# The puzzle of bicriticality in the XXZ antiferromagnet

Amnon Aharony<EMAIL_ADDRESS>School of Physics and Astronomy, Tel Aviv
University, Tel Aviv 6997801, Israel  Ora Entin-Wohlman<EMAIL_ADDRESS>School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel

###### Abstract

Renormalization-group theory predicts that the XXZ antiferromagnet in a
magnetic field along the easy Z-axis has asymptotically either a tetracritical
phase-diagram or a triple point in the field-temperature plane. Neither
experiments nor Monte Carlo simulations procure such phase diagrams. Instead,
they find a bicritical phase-diagram. Here this discrepancy is resolved: after
generalizing a ubiquitous condition identifying the tetracritical point, we
employ new renormalization-group recursion relations near the isotropic fixed
point, exploiting group-theoretical considerations and using accurate
exponents at three dimensions. These show that the experiments and simulations
results can only be understood if their trajectories flow towards the
fluctuation-driven first order transition (and the associated triple point),
but reach this limit only for prohibitively large system sizes or correlation
lengths. In the crossover region one expects a bicritical phase diagram, as
indeed is observed. A similar scenario may explain puzzling discrepancies
between simulations and renormalization-group predictions for a variety of
other phase diagrams with competing order parameters.

Introduction. Natural systems show behaviors ascribed to fluctuations on many
length scales (e.g., critical phenomena, fully-developed turbulence, quantum
field-theory, the Kondo effect, and polymers described by self-avoiding
walks). These behaviors can be treated by the renormalization group (RG)
theory [1, 2, 3]: gradually eliminating short-range details, during which the
system size $L$ and the correlation length $\xi$ rescale to $L\rightarrow
L(\ell)=L/e^{\ell}$ and $\xi\rightarrow\xi(\ell)=\xi/e^{\ell}$ ($\ell$ is the
number of RG iterations), the parameters characterizing the system can ‘flow’
to a ‘stable’ fixed point (FP), which determines universal power-laws
describing physical quantities. Varying the parameters can lead to an
instability of a FP (with one or more parameters becoming ’relevant’ and
’flowing’ away from it, as $e^{\lambda\ell}$, with a positive ‘stability
exponent’ $\lambda$), generating transitions between different universality
classes. Although in most cases the predictions of the RG have been confirmed
experimentally and/or by numerical simulations, some puzzling discrepancies
still await explanations. Here we resolve one such puzzle, involving the phase
transitions between competing ordered phases. As listed e.g. in Refs. 4 and 5,
phase diagrams with competing order parameters arise in a variety of physical
examples. Some of these are mentioned below, after analyzing the phase diagram
of the anisotropic antiferromagnet in a magnetic field.

A uniaxially anisotropic XXZ antiferromagnet has long-range order (staggered
magnetization) along its easy axis, Z. A magnetic field $H_{\parallel}$ along
that axis causes a spin-flop transition into a phase with order in the
transverse plane, plus a small ferromagnetic order along Z. Experiments [6, 7]
and Monte Carlo simulations on three-dimensional lattices [8, 9, 10] typically
find a bicritical phase diagram in the temperature-field $T-H_{\parallel}$
plane [Fig. 1(a)]: a first-order transition line between the two ordered
phases, and two second-order lines between these phases and the disordered
(paramagnetic) phase, all meeting at a bicritical point. Recently, the spin-
flop transition in XXZ antiferromagnets has raised renewed interest [11],
related to possible spintronic applications of the Seebeck effect near that
transition. Simulations in that paper also seem to find a bicritical phase
diagram.

Figure 1:  Possible phase-diagrams for the XXZ antiferromagnet in a
longitudinal magnetic field. (a) Bicritical phase diagram. (b) Tetracritical
phase diagram. (c) Diagram with a triple point. Thick lines - first-order
transitions. Thin lines - second-order transitions. The first-order transition
lines between the ordered phases and the disordered paramagnetic phase end at
tricritical points (small empty circles). After Refs. 12, 14. $T_{1}$ and
$T^{\prime}_{1}$ are the transition lines between the ordered phases and the
paramagnetic phase. $T_{2}$ and $T^{\prime}_{2}$ are the lines second-order
lines which border the mixed phase.

History. The early RG calculations [4] were based on low-order expansions in
$\epsilon=4-d$, where $d$ is the spatial dimensionality. These calculations
found that the (rotationally-invariant)isotropic FP is stable at $d=3$,
yielding asymptotically the bicritical phase diagram. These calculations also
found that the isotropic FP becomes unstable as the total number of spin
components $n$ ($=3$ in our case) increases beyond a threshold $n_{c}(d)$, and
estimated that $n_{c}(3)>3$. For $n>n_{c}(d)$ they found a stable biconical
FP. Had the RG trajectories flown to that FP, the first-order line between the
two ordered phases would be replaced by an intermediate (mixed) phase, bounded
by two second-order lines, and all four second-order lines would have met at a
tetracritical point [Fig. 1(b)] [4, 12, 13]. In addition, if the system
parameters are initially outside the region of attraction of that PF, the
bicritical point turns into a triple point, and the transitions between the
ordered phases and the disordered paramagnetic phase become first-order near
that point, turning second-order only at finite distances from it [Fig. 1(c)]
[14].

However, the $\epsilon-$expansions diverge, and low-order calculations are not
reliable [15]. One way to overcome this divergence is to use resummation
techniques, e.g., by taking into account the singularities of the series’
Borel transforms [16], and extrapolating the results to $\epsilon=1$. These
yielded three stability exponents for the isotropic FP, $\lambda_{0,2,4}$. The
small exponent $\lambda_{4}$ also describes the (in)stability against a cubic
perturbation [17, 13], and it vanishes at $n=n_{c}(d)$. The same resummation
techniques (carried out on sixth-order $\epsilon-$expansions) have been
applied to the latter problem [18]. The results were compared with a
resummation of the sixth-order perturbative (divergent) expansions in the
original field-theory coefficients at $d=3$ [19], with recent bootstrap
calculations [20], with Monte Carlo simulations [21] and with high-temperature
series (for $\lambda_{0}$) [22]. An updated table of these results appears in
Ref. 20. The agreement between all the techniques indicates the accuracy of
the exponents:

$\displaystyle\lambda_{0}\approx-0.78,\ \ \lambda_{2}\approx-0.55,\ \
\lambda_{4}\approx 0.01.$ (1)

Since $\lambda_{4}>0$, the isotropic fixed point is unstable at $d=3$, and
$n_{c}(3)<3$, contradicting previous estimates [4, 13]. Therefore, as
explained below, the bicritical phase diagram should be replaced by the
tetracritical or the triple one, but neither of these agrees with the
experiments or the simulations.

