# Nonreciprocal transport in $U(1)$ gauge theory of high-$T_{c}$ cuprates

Taekoo Oh  RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama
351-0198, Japan  Naoto Nagaosa<EMAIL_ADDRESS>RIKEN Center for Emergent
Matter Science (CEMS), Wako, Saitama 351-0198, Japan

###### Abstract

The nature of the charge carriers in high-$T_{c}$ cuprates is an essential
issue to reveal their novel physical properties and the mechanism of their
superconductivity. However, the experimental probes and the theoretical
analysis have been mostly restricted to the linear responses. On the other
hand, recent observations of Rashba-type spin-orbit coupling (SOC) on the
surface of high-$T_{c}$ cuprates imply the possible nonlinear and
nonreciprocal transport phenomena under in-plane magnetic fields. In this
paper, we study the nonreciprocal transport properties on the surface of
cuprates by employing a $U(1)$ gauge theory framework, where the electrons are
considered to be fractionalized. Our investigation highlights the intricate
variations in nonreciprocal transport with respect to temperatures and
dopings. First, it reveals contrasting behavior of nonreciprocity in each
normal phase. Second, it discerns the tendencies between underdoped and
overdoped superconducting states and their paraconductivity. The complex
behaviors of nonreciprocal transport originate from the spinful and spinless
nature of the charge carriers and their kinetic energy scales. These findings
pave a new avenue to explore the electronic states in high-$T_{c}$ cuprates in
terms of nonreciprocal transport phenomena.

Introduction.— Several decades have passed since the groundbreaking discovery
of high-$T_{c}$ superconductors in doped cuprates Bednorz and Müller (1986).
The parent phase of undoped cuprates is the antiferromagnetic Mott insulator,
and electron or hole doping leads to a complex phase diagram, encompassing the
pseudogap, strange metal, Fermi liquid, and $d$-wave superconductor phases.
[See Fig. 1(a).] Despite extensive research efforts Scalapino (1995); Kastner
_et al._ (1998); Timusk and Statt (1999); Orenstein and Millis (2000);
Damascelli  _et al._ (2003); Campuzano  _et al._ (2004); Lee  _et al._ (2006);
Fischer  _et al._ (2007); Alloul  _et al._ (2009); Armitage  _et al._ (2010),
achieving a consensus on the origin of the phase diversity in cuprates remains
elusive.

Notably, while cuprates are known for their weak spin-orbit coupling (SOC)
compared with strong electronic correlations, recent experiments using spin-
angle-resolved photoemission spectroscopy Gotlieb  _et al._ (2018) unveiled
Rashba-type spin-orbit coupling on the surface of Bi2Sr2CaCu2O8+d. This arises
from local inversion symmetry breaking at the surface of CuO2 layers. Hence,
the potential emergence of nonreciprocal transport in high-$T_{c}$ cuprates is
anticipated when subjected to external magnetic fields, owing to both
inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetry breaking.
Hence, the nonreciprocal transport could be the useful probe to investigate
the physical properties of high-$T_{c}$ cuprates.

In metals, nonlinear nonreciprocal transport manifests through current-
dependent resistance $R$ Rikken  _et al._ (2001); Tokura and Nagaosa (2018);
Wakatsuki  _et al._ (2017); Hoshino  _et al._ (2018), expressed as:

$\displaystyle R=R_{0}(1+\gamma hI).$ (1)

Here, $R_{0}$ represents current-independent resistance, $\gamma$ denotes
nonreciprocity, $h$ stands for the magnetic field, and $I$ corresponds to the
current. This leads to asymmetric nonlinear $I-V$ curves in Fig. 1(b), a
phenomenon known as magnetochiral anisotropy (MCA).

Conversely, in superconductors, nonreciprocal transport encompasses the
nonlinear paraconductivity and the superconducting (SC) diode effect.
Paraconductivity means a rapid increase in conductivity just above the
transition temperature due to SC fluctuations, whose nonlinear part is also
measured by $\gamma$ Wakatsuki  _et al._ (2017); Wakatsuki and Nagaosa (2018).
The SC diode effect measures the disparity in critical currents depending on
the direction of the current flow, quantified by the quality factor $Q$ Cui
_et al._ (2019); Ando  _et al._ (2020); Itahashi  _et al._ (2020); Schumann
_et al._ (2020); Miyasaka  _et al._ (2021); Kawarazaki  _et al._ (2022);
Narita  _et al._ (2022); Masuko  _et al._ (2022); Lin  _et al._ (2022);
Scammell  _et al._ (2022); He  _et al._ (2022); Du  _et al._ (2023); Nadeem
_et al._ (2023),

$\displaystyle Q=\frac{I_{c+}-I_{c-}}{(I_{c+}+I_{c-})/2}.$ (2)

Here, $I_{c\pm}$ represent the critical currents for positive and negative
current directions in Fig. 1(c), respectively.

Figure 1: (a) The phase diagram of high-$T_{c}$ cuprates in $\delta$ and $T$
is divided into four regions. The schematics of (b) MCA and (c) the SC diode
effect. (d) The lattice structure of single CuO2 layer and directions of
$\vec{D}_{ij}\parallel\vec{d}_{ij}$ and $\vec{h}$.

In this Letter, we systematically explore the surface nonreciprocal transport
across the complete phase diagram by employing the Slave boson $U(1)$ gauge
theory of cuprates, where the electrons are considered be fractionalized into
particles, i.e., fermionic spinons and bosonic holons. Upon the introduction
of a magnetic field ($h$) into the system, we find the rich and nontrivial
behaviors of the nonreciprocal transports in the plane of temperature ($T$)
and doping ($\delta$). The phase diagram is partitioned into three regions:
(I) the normal phases, including antiferromagnet (AF), pseudogap (PG), strange
metal (SM), and Fermi liquid (FL) phases observed at high temperatures ($T\gg
T_{c}$); (II) the phase marked by superconducting (SC) fluctuations at
temperatures around $T\gtrsim T_{c}$; and (III) the $d$-wave superconducting
phase existing at $T<T_{c}$ [refer to Fig. 1(a)]. In the AF, FL, and overdoped
SC phases, the carriers are predominantly electrons, while in PG, SM, and
underdoped SC phases, the carriers manifest as bosonic holons Lee and Nagaosa
(1992); Nagaosa and Lee (1992).

Each phase of normal states shows contrasting behavior of nonreciprocal
transport, as for magnitude and dependence on $T$ and $\delta$. Moreover, in
both the SC fluctuation and $d$-wave SC phases, we find the significant
increase of both $\gamma$ for paraconductivity and the quality factor $Q$ for
the SC diode effect in the overdoped regions. Based on the observations, we
provide a global picture for nonreciprocal transport as well as practical
predictions.

These observations are elucidated by the distinct characteristics of the
carrier type and its kinetic energy. The escalation of kinetic energy
amplifies linear conductivity, diminishing nonreciprocal transport.
Additionally, the absence of spin in holons results in an indirect Zeeman
effect, and the symmetry breaking is weaker than electron case. Accordingly,
the magnitude of nonreciprocal transport in normal states is intricately
linked to the kinetic energy of carriers, while its dependence on $T$ and
$\delta$ predominantly comes from their symmetry breaking energy scales.
Similarly, the overdoped SC and its vicinity shows a sharp increase in
nonreciprocal transports compared to the underdoped SC and its vicinity, owing
to the carrier type and the fermion pairing energies.

