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We study $SU(2)$ Chern-Simons theories at level $k$ coupled to a scalar on
$T^2\times \mathbb{R}$ at large baryon number. We find a homogeneous but
anisotropic ground state configuration for any values of $k$ on the IR
fixed-point of those models. This classical analysis is valid as long as we
take the baryon number large. As a corollary, the result negates the na\"ive
folklore that Chern-Simons-matter theories reduce to ungauged theories
projected onto the singlet sector, on the torus spatial slice. This system is
also an interesting example where persistent currents are present in the ground
state, without possibly violating the Bloch's theorem. This paper will be one
primitive step towards quantitative analysis of Chern-Simons-matter dualities
using the large charge expansion.

DOI： 10.1007/JHEP10(2021)245

arXiv

Authorship：Lead author,　Last author,　Corresponding author   Language：English  

As a sequel to previous work, we extend the study of the ground state
configuration of the $D=3$, Wilson-Fisher conformal $O(4)$ model. In this work,
we prove that for generic ratios of two charge densities, $\rho_1/\rho_2$, the
ground-state configuration is inhomogeneous and that the inhomogeneity
expresses itself towards longer spatial periods. This is the direct extension
of the similar statements we previously made for $\rho_1/\rho_2\ll 1$. We also
compute, at fixed set of charges, $\rho_1,\, \rho_2$, the ground state energy
and the two-point function(s) associated with this inhomogeneous configuration
on the torus. The ground state energy was found to scale
$(\rho_1+\rho_2)^{3/2}$, as dictated by dimensional analysis and similarly to
the case of the $O(2)$ model.
Unlike the case of the $O(2)$ model, the ground also strongly violates
cluster decomposition in the large-volume, fixed-density limit, with a
two-point function that is negative definite at antipodal points of the torus
at leading order at large charge.

DOI： 10.1007/JHEP08(2021)079

arXiv

Authorship：Lead author,　Last author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

We study the ratio of pairs of adjacent correlators of Coulomb-branch
operators in $SU(2)$ $\mathcal{N}=2$ SQCD with four flavors within the
framework of the Large Quantum Number Expansion. Capitalizing on the
order-by-order S-duality invariance of the large-R-charge expansion we compute
ab initio the dependence of the leading large-$\mathcal{J}$ behavior on the
marginal coupling $\tau$ and we find excellent agreement with numerical
estimates from localization.

DOI： 10.1007/JHEP04(2021)287

arXiv

Authorship：Lead author,　Last author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

We compute the lowest operator dimension $\Delta(J;D)$ at large global charge
$J$ in the $O(2)$ Wilson-Fisher model in $D=4-\epsilon$ dimensions, to leading
order in both $1/J$ and $\epsilon$. The final result for $\Delta(J;D)$ in the
(resummed) $\epsilon$-expansion, valid when $J\gg 1/\epsilon \gg 1$, turns out
to be \begin{equation*}
\Delta(J;D)=\left[\frac{2(D-1)}{3(D-2)}\left(\frac{9(D-2)\pi}{5D}\right)^{\frac{D}{2(D-1) } }\left[\frac{5\Gamma\left(\frac{D}{2}\right)}{24\pi^2}\right]^{\frac{1}{D-1 } }
\epsilon^{\frac{D-2}{2(D-1) } }\right]\times
J^{\frac{D}{D-1 } }+O\left(J^{\frac{D-2}{D-1 } }\right) \end{equation*} where
next-to-leading order onwards were not computed here due to technical
cumbersomeness, despite there are no fundamental difficulties. We also compare
the result at $\epsilon=1$, \begin{equation*} \Delta(J)=0.293\times
J^{3/2}+\cdots \end{equation*} to the actual data from the Monte-Carlo
simulation in three dimensions \cite{Banerjee:2017fcx}, and the discrepancy of
the coefficient $0.293$ from the numerics turned out to be $13\%$.
Additionally, we also find a crossover of $\Delta(J;D)$ from $\Delta(J)\propto
J^{\frac{D}{D-1 } }$ to $\Delta(J)\propto J$, at around $J\sim 1/\epsilon$, as
one decreases $J$ while fixing $\epsilon$ (or vice versa), reflecting the fact
that there are no interacting fixed-point at $\epsilon=0$. Based on this
behaviour, we propose an interesting double-scaling limit which fixes
$\lambda\equiv J\epsilon$, suitable for probing the region of the crossover. I
will give $\Delta(J;D)$ to next-to-leading order in perturbation theory, either
in $1/\lambda$ or in $\lambda$, valid when $\lambda\gg 1$ and $\lambda\ll 1$,
respectively.