The field theoretical analysis is based on the Ginzburg-Landau-Wilson (GLW)
Hamiltonian density [4],

$\displaystyle{\cal H}({\bf r})=$
$\displaystyle\big{(}|{\boldmath{\nabla}}{\bf S}|^{2}+t|{\bf
S}|^{2}\big{)}/2+U_{2}+U_{4},$ (2) $\displaystyle U_{2}$
$\displaystyle=g\big{[}|S_{\parallel}|^{2}-|{\bf S}|^{2}/3\big{]},$ (3)
$\displaystyle U_{4}$
$\displaystyle=u_{\parallel}|S_{\parallel}|^{4}+u_{\perp}|{\bf
S}_{\perp}|^{4}+2u_{\times}|S_{\parallel}|^{2}|{\bf S}_{\perp}|^{2},$ (4)

with the local three-component ($n=3$) staggered magnetization, ${\bf S}({\bf
r})\equiv\big{(}S_{\parallel}({\bf r}),{\bf S}_{\perp}({\bf r})\big{)}$. For
$g=0$ and $u_{\parallel}=u_{\perp}=u_{\times}=u$, ${\cal H}$ reduces to the
isotropic Wilson-Fisher Hamiltonian [1, 2, 3], which has an (isotropic) FP at
$u=u^{I}$. [23]

Group theory. A priori, at $g=0$, the stability of the isotropic FP against
symmetry-breaking perturbations requires an analysis of 15 terms in the GLW
Hamiltonian, which are quartic in the spin components,
$S_{\alpha}S_{\beta}S_{\gamma}S_{\delta}$. Group-theoretical arguments showed
that these terms split into subsets of $1+5+9$ terms, and all the terms within
a subgroup have the same stability exponent, listed in Eq. (1) [24, 25, 16,
21, 26]. In our case, [$O(3)\Rightarrow O(1)\bigoplus O(2)$], the three
exponents are associated with the following combinations of quartic terms:

$\displaystyle{\cal P}_{4,0}$ $\displaystyle\equiv|{\bf S}|^{4},\ \ \ {\cal
P}_{4,2}\equiv|{\bf S}|^{4}[x-1/3],$ $\displaystyle{\cal P}_{4,4}$
$\displaystyle\equiv|{\bf S}|^{4}\big{[}x(1-x)-(1+x)/7+2/35\big{]},$ (5)

where $x=S^{2}_{\parallel}/|{\bf S}|^{2}$. The largest (negative) exponent
$\lambda_{0}$ corresponds to the stability within the $O(3)-$symmetric case,
${\cal P}_{4,0}$. In our case, the exponent $\lambda_{2}$ corresponds the a
term which splits the $O(3)$ isotropic symmetry group into $O(1)\bigoplus
O(2)$. Similar to $U_{2}$, ${\cal P}_{4,2}$ ‘prefers’ ordering of
$S_{\parallel}$ or of ${\bf S}_{\perp}$. The smallest exponent $\lambda_{4}$
describes the crossovers away from the isotropic FP, towards either the
biconical or the cubic FP. Writing the quartic terms as

$\displaystyle U_{4}=(u^{I}+p_{0}){\cal P}_{4,0}+p_{2}{\cal
P}_{4,2}-p_{4}{\cal P}_{4,4},$ (6)

with arbitrary coefficients $p_{i},~{}i=0,2,4$ (which vanish at the isotropic
FP), implies the linear recursion relations near the isotropic FP,

$\displaystyle dp_{i}/d\ell\approx\lambda_{i}p_{i}\ \ \ \ \Rightarrow\ \ \
p_{i}(\ell)=p_{i}(0)e^{\lambda_{i}\ell}.$ (7)

Finite sizes. The calculations of the stability exponents, Eqs. (1), apply
only in the asymptotic limit, for infinite samples and very close to the
multicritical point, i.e., at very large $\ell$. The explanation of the
experiments (carried out at a finite $\xi$) and simulations (accomplished at a
finite $L$) requires the usage of a finite number of RG iterations,
$\ell=\ell_{f}$, at which the fluctuations have been eliminated: The
renormalized correlation length $\xi(\ell_{f})={\cal O}(1)$, with
$\xi(0)\sim|t|^{-\nu}$ ($t=T/T_{c}-1$ measures the distance from the
transition temperature $T_{c}$, and $\nu\approx.711$ is the critical
exponent), or the system size $L(\ell_{f})={\cal O}(1)$ [2] (lengths are
measured in units of the lattice constant). $\ell_{f}$ increases with the
system’s size $L$ (at criticality), or when the initial parameters are closer
to criticality (i.e., a larger initial correlation length). At this stage, one
can solve the problem using the mean-field Landau theory [2]. An analysis of
this situation requires the full RG flow of the system’s Hamiltonian [27].
Such an analysis, based on resummation of (approximate) second-order
$\epsilon-$expansions, was performed by Folk et al. [28]. That paper presented
numerical RG flows in the parameter space, and observed the slow flow close to
the isotropic and biconical FP’s.

Our calculation. This Letter presents a more precise way to perform this
analysis, based on the following steps. (1) Using the stability exponents of
the isotropic FP at three dimensions, Eq. (1), we construct flow recursion
relations near that FP. (2) Equating Eq. (4) with Eq. (6), the initial quartic
parameters $\\{u_{i}\\}$ are expressed in terms of the $p_{i}$’s, with
coefficients true to all orders in $\epsilon$ [see Eq. (11) below]. (3) Since
$p_{0}$ and $p_{2}$ are strongly irrelevant ($\lambda_{0}$ and $\lambda_{2}$
are negative and large [Eq. (1)]) near the isotropic FP, they decay after a
small number $\ell_{1}$ of ‘transient’ RG iterations (irrespective of non-
linear terms in their recursion relations). After that, the RG iterations
continue on a single universal straight line in the three-dimensional
parameter space, given in Eq. (12). In a way, this line generalizes the
concept of universality. (4) On this universal line, Eq. (7) for $p_{4}$
yields a slow flow [as $p_{4}(\ell)\sim e^{\lambda_{4}\ell}$] away from the
isotropic FP for both positive and negative $p_{4}$. The smallness of
$\lambda_{4}$ allows us to expand in powers of $p_{4}$ around the isotropic FP
[instead of the ‘usual’ expansion in all the $u$’s near the Gaussian FP]. To
second order in $p_{4}$ [for $\ell>\ell_{1}$],

$\displaystyle dp_{4}/d\ell=\lambda_{4}p_{4}-Bp^{2}_{4},$ (8)

where the (positive) coefficient $B$ (the only unknown parameter) is
presumably of order $1$. This yields explicit solutions for $p_{4}(\ell)$, Eq.
(13), and typical solutions are shown in Fig. 2. (5) For $p_{4}>0$ the
trajectories flow to the stable biconical FP, and the stability exponents at
that point agree (approximately) with the full calculation in Ref. 16 – adding
credibility to our approximate expansion. On these trajectories the
coefficients are shown to yield a tetracritical phase diagram. (6) For
$p_{4}<0$ the trajectories eventually flow to a fluctuation-driven first-order
transition, which occurs when $p_{4}(\ell)$ crosses the horizontal line in
Fig. 2. In the wide intermediate range of $\ell$, before that crossing, the
parameters yield a bicritical phase diagram. Beyond that crossing, for very
large $\ell$ (corresponding to very large $L$ or $\xi$) the bicritical point
turns into a triple point. The bicritical phase-diagrams observed in the
experiments/simulations apparently occur at this intermediate range.