MCA in normal phases.— To simulate the surface of doped cuprates under
magnetic fields for normal phases at $T\gg T_{c}$, we employ the extended
$t-J$ model $H_{0}=H_{t}+H_{SO}+H_{Z}+H_{S}$ and its associates in a 2D square
lattice. The components can be expressed as:

$\displaystyle H_{t}=$ $\displaystyle-t\sum_{\langle
ij\rangle}[c_{i\sigma}^{\dagger}c_{j\sigma}+h.c.],$ $\displaystyle H_{SO}=$
$\displaystyle i\lambda_{SO}\sum_{\langle
ij\rangle}[c_{i\alpha}^{\dagger}(\vec{d}_{ij}\cdot\vec{\sigma}_{\alpha\beta})c_{j\beta}+h.c.],$
$\displaystyle H_{Z}=$
$\displaystyle-\frac{\vec{h}}{2}\cdot\sum_{i}c_{i\alpha}^{\dagger}\vec{\sigma}_{\alpha\beta}c_{i\beta},\text{
and}$ $\displaystyle H_{S}=$ $\displaystyle\sum_{\langle
ij\rangle}[J\vec{S}_{i}\cdot\vec{S}_{j}+\vec{D}_{ij}\cdot\vec{S}_{i}\times\vec{S}_{j}]-\vec{h}\cdot\sum_{i}\vec{S}_{i}.$
(3)

Here, $H_{t}$ represents the spin-independent hopping, $H_{SO}$ includes spin-
dependent hopping due to SOC, and $H_{Z}$ accounts for the Zeeman effect of
electrons. $H_{S}$ encompasses exchange, Dzyaloshinskii-Moriya (DM), and
Zeeman interactions within the spin system. In Fig. 1(d), we illustrate the 2D
square lattice with the directions of $\vec{D}_{ij}\parallel\vec{d}_{ij}$ and
$\vec{h}\parallel\hat{y}$. The $U(1)$ Slave-boson theory is employed to solve
$H_{0}$ for PG and SM phases. The model for AF and FL phases is derived from
$H_{0}$. The spin exchanges are replaced with local electron spin couplings to
magnetic order, $H_{AF}=-2u\sum_{i}\vec{m_{i}}\cdot\vec{j_{i}}$, where
$\vec{m_{i}}$ is the local magnetic moment and $\vec{j}_{i}$ is the electron
spin.

To obtain the order parameters in the equilibrium state, we utilize self-
consistent mean-field theory. The order parameters in the AF phase are
staggered magnetic moments at A and B sublattices ($m_{A}$, $m_{B}$), while in
the PG or SM phase, they include the spinonic singlet ($\chi$) and triplet
($\xi_{x},\xi_{y}$) hoppings, singlet pairing ($\Delta$), and spinonic/holonic
chemical potentials ($\mu_{F}/\mu_{B}$). Notably, $\vec{h}$ is instrumental in
creating the asymmetry of the electronic band in AF and FL phases, while
$\xi_{x}$ and $\xi_{y}$ contribute significantly to the asymmetry of the
holonic band in PG and SM phases [see Supplementary Information (SI) for
details].

Figure 2: The nonreciprocity $\gamma~{}($T-1A${}^{-1})$ in (a) AF
($\delta\in[0,0.02]$), (b) PG ($\delta\in[0.02,0.12]$), (c) SM
($\delta\in[0.07,0.27]$), and (d) FL ($\delta\in[0.22,0.3]$) phases.

For normal phases, we consider the longitudinal current up to the second order
in an external electric field $\vec{\mathcal{E}}=Re[\vec{E}e^{i\omega t}]$ in
a semiclassical manner. Accounting for SOC and $\vec{h}$, the magnetic space
group reduces to $Pma^{\prime}2^{\prime}$, allowing for the second-order
current along the $x$-axis. The current is defined as
$J_{x}=Re[\sigma_{1}E_{x}e^{i\omega
t}+\sigma_{20}|E_{x}|^{2}+\sigma_{22}E_{x}^{2}e^{2i\omega t}]$, and the
conductivities can be expressed as follows Nakai and Nagaosa (2019):

$\displaystyle\sigma_{1}=$ $\displaystyle-\frac{e^{2}\tau}{1+i\omega\tau}\int
d\epsilon~{}\frac{\partial
f_{0}}{\partial\epsilon}\int_{k}v_{x}^{2}\delta(\epsilon-\epsilon_{k}),$ (4)
$\displaystyle\sigma_{20}=$
$\displaystyle\frac{e^{3}\tau^{2}}{4(1+i\omega\tau)}\int
d\epsilon~{}\frac{\partial^{3}f_{0}}{\partial\epsilon^{3}}\int_{k}v_{x}^{3}\Theta(\epsilon-\epsilon_{k}),$
(5) $\displaystyle\sigma_{22}=$
$\displaystyle\frac{\sigma_{20}}{1+2i\omega\tau}.$ (6)

Here, $\int_{k}\equiv d^{2}k/(2\pi)^{2}$, $\tau$ represents the relaxation
time, $f_{0}$ is the distribution function, $\epsilon_{k}$ is the energy
dispersion, and $v_{x}$ is the $x$-direction of group velocity of the
carriers. When $\omega=0$, nonreciprocity can be approximated as
$\gamma\approx-\frac{\sigma_{22}}{\sigma_{1}^{2}Wh}$, where $W$ represents the
sample width, which is assumed to be $\sim 100~{}\mu m$. Remarkably, $\gamma$
is independent of the relaxation time $\tau$. Based on the measured values
Zhang and Rice (1988); Kastner  _et al._ (1998); Lee  _et al._ (2006); Gotlieb
_et al._ (2018), the parameters are fixed as $J=1$, $t=2$,
$|\mathbf{D}_{ij}|=0.033$, $\lambda_{SO}=0.067$, and $u=10$. The energy unit
is $\sim 0.2~{}eV$, and the lattice constant is $4~{}\AA$. For the AF and FL
phases, the electronic conductivity is computed while for the PG and SM
phases, the holonic conductivity is computed. The magnetic field
$|\vec{h}|\sim 7~{}$T is fixed since $\gamma$ is almost constant within
$10~{}$T. The range of temperatures is from $170$ to $250~{}$K. According to
the phase diagram in Fig. 1(a), we set the doping ranges from $0$ to $0.02$ in
AF phase, from $0.02$ to $0.12$ in PG phase, from $0.07$ to $0.27$ in SM
phase, and from $0.22$ to $0.3$ in FL phase. The phase transition for AF, PG,
and SM phases occurs at $T>300~{}$K in the doping range.

Figure 3:  (a) A guide map for nonreciprocity
$\log_{10}[\gamma~{}($T-1A${}^{-1})]$ in temperatures and dopings for normal
phases. (b) The nonreciprocity $\gamma$ of paraconductivity in $T\gtrsim
T_{c}$ and the quality factor $Q$ of SC diode effect in $T<T_{c}$. The unit of
$h$ is tesla, and $\epsilon=|\epsilon^{\prime}|$.

In Fig. 2, we present the change of $\gamma~{}$(T-1A-1) in temperatures and
dopings for AF, PG, SM, and FL phases, respectively. We observe the
followings. $\gamma$ exhibits a sharp contrast for each phase in magnitude and
the dependence on $T$ and $\delta$. Specifically, the magnitude of $\gamma$ in
PG and SM phases is much larger than that in FL phases, but much smaller than
that in AF phases. As $T$ is lowered, $\gamma$ grows in PG, SM, and AF phases
while it slightly diminishes in FL phase. As $\delta$ increases, $\gamma$
grows in PG and SM phases while it diminishes in AF and FL phases. Figure 3(a)
shows the integrated diagram of $\log_{10}\gamma$ relevant to the phase
diagram in Fig. 1(a).

Paraconductivity and SC diode effect.— Meanwhile, to describe the
paraconductivity and superconducting diode effect, we must formulate the
Ginzburg-Landau (GL) free energy of SC phase. Let us assume that the GL free
energy has the form: $F=\int_{k}(\eta(k)|\phi_{k}|^{2}+\nu(k)|\phi_{k}|^{4})$
with the order parameter $\phi_{k}$. From this free energy, we can compute the
current due to paraconductivity above $T_{c}$ using the following equation
Schmid (1969); Bennemann and Ketterson (2008); Wakatsuki and Nagaosa (2018),

$\displaystyle
J_{x}=\frac{2T_{c}}{V\Gamma}\sum_{k}\int_{-\infty}^{t}dt^{\prime}j_{x}(t)\exp(-\frac{2}{\Gamma}\int_{t^{\prime}}^{t}dt^{\prime\prime}\eta(t^{\prime\prime})).$
(7)

Here, $\Gamma$ represents the coefficient in the time-dependent GL equation,
$V$ is the system volume, $\eta(t)=\eta(\vec{k}-2e\vec{E}t)$, and
$j_{x}(t)\equiv-\partial\eta(t)/\partial A_{x}$ Wakatsuki and Nagaosa (2018).
Contrarily, the supercurrent below $T_{c}$ is computed using the derivative of
the integral kernel of the free energy with respect to momentum, denoted as
$J(k)=-2e[\eta^{\prime}(k)|\phi_{k0}|^{2}+\nu^{\prime}(k)|\phi_{k0}|^{4}]$ He
_et al._ (2022). Here, $\phi_{k0}$ is the order parameter at the momentum $k$
corresponding to the energy minimum He  _et al._ (2022). The critical currents
$I_{c+}$ and $-I_{c-}$ can be achieved at the maximum of $J(k)$ for positive
momentum and the minimum for negative momentum, respectively. The quality
factor is then computed as per Eq. 2.

