statistics. We find that the anyonic statistics is a consequence of emergent two-form symmetry, generated by color Wilson loop. The logarithmic confinement of vortices suggest that this emergent symmetry is unbroken, showing that hadronic and quark superfluids have the same symmetry breaking including generalized global symmetry.
[1807.07666[hep-th]] Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking
When I gave a seminar at Toronto University about anomaly matching, Erich pointed out the possibility of application of anomaly to exotic scenarios of chiral symmetry breaking in QCD. This paper is motivated by the discussion at that time, and we exclude the Stern phase based on symmetry and the discrete 't Hooft anomaly of massless QCD.
Stern phase is the chiral-symmetry broken phase with the ordinary pattern of continuous chiral symmetry breaking, but the discrete axial symmetry remains unbroken there. We prove that the baryon current is anomalously broken for massless QCD if we introduce the background gauge fields for discrete axial symmetry and the projective vector-like flavor symmetry. In the ordinary chiral broken phase, the 3-form Wess-Zumino current (Skyrmion current) reproduces this anomalous breaking, and this result is very interesting to me that the Skyrmions play a pivotal role for anomaly matching. In the Stern phase, however, the anomalous violation of the Skyrmion current takes a different form, and the anomaly matching is not satisfied. We, therefore, rule out the Stern phase from possible QCD vacua.
The Stern phase was previously excluded from the zero-density QCD phases using the QCD inequality, and our no-go theorem is consistent with that previous result. Moreover, our result extends its applicability to the finite-density zero-temperature QCD.
[1805.11423[cond-mat.str-el]] Anomaly and global inconsistency matching: θ-angles, SU(3)/U(1)^2 nonlinear sigma model, SU(3) chains and its generalizations
with Tin Sulejmanpasic
In a recent paper (1706.06598), Lajko, Wamer, Mila, and Affleck has shown that the SU(3)/[U(1)*U(1)] nonlinear sigma model describes 1D SU(3) spin chains, and it generalizes the Haldane conjecture about SU(2) spin chains. While discussing the application of anomaly matching to spin systems, Tin told me about this paper. The system shows a very interesting nature, and we, therefore, want to revisit this model from the viewpoint of recent developments of 't Hooft anomaly matching.
We first rediscover the Lieb-Schultz-Mattis theorem for projective spin rotation and lattice translation as a mixed 't Hooft anomaly. In addition, we find the global inconsistency (or secondary anomaly) for projective spin rotation and charge conjugation. Therefore, we can obtain a stronger consequence about the phase diagram than before by applying not only anomaly matching but also global inconsistency matching, and we argue that the various phase transitions must exist in the space of θ-angles. This gives the new constraints on the phase diagram that goes beyond LSM, so it's very interesting.
At the special theta angle, the model is believed to show the scale-invariant behavior. It is therefore interesting to ask which CFT can match the anomaly. To investigate it, we consider the Wess-Zumino-Witten model and compute its anomaly. We find the correspondence of anomalies between those models, so we show that the anomaly matching can be satisfied by the level 1 (or -1) SU(3) Wess-Zumino-Witten model.
with Gerald Dunne and Mithat Ünsal
When I found the construction of persistent anomaly under S^1 compactification with the symmetry-twisted boundary condition (1710.08923), I was wondering its meaning from the viewpoint of physical states of QFT since the construction was quite formal. With Gerald and Mithat, we discuss the role of the symmetry-twisted boundary condition at the KITP workshop to revisit the volume independence via new perspectives.
In this paper, we emphasize that the choice of the symmetry-twisted boundary condition does not change the theory. Our interpretation is that it is a procedure that makes the ground-state contribution dominant to the partition function, which cannot happen for the usual thermal partition function, and we name this the Quantum Distillation. Since the 't Hooft anomaly constrains the property of ground states by anomaly matching, it is quite reasonable for the persistence of anomaly under S^1 compactification when the symmetry-twisted boundary condition distills the ground state.
with Tin Sulejmanpasic
At the KITP workshop about resurgence theory, Tin explained to me his ongoing work about spin systems, and it is very interesting. I was wondering whether the gapless phase of the spin-1/2 antiferromagnetic chain is related to Kramers doublet, which is the consequence of time-reversal symmetry. Such gapless phase is conventionally shown by the LSM theorem as a consequence of the projective representation of SO(3) spin symmetry on each site, and the rough idea here is to use the projective nature of the time-reversal symmetry instead. Then, Tin showed me that the topological charge of the O(3) sigma model becomes half-integers after gauging both parity P and time-reversal T. This implies that there is a mixed 't Hooft anomaly between C, P, and T.