Figure 2: (color online) The function $p_{4}(\ell-\ell_{1})$ (blue) for $B=1$
and $p_{4}(\ell_{1})=.3$ and $-.1$. Below the horizontal (orange) line at
$p_{4}=-35u^{I}/8=-1.75$, the transition becomes first order and the
bicritical point becomes a triple point.

Criteria for tetracriticality. Eliminating the small (non-critical)
paramagnetic moment (generated by $H_{\parallel}$) from the free energy
renormalizes the three $u$’s in Eq. (4), with corrections of order
$H^{2}_{\parallel}$ [4]. Although these corrections are small, so that the new
coefficients remain close to the isotropic $u$, they are important because
they determine the ultimate shape of the phase diagram. The tetracritical
phase diagram [Fig. 1(b)] requires that on the line $g=0$ both order
parameters are non-zero, implying that the mean-field free energy has a
minimum at $0<x<1$ [29]. Presenting Eq. (4) as

$\displaystyle U_{4}=|{\bf
S}|^{4}\big{[}u_{\parallel}x^{2}+u_{\perp}(1-x)^{2}+2u_{\times}x(1-x)\big{]}\
,$ (9)

this minimum is at
$x=(u_{\perp}-u_{\times})/(u_{\parallel}+u_{\perp}-2u_{\times})$, provided
that

$\displaystyle u_{\times}<u_{\parallel}\ \ {\rm and}\ \ u_{\times}<u_{\perp}.$
(10)

These conditions for tetracriticality are more restrictive than the condition
found before, $u_{\parallel}u_{\perp}-u^{2}_{\times}>0$ [4]. When even one of
them is violated, the minimum of $U_{4}$ is at $x=1$ or at $x=0$, implying
that the mixed phase does not exist; it is replaced by a first-order
transition line, as in Figs. 1(a,c).

Renormalization group. Comparing Eqs. (4) and (6) for $U_{4}$ one finds

$\displaystyle\delta u_{\parallel}=p_{0}+(70p_{2}+24p_{4})/105,$
$\displaystyle\delta u_{\perp}=p_{0}-(35p_{2}-9p_{4})/105,$
$\displaystyle\delta u_{\times}=p_{0}+(35p_{2}-72p_{4})/210,$ (11)

with $\delta u_{i}=u_{i}-u^{I}$. According to Eq. (10), the multicritical
point is tetracritical if both anisotropy parameters
$u_{\parallel}-u_{\times}=p_{2}/2+4p_{4}/7$ and
$u_{\perp}-u_{\times}=-p_{2}/2+3p_{4}/7$ are positive, i.e., when
$|p_{2}(\ell)|<6p_{4}(\ell)/7$. Since $p_{2}(\ell)\approx
p_{2}(0)e^{\lambda_{2}\ell}$ decays rather quickly, and $p_{4}(\ell)$ varies
slowly (see below), this will happen when
$e^{\lambda_{2}\ell}<6p_{4}(0)/[7|p_{2}(0)]$. Assuming that
$p_{4}(0)[=u_{\parallel}+u_{\perp}-2u_{\times}]$ and
$p_{2}(0)[=2(3u_{\parallel}-4u_{\perp}+u_{\times})/7]$ are small and of the
same order, this happens for a small $\ell<\ell_{1}$. We conclude that the
phase diagram is in fact tetracritical whenever $p_{4}(0)>0$, for practically
all $\ell$, irrespective of the value of $B$. Since the experiments and
simulations do not exhibit this phase diagram, we conclude that they probably
have $p_{4}(0)<0$.

To complete the RG analysis, we note that both $p_{0}$ and $p_{2}$ decay
quickly, so there is no need to add higher-order terms for them in Eq. (7).
They can be neglected in Eq. (11) after a transient stage of $\ell_{1}$
iterations [30], and then all the flows continue on the universal semi-
asymptotic line,

$\displaystyle\big{(}\delta u_{\parallel},~{}\delta u_{\perp},~{}\delta
u_{\times}\big{)}=\big{(}8,~{}3,~{}-12\big{)}p_{4}/35.$ (12)

Higher-order terms in the RG recursion relations may turn this line non-linear
[5].

For $\ell>\ell_{1}$ the recursion relation for $p_{4}$, Eq. (8), gives the
solution [5]

$\displaystyle
p_{4}(\ell)=\frac{p_{4}(\ell_{1})e^{\lambda_{4}(\ell-\ell_{1})}}{1+Bp_{4}(\ell_{1})(e^{\lambda_{4}(\ell-\ell_{1})}-1)/\lambda_{4}}.$
(13)

For $p_{4}(\ell_{1})>0$, the flow approaches the biconical FP,
$p_{4}(\ell)\rightarrow p^{B}_{4}=\lambda_{4}/B$, with $p^{B}_{4}\ll 1$ –
justifying stopping the expansion in Eq. (8) at second order [31, 32]. Near
the biconical FP one finds that (to linear order in $p_{4}-p^{B}_{4}$)
$d[p_{4}-p^{B}_{4}]/d\ell=-\lambda_{4}[p_{4}-p^{B}_{4}]$, identifying the
stability exponent at this FP as $\lambda^{B}_{4}=-\lambda_{4}\approx-0.01$,
independent of $B$, and the biconical FP is indeed stable. Within our
approximate recursion relations for $p_{0}$ and $p_{2}$, the other two
exponents approximately remain unchanged,
$\lambda^{B}_{0,2}\approx\lambda_{0,2}$. All three values are close to those
found near the biconical FP by the full sixth-order calculation in Ref. 16,
confirming the validity of our approximate expansion near the isotropic FP.

For $p_{4}(\ell_{1})<0$, Eq. (8) implies that $p_{4}(\ell)$ grows more and
more negative (note: both $B$ and $\lambda_{4}$ were assumed to be positive).
At $\ell=\ell_{f}$, Eq. (10) is not obeyed, the minimum of $U_{4}$ is at
$x=1$, with $U_{4,min}=|{\bf S}|^{4}u_{\parallel}=|{\bf
S}|^{4}[u^{I}+8p_{4}(\ell_{f})/35]$, where we used Eq. (12). This becomes
negative when $p_{4}(\ell_{f})<-35u^{I}/8$. The resummation of the
$\epsilon-$expansion gives $u^{I}\sim 0.4$ [5], leading to $35u^{I}/8\sim
1.75$ [the orange horizontal line in Fig. 2], which is quite large compared to
reasonable values of $p_{4}(\ell_{1})$, and probably out of the region of
applicability of the quadratic approximation which yielded Eq. (13). However,
it may still be reasonable for intermediate values of $\ell$ (e.g.,
$\ell-\ell_{1}<8$ in Fig. 2). Equation (13) diverges at a large
$\ell=\ell_{2}$ [33], and we expect $p_{4}(\ell)$ to cross the value $-1.75$
not very far below $\ell_{2}$. With the parameters used in Fig. 2, the
divergence occurs at
$\ell_{2}-\ell_{1}\sim\log[1-\lambda_{4}/(Bp_{4}(\ell_{1})]/\lambda_{4}\sim
9.5$, and the transition to first-order occurs at $\ell_{x}-\ell_{1}\sim 9$.
These numbers become smaller for larger values of $Bp_{4}(\ell_{1})$. In this
example, the bicritical point turns into a triple point at $\xi\sim
e^{\ell_{x}}\sim e^{8+9}\sim 10^{7}$, which cannot be reached experimentally.
Even if this approximation is improved, and if $Bp_{4}(0)$ increases (see the
end of the paper), there will still be a wide range of parameters where
experiments and simulations will follow the bicritical phase-diagram. In this
range, the effective exponents near the bicritical point may depend on
$\ell_{f}$ and differ significantly from their isotropic-FP values [5].