The free energy for underdoped and overdoped SC phases should be different due
to their distinct carriers and order parameters. In underdoped region, the
carriers are predominantly holons near $T_{c}$. Considering that the
superconducting phase transition occurs due to holon condensation, the free
energy is as follows Nagaosa and Lee (1992):

$\displaystyle
F_{1}=\int_{k}(\alpha_{1}+\omega_{k})|\phi_{k}|^{2}+\frac{\beta_{1}}{2}|\phi_{k}|^{4},$
(8)

Here, $\alpha_{1}=\alpha_{10}T_{c}\epsilon^{\prime}$ and $\beta_{1}$ are the
conventional Ginzburg-Landau parameters. $\phi_{k}$ is the superconducting
order parameter, $\epsilon^{\prime}=(T-T_{c})/T_{c}$, and $\omega_{k}\equiv
a_{1}k^{2}-b_{1}(k_{x}-k_{x}^{3}/6)$ is the holonic energy up to third order
of momentum. By applying proper algebra, we find that the nonreciprocity of
paraconductivity is $\gamma\sim 10-100~{}$T-1A-1, and the quality factor of
the SC diode effect is $Q\sim 10^{-9}h\sqrt{|\epsilon^{\prime}|}$ [see SI for
details].

On the other hand, in overdoped region, the carriers are mostly electrons near
$T_{c}$. The superconducting phase transition here occurs due to the
condensation of electron pairs. Furthermore in an $s$-wave SC, the SC diode
effect can be present due to Rashba SOC He  _et al._ (2022) while the
nonlinear paraconductivity mainly arises from even and odd parity mixing by
$\mathcal{P}$ symmetry breaking Wakatsuki and Nagaosa (2018). Although
cuprates exhibit a $d$-wave SC, the underlying physics remains the same. Thus,
we employ two distinct free energies $F_{2}$ and $F_{3}$ to represent
$T\gtrsim T_{c}$ and $T<T_{c}$, respectively Mineev  _et al._ (1999); Bauer
and Sigrist (2012); Wakatsuki and Nagaosa (2018):

$\displaystyle F_{2}=$
$\displaystyle\int_{k}\sum_{AB}\psi_{Ak}^{*}[g^{-1}_{AB}+L_{Ak}\delta_{AB}]\psi_{Bk},$
(9) $\displaystyle F_{3}=$
$\displaystyle\int_{k}\eta_{3}(k)|\psi_{k}|^{2}+\nu_{3}(k)|\psi_{k}|^{4}.$
(10)

Here, $\psi_{k}$ is the SC order parameter, $A$ and $B$ represent the matrix
elements for even and odd parity order parameters, $g$ is the matrix for even
and odd-parity electron interactions, $\eta_{3}(k)$ is a function of fourth
order in momentum, and $L_{Ak}$ and $\nu_{3}(k)$ are the functions of second
order in momentum. The reciprocity of paraconductivity, $\gamma\sim
10^{5}~{}T^{-1}A^{-1}$ and, the quality factor, $Q\sim
10^{-1}h\sqrt{|\epsilon^{\prime}|}$, is obtained [see SI for details].
Compared to the underdoped case, the nonreciprocal transports sharply
increase. The results are summarized in Fig. 3(b).

Discussion.— Nonreciprocal transport phenomena in high-$T_{c}$ cuprates are
chiefly influenced by two factors: the kinetic energy and the type of
carriers. As the kinetic energy increases, the magnitude of linear
conductivity becomes more prominent, so the nonreciprocal transport
diminishes. On the other hand, considering the spinlessness and indirect
Zeeman effect of holons, nonreciprocal transport of holons will be small due
to the weak symmetry breaking. Specifically, in normal states, the magnitude
of nonreciprocity $\gamma$ is large in AF, intermediate in PG and SM, and
small in FL, because the carriers gain more kinetic energy at a higher doping.
Upon the kinetic energy, the drop of $\gamma$ at the transition from AF to PG
phases can be attributed to the fact that the carrier changes from electron to
holon. On the other hand, $\gamma$ drops significantly at the transition from
SM to FL despite electron carriers in FL phase, since the kinetic energy
increases more rapidly.

Moreover, the disparity in energy scales of symmetry breaking of carriers
dictates the dependence of $\gamma$ on $T$ and $\delta$. In FL phase, the
symmetry breaking energy scale of $h$ and $\lambda_{SO}$ remains constant with
changes in $T$ and $\delta$, leading to a reduction in $\gamma$ as $T$
decreases and $\delta$ increases due to kinetic energy. Similarly, in AF
phase, upon $h$ and $\lambda_{SO}$, the symmetry breaking is governed by the
magnetization which decreases as $T$ increases and $\delta$ decreases. This
results in the enhancement in $\gamma$ as $T$ increases, but $\gamma$ still
reduces in increasing $\delta$ because of the kinetic energy. In contrast, in
the PG and SM phases, the symmetry breaking is attributed to the energy scale
of $\lambda_{SO}\xi_{y}$. This increases as $T$ decreases and $\delta$
increases, resulting in the enhancement of $\gamma$ [see SI for details].

In the superconducting (SC) phase and its vicinity, where the system is
regarded as a boson condensate, the energy scale of symmetry breaking assumes
paramount importance. The breaking of $\mathcal{T}$ and $\mathcal{P}$
symmetries is attributed to $h$ and $\lambda_{SO}$ in electronic systems,
whereas it is linked to $\lambda_{SO}\xi_{y}$ in holonic systems. Notably,
since $h\sim 1~{}meV$ and $\lambda_{SO}\sim 10~{}meV$ are much larger than
$\lambda_{SO}\xi_{y}\sim 1~{}\mu eV$, the symmetry breaking is strong in the
overdoped region. Additionally, as the energy scale is introduced by the
electron pairing $T_{c}\lesssim 70~{}K$, both $\gamma$ for paraconductivity
and the quality factor for the SC diode effect significantly enhance in the
overdoped region.

In light of the theoretical analysis depicted in Figure 3, we propose several
predictions for experimental exploration. Discontinuities, suggested by the
theoretical computations, might be replaced by a smooth but rapid change in
reality, owing to the inherent approximations of the mean-field theory.
Accordingly, we envisage distinctive nonmonotonous $\gamma$ profiles, which
begins with a $U$-shape in underdoped region and ends with a $\lambda$-shape
in overdoped region as a function of $\delta$. Moreover, the spinon pairing
reduces the kinetic energy of holons, PG has a slightly larger nonreciprocity
than SM. This might result in a weak two-hump shape for $\gamma$ in $\delta$
on optimally doped region. Close to $T_{c}$, the surge in $\gamma$ due to
paraconductivity is expected to be more pronounced in the overdoped region
compared to the underdoped region. In the SC phase, the rapid escalation of
$Q$ in the SC diode effect driven by additional doping, is also foreseen.
Lastly, we address a caveat that our argument is not applicable to the bulk 3D
system, since we only consider the 2D surface.

Our extensive theoretical examination underscores the intriguing aspects of
surface nonreciprocal transport in diverse phases of high-$T_{c}$ cuprates and
offers practical insights for future investigations. From the MCA in the AF
phase to the nonlinear paraconductivity and SC diode effect in the SC phase,
these findings illuminate the intricate behavior of high-$T_{c}$ cuprates. We
are confident that these practical expectations will pave the way for
pioneering discoveries in cuprates, serving as valuable tools in advancing the
field of high-$T_{c}$ superconductors.

###### Acknowledgements.

This work was supported by JST, CREST Grant Number JPMJCR1874, Japan.