We did the various consistency check of its implications from exactly solvable models and semiclassical analysis, and everything is cleared so far. It is quite amusing and also surprising that the trivial mass gap is ruled out for spin-1/2 chains without any subgroup of SO(3) spin symmetry. For further nontrivial checks in the future, we propose the spin-1/2 chain which completely breaks SO(3) while keeping C, P, and T, which we name the skew-XYZ spin chain.
[1801.10283[cond-mat.mes-hall]] Giant photocurrent in asymmetric Weyl semimetals from the helical magnetic effect
with Dmitri E. Kharzeev, Yuta Kikuchi, Rene Meyer
One day, Dima suggested to us that there may be a photocurrent induced similarly with the Chiral Magnetic Effect (CME). Using the chiral kinetic theory, we can immediately find that such current vanishes for usual Weyl semimetals, but it also becomes evident that such current exists when both the particle-hole and parity symmetries are explicitly broken. Since the current is induced thanks to helicity imbalance, we call it the Helical Magnetic Effect (HME). What is more interesting is that Yuta found the HME current becomes quite huge compared with the ordinary photocurrent in the infrared region. Therefore, we evaluate its magnitude based on the chiral kinetic theory and compute its frequency dependence.
with Yuta Kikuchi, Tatsuhiro Misumi & Norisuke Sakai
This paper is written for the application of the technique developed in the previous paper to study the phase structure of massless ℤN-QCD. Massless ℤN-QCD is defined as SU(N) Yang-Mills theory with N-flavor massless Dirac fermions with ℤN flavor twisted boundary condition, and it turns out to have a beautiful structure from the topological viewpoint. It has an 't Hooft anomaly at any circle compactification radius and any quark chemical potentials, and the phase diagram is constrained by the anomaly matching condition.
Application of anomaly matching to ℤN-QCD was a strong motivation for Tatsuhiro and me to study recent developments of 't Hooft anomalies involving two-form gauge fields, and I am very satisfied. Asking Yuta to join this collaboration, we carefully check possible anomalies and the implications of anomaly matching to ℤN-QCD.
with Tatsuhiro Misumi & Norisuke Sakai
We develop a systematic procedure that derives an 't Hooft anomaly under circle compactifications (or, at finite temperatures) starting from that of the original uncompactified theory (i.e., at the zero temperature). Since an 't Hooft anomaly is the obstruction of gauging symmetries coming out of topology related to symmetry, that information is usually eliminated by thermal fluctuations at finite temperatures. We find that, under a certain twisted boundary condition, this difficulty does not appear and the phase structure of the circle-compactified theory obeys the constraint by 't Hooft anomaly matching. This applies to the ℤN-twisted ℂℙN-1 model and also to massless ℤN-QCD. We discuss the consistency of our computations with previous results and consider possible implications on the phase diagram of massless ℤN-QCD. Although a special limit or idealization is taken, we have a new rigorous result on the fundamental theory of strong interaction (QCD).
This work is started through the discussion with Tatsuhiro about QCD with flavor-twisted boundary condition (Z(N)-QCD) at the Resurgence conference in Kobe, Japan. Indeed, my original motivation to study bifundamental QCD via anomaly is to understand ℤN-QCD from the viewpoint of recent developments on the anomaly, and this was a nice opportunity for me to understand the properties of the system very deeply. Through the discussion on the resurgence of the ℤN-twisted ℂℙN-1 model with Sakai-san in the workshop at KITP, I got the idea of the general structure of anomaly under circle compactifications.
[1708.01962[hep-th]] Global inconsistency, 't Hooft anomaly, and level crossing in quantum mechanics
with Yuta Kikuchi
An 't Hooft anomaly is a useful tool to constrain possible low-energy dynamics of strongly-coupled quantum field theories. This paper focuses on the global inconsistency condition, which is similar to the 't Hooft anomaly but is applicable by comparing different high symmetry points even when the 't Hooft anomaly is absent. The global inconsistency condition is proposed in a recent paper, and the outcome from it needs detailed studies. Yuta and I clarified, by explicit computation of several quantum systems, that the global inconsistency may be satisfied if either of the high symmetry points has nontrivial ground states or those high symmetry points are separated by phase transition.