Other examples. Similar phase diagrams pertain to the structural transitions
in uniaxially stressed perovskites, which are described by the cubic model [5,
17, 12]. Similarly to the XXZ antiferromagnet, the almost isotropic SrTiO3
(with $p_{4}\lessapprox 0$) yielded an apparent bicritical phase diagram.
However, the more anisotropic RbCaF3 did yield the diagram 1(c), as expected
by the RG calculations [5].

In reality, cubic anisotropic antiferromagnets are subjected to both the
anisotropic and cubic terms, $U_{4}$ and $U_{c}$ (or other crystal-field
terms). In most magnetic cases, the cubic terms are small [7]. Since both
${\cal P}_{4,4}$ and $U_{c}$ scale with the same small exponent $\lambda_{4}$,
we expect the same qualitative flow diagrams as discussed above. However, the
competition (within this subgroup) between the biconical and the cubic FP’s
(which are degenerate at linear order), can only be settled by including
higher-order terms in the RG recursion relations, still awaits further
analysis. Studies with other crystal symmetries (e.g., tetragonal), and
detailed studies of the sixth-order terms which dominate the fluctuation-
driven tricritical point, also await a detailed analysis (and corresponding
dedicated experiments).

For larger values of $n=n_{1}+n_{2}>3$, the biconical FP becomes unstable,
being replaced by the decoupled FP, at which $u^{D}_{\times}=0$ [34], implying
a tetracritical phase diagram. This has been particularly expected for the
SO(5) theory aimed to describe the competition between superconductivity
($n_{1}=2$) and antiferromagnetism ($n_{2}=3$) in the cuprates [35]. In
contrast, Monte Carlo simulations of this model gave a bicritical phase
diagram, with isotropic $n=5$ critical exponents [36]. Similar results were
reported for the iron pnictides [37]. Assuming that the parameters of these
materials obey $u_{\times}(\ell_{f})\nless
u_{\parallel}(\ell_{f}),~{}u_{\perp}(\ell_{f})$, preferring the bicritical
scenario, and that the RG trajectories stay close to the isotropic FP, could
also resolve that long-standing puzzle.

A very recent experiment [38] studied a critical pressure-temperature phase
diagram, with competing ferromagnetic and antiferromagnetic phases, is also
apparently in contrast to the RG results, which predict for $n_{1}=n_{2}=3$ an
asymptotic decoupled tetracritical phase diagram (or a triple point). It would
be interesting to study the RG trajectories for these experiments.

Competing order parameters, with larger values of $n$, also arise in certain
field-theory models [20, 39], which are similar in structure to the standard
model of particle interactions. It would be interesting to see whether those
theories yield puzzles of the sort discussed here.

Summary. In conclusion, experiments and simulations do not contradict the
renormalization-group predictions. The new system of recursion relations
presented here, which is based on group-theoretical exact coefficients for an
expansion near the isotropic fixed point, clearly shows that the simulations
and experiments are in a crossover regime, between the bicritical point and
the triple point. Our quantitative estimates show that it will probably be
very difficult to reach the triple point experimentally. However, in principle
the renormalization-group also supplies intermediate effective exponents [5],
whose measurements can confirm its validity. Dedicated experiments (carried
out on larger samples, at temperatures closer to the multicritical point), and
exploiting a wider range of the initial Hamiltonians, which will allow
increasing $p_{4}(0)$ by moving away from the parameters characterizing the
isotropic fixed point (e.g., by adding single-ion anisotropies [40]), may find
the tetracritical or the triple point, or – at least – detect the variation of
the non-asymptotic (effective) critical exponents.

Acknowledgement: We thank Andrey Kudlis, Walter Selke, David Landau and Andrea
Pelissetto for helpful correspondence.

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[width=0.9]Sections/figure2.pdf

Figure : Qualitative examples on iVQA videos. Single frame and ten-word
summary is generated from the original video for Video Question Answering
task. First two examples demonstrate successful cases where both visual and
textual signals signals are able to capture the question-relevant information.
The last two examples show some failure cases where visual and/or textual
signals are distracted from the question.Figure could be improved by clearer
labels for what is going on and thicker lines, etc.

##  1 Experiments

section summary here In this section, we provide our experiment results. First
we detail our implementation and VideoQA datasets; then we provide VideoQA
results under different input sparsities, followed by multi-modal results.
Finally, we offer some qualitative visualization to analyze our approach.

Table : Effect of the temperature $\tau$. Smaller $\tau$ leads to more exploitation while higher leads to more exploration. We observe that more explorative selection is beneficial for denser inputs. | VLEP | VIOLIN
---|---|---
Input Percentage | $\tau=0.01$ | $\tau=0.1$ | $\tau=0.5$ | $\tau=0.01$ | $\tau=0.1$ | $\tau=0.5$
10% | 60.25 | 56.01 | 58.94 | 56.25 | 60.57 | 58.80
30% | 60.95 | 63.52 | 59.13 | 61.72 | 57.64 | 62.34
50% | 63.05 | 63.64 | 64.30 | 65.57 | 64.48 | 66.06
70% | 63.73 | 65.14 | 65.32 | 65.94 | 66.52 | 67.06
|  |  |  |  |  |
Table : Effect of the balancing weight $\lambda$. $\lambda$ balances the selection loss and task loss as specificed in eq. LABEL:eq:loss. We report results on VLEP dataset and observe that $\lambda=1.0$ yields the best balance. Higher $\lambda$ might lead to distraction of task, while lower $\lambda$ might lead to insufficient sparsification. We pick $\lambda=1.0$ based on the following ablation. Input Percentage | $\lambda=0.01$ | $\lambda=0.1$ | $\lambda=1.0$ | $\lambda=10.0$
---|---|---|---|---
10% | 59.12 | 59.85 | 60.25 | 59.90
70% | 65.23 | 65.32 | 65.32 | 65.11
|  |  |  |
Table : Comparison of our two Gumbel variants. Overall the first variants perform slightly better. The second variant is superior at highly sparsified level ($10\%$) as it adds more flexibility in individual sparsity levels across different videos.  Input Percentage | $10\%$ | $30\%$ | $50\%$ | $70\%$
---|---|---|---|---
Gumbel-TopK Selection | 60.25 | 63.52 | 64.30 | 65.32
Ratio-controlled Gumbel | 61.43 | 63.42 | 63.49 | 65.01
|  |  |  |