## Appendices to ”Nonreciprocal transport in $U(1)$ gauge theory of
high-$T_{c}$ cuprates”

###### Contents

  1. .1 $t-J$ model
  2. .2 The associated model derived from $t-J$ model
  3. .3 Nonreciprocal transport
  4. A The free energies of superconductors
    1. A.1 Paraconductivity and SC diode effect
    2. A.2 Underdoped
    3. A.3 Overdoped

The $t-J$ model and its associates

###  .1 $t-J$ model

The surface of doped cuprates is well described by the extended $t-J$ model,
expressed as:

$\displaystyle H=H_{S}+H_{t}+H_{SO}+H_{Z},$ (1)

where

$\displaystyle H_{S}=$ $\displaystyle\sum_{\langle
ij\rangle}[J\vec{S}_{i}\cdot\vec{S}_{j}+\vec{D}_{ij}\cdot\vec{S}_{i}\times\vec{S}_{j}]-\vec{h}\cdot\sum_{i}\vec{S}_{i},$
$\displaystyle H_{t}=$ $\displaystyle-t\sum_{\langle
ij\rangle}[c_{i\alpha}^{\dagger}c_{j\alpha}+h.c.],$ $\displaystyle H_{SO}=$
$\displaystyle i\lambda_{SO}\sum_{\langle
ij\rangle}[c_{i\alpha}^{\dagger}(\vec{d}_{ij}\cdot\vec{\sigma}_{\alpha\beta})c_{j\beta}+h.c.],$
$\displaystyle H_{Z}=$
$\displaystyle-\frac{\vec{h}}{2}\cdot\sum_{i}c_{i\alpha}^{\dagger}\vec{\sigma}_{\alpha\beta}c_{i\beta}.$
(2)

We define the vectors

$\displaystyle\vec{D}_{ij}=-D\hat{y},(j=i+x),D\hat{x},(j=i+y);$
$\displaystyle\vec{d}_{ij}=-\hat{y},(j=i+x),\hat{x},(j=i+y);\text{ and}$
$\displaystyle\vec{h}=h\hat{y}.$ (3)

We employ the Slave-boson U(1) theory as follows,

$\displaystyle
c_{i\alpha}^{\dagger}=f_{i\alpha}^{\dagger}b_{i},c_{i\alpha}=b_{i}^{\dagger}f_{i\alpha},\vec{S}_{i}=\frac{1}{2}f_{i\alpha}^{\dagger}\vec{\sigma}_{\alpha\beta}f_{i\beta}.$
(4)

Here, $f_{i\alpha}$ is fermionic, $b_{i}$ is bosonic. Accordingly, we have

$\displaystyle H_{S}=$
$\displaystyle-\tilde{J}\sum_{i}[A_{i,i+x}^{\dagger}A_{i,i+x}+B_{i,i+x}^{\dagger}B_{i,i+x}$
$\displaystyle+A_{i,i+y}^{\dagger}A_{i,i+y}+B_{i,i+y}^{\dagger}B_{i,i+y}]$
$\displaystyle-\tilde{D}\sum_{i}[-C_{y,i,i+x}^{\dagger}B_{i,i+x}-B_{i,i+x}^{\dagger}C_{y,i,i+x}$
$\displaystyle+C_{x,i,i+y}^{\dagger}B_{i,i+y}+B_{i,i+y}^{\dagger}C_{x,i,i+y}]$
$\displaystyle-\tilde{h}\sum_{i}(-i)(f_{i\uparrow}^{\dagger}f_{i\downarrow}-f_{i\downarrow}^{\dagger}f_{i\uparrow})$
$\displaystyle+\sum_{i}\lambda_{i}(\sum_{\alpha}f_{i\alpha}^{\dagger}f_{i\alpha}+b_{i}^{\dagger}b_{i}-1),$
$\displaystyle H_{t}=$
$\displaystyle-t\sum_{i,\alpha}(f_{i\alpha}^{\dagger}b_{i}b_{i+x}^{\dagger}f_{i+x\alpha}+f_{i\alpha}^{\dagger}b_{i}b_{i+y}^{\dagger}f_{i+y\alpha}+\text{h.c.}),$
$\displaystyle H_{SO}=$
$\displaystyle\lambda_{SO}\sum_{i}(-b_{i+x}^{\dagger}b_{i}(f_{i\uparrow}^{\dagger}f_{i+x\downarrow}-f_{i\downarrow}^{\dagger}f_{i+x\uparrow})$
$\displaystyle+b_{i+y}^{\dagger}b_{i}i(f_{i\uparrow}^{\dagger}f_{i+y\downarrow}+f_{i\downarrow}^{\dagger}f_{i+y\uparrow})+h.c.),\text{
and}$ $\displaystyle H_{Z}=$
$\displaystyle-\tilde{h}\sum_{i}(-i)(f_{i\uparrow}^{\dagger}f_{i\uparrow}-f_{i\downarrow}^{\dagger}f_{i\downarrow})b_{i}^{\dagger}b_{i}.$
(5)

Here, $\tilde{J}=J/4$, $\tilde{D}=D/4$, $\tilde{h}=h/2$, and

$\displaystyle
A_{ij}^{\dagger}=\epsilon_{\alpha\beta}f_{i\alpha}^{\dagger}f_{j\beta}^{\dagger},A_{ij}=\epsilon_{\alpha\beta}f_{j\beta}f_{i\alpha}$
$\displaystyle
B_{ij}^{\dagger}=f_{i\alpha}^{\dagger}f_{j\alpha},B_{ij}=f_{j\alpha}^{\dagger}f_{i\alpha},$
$\displaystyle-
iC_{x,ij}^{\dagger}=f_{i\uparrow}^{\dagger}f_{j\downarrow}+f_{i\downarrow}^{\dagger}f_{j\downarrow},iC_{x,ij}=f_{j\uparrow}^{\dagger}f_{i\downarrow}+f_{j\downarrow}^{\dagger}f_{i\downarrow},$
$\displaystyle
C_{y,ij}^{\dagger}=(f_{i\uparrow}^{\dagger}f_{j\downarrow}-f_{i\downarrow}^{\dagger}f_{j\uparrow}),C_{y,ij}=f_{j\downarrow}^{\dagger}f_{i\uparrow}-f_{j\uparrow}^{\dagger}f_{i\downarrow}.$
(6)

Mean-field theory gives $\Delta=\langle A_{i,i+x}\rangle=-\langle
A_{i,i+y}\rangle=\Delta^{*},\chi=\langle
B_{i,i+x}\rangle=\chi^{*},\xi_{x}=-i\langle
C_{x,ij}^{\dagger}\rangle=\xi_{x}^{*},\xi_{y}=-i\langle
C_{y,ij}^{\dagger}\rangle=\xi_{y}^{*}$, and $m_{y}=-i\langle
f_{i\uparrow}^{\dagger}f_{i\downarrow}-f_{i\downarrow}^{\dagger}f_{i\uparrow}\rangle$.
Here, $\Delta$ denotes the $d$-wave singlet spinon coupling, $\chi$ is the
singlet spinon hopping, $\xi_{x}$ and $\xi_{y}$ are the triplet spinon
hopping, and $m_{y}$ is the magnetization. We arrive at

$\displaystyle H_{F}=$
$\displaystyle-\tilde{J}\sum_{i}[\Delta(A_{i,i+x}^{\dagger}+A_{i,i+x}-A_{i,i+y}^{\dagger}-A_{i,i+y})$
$\displaystyle+\chi(B_{i,i+x}^{\dagger}+B_{i,i+x}+B_{i,i+y}^{\dagger}+B_{i,i+y})]$
$\displaystyle-i\tilde{D}\sum_{i}[-\chi\tilde{C}_{y,i,i+x}^{\dagger}-\xi_{y}B_{i,i+x}+\xi_{y}B_{i,i+x}^{\dagger}$
$\displaystyle+\chi\tilde{C}_{y,i,i+x}+\chi\tilde{C}_{x,i,i+y}^{\dagger}+\xi_{x}B_{i,i+y}-\xi_{x}B_{i,i+y}^{\dagger}$
$\displaystyle-\chi\tilde{C}_{x,i,i+y}]-\tilde{h}\sum_{i}(-i)(f_{i\uparrow}^{\dagger}f_{i\downarrow}-f_{i\downarrow}^{\dagger}f_{i\uparrow})$
$\displaystyle+\mu_{F}\sum_{i}(\sum_{\alpha}f_{i\alpha}^{\dagger}f_{i\alpha}+x-1),$
$\displaystyle H_{B}=$
$\displaystyle-t\chi\sum_{i}(b_{i+x}^{\dagger}b_{i}+b_{i+y}^{\dagger}b_{i}+h.c.)$
$\displaystyle+i\lambda_{SO}\sum_{i}(-\xi_{y}b_{i+x}^{\dagger}b_{i}+\xi_{x}b_{i+y}^{\dagger}b_{i}+h.c.)$
$\displaystyle-\tilde{h}\sum_{i}m_{y}b_{i}^{\dagger}b_{i}+\mu_{B}\sum_{i}(b_{i}^{\dagger}b_{i}-x).$
(7)

where $\tilde{C}^{\dagger}=-iC^{\dagger},\tilde{C}=iC$, and $x=\langle
b_{i}^{\dagger}b_{i}\rangle$ is the hole doping.