with Hiromichi Nishimura & Jacobus J. M. Verbaarschot
The Lefschetz-thimble approach to the sign problem uses a gradient flow in the complexified field configurations to find appropriate stationary-phase cycles. We noticed that the gradient flow generically blows up in a finite time, so one must control its behavior very carefully in the numerical computation. Instead of paying attention to such blow-ups, we propose a new gradient flow that equally defines Lefschetz thimbles and does not suffer from blow-ups. We give its theoretical foundations, and also checked its properties numerically in simple examples.
with Yuta Kikuchi
Recently, a new topological technique has been developed to put a rigorous constraint on the possible dynamics in the nonperturbative gauge theories at finite theta angles. Yuta and I get interested in this subject, and we decided to apply that technique to study bifundamental gauge theories. Especially, SU(n)×SU(n) Yang-Mills theory with one bifundamental Dirac fermion (bifundamental QCD) is an interesting theory, since it may be equivalent to N=1 supersymmetric Yang-Mills theory through the orbifold equivalence in the large-n limit, and their nonperturbative equivalence is an important and interesting question. Our constraint itself does not answer this question, but it gives a strong constraint on the phase diagram of this theory so we hope that it becomes useful in future studies.
We have shown the possible phase boundaries at finite topological angles, and understand the result from the viewpoint of the monopole condensation scenario of confinement. It is quite surprising for me that we can put a rigorous and nonperturbative constraint on the first-order phase transition lines without computing microscopic details of the dynamics.
[1703.02040[cond-mat.mes-hall]] Topological Properties of the Chiral Magnetic Effect in Multi-Weyl Semimetals
with Yuta Kikuchi & Tomoya Hayata
Multi-Weyl semimetals are semimetals described by two-component fermions with non-linear dispersion, and they have the monopole charge greater than 1. We elucidate how their topological property appears in static and dynamic chiral magnetic effect. We also propose an experimental measurement to confirm the multiple monopole charge.
This work was initiated by Yuta and Tomoya, and Yuta explained this work to me on the blackboard when I visited his office at Stony Brook. I realized that what they want to compute can be readily obtained and generalized by using differential forms. This mathematical simplification was quite helpful for us to realize that the trace part of dynamic chiral magnetic conductivity is always topologically determined, and the proposal for the experimental setup is based on this observation.
with Motoi Tachibana
We study the phase diagram of multi-flavor massless Schwinger model by using the mean-field calculation with the complex saddle points. Multi-flavor massless Schwinger model suffers from the sign problem in general, but we prove that the deformation of path-integration cycles into Lefschetz thimbles weaken the sign problem sufficiently. After this deformation, we show that the saddle-point approximation becomes exact in the zero-temperature limit, and we indeed checked that the exact result obtained by Lohmayer and Narayanan (2013) is correctly reproduced. Our calculation shows that the Lefschetz-thimble Monte Carlo simulation solves the exponential complexity of multi-flavor massless QED2 at finite densities, and this is highly nontrivial.
Motoi asked me whether this method is applicable to analytic studies of two-dimensional gauge theories when I visited Saga University, and then we looked for the simplest case and started this collaboration.
with Can Kozçaz, Tin Sulejmanpasic & Mithat Ünsal
There is a lot of attention to the asymptotic nature of the perturbation theory under the name of resurgence theory. In the case of quantum mechanics, the asymptotic nature is caused by the instanton–anti-instanton correlated event or the complex bion solution. In this study, we consider the impact of resurgence to the quasi-exact-solvability (QES), which is also known as the non-linear/multifold supersymmetry. In the QES system, the perturbation theory for first several states converges to give the exact result despite the fact that there exist instanton-like solutions (real bions) giving nonperturbative corrections.
We showed that the complex bion describes a new nonperturbative contribution of QES systems, by looking at the asymptotic perturbative series after a tiny deformation of QES. We name this phenomenon as the Cheshire Cat Effect of Resurgence, which plays the key role to understand the absence of nonperturbative corrections in QES: Complex-bion contributions exactly cancel the nonperturbative correction from real bions. Furthermore, we extend this result to a “pseudo-QES” system that suffers from the nonperturbative correction to the convergent perturbative solution.
We started this collaboration when we gathered at Harvard University in the last November. Mithat introduced relevant works to us there and brought my interest to the resurgence theory, which has a close relationship with the Lefschetz-thimble method of path integral.
with Tomoya Hayata & Yoshimasa Hidaka
In this paper, we explicitly show that the complex Langevin method is NOT consistent with complex semiclassical analysis in generic cases by relating the complex Langevin method with the Lefschetz-thimble integration. Unfortunately, we must conclude that the original complex Langevin method cannot tackle the high-density nuclear matter because of this limitation. We propose a modified complex Langevin method as a working hypothesis in order to evade the semiclassical inconsistency. Although this modification introduces a sign problem again, this is weaker than the original one. I guess this would be the weakest sign problem so long as using the coherent-state expression.