[width=]Sections/qa_curve_full.pdf

Figure : Sparsified VideoQA results on VLEP and VIOLIN datasets. Accuracy at
the $100\%$ level refers to the original full input baseline result. We can
conclude that learnable sparsification is better than fixed sampling
(Uniform), and that stochasitic sampling is better than deterministic
selection (TopK). Our Multi-Gumbel estimator achieves the best result overall.
Table : VideoQA results on iVQA. We apply our approach on the state-of-the-art
method [Yang_2021_ICCV]. We consider multi-modal sparsification where we
sparsify both visual (i.e., frames) and textual (i.e., words) inputs. Compared
to single-modality, multi-modal performance is stronger at different
sparsification levels. With additional extracted words, we also outperform the
state-of-the-art result on iVQA (last column).

|  | Visual (Snippets)
---|---|---
|  | 0 | 1 snippet | 2 snippets | 5 snippets | 20 snippets
5*  | Textual
---
(Words)
0 | 14.6 (Q-only) | 28.65 | 30.24 | 31.26 | 35.43 [Yang_2021_ICCV]
| 5 words | 17.5 | 28.68 | 30.31 | 31.70 | 35.43
| 10 words | 18.22 | 29.87 | 31.43 | 31.88 | 36.01
| 25 words | 20.14 | 30.16 | 31.59 | 32.03 | 36.09
| 100 words | 26.75 | 31.47 | 32.11 | 33.21 | 36.42
|  |  |  |  |  |

###  Implementation Details

To verify our idea, we experimented on two state-of-the-art video-and-language
models VQA-T [Yang_2021_ICCV] and HERO [li2020hero]. HERO considers multi-
channel videos where videos come with subtitles as additional channel of
inputs. HERO follows a hierarchical transformer architecture to first exploit
the information within video modalities and contexts, and then has another
task head to operate the task. VQA-T simply consists of two Distill-BERT
models to deal with video+question inputs and answer candidates, and computes
the answer based on embedding similarity. For extracting the video features,
we follow [Yang_2021_ICCV] to use the S3D model pre-trained on Howto100M
dataset. For extracting the key word candidates, we use the model offered by
[Yang_2021_ICCV] and the vocabulary from the training split of the dataset to
extract the words/phrases based on feature similarity.

###  Datasets and Metrics

We evaluate our idea on public VideoQA benchmarks including VLEP
[lei2020vlep], VIOLIN [liu2020violin] and iVQA [Yang_2021_ICCV]. For VLEP and
VIOLIN, we follow [li2020hero] to build our method on top of HERO. VLEP and
VIOLIN provide both raw videos and subtitles as inputs. Our selection is then
based on the multi-modal inputs. For iVQA, we follow [Yang_2021_ICCV] to build
our method on top of VQA-T. We report VideoQA accuracies across different
input sparsity level: $10\%$, $30\%$, $50\%$, $70\%$ and full ($100\%$)
inputs.

###  VideoQA Experiments

Introduction for this experiment, what is it meant to show? We present our
single modality sparsified VideoQA results here. First, we study the design
choices of our two multi-gumbel estimator variants, followed by the comparison
between our approach and other token sparsification baselines. Effect of
temperature $\tau$. In our experiments, we found that varying $\tau$ could
result in very different performance. We elaborate the result in Table 1 with
our Gumbel-TopK selection variant, where we choose $\tau=(0.01,0.1,0.5)$ for
each sparsity level, and fix $\lambda=1.0$. A smaller $\tau$ means the model
focuses more on exploitation, while a larger $\tau$ makes the model focus more
on exploration. We can observe that on both datasets, the model that is more
explorative with denser inputs gives a better results; but on sparser inputs,
the model tends to stick to exploitation. Effect of loss balancing weight
$\lambda$. We also study how the balancing weight $\lambda$ affects the
performance with our ratio-controlled Gumbel estimator. In Table 1, we choose
$\lambda=0.01,0.1,1.0$ and $10.0$, and then report the results at highly
sparsified ($10\%$) and lowerly sparsified ($70\%$) levels on VLEP dataset. We
fix $\tau=0.01$ for $10\%$ level and $\tau=0.5$ for $70\%$ level. $\lambda$
has slightly larger impact on highly sparsified setting. We observe that
$\lambda=1.0$ yields the best balance and hence choose it for all the other
performance. Comparison of two multi-gumbel variants. We compare the two
variants of Gumbel estimator for token sparsification. In Table 1, we compare
these two variants at different sparsity levels on VLEP. Our Gumbel-TopK
selection variant is better than ratio-controlled Gumbel overall. Ratio-
controlled Gumbel is superior at highly sparsified level ($10\%$) as it adds
more flexibility in individual sparsity levels across different videos.
Comparison with other sparsification approaches. To our knowledge, no prior
work has studied the same topic on VideoQA before, so there is no direct
comparison. To validate our approach of Multi-Gumbel Estimator, we define the
baselines on our own: 1. Uniform(Fixed): Fixed uniform sampling of inputs
w.r.t. different sparsity levels. 2. TopK: During training, directly select
inputs with higher keeping probability $s_{i}$ after softmax step, without
noise perturbing. 3. Multi-Gumbel(Ours): Our approach which stochastically
sparsifies tokens with Gumbel perturbing, we plot the better result from the
two variants we introduced. We show the accuracy vs. density curve in Figure
1. We can see that our Multi-Gumbel approach module is able to achieve the
best performance across different sparsity levels. Compared to learnable
selection, fixed uniform sampling is weaker as it does not contain any form of
task adaptive selection. A direct TopK selection training performs weaker than
training with stochastic sampling, as we observe that the deterministic
selection tends to a local optimal choice, while our stochastic Multi-Gumbel
approach gives more flexibility of by adding noises while learning. One
noticeable observation is that, at $10\%$ level, which corresponds to very few
frames (2 frames for VLEP and 4 frames for VIOLIN), the performance is still
quite good. It implies the potential of accomplishing the task with very few
inputs.

[width=]Sections/figure4.pdf

Figure : Frame importance visualization. Darker color means the corresponding
word/frame is of more importance to predict the answer. We can see that the
model is able to discard some repetitive frames or frames that are not
relevant. Add labels to the image? I.e. Question, video, etc?

###  Multi-modal Sparsification Results on iVQA

In the multi-modal experiments, we would like to study the relation between
visual and textual modalities under a controlled input setting. In order to do
that, we extend our learnable selection module to the multi-modal setting
following Section LABEL:sec:multimodal to generate key frames and key words
from the original video inputs. We first get a pool of candidate inputs from
the raw video. The candidate frames are directly sampled from the videos,
while the candidate key words are extracted using CLIP-based model, which
finds the closest words or phrases using nearest embedding matching. We use
all the phrases and words from the iVQA training set as the vocabulary
dictionary to choose words from. To better demonstrate the results, we use the
format of few-word or few-frame inputs. For visual frame inputs, we process
with the same method as before. For textual inputs, we treat 5-word as one
unit. 5-word/sec is the average reading speed for adults, which consumes
similar attention from watching a frame. So 5-word and one single frame could
be thought of as equivalent in consuming user attention. We combine the units
into a sequence, then apply the same selection method for word selection. For
multi-modal setting, we concatenate the frames and word units as a multi-modal
sequence and select from both. We fix $\tau=0.1$ for training the models. Our
results are shown in Table 1. For single-modality inputs, we similarly observe
an increasing performance trend with increasing number of inputs. Even with
very few inputs, the VideoQA performance is very close to the upper bound from
dense inputs. We can also observe a boost of performance from increasing
density of inputs on both modalities at sparsified levels, which validates the
effectiveness of our sparsification techniques. In general, the visual inputs
perform stronger than textual inputs, which is mainly due to the fact that
visual signals are much more informative. On the other hand, we can still
observe an increase of performance from adding even very few multi-modal
inputs. For example, adding only 5-word to the visual snippet could still get
some performance gains. This implies the complimentary manner from different
modalities from the perspective from strictly controlled inputs. Noticeably,
as an intermediate output from our learnable selection, we can get a few-frame
and few-word summarization of the original video, which is human-
interpretable. We provide more examples and analysis in the following section
to demonstrate this advantage.