When we define $u_{i}=f_{i\uparrow}$ and $d_{i}=f_{i\downarrow}$, the
fermionic Hamiltonian is explicitly

$\displaystyle H_{F}=$
$\displaystyle-\tilde{J}\sum_{i}[\Delta(u_{i}^{\dagger}d_{i+x}^{\dagger}-d_{i}^{\dagger}u_{i+x}^{\dagger}+d_{i+x}u_{i}-u_{i+x}d_{i}$
$\displaystyle-
u_{i}^{\dagger}d_{i+y}^{\dagger}+d_{i}^{\dagger}u_{i+y}^{\dagger}-d_{i+y}u_{i}+u_{i+y}d_{i})$
$\displaystyle+\chi(u_{i}^{\dagger}u_{i+x}+d_{i}^{\dagger}d_{i+x}+u_{i}^{\dagger}u_{i+y}+d_{i}^{\dagger}d_{i+y}+h.c.)]$
$\displaystyle+\tilde{D}\sum_{i}\\{\chi(u_{i}^{\dagger}d_{i+x}-d_{i}^{\dagger}u_{i+x}-u_{i+x}^{\dagger}d_{i}+d_{i+x}^{\dagger}u_{i})$
$\displaystyle+\xi_{y}(-i)(u_{i}^{\dagger}u_{i+x}+d_{i}^{\dagger}d_{i+x}-u_{i+x}^{\dagger}u_{i}-d_{i+x}^{\dagger}d_{i}))$
$\displaystyle+(-i)[\chi(u_{i}^{\dagger}d_{i+y}+d_{i}^{\dagger}u_{i+y}-u_{i+y}^{\dagger}d_{i}-d_{i+y}^{\dagger}u_{i})$
$\displaystyle+\xi_{x}(u_{i+y}^{\dagger}u_{i}+d_{i+y}^{\dagger}d_{i}-u_{i}^{\dagger}u_{i+y}-d_{i}^{\dagger}d_{i+y})]\\}$
$\displaystyle-\tilde{h}\sum_{i}(-i)(u_{i}^{\dagger}d_{i}-d_{i}^{\dagger}u_{i})$
$\displaystyle+\mu_{F}\sum_{i}(u_{i}^{\dagger}u_{i}+d_{i}^{\dagger}d_{i}+x-1),$
(8)

Figure S1: The order parameters of pseudogap phase. (a) The singlet hopping
($\chi$), (b) the singlet coupling ($\Delta$), (c) the triplet coupling
($\xi_{y}$), (d) the magnetization ($M_{y}$), (e) the fermionic chemical
potential ($\mu_{F}$), and (f) the bosonic chemical potential ($\mu_{B}$).

The Fourier transform of fermionic Hamiltonian is

$\displaystyle H_{F}=$
$\displaystyle-2\tilde{J}\sum_{k}[\Delta(u_{k}^{\dagger}d_{-k}^{\dagger}+d_{-k}u_{k})(\cos
k_{x}-\cos k_{y})$
$\displaystyle+\chi(u_{k}^{\dagger}u_{k}+d_{k}^{\dagger}d_{k})(\cos k_{x}+\cos
k_{y})]$
$\displaystyle+2\tilde{D}\sum_{k}\\{\chi[i(u_{k}^{\dagger}d_{k}-d_{k}^{\dagger}u_{k})\sin
k_{x}$ $\displaystyle+(u_{k}^{\dagger}d_{k}+d_{k}^{\dagger}u_{k})\sin k_{y}]$
$\displaystyle+(u_{k}^{\dagger}u_{k}+d_{k}^{\dagger}d_{k})(\xi_{y}\sin
k_{x}-\xi_{x}\sin k_{y})\\}$
$\displaystyle-\tilde{h}\sum_{k}(-i)(u_{k}^{\dagger}d_{k}-d_{k}^{\dagger}u_{k})$
$\displaystyle+\mu_{F}\sum_{k}(u_{k}^{\dagger}u_{k}+d_{k}^{\dagger}d_{k}).$
(9)

In matrix form,

$\displaystyle
H_{F}=\frac{1}{2}\sum_{k}\psi_{k}^{\dagger}\left(\begin{matrix}H_{0}(k)&\Delta_{gap}(k)\\\
\Delta_{gap}^{\dagger}(k)&-H_{0}(-k)\end{matrix}\right)\psi_{k},$ (10)

where $\psi_{k}=[u_{k},d_{k},u_{-k}^{\dagger},d_{-k}^{\dagger}]^{T}$,

$\displaystyle H_{0}(k)=$ $\displaystyle[-2\tilde{J}\chi(\cos k_{x}+\cos
k_{y})+2\tilde{D}(\xi_{y}\sin k_{x}$ $\displaystyle-\xi_{x}\sin
k_{y})+\mu_{F}]\tau_{0}+2\tilde{D}\chi\sin k_{y}\tau_{1}$
$\displaystyle-(\tilde{h}+2\tilde{D}\chi\sin k_{x})\tau_{2},$
$\displaystyle\Delta_{gap}(k)=$
$\displaystyle\left(\begin{matrix}0&\Delta_{d}(k)\\\
-\Delta_{d}(-k)&0\end{matrix}\right),\text{ and}$
$\displaystyle\Delta_{d}(k)=$ $\displaystyle-2\tilde{J}\Delta(\cos k_{x}-\cos
k_{y}).$ (11)

Here, $\tau$ is the Pauli matrix representing the spin degrees of freedom.
When $\sigma$ are the Pauli matrices representing particle-hole degrees of
freedom, we represent

$\displaystyle H(k)=$ $\displaystyle\frac{1}{2}\\{[-2\tilde{J}\chi(\cos
k_{x}+\cos k_{y})+\mu_{F}]\tau_{0}\sigma_{3}$
$\displaystyle+[2\tilde{D}(\xi_{y}\sin k_{x}-\xi_{x}\sin k_{y})]\tau_{0}$
$\displaystyle+2\tilde{D}\chi(\sin k_{y}\tau_{1}-\sin
k_{x}\tau_{2})-\tilde{h}\tau_{2}\sigma_{3}$
$\displaystyle-\Delta_{d}(k)\tau_{2}\sigma_{2}\\}.$ (12)

On the other hand, the Fourier transform for bosonic system gives

$\displaystyle H_{B}=$ $\displaystyle-2t\chi\sum_{k}b_{k}^{\dagger}b_{k}(\cos
k_{x}+\cos k_{y})$
$\displaystyle+\lambda_{SO}\sum_{k}(-\xi_{y}b_{k}^{\dagger}b_{k}\sin
k_{x}+\xi_{x}b_{k}^{\dagger}b_{k}\sin k_{y}))$
$\displaystyle+(-\tilde{h}m_{y}+\mu_{B})\sum_{i}b_{k}^{\dagger}b_{k}.$ (13)

This is a single band system. The bosonic energy is

$\displaystyle\epsilon_{k}=$ $\displaystyle-2t\chi(\cos k_{x}+\cos
k_{y})+\lambda_{SO}(\xi_{x}\sin k_{y}-\xi_{y}\sin k_{x})$
$\displaystyle+\mu_{B}-\tilde{h}m_{y}.$ (14)

Notably, the symmetry is broken by $\lambda_{SO}\xi_{x}$ and
$\lambda_{SO}\xi_{y}$ while the role of $h$ and $m_{y}$ is to shift the energy
band.

Figure S2: The order parameters of strange metal phase. (a) The singlet
hopping ($\chi$), (b) the triplet coupling ($\xi_{y}$), (c) the magnetization
($M_{y}$), (d) the fermionic chemical potential ($\mu_{F}$), and (e) the
bosonic chemical potential ($\mu_{B}$).