DOI： 10.1007/JHEP04(2021)264

arXiv

Authorship：Lead author,　Last author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

We study the transition between phases at large $R$-charge on a conformal manifold. These phases are characterized by the behaviour of the lowest operator dimension $\Delta(Q_R)$ for fixed and large $R$-charge $Q_R$. We focus, as an example, on the $D=3$, $\mathcal{N}=2$ Wess-Zumino model with cubic superpotential $W=XYZ+\frac{\tau}{6}(X^3+Y^3+Z^3)$, and compute $\Delta(Q_R,\tau)$ using the $\epsilon$-expansion in three interesting limits. In two of these limits the (leading order) result turns out to be \begin{equation*} \Delta(Q_R,\tau)= \begin{cases} \left(\text{BPS bound}\right)\left[1+O(\epsilon |\tau|^2Q_R)\right ] , & Q_R\ll \left\{ \frac{1}{\epsilon},\, \frac{1}{\epsilon|\tau|^2}\right\}\\ \frac{9}{8}\left(\frac{\epsilon|\tau|^2}{2+|\tau|^2}\right)^{\frac{1}{D-1 } }Q_R^{\frac{D}{D-1 } } \left[1+O\left(\left(\epsilon |\tau|^2Q_R\right)^{-\frac{2}{D-1 } }\right)\right ] , & Q_R\gg \left\{ \frac{1}{\epsilon},\, \frac{1}{\epsilon|\tau|^2}\right\} \end{cases} \end{equation*} which leads us to the double-scaling parameter, $\epsilon |\tau|^2Q_R$, which interpolates between the "near-BPS phase" ($\Delta(Q)\sim Q$) and the "superfluid phase" ($\Delta(Q)\sim Q^{D/(D-1)}$) at large $R$-charge. This smooth transition, happening near $\tau=0$, is a large-$R$-charge manifestation of the existence of a moduli space and an infinite chiral ring at $\tau=0$. We also argue that this behavior can be extended to three dimensions with minimal modifications, and so we conclude that $\Delta(Q_R,\tau)$ experiences a smooth transition around $Q_R\sim 1/|\tau|^2$. Additionally, we find a first-order phase transition for $\Delta(Q_R,\tau)$ as a function of $\tau$, as a consequence of the duality of the model. We also comment on the applicability of our result down to small $R$-charge.

DOI： 10.1007/JHEP01(2021)068

arXiv

Authorship：Lead author,　Last author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