###  Qualitative Analysis

figure 4 is mentioned before figure 3 Here we provide visualizations on the
selected frame and/or key words from iVQA dataset. For illustration purposes,
we present the result for single frame selection and 10-word extraction in
Figure , along with their associated questions and the predicted answer.
Answer in green color means the system correctly predicts the answer, while
red color means the system predicts the wrong answer and the ground truth is
in parenthesis. In the successful cases, the sparsified output is able to
capture an appropriate figure for the topic, and the texts also contain words
related to the answer, which leads to the correct answer. In the first failure
case, even though the selected frame contains information related to the
answer “shirt”, the textual component is a distraction, and the system
generates an answer more related to the key words which are closely describing
pipes. In the second failure case, the generated key frame and words are both
irrelevant to the question. This is probably because the question itself is
asking something minor (since most of the video contents are about the
architectures and surroundings) while the model is trained to get information
that is of major interest for the overall dataset and task. Additionally, we
analyze the token importance using the tool provided by [Chefer_2021_CVPR]
which calculates the importance score of each input token w.r.t. to the task
prediction. In Figure 1, we provide some visualization examples where question
words and video frame inputs are highlighted according to their importance
scores. For illustration purposes, we only sample 10 frames in each sample.
Words or frames that are of darker green color means they contribute more to
the prediction. From the given examples, we can see that not every video frame
is of significant importance. The model is able to discard frames that do not
contain any useful information for the question (e.g., in the second example,
only the frames showing the fingers are contributing). On the other hand, in
the example where the scene is relatively stable, we can also observe that the
model focuses mostly on one of these similar frames (as in the third example),
while the rest seems to be diverging. These observations show the potential of
dropping unnecessary video inputs to improve the efficiency, which validates
our motivation.


# Non-normable spaces of analytic functions

Iván Jiménez  Departamento de Matemáticas, Universidad Autónoma de Madrid,
28049 Madrid, Spain<EMAIL_ADDRESS>and  Dragan Vukotić
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid,
Spain<EMAIL_ADDRESS>
(Date: 01 August, 2024.)

###### Abstract.

For each value of $p$ such that $0<p<1$, we give a specific example of two
functions in the Hardy space $H^{p}$ and in the Bergman space $A^{p}$ that do
not satisfy the triangle inequality. For Hardy spaces, this provides a much
simpler proof than the one due to Livingston that involves abstract functional
analysis arguments and an approximation theorem. For Bergman spaces, we have
not been able to locate any examples or proofs in the existing literature.

###### Key words and phrases:

Hardy spaces, Bergman spaces, normable spaces

###### 2020 Mathematics Subject Classification:

30H05

##  1\. Introduction

It is well known that, for a given positive measure $\mu$, the Lebesgue space
$L^{p}(X,\mu)$ is a complete (Banach) space when equipped with the usual norm
$\|f\|_{p}=\left(\int_{X}|f|^{p}\,d\mu\right)^{1/p}$ whenever $1\leq
p<\infty$, while the above expression $\|\cdot\|_{p}$ in general does not
define a norm when $0<p<1$ since it fails to satisfy the triangle inequality.

The standard Hardy and Bergman spaces of analytic functions in the unit disk
${\mathbb{D}}$, denoted respectively by $H^{p}$ and $A^{p}$, can be seen as
closed subspaces of the spaces $L^{p}({\mathbb{T}},dm)$ and
$L^{p}({\mathbb{D}},dA)$, where ${\mathbb{T}}$ denotes the unit circle,
$dm(\theta)=d\theta/(2\pi)$ the normalized arc length measure on it, and
$dA(z)=dx\,dy/\pi$ the normalized area measure on ${\mathbb{D}}$. These spaces
are also complete with respect to their respective $L^{p}$-type norms when
$1\leq p<\infty$, and it is also known that they are not normed space when
$0<p<1$. There are many known monographs or texts that treat Hardy spaces or
Bergman spaces [1, 2, 3, 4, 5, 6, 7, 8, 10] and this fact is mentioned in
passing in many of them.

However, it seems that the proof of this “obvious” fact is not contained in
any of the texts quoted, not even among the exercises. The likely reason is
that this is not so easy to prove in a direct way. The spaces $H^{p}$ and
$A^{p}$ consist of holomorphic functions, which have many rigidity properties
and therefore they cannot be varied in any flexible way, not even on very
small sets. Thus, specific examples are not nearly as easy to construct as in
the context of $L^{p}$ spaces where we have all the freedom of modifying
measurable functions at our pleasure. Also, it is complicated in general to
compute or estimate precisely the norms of functions in such spaces when
$p\neq 2$.

This specific issue was discussed from a different point of view in
Livingston’s paper [9] from the 1950’s. By a well-known theorem of Kolmogorov
from 1934 [11, Theorem 1.39], a topological vector space is normable (has an
equivalent normed topology) if and only if its origin has a convex bounded
neighborhood. It was shown in [9] that, for $0<p<1$, the open unit ball of
$H^{p}$ contains no convex neighborhood of the origin, which implies the non-
normability of the space. To the best of our knowledge, we have not been able
to locate a proof of the analogous well-known fact for $A^{p}$ spaces in the
literature.

It seems useful to have a ‘hard analysis’ proof that the usual expression
$\|\cdot\|_{p}$ is not a norm when $0<p<1$; of course, a ‘soft analysis’
argument seems to be called for in order to prove the stronger fact that
actually no norm can be defined on the same space defining an equivalent
topology. The purpose of this note is to fill this gap in the literature by
giving specific examples of two functions, in both $H^{p}$ and $A^{p}$ spaces
with $0<p<1$, that do not satisfy the triangle inequality. We hope that
graduate students and other young researchers may find useful the examples
given in this note.

##  2\. Preliminary facts

###  2.1. Hardy spaces

Let ${\mathbb{D}}$ denote the unit disc in the complex plane. It is well known
[2, Chapter 1] that for any function $f$ analytic in ${\mathbb{D}}$ and
$0<p<\infty$, the integral means of order $p$ of $f$:

$M_{p}(r;f)=\left(\int_{0}^{2\pi}|f(re^{i\theta})|^{p}\frac{d\theta}{2\pi}\right)^{1/p}$

are increasing functions of $r\in(0,1)$. The Hardy space $H^{p}$ is the set of
all analytic functions in ${\mathbb{D}}$ for which these means have finite
limits: $\|f\|_{H^{p}}=\lim_{r\to 1^{-}}M_{p}(r;f)<\infty$. This is not a true
norm if $0<p<1$ but the same notation and the term “norm” will still be used
in this case. We list several properties of Hardy spaces that will be needed
in the sequel.