The Green function for BdG fermionic Hamiltonian is

$\displaystyle G(i\omega_{n},k)=$
$\displaystyle[i\omega_{n}-H(k)]^{-1}=\sum_{a}\frac{|a,k\rangle\langle
a,k|}{i\omega_{n}-\omega_{k,a}}$ $\displaystyle=$
$\displaystyle\sum_{a}\frac{U_{\alpha,a}U^{\dagger}_{a,\beta}}{i\omega_{n}-\omega_{k,a}}|\alpha,k\rangle\langle\beta,k|,$
(15)

where $\omega_{k,a}$ is the eigenenergy, $a=1,2,3,4$, and $U^{\dagger}HU=D$.
($U_{\alpha,a}=\langle\alpha,k||a,k\rangle,U^{\dagger}_{a,\alpha}=\langle
a,k||\alpha,k\rangle$) The Green function can be represented as

$\displaystyle G_{\alpha\beta}(\tau,k)=$ $\displaystyle-\langle
T_{\tau}\psi_{k,\alpha}(\tau)\psi_{k,\beta}^{\dagger}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\sum_{i\omega_{n}}e^{-i\omega_{n}\tau}G_{\alpha\beta}(i\omega_{n},k).$
(16)

Then,

$\displaystyle G_{\alpha\beta}(0^{-},k)=$
$\displaystyle\langle\psi_{k,\beta}^{\dagger}\psi_{k,\alpha}\rangle=\frac{1}{\beta}\sum_{i\omega_{n}}e^{i\omega_{n}0^{+}}G_{\alpha\beta}(i\omega_{n},k)$
$\displaystyle=$ $\displaystyle\oint\frac{dz}{2\pi
i}n_{F}(z)G_{\alpha\beta}(z,k)$ $\displaystyle=$
$\displaystyle\oint\frac{dz}{2\pi
i}n_{F}(z)\sum_{a}\frac{U_{\alpha,a}U^{\dagger}_{a,\beta}}{z-\omega_{k,a}}$
$\displaystyle=$
$\displaystyle\sum_{a}U_{\alpha,a}n_{F}(\omega_{k,a})U^{\dagger}_{a,\beta}$
(17)

The self-consistent equations for mean-field parameters are following. When we
denote $\int_{k}=\int\frac{d^{2}k}{(2\pi)^{2}}$, the singlet hopping is

$\displaystyle\chi=$ $\displaystyle\frac{1}{4N}\sum_{i}\langle
B_{i,i+x}+B_{i,i+x}^{\dagger}+B_{i,i+y}+B_{i,i+y}^{\dagger}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{2N}\sum_{k}\langle
u_{k}^{\dagger}u_{k}+d_{k}^{\dagger}d_{k}\rangle(\cos k_{x}+\cos k_{y})$
$\displaystyle=$
$\displaystyle\frac{1}{4}\int_{k}\Tr[G(0^{-},k)\tau_{0}\sigma_{3}](\cos
k_{x}+\cos k_{y}).$ (18)

Also, the singlet coupling is

$\displaystyle\Delta=$ $\displaystyle\frac{1}{4N}\sum_{i}\langle
A_{i,i+x}+A_{i,i+x}^{\dagger}-A_{i,i+y}-A_{i,i+y}^{\dagger}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{2N}\sum_{k}\langle
u_{k}^{\dagger}d_{-k}^{\dagger}+d_{-k}u_{k}\rangle(\cos k_{x}-\cos k_{y})$
$\displaystyle=$
$\displaystyle\frac{1}{4}\int_{k}\Tr[G(0^{-},k)(-\tau_{2}\sigma_{2})](\cos
k_{x}-\cos k_{y}).$ (19)

In addition, the triplet hopping is

$\displaystyle\xi_{x}=$
$\displaystyle\frac{1}{2N}\sum_{i}\langle\tilde{C}_{x,i,i+y}+\tilde{C}_{x,i,i+y}^{\dagger}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{N}\sum_{k}\langle
u_{k}^{\dagger}d_{k}+d_{k}^{\dagger}u_{k}\rangle(\cos k_{y})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{k}\Tr[G(0^{-},k)\tau_{1}\sigma_{3}](\cos
k_{y}).$ (20)

Similarly, the other part is

$\displaystyle\xi_{y}=$
$\displaystyle\frac{1}{2N}\sum_{i}\langle\tilde{C}_{y,i,i+x}+\tilde{C}_{y,i,i+x}^{\dagger}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{2N}\sum_{i}(-i)\langle
u_{i}^{\dagger}d_{i+x}-d_{i}^{\dagger}u_{i+x}+u_{i+x}^{\dagger}d_{i}-d_{i+x}^{\dagger}u_{i}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{N}\sum_{k}(-i)\langle
u_{k}^{\dagger}d_{k}-d_{k}^{\dagger}u_{k}\rangle\cos k_{x}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{k}\Tr[G(0^{-},k)\tau_{2}\sigma_{3}]\cos k_{x}.$
(21)

The magnetization is

$\displaystyle m_{x}=$ $\displaystyle\frac{1}{N}\sum_{i}\langle
u_{i}^{\dagger}d_{i}+d_{i}^{\dagger}u_{i}\rangle,$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{k}\langle
u_{k}^{\dagger}d_{k}+d_{k}^{\dagger}u_{k}\rangle,$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{k}\Tr[G(0^{-},k)\tau_{1}\sigma_{3}],$
$\displaystyle m_{y}=$ $\displaystyle\frac{1}{N}\sum_{i}(-i)\langle
u_{i}^{\dagger}d_{i}-d_{i}^{\dagger}u_{i}\rangle,$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{k}(-i)\langle
u_{k}^{\dagger}d_{k}-d_{k}^{\dagger}u_{k}\rangle,$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{k}\Tr[G(0^{-},k)\tau_{2}\sigma_{3}].$ (22)

Finally, the number density constraint is

$\displaystyle 1-x=$ $\displaystyle\frac{1}{N}\sum_{i}\langle
u_{i}^{\dagger}u_{i}+d_{i}^{\dagger}d_{i}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{k}\langle
u_{k}^{\dagger}u_{k}+d_{k}^{\dagger}d_{k}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{k}(\Tr[G(0^{-},k)\tau_{0}\sigma_{3}]+2),$
$\displaystyle-x=$
$\displaystyle\frac{1}{2}\int_{k}\Tr[G(0^{-},k)\tau_{0}\sigma_{3}].$ (23)

On the other hand, the bosonic Green function is given by

$\displaystyle G_{B}(i\epsilon_{n},k)=(i\epsilon_{n}-\epsilon_{k})^{-1}.$ (24)

Then,

$\displaystyle G_{B}(0^{-},k)=$
$\displaystyle\frac{1}{\beta}\sum_{i\epsilon_{n}}e^{i\epsilon_{n}0^{+}}G_{B}(i\omega_{n},k)$
$\displaystyle=$ $\displaystyle-\oint\frac{dz}{2\pi i}n_{B}(z)G_{B}(z,k),$
$\displaystyle=$ $\displaystyle-\oint\frac{dz}{2\pi
i}\frac{n_{B}(z)}{z-\epsilon_{k}}=-n_{B}(\epsilon_{k}).$ (25)

Thus,

$\displaystyle x=\frac{1}{N}\sum_{i}\langle
b_{i}^{\dagger}b_{i}\rangle=\frac{1}{N}\sum_{k}\langle
b_{k}^{\dagger}b_{k}\rangle=\int_{k}n_{B}(\epsilon_{k}).$ (26)

Typically, the energy is known to be $t\sim 400~{}$meV, $J\sim 200$ meV,
$\lambda_{SO}\sim 0.03t$. The parameters we use here are
$t=3,J=1.5,\lambda_{SO}=0.1,|\vec{D}|=0.05$, and $h=0.005$ ($\sim 7-8$ T),
where the unit is $\sim 0.12~{}eV$. The temperature varies from $0.100$ to
$0.168$ ($\sim 150-250$ K). The doping range is $0.02-0.12$ for pseudogap
phase, and $0.07-0.27$ for strange metal phase. The order parameters for
pseudogap and strange metal phases are in Figs. S1 and S2. Notably, for both
cases, $M_{x}$ and $\xi_{x}$ are negligibly small due to the symmetry.

Figure S3: The order parameters of Hubbard model. (a) The antiferromagnetic
order ($M_{A}$), (b) The ferromagnetic order ($M$).

###  .2 The associated model derived from $t-J$ model

The Fermi liquid and antiferromagnet phases are described by the following
model derived from $t-J$ model:

$\displaystyle H=H_{AF}+H_{t}+H_{SO}+H_{z}.$ (27)

Since local electron spin couplings to the magnetic order becomes significant,
we replace $H_{S}$ with

$\displaystyle H_{AF}=-2u\sum_{i}\vec{m}_{i}\cdot\vec{j}_{i}.$ (28)

Here,
$\vec{j}_{i}=\frac{1}{2}c_{i\alpha}^{\dagger}\vec{\sigma}_{\alpha\beta}c_{i\beta}$
and $\vec{m}_{i}=\langle\vec{j}_{i}\rangle$.