Carrying to higher precision the large-$\mathcal{J}$ expansion of Hellerman
and Maeda, we calculate to all orders in $1/\mathcal{J}$ the power-law
corrections to the two-point functions $\mathcal{Y}_n \equiv |x -
y|^{2n\Delta_{\mathcal{O } } } \langle {\mathcal{O } }_n(x) \bar{\mathcal{O } }_n(y)
\rangle$ for generators $\mathcal{O}$ of Coulomb branch chiral rings in
four-dimensional $\mathcal{N} =2$ superconformal field theories. We show these
correlators have the universal large-$n$ expansion \[ \log(\mathcal{Y}_n)
\simeq \mathcal{J} \mathbf{A} + \mathbf{B} + \log(\Gamma( \mathcal{J} + \alpha
+ 1)) , \] where $\mathcal{J} \equiv 2 n \Delta_{\mathcal{O } }$ is the total
$R$-charge of $\mathcal{O}_n$, the $\mathbf{A}$ and $\mathbf{B}$ are
theory-dependent coefficients, $\alpha$ is the coefficient of the Wess-Zumino
term for the Weyl $a$-anomaly, and the $\simeq$ denotes equality up to terms
exponentially small in $\mathcal{J}$. Our methods combine the structure of the
Coulomb-branch effective field theory (EFT) with the supersymmetric recursion
relations. However, our results constrain the power-law corrections to all
orders, even for non-Lagrangian theories to which the recursion relations do
not apply. For the case of $\mathcal{N} = 2$ SQCD, we also comment on the
nature of the exponentially small corrections, which can be calculated to high
precision in the double-scaling limit recently discussed by Bourget et al. We
show the exponentially small correction is consistent with the interpretation
of the EFT breaking down due to the propagation of massive BPS particles over
distances of order of the infrared scale $|x - y|$.

DOI： 10.1007/JHEP12(2019)047

arXiv

Authorship：Lead author,　Last author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

In this note we search for the ground state, in infinite volume, of the $D=3$
Wilson-Fisher conformal $O(4)$ model, at nonzero values of the two independent
charge densities $\rho_{1,2}$. Using an effective theory valid on scales longer
than the scale defined by the charge density, we show that the ground-state
configuration is inhomogeneous for generic ratios $\rho_1 / \rho_2$. This
result confirms, within the context of a well-defined effective theory, a
recent no-go result of Alvarez-Gaume' et al. We also show that any spatially
periodic ground state solutions have an energetic preference towards longer
periods, within some range of $\rho_1 / \rho_2$ containing a neighborhood of
zero. This suggests that the scale of variation of the ground state solution in
finite volume will be the infrared scale, and that the use of the effective
theory at large charge in finite volume is self-consistent.

DOI： 10.1007/JHEP10(2019)038

arXiv

Authorship：Lead author,　Last author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

We calculate in detail the Renyi entanglement entropies of cTPQ states as a
function of subsystem volume, filling the details of our prior work [Nature
Communications 9, 1635 (2018)], where the formulas were first presented.
Working in a limit of large total volume, we find universal formulas for the
Renyi entanglement entropies in a region where the subsystem volume is
comparable to that of the total system. The formulas are applicable to the
infinite temperature limit as well as general interacting systems. For example
we find that the second Renyi entropy of cTPQ states in terms of subsystem
volume is written universally up to two constants, $S_2(\ell)=-\ln
K(\beta)+\ell\ln a(\beta)-\ln\left(1+a(\beta)^{-L+2\ell}\right)$, where $L$ is
the total volume of the system and $a$ and $K$ are two undetermined constants.
The uses of the formulas were already presented in our prior work and we mostly
concentrate on the theoretical aspect of the formulas themselves. Aside from
deriving the formulas for the Renyi Page curves, the expression for the von
Neumann Page curve is also derived, which was not presented in our previous
work.

DOI： 10.1007/JHEP12(2018)112

arXiv

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

A pure quantum state can fully describe thermal equilibrium as long as one
focuses on local observables. Thermodynamic entropy can also be recovered as
the entanglement entropy of small subsystems. When the size of the subsystem
increases, however, quantum correlations break the correspondence and cause a
correction to this simple volume-law. To elucidate the size dependence of the
entanglement entropy is of essential importance in linking quantum physics with
thermodynamics, and in addressing recent experiments in ultra-cold atoms. Here
we derive an analytic formula of the entanglement entropy for a class of pure
states called cTPQ states representing thermal equilibrium. We further find
that our formula applies universally to any sufficiently scrambled pure states
representing thermal equilibrium, i.e., general energy eigenstates of
non-integrable models and states after quantum quenches. Our universal formula
can be exploited as a diagnostic of chaotic systems; we can distinguish
integrable models from chaotic ones and detect many-body localization with high
accuracy.