It is a well-known fact that $H^{p}$ functions have radial limits:
$\tilde{f}(e^{i\theta})=\lim_{r\to 1^{-}}f(re^{i\theta})$ almost everywhere on
the unit circle and the norm can be computed in terms of these limits (see
[2], [4], or [5]):

$\|f\|_{H^{p}}=\left(\int_{0}^{2\pi}|\tilde{f}(e^{i\theta})|^{p}\frac{d\theta}{2\pi}\right)^{1/p}\,.$

A direct computation shows that the $H^{p}$ norm is invariant under rotations;
in particular, if $g(z)=f(-z)$, then $\|g\|_{H^{p}}=\|f\|_{H^{p}}$.

It is also well known that the composition with the function $z\mapsto z^{2}$
is bounded on any Hardy space. The following useful fact quantifies this.

######  Lemma 1.

Let $f\in H^{p}$ and $h(z)=f(z^{2})$, for $z\in{\mathbb{D}}$. Then $h\in
H^{p}$ and $\|h\|_{H^{p}}=\|f\|_{H^{p}}$.

###### Proof.

Follows by a simple change of variable $t=2\theta$ and periodicity:

$\|h\|_{H^{p}}^{p}=\int_{0}^{2\pi}|\tilde{f}(e^{2i\theta})|^{p}\frac{d\theta}{2\pi}=\frac{1}{2}\int_{0}^{4\pi}|\tilde{f}(e^{it})|^{p}\frac{dt}{2\pi}=\int_{0}^{2\pi}|\tilde{f}(e^{it})|^{p}\frac{dt}{2\pi}=\|f\|_{H^{p}}^{p}\,.$

∎

###  2.2. Bergman spaces

The Bergman “norm” is defined as

(1)
$\|f\|_{A^{p}}^{p}=\int_{\mathbb{D}}|f(z)|^{p}\,dA(z)=\int_{0}^{1}2rM_{p}^{p}(r,f)\,dr\,,$

where $dA$ denotes the normalized Lebesgue area measure on ${\mathbb{D}}$:

$dA(z)=\frac{dx\,dy}{\pi}=\frac{r\,dr\,d\theta}{\pi}\,,\qquad
z=x+iy=re^{i\theta}\,.$

In the most special case when $p=2$ and $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\/$
in ${\mathbb{D}}$, using orthogonality, the norm can be computed explicitly in
terms of the Taylor coefficients:

(2) $\|f\|_{A^{2}}^{2}=\sum_{n=0}^{\infty}\frac{|a_{n}|^{2}}{n+1}\,.$

We refer the readers to [3] or [6] for these basic facts.

Of course, the expression in (1) is a norm only when $1\leq p<\infty$ but we
shall again use the term “norm” also when $0<p<1$. To show explicitly that
this is not a norm for small exponents $p$ becomes more involved than for the
Hardy spaces. This is due to the lack of boundary values of functions in
$A^{p}$ and to the fact that computing the norm involves integration with
respect to area.

It is readily checked that the Bergman “norm” is invariant under rotations,
hence $f(z)$ and $f(-z)$ have the same norm. Again, it is well known that the
composition with the function $z\mapsto z^{2}$ is bounded on any Bergman
space. The following related exact formula will be useful.

######  Lemma 2.

If $h\in A^{p}$, then

$\int_{\mathbb{D}}|h(z)|^{p}\,dA(z)=2\int_{\mathbb{D}}|h(z^{2})|^{p}|z|^{2}\,dA(z)\,.$

###### Proof.

By the obvious change of variable $2\theta=\varphi$ and periodicity, followed
by another change of variable $r^{2}=\rho$, we obtain

$\displaystyle\int_{\mathbb{D}}|h(z^{2})|^{p}|z|^{2}\,dA(z)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}2r^{3}\int_{0}^{2\pi}|h(r^{2}e^{2i\theta})|^{p}\frac{d\theta}{2\pi}\,dr$
$\displaystyle=$
$\displaystyle\int_{0}^{1}2r^{3}\int_{0}^{2\pi}|h(r^{2}e^{i\varphi})|^{p}\frac{d\varphi}{2\pi}\,dr$
$\displaystyle=$ $\displaystyle\int_{0}^{1}\rho M_{p}^{p}(\rho,h)\,d\rho\,,$

and the statement follows. ∎

The following fact is well known. We include an indication of a proof for the
sake of completeness.

######  Lemma 3.

Let $h(z)=(1-z)^{-\alpha}$, $\alpha>0$. Then $h\in A^{p}$ if and only if
$p\alpha<2$.

###### Proof.

This is easily established by integrating in polar coordinates centered at
$z=1$ rather than at the origin: write $z=1-re^{i\theta}$, where
$-\pi/2<\theta<\pi/2$ and $0<r<2\cos\theta$. The rest is also elementary
calculus. ∎

##  3\. Examples

###  3.1. A Hardy space example

It seems intuitively clear that the nice properties that $H^{p}$ spaces enjoy
should allow us to present simple examples. Actually, it turns out that there
is one single example that works for every $p$ with $0<p<1$.

######  Theorem 4.

Let $0<p<1$. Then the functions $f$ and $g$, defined respectively by

$f(z)=\frac{1+z}{1-z}\,,\quad g(z)=-f(-z)=-\frac{1-z}{1+z}\,,$

both belong to $H^{p}$ but fail to satisfy the triangle inequality for
$\|\cdot\|_{H^{p}}$.

###### Proof.

It is a well-known exercise [2, Chapter 1, Problem 1] that, for
$h(z)=\frac{1}{1-z}$, we have $h\in H^{p}$ if and only if $0<p<1$, and the
same is easily seen for the closely related function $f$. By the basic
properties, $\|f\|_{H^{p}}=\|g\|_{H^{p}}$. Also, a direct computation shows
that

$f(z)+g(z)=\frac{4z}{1-z^{2}}\,.$

Taking into account that $|z|=1$ on the unit circle, as well as Lemma 1 and
the inequality $|1+z|\leq 2$ which is actually strict for all
$z\in{\mathbb{T}}\setminus\\{1\\}$, we obtain that

$\|f+g\|_{H^{p}}=\left\|\frac{4}{1-z^{2}}\right\|_{H^{p}}=4\left\|\frac{1}{1-z}\right\|_{H^{p}}>2\left\|\frac{1+z}{1-z}\right\|_{H^{p}}=2\|f\|_{H^{p}}=\|f\|_{H^{p}}+\|g\|_{H^{p}}\,,$

showing that the triangle inequality fails in this case. ∎

Once discovered, the last example may even look trivial. However, it should be
mentioned that it was not the first example of this kind that we found; the
earlier examples required a lot more involved calculations and estimates.
Moreover, the “obvious” example one would first think of does not work: for
the function $h$ mentioned in the proof and the related modified function
$k(z)=-h(-z)$, it can be easily checked by a completely similar argument that
we have equality in the triangle inequality:
$\|h+k\|_{H^{p}}=\|h\|_{H^{p}}+\|k\|_{H^{p}}$.