For Fermi liquid phase, we let $u=0$. For antiferromagnetic phase, we let
$u=15$ ($u\sim 2$ eV), and acquire $\langle\vec{j}_{i}\rangle$ for two
distinct sublattices. The doping range for Fermi liquid phase is $0.22-0.30$,
while that for antiferromagnetic phase is $0-0.02$. For AF phase, the order
parameters are $M_{A}=|(\vec{m}_{1}-\vec{m}_{2})/2|$ and
$M=|(\vec{m}_{1}+\vec{m}_{2})/2|$. Notably, $\mathcal{T}$ and $\mathcal{P}$
symmetries of the energy bands are broken by $\lambda_{SO}$, $h$, and $M$. We
represent the order parameters in Fig. S3.

Figure S4: Linear (left panels; $V_{x}$) and second-order transport (right
panels; $-C_{x}$) for (a-b) AF, (c-d) PG, (e-f) SM, (g-h) FL phases.

###  .3 Nonreciprocal transport

When the system is subject to the electric field $Re[\vec{E}e^{i\omega t}]$,
the current is represented as $J_{x}=Re[\sigma_{1}E_{x}e^{i\omega
t}+\sigma_{20}|E_{x}|^{2}+\sigma_{22}E_{x}^{2}e^{2i\omega t}]$. Then, in a
semiclassical way, the conductivities are

$\displaystyle\sigma_{1}(\omega)=$
$\displaystyle\frac{e^{2}\tau}{1+i\omega\tau}V_{x},$
$\displaystyle\sigma_{20}(\omega)=$
$\displaystyle\frac{e^{3}\tau^{2}}{4(1+i\omega\tau)}C_{x},\text{ and}$
$\displaystyle\sigma_{22}(\omega)=$
$\displaystyle\frac{\sigma_{20}}{1+2i\omega\tau},$ (29)

where

$\displaystyle V_{x}=$ $\displaystyle\int d\epsilon~{}(-\frac{\partial
f_{0}}{\partial\epsilon})\int_{k}v_{x}^{2}\delta(\epsilon-\epsilon_{k}),\text{
and}$ $\displaystyle C_{x}=$ $\displaystyle\int
d\epsilon~{}\frac{\partial^{3}f_{0}}{\partial\epsilon^{3}}\int_{k}v_{x}^{3}\Theta(\epsilon-\epsilon_{k}).$
(30)

Here, $\int_{k}=\int d^{2}k/(2\pi)^{2}$, $\epsilon_{k}$ is the energy
dispersion, $v_{x}=\frac{1}{\hbar}\frac{\partial\epsilon_{k}}{\partial k_{x}}$
is the group velocity, $f_{0}$ is the distribution function, and $\tau$ is the
lifetime of carriers. We use Fermi-Dirac distribution for AF and FL, but Bose-
Einstein distribution for PG and SM phases. By letting $\hbar$ and the lattice
constant to be a unit, we compute $V_{x}$ and $C_{x}$ for each phase and
represent it in Fig. S4. The nonreciprocity is defined as:

$\displaystyle\gamma\approx-\frac{\sigma_{22}(0)}{\sigma_{1}^{2}(0)Wh}=-\frac{C_{x}}{4eV_{x}^{2}Wh}.$
(31)

Here, $W$ is the sample width which is assumed to be $\sim 100~{}\mu m.$

##  Appendix A The free energies of superconductors

###  A.1 Paraconductivity and SC diode effect

We here describe the Ginzburg-Landau free energies and the nonreciprocal
transports for superconductor phase. The basic form of free energy is

$\displaystyle F=\int_{k}(\eta(k)|\phi_{k}|^{2}+\nu(k)|\phi_{k}|^{4}).$ (32)

Using this free energy, we obtain the nonreciprocity $\gamma$ by computing the
paraconductivity, and the quality factor for SC diode effect. The current from
paraconductivity is defined as:

$\displaystyle
J_{x}=\frac{2T_{c}}{V\Gamma}\sum_{k}\int_{-\infty}^{t}dt^{\prime}j_{x}(t)\exp(-\frac{2}{\Gamma}\int_{t^{\prime}}^{t}dt^{\prime\prime}\eta(t^{\prime\prime})).$
(33)

Here, $\Gamma$ is the coefficient in the time-dependent Ginzburg-Landau
equation, $V$ is the volume of system, $\eta(t)=\eta(\vec{k}-2e\vec{E}t)$, and
$j_{x}(t)\equiv-\partial\eta(t)/\partial A_{x}$.

On the other hand, the supercurrent is computed by derivative of the integral
kernel of the free energy with respect to momentum,

$\displaystyle
J(k)=-2e[\eta^{\prime}(k)|\phi_{k0}|^{2}+\nu^{\prime}(k)|\phi_{k0}|^{4}].$
(34)

Here, $\phi_{k0}$ is the order parameter at the energy minimum for momentum
$k$. The critical currents $I_{c+}$ and $-I_{c-}$ are found at the maximum of
$J(k)$ for positive momentum and the minimum for negative momentum,
respectively. The quality factor is $Q=2(I_{c+}-I_{c-})/(I_{c+}+I_{c-})$.

###  A.2 Underdoped

For underdoped case, the holonic transport is dominant. Because the holon is
bosonic, the superconductor arises when the holonic condensation occurs. The
order parameter for bose condensation is $\langle b\rangle$, where $b$ is the
boson operator. Accordingly, the free energy for holonic superconductor is
simply

$\displaystyle
F_{1}=\int_{k}(\alpha_{1}+\epsilon_{k})|\phi_{k}|^{2}+\frac{\beta_{1}}{2}|\phi_{k}|^{4}.$
(35)

Here, $\phi_{k}$ is the order parameter, $\alpha_{1}=\alpha_{0}T_{c}\epsilon$
and $\beta_{1}$ are conventional Ginzburg-Landau order parameters,
$\alpha_{0}\sim H_{cB}^{2}/8\pi\sim J\delta^{2}$, $\epsilon=(T-T_{c})/T_{c}$,
and $\epsilon_{k}$ is the holonic energy. From Eq. 14, we approximate
$\epsilon_{k}$ up to the third order of momentum near $\Gamma$ point,
$\epsilon_{k}=ak^{2}-\lambda(k_{x}-k_{x}^{3}/6)$, where $a\sim t\chi$,
$\lambda\sim\lambda_{SO}\xi_{y}$.

Using Eq. 33, the current from paraconductivity is given by

$\displaystyle
J=\frac{e^{2}}{16\epsilon}E-\frac{e^{3}\pi}{1536T_{c}\epsilon^{2}}\frac{\lambda}{a}E^{2}.$
(36)

Thus, the nonreciprocity is

$\displaystyle\gamma=\frac{\pi\lambda}{6aheT_{c}W}.$ (37)

The quality factor from SC diode effect, on the other hand, is given by

$\displaystyle Q=\frac{\sqrt{-\alpha_{1}}\lambda}{3\sqrt{3}a^{3/2}}.$ (38)

When we apply $t\sim 3,\chi\sim 1,\lambda\sim 0.1,\xi_{y}\sim
10^{-2}-10^{-1}h,T_{c}\sim 10^{-2}$, $\alpha_{0}\sim J\delta^{2},$ and
$\delta\sim 0.1$, we got $\gamma\sim 10-100~{}T^{-1}A^{-1}$ and $Q\sim
10^{-9}-10^{-8}h\sqrt{-\epsilon}$.

###  A.3 Overdoped

For overdoped case, the electronic transport is dominant. The superconductor
arises when the Cooper pair is formed. The order parameter here is $\langle
cc\rangle$, where $c$ is the fermionic operator. Hence, the free energy could
be obtained by computing the bubble diagrams.

Here, while the SC diode effect comes from Rashba spin-orbit coupling, the
nonlinear paraconductivity is attributed to the parity mixing by $\mathcal{P}$
symmetry breaking. Therefore, we employ two different free energies as for the
temperature range.

Basically, we suppose that the Hamiltonian in spinor basis is
$H_{0}=\sum_{k}c_{k\alpha}^{\dagger}H_{k,\alpha\beta}c_{k\beta}$, where

$\displaystyle
H_{k}=\xi_{k}+a(k_{x}\sigma_{y}-k_{y}\sigma_{x})-\vec{h}\cdot\vec{\sigma}.$
(39)

Here, $\xi_{k}=k^{2}/2m-\mu$. Comparing with Eq. 11, we could let $m^{-1}\sim
J\chi,a\sim D\chi$. The energy dispersion is

$\displaystyle E_{kA}=\xi_{k}+A\sqrt{(ak_{x}-h)^{2}+(ak_{y})^{2}},$ (40)

where $A=\pm 1$.