DOI： 10.1038/s41467-018-03883-9

arXiv

Authorship：Lead author,　Last author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

We consider the operator spectrum of a three-dimensional ${\cal N} = 2$
superconformal field theory with moduli spaces of one complex dimension, such
as the fixed point theory with three chiral superfields $X,Y,Z$ and a
superpotential $W = XYZ$. By using the existence of an effective theory on each
branch of moduli space, we calculate the anomalous dimensions of certain
low-lying operators carrying large $R$-charge $J$. While the lowest primary
operator is a BPS scalar primary, the second-lowest scalar primary is in a
semi-short representation, with dimension exactly $J+1$, a fact that cannot be
seen directly from the $XYZ$ Lagrangian. The third-lowest scalar primary lies
in a long multiplet with dimension $J+2 - c_{-3} \, J^{-3} + O(J^{-4})$, where
$c_{-3}$ is an unknown positive coefficient. The coefficient $c_{-3}$ is
proportional to the leading superconformal interaction term in the effective
theory on moduli space. The positivity of $c_{-3}$ does not follow from
supersymmetry, but rather from unitarity of moduli scattering and the absence
of superluminal signal propagation in the effective dynamics of the complex
modulus. We also prove a general lemma, that scalar semi-short representations
form a module over the chiral ring in a natural way, by ordinary multiplication
of local operators. Combined with the existence of scalar semi-short states at
large $J$, this proves the existence of scalar semi-short states at all values
of $J$. Thus the combination of ${\cal N}=2$ superconformal symmetry with the
large-$J$ expansion is more powerful than the sum of its parts.

DOI： 10.1007/JHEP10(2017)089

arXiv

Authorship：Lead author,　Last author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

We calculate the anomalous dimensions of operators with large global charge
$J$ in certain strongly coupled conformal field theories in three dimensions,
such as the O(2) model and the supersymmetric fixed point with a single chiral
superfield and a $W = \Phi^3$ superpotential. Working in a $1/J$ expansion, we
find that the large-$J$ sector of both examples is controlled by a conformally
invariant effective Lagrangian for a Goldstone boson of the global symmetry.
For both these theories, we find that the lowest state with charge $J$ is
always a scalar operator whose dimension $\Delta_J$ satisfies the sum rule $
J^2 \Delta_J - \left( \tfrac{J^2}{2} + \tfrac{J}{4} + \tfrac{3}{16} \right)
\Delta_{J-1} - \left( \tfrac{J^2}{2} - \tfrac{J}{4} + \tfrac{3}{16} \right)
\Delta_{J+1} = 0.035147 $ up to corrections that vanish at large $J$. The
spectrum of low-lying excited states is also calculable explcitly: For example,
the second-lowest primary operator has spin two and dimension $\Delta\ll J +
\sqrt{3}$. In the supersymmetric case, the dimensions of all half-integer-spin
operators lie above the dimensions of the integer-spin operators by a gap of
order $J^{1/2}$. The propagation speeds of the Goldstone waves and heavy
fermions are $\frac{1}{\sqrt{2 } }$ and $\pm \frac{1}{2}$ times the speed of
light, respectively. These values, including the negative one, are necessary
for the consistent realization of the superconformal symmetry at large $J$.

DOI： 10.1007/JHEP12(2015)071

arXiv

Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

Massless Dirac particles on the helicoid are theoretically investigated. With
its possible application being helical graphene, we explore how the
peculiarities of Dirac particles appear on the curved, screw-symmetric surface.
The zweibein is used to derive the massless Dirac equation on the helicoid, as
well as general curved surfaces. Bound states of massless Dirac electrons are
shown to be absent on the helicoid, and then the scattering probabilities and
the phase shifts on the surface are obtained from numerically calculated wave
functions. We find the local density of states and the phase shifts
idiosyncratic especially around the axis of the helicoid. Bound states of
massive Dirac electrons on the surface are also shown to be absent as an
extension of the above result on massless Dirac electrons.

DOI： 10.1103/PhysRevB.92.205425

arXiv