###  3.2. Bergman space examples

In this case, we do not have one single example covering the entire range of
values of the exponent $p$. Instead, we exhibit two different types of
examples, depending on the value of $p$. The first example is a modification
of the Hardy space example given earlier.

######  Theorem 5.

Let $\frac{1}{2}\leq p<1$ and let $\varepsilon\leq 1$ and
$\/(1-p)/p\leq\varepsilon<2(1-p)/p$. Then the functions $f$ and $g$, given by

$f(z)=\frac{(1+z)^{2-\varepsilon}}{(1-z)^{2+\varepsilon}}\,,\quad
g(z)=-f(-z)=-\frac{(1-z)^{2-\varepsilon}}{(1+z)^{2+\varepsilon}},$

fail to satisfy the triangle inequality for $\|\cdot\|_{A^{p}}$.

###### Proof.

Note that $\varepsilon=1$ if and only if $p=1/2$; otherwise we have a whole
interval to choose the value of $\varepsilon$. Also note that the numerator in
the expression for $f$ is bounded in view of the condition $\varepsilon\leq 1$
while $\varepsilon<2(1-p)/p$ implies that $p(2+\varepsilon)<2$. Thus, $f\in
A^{p}$ by Lemma 3.

We already know that $\|f\|_{A^{p}}=\|g\|_{A^{p}}$. Since

$f(z)+g(z)=\frac{(1+z)^{4}-(1-z)^{4}}{(1-z^{2})^{2+\varepsilon}}=\frac{8z(1+z^{2})}{(1-z^{2})^{2+\varepsilon}}\,,$

the desired inequality $\|f+g\|_{A^{p}}>\|f\|_{A^{p}}+\|g\|_{A^{p}}$ is
equivalent to the statement that $\|f+g\|_{A^{p}}^{p}>2^{p}\|f\|_{A^{p}}^{p}$,
that is,

$2^{3p}\int_{\mathbb{D}}\frac{|z|^{p}|1+z^{2}|^{p}}{|1-z^{2}|^{(2+\varepsilon)p}}\,dA(z)>2^{p}\int_{\mathbb{D}}\frac{|1+z|^{(2-\varepsilon)p}}{|1-z|^{(2+\varepsilon)p}}\,dA(z)=2^{p+1}\int_{\mathbb{D}}\frac{|z|^{2}|1+z^{2}|^{(2-\varepsilon)p}}{|1-z^{2}|^{(2+\varepsilon)p}}\,dA(z)\,,$

by Lemma 2 applied to the function $f$. This is clearly equivalent to

$\int_{\mathbb{D}}\frac{|z|^{p}|1+z^{2}|^{p}\left(2^{2p-1}-|z|^{2-p}|1+z^{2}|^{(1-\varepsilon)p}\right)}{|1-z^{2}|^{(2+\varepsilon)p}}\,dA(z)>0.$

By our choice of $\varepsilon$ and restrictions on $p$, we have
$(2p-1)-(1-\varepsilon)p=p+\varepsilon p-1\geq 0$ and $(1-\varepsilon)p\geq
0$, hence

$2^{2p-1}-|z|^{2-p}|1+z^{2}|^{(1-\varepsilon)p}>2^{2p-1}-2^{(1-\varepsilon)p}\geq
0$

for all $z\in{\mathbb{D}}$, and the desired integral inequality follows. ∎

We now turn to the remaining range of exponents. To this end, the following
simple inequality will be useful.

######  Lemma 6.

If $a$, $b>0$ and $q>1$, then $|a^{q}-b^{q}|\geq|a-b|^{q}$.

###### Proof.

Without loss of generality, we may assume that $a\geq b>0$. Then, writing
$x=a/b$, the inequality reduces to $(x^{q}-1)-(x-1)^{q}\geq 0$, for $x\geq 1$,
and this is easily proved by elementary calculus since the function on the
left-hand side is non-decreasing in $[1,+\infty)$. ∎

Our next example covers the remaining range of values of $p$ and, actually, a
somewhat larger interval of values.

######  Theorem 7.

Let $0<p<\frac{1}{2}$ and define

$f(z)=(1+z)^{4/p}\,,\quad g(z)=-f(-z)=-(1-z)^{4/p}\,,$

choosing the appropriate branch of the complex logarithm so that, say, $\log
1=0$. Then the functions $f$ and $g$ both belong to $A^{p}$ but fail to
satisfy the triangle inequality for $\|\cdot\|_{A^{p}}$.

###### Proof.

Again, it is clear that $\|f\|_{A^{p}}=\|g\|_{A^{p}}$. We can compute this
value by using the formula (2) to obtain

$\|f\|_{A^{p}}^{p}=\int_{\mathbb{D}}|1+2z+z^{2}|^{2}\,dA(z)=1+2+\frac{1}{3}=\frac{10}{3}\,.$

Next, we use the formulas defining the functions $f$ and $g$, the standard
triangle inequality for complex numbers, then employ the elementary inequality
from Lemma 6 with

$q=1/p\,,\quad a=|1+z|^{4}\,,\quad b=|1-z|^{4}\,,$

then some basic algebra of complex numbers, and afterwards express the
integral obtained in polar coordinates and use Fubini’s theorem to obtain:

$\displaystyle\|f+g\|_{A^{p}}^{p}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{D}}|(1+z)^{4/p}-(1-z)^{4/p}|^{p}dA(z)$
$\displaystyle\geq$
$\displaystyle\int_{\mathbb{D}}\left||1+z|^{4/p}-|1-z|^{4/p}\right|^{p}dA(z)$
$\displaystyle\geq$
$\displaystyle\int_{\mathbb{D}}\left||1+z|^{4}-|1-z|^{4}\right|dA(z)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{D}}\left|(1+|z|^{2}+2\operatorname{Re}z)^{2}-(1+|z|^{2}-2\operatorname{Re}z)^{2}\right|dA(z)$
$\displaystyle=$ $\displaystyle
8\int_{\mathbb{D}}(1+|z|^{2})|\operatorname{Re}z|dA(z)$ $\displaystyle=$
$\displaystyle\frac{8}{\pi}\int_{0}^{1}r^{2}(1+r^{2})\,dr\cdot
2\int_{-\pi/2}^{\pi/2}\cos\theta\,d\theta$ $\displaystyle=$
$\displaystyle\frac{2^{8}}{15\pi}$ $\displaystyle>$ $\displaystyle
2^{p}\frac{10}{3}$ $\displaystyle=$
$\displaystyle(\|f\|_{A^{p}}+\|g\|_{A^{p}})^{p}\,,$

whenever $p<\frac{1}{2}$, as is easily seen by computation. Actually, the last
inequality holds for a larger range of values of $p$ which we do not need for
our purpose. ∎

For the sake of simplicity, we have avoided discussing the weighted Bergman
spaces $A^{p}_{\alpha}$ with standard radial weights. It does not appear too
difficult to find related examples in such cases as well.

Acknowledgments. The authors would like to thank Ole F. Brevig and Raymond
Mortini for their interest in the first draft of the paper and for some useful
comments.

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