In order to check its usefullness, we apply it to the one-site Hubbard model. When we apply the naive complex Langevin method to the path-integral expression of this toy model, it becomes consistent with a kind of phase-quenched approximations as we expected, and the complex Langevin method gives an incorrect result. The modified one gives a significant improvement and shows the non-analytic behavior of this model.
with Yoshimasa Hidaka & Tomoya Hayata
Lefschetz-thimble approach is recently getting much attention, and its properties in practical applications need to be understood in more detail. In this paper, the one-site Hubbard model is studied using the Lefschetz-thimble integration method after the Hubbard-Stratonovich transformation. This toy model has a "quantum phase transition" as the chemical potential increases, and its partition function has the sign problem in the path integral expression. Therefore, this is a good playground of theoretical approaches to the sign problem.
One day, Tomoya and Yoshimasa asked me whether this "phase transition" of the toy model can be nicely explained using the Lefschetz-thimble approach to the path integral. (Original question was about the Bose Hubbard model, but we solved the Fermi Hubbard model in this paper because of some inessential complexity, instead.) The computation showed that interference among Lefschetz thimbles is of great significance in order to explain such behaviors, and this fact surprised us a lot. This model has a remarkable similarity with the finite-density QCD, and we expect that a similar thing would happen also there.
[1507.07351[hep-th]] Hamilton dynamics for the Lefschetz thimble integration akin to the complex Langevin method
with Kenji Fukushima
Original motivation of this study is to explore the possible connection between two seemingly-similar approaches, complex Langevin method and Lefschetz-thimble integral, to the sign problem. Kenji questioned the difference of dimensions between complex Langevin distribution and Lefschetz thimbles.
It has been relatively less known that the Lefschetz thimbles can be obtained as ground states of N=2 supersymmetric quantum mechanics, although it was already pointed out in the original paper by Witten. Each ground state of N=2 quantum mechanics corresponds to the Lefschetz thimble, and the Q-exact deformation of the action makes the Lefschetz thimble broadened. This interesting reinterpretation turn out to provide another computational tool for Lefschetz thimbles, and it is studied numerically through zero-dimensional interacting model. The Hamiltonian looks quite similar to the Fokker–Planck operator of the complex Langevin equation and the same computational technique can be used to find the ground state, although there are still many differences in analytical properties if one compares these two formalisms carefully.
Kenji made a nice cartoon to show the Fokker–Planck equation converges to the two-component ground-state wave functions very intuitively; convergence of first-component of the wave function and the second-component.
[1504.07268[hep-ph]] From quantum to classical dynamics: The relativistic O(N) model in the framework of the real-time functional renormalization group
with David Mesterházy & Jan H. Stockemer
By using nonperturbative RG to the Schwinger-Keldysh formalism of real-time dynamics, the model A dynamics and the microscopic theory are shown to be smoothly connected. We derive the RG equation within the local interaction approximation, and the nontrivial IR fixed point is discussed within this approximation. It elaborates how the dimensionally-reduced classical system emerges for static behaviors at the second-order phase transition, and allows us to address dynamical phenomena. We also discuss discrepancies between our result and the result obtained from Langevin dynamics, and it will tell a missing part of our approximation in order to complete our microscopic description of classical Langevin dynamics in future studies.
David was a postdoc at University of Illinois at Chicago when I visited there, and we frequently talked about the related topic during my visit. David is quite familiar with the real-time formalism of quantum field theories and its nonperturbative RG, and he led this collaboration powerfully.
[1504.02979[hep-th]] Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral
with Hiromichi Nishimura & Kouji Kashiwa
We recently found a systematic strategy to solve the fermion sign problem appearing in the mean field approximation. The Lefschetz-thimble method is shown to manifestly respect the reflection symmetry, so that physical quantities take real values, even when the dominant configuration accumulates around the complex saddle point. This method can be applied to broad physical systems, ranging from the Hubbard model to effective theories of dense QCD.
Kouji and I discussed this possible application of Lefschetz thimbles, and this collaboration started when Hiromich visited RIKEN for introducing his recent research in the last December. It was also very lucky for me to have a chance of discussion with Yoshimasa Hidaka and Rob Pisarski, which deepened my comprehension on the sign problem.
with Takuya Kanazawa
Takuya is conversant with chiral symmetry, and he indicated me new possible connections of Lefschetz-thimble methods with other techniques used in random matrix models. Especially, the close relation between Lefschetz-thimble decomposition and Lee-Yang zeros is a big finding in this research. In this paper, we concentrate on simple models including fermionic degrees of freedom, and structure of Lefschetz thimbles are scrutinized carefully in order to demonstrate those new insights. In the part of QCD-like chiral symmetric models, techniques developed in the paper below are employed.
[1412.1891[hep-th]] Lefschetz-thimble techniques for path integral of zero-dimensional O(n) sigma models
I had a chance to discuss Lefschetz-thimble methods with Takuya Kanazawa, and it stimulated me a lot and brought my interest to its application for systems with continuous symmetries: I would like to understand general features of the gradient flow under the small symmetry-breaking perturbations, and how they affect the symmetry-breaking phenomena. In this paper, I consider a simple model as a first step, and show importance of the slow motion induced by the perturbation.
The physically important question, "How it affects SSB", is not yet solved, so I would like to find its solution, by proceeding research further.
[1406.2386[math-ph]] Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling
with Takayuki Koike
In this paper, we discuss properties of real-time path integrals from the viewpoint of Picard–Lefschetz theory. Recently, this technique starts to gather much attention in the context of resurgence trans-series for perturbations in QFT, and also of the sign problem in numerical Monte Carlo simulation. I invited my mathematician friend, Takayuki, to start this collaborating work, and asked to tell me its basic mathematical structure. We applied it to the 'real-time' description of quantum tunneling in the double-well potential, and discuss how it can be useful and what is the problem to be solved for practical computations.
with Kouji Kashiwa
Since I had a chance to visit RBRC for two months, I asked Kouji to start a new collaboration. First we tried to find a new phenomenon without introducing Polyakov loop background, but it was difficult because there are already many previous studies on this topic. When Polyakov loop was introduced, we found that many metastable states appear, and they were not discussed in previous studies. It must be interesting since they could affect dynamics of phase transitions of gauge-Higgs unification models.
[1402.0283[cond-mat.quant-gas]] Multi-Regulator Functional Renormalization Group for Many-Fermion Systems
with Tetsuo Hatsuda
Since we succeeded to describe the BCS theory and BEC of composite particles using fermionic FRG, we tried to combine those analyses to describe the whole region of the BCS-BEC crossover. This requires to control both the fermionic one-particle excitations and the bosonic composite-particle excitations without introducing the bosonic auxiliary fields. For this purpose, we constructed the fermionic functional renormalization group with multiple regulators, which we call Multi-Regulator FRG. The Nozieres–Schmitt-Rink theory turns out to be derived as a leading-order approximation in this theoretical formalism. We hope to apply it to strongly-coupled systems, such as nuclear matters, in future.
[1311.4157[cond-mat.quant-gas]] Fermionic functional renormalization group approach to Bose-Einstein Condensation of dimers
This is the second paper including a main topic in master's thesis. BEC of composite particles is described in the language of the original field of elementary particles. For that purpose, a new kind of IR regulators, which are called vertex IR regulators, is first introduced in this paper.
In order to describe composite-particle excitations using the language of elementary fields, non-local terms emerge in the effective Lagrangian, and they play an important role to describe BEC. Such non-locality is difficult to control, but the vertex IR regulator turns out to provide a nice way to treat it in the deep BEC limit.
[1310.5800[cond-mat.quant-gas]] Fermionic functional renormalization group approach to superfluid phase transition
with Gergely Fejős & Tetsuo Hatsuda
This is the first paper including a main topic in my master's thesis. For an unbiased study of interacting fermions, superfluid phase transition is discussed within "fermionic" FRG. After reviewing its connection to the BCS theory, we reproduce the Gorkov & Melik-Barkhudarov correction and discuss effects of the fermion self-energy correction to the critical temperatures and chemical potentials.
[1304.3286[cond-mat.quant-gas]] Flow equation of functional renormalization group for three-body scattering problems
When writing master's thesis, I noticed inconsistency between the conventional flow equation of FRG and the Dyson-Schwinger equation in few-body physics. The inconsistency is solved by analyzing the flow equation using all order perturbation theory. The flow equation proposed in this paper is closed, but loses its original one-loop property of Wetterich's equation in representing the "feedback" term in a closed form.