When $T\gtrsim T_{c}$, in order to consider the parity mixing, we include the
following electron interactions in spinor basis:

$\displaystyle H_{int}^{g}=$
$\displaystyle\frac{1}{2\mathcal{V}}\sum_{\vec{k},\vec{k}^{\prime}}V^{g}(\vec{k},\vec{k}^{\prime})c_{k,\uparrow}^{\dagger}c_{-k,\downarrow}^{\dagger}c_{-k^{\prime},\downarrow}c_{k^{\prime},\uparrow}$
$\displaystyle H_{int}^{u}=$
$\displaystyle\frac{1}{2V}\sum_{\vec{k}\vec{k}^{\prime}}V^{u}_{ij}(\vec{k},\vec{k}^{\prime})(i\sigma_{i}\sigma_{2})_{\alpha\beta}(i\sigma_{j}\sigma_{2})^{\dagger}_{\gamma\delta}$
$\displaystyle\times
c_{k\alpha}^{\dagger}c_{-k\beta}^{\dagger}c_{-k^{\prime}\gamma}c_{-k^{\prime}\delta},$
(41)

Here,
$V_{g}(\vec{k},\vec{k}^{\prime})=-V_{g}(k_{x}^{2}-k_{y}^{2})(k_{x}^{\prime
2}-k_{y}^{\prime 2})/2k^{2}k^{\prime 2}$ and
$V^{u}_{ij}(\vec{k},\vec{k}^{\prime})=V_{u}g_{i}(\vec{k})g_{j}(\vec{k}^{\prime})$,
where $\vec{g}(\vec{k})=(-k_{y},k_{x})/k$.

By transforming the Hamiltonians to the band representation and perform the
calculation for bubble diagram, we acquire the free energy.

$\displaystyle F_{2}=$
$\displaystyle\int_{k}\sum_{AB}\psi_{Ak}^{*}[g^{-1}_{AB}+L_{Ak}\delta_{AB}]\psi_{Bk}.$
(42)

Here,

$\displaystyle g^{-1}=$
$\displaystyle\frac{1}{r_{t}V_{g}}[(\sigma_{0}-\sigma_{x})+r_{t}\sigma_{x}],\text{
and}$ $\displaystyle L_{Ak}=$ $\displaystyle-
N_{A}(S_{1}(T)-K(T)k^{2}+AR(T)hk_{x}).$ (43)

Also,

$\displaystyle S_{1}(T)=$ $\displaystyle\log\frac{2e^{\gamma_{E}}E_{c}}{\pi
T}\approx S_{1}(T_{c})-\epsilon,$ $\displaystyle K(T)=$
$\displaystyle\frac{1}{2}S_{3}(T)\frac{(E_{R}+\mu)}{2m},$ $\displaystyle
R(T)=$ $\displaystyle\frac{1}{2}S_{3}(T)\sqrt{\frac{E_{R}+\mu}{2m}},\text{
and}$ $\displaystyle S_{3}(T)=$
$\displaystyle\frac{7\zeta(3)}{4\pi^{2}T^{2}}.$ (44)

$r_{t}=\frac{2V_{u}}{V_{g}+V_{u}}$, $\gamma_{E}$ is Euler number, $E_{c}$ is
the cutoff energy, $N_{A}$ is the density of states at band $A$, $\zeta(x)$ is
Riemann zeta function, and $E_{R}=ma^{2}/2$ is the Rashba energy. Since the
free energy is a $2\times 2$ matrix, we have two distinct eigenvalues. Only
one of them is physical for small $r_{t}$, so we have

$\displaystyle F_{2}^{\prime}=\int_{k}\psi_{1k}^{*}P_{1}(k)\psi_{1k},$ (45)

where

$\displaystyle P_{1}(k)=$
$\displaystyle\frac{1}{4}(L_{+k}(N_{S}(k)+N_{U}(k))+L_{-k}(N_{S}(k)-N_{U}(k))$
$\displaystyle+2N_{S}(k)S_{1}(T_{c}))+\frac{r_{t}}{16N_{S}(k)S_{1}(T_{c})}$
$\displaystyle\times[\\{L_{+k}(N_{S}(k)+N_{U}(k))+L_{-k}(N_{U}(k)-N_{S}(k))\\}^{2}$
$\displaystyle-4N_{U}^{2}(k)S_{1}^{2}(T_{c})].$ (46)

Here, $N_{S}=N_{+}+N_{-},N_{U}=N_{+}-N_{-}$.

Using Eq. 33 with $F_{2}^{\prime}$, we can obtain the current from
paraconductivity for $\mu>0$,

$\displaystyle J=\frac{e^{2}}{16\epsilon}E-\frac{e^{3}\pi
R(T_{c})r_{t}}{256S_{1}(T_{c})T_{c}\epsilon^{2}}\frac{\mu^{\prime}}{(1+\mu^{\prime})^{3/2}}hE^{2}.$
(47)

However, for $\mu<0$,

$\displaystyle J=\frac{e^{2}}{16\epsilon}E.$ (48)

Here, $\mu^{\prime}=\mu/E_{R}$. Thus, for $\mu>0$,

$\displaystyle\gamma=\frac{\sqrt{E_{R}}\pi
r_{t}S_{3}(T_{c})}{2\sqrt{2}eT_{c}S_{1}(T_{c})W}\frac{\mu^{\prime}}{1+\mu^{\prime}},$
(49)

while for $\mu<0$, $\gamma=0$. Notably, this model cannot be used to obtain
the critical currents, because we employ only up to second order of momentum.

Figure S5: For overdoped case, (a) Normalized $\gamma$ and (b) Normalized $Q$
as a function of $\mu^{\prime}$.

On the other hand, for $T<T_{c}$, we only consider $d$-wave interaction and
obtain the free energy by bubble diagrams as follows:

$\displaystyle F_{3}=$
$\displaystyle\int_{k}\eta_{3}(\vec{k})|\psi_{k}|^{2}+\nu_{3}(\vec{k})|\psi_{k}|^{4}.$
(50)

Here,

$\displaystyle\eta_{3}(k)=$
$\displaystyle~{}\alpha+\gamma_{2}k^{2}+\gamma_{4}k^{4}+h(\kappa_{1}k+\kappa_{3}k^{3}),$
$\displaystyle\nu_{3}(k)=$
$\displaystyle~{}\frac{1}{2}(\beta+h\beta_{1}k+\beta_{2}k^{2}).$ (51)

From the bubble diagram computation, each coefficient is expressed as follows.
For $\mu>0$,

$\displaystyle\alpha_{+}=$ $\displaystyle\frac{m}{\pi}\epsilon,$
$\displaystyle\gamma_{2+}=$
$\displaystyle\frac{7\zeta(3)}{16\pi^{3}}\frac{E_{R}+\mu}{T_{c}^{2}},$
$\displaystyle\kappa_{1+}=$
$\displaystyle-\frac{7\zeta(3)\sqrt{mE_{R}}}{4\sqrt{2}\pi^{3}T_{c}^{2}},$
$\displaystyle\kappa_{3+}=$
$\displaystyle\frac{217\zeta(5)\sqrt{E_{R}}(E_{R}+\mu)}{128\sqrt{2m}\pi^{5}T_{c}^{4}},$
$\displaystyle\gamma_{4+}=$
$\displaystyle-\frac{217\zeta(5)(E_{R}+\mu)^{2}}{1024m\pi^{5}T_{c}^{4}},$
$\displaystyle\beta_{+}=$
$\displaystyle\frac{21\zeta(3)m}{32\pi^{3}T_{c}^{2}},$
$\displaystyle\beta_{1+}=$
$\displaystyle\frac{279\zeta(5)\sqrt{mE_{R}}}{64\sqrt{2}\pi^{5}T_{c}^{4}},$
$\displaystyle\beta_{2+}=$
$\displaystyle-\frac{279\zeta(5)(E_{R}+\mu)}{256\pi^{5}T_{c}^{4}}.$ (52)

For $\mu<0$,

$\displaystyle\alpha_{-}=$
$\displaystyle\frac{\alpha_{+}}{\sqrt{1+\mu^{\prime}}},$
$\displaystyle\gamma_{2-}=$
$\displaystyle\frac{7\zeta(3)}{16\pi^{3}}\frac{\sqrt{E_{R}(E_{R}+\mu)}}{T_{c}^{2}},$
$\displaystyle\kappa_{1-}=$
$\displaystyle-\frac{7\zeta(3)\sqrt{m(E_{R}+\mu)}}{4\sqrt{2}\pi^{3}T_{c}^{2}},$
$\displaystyle\kappa_{3-}=$
$\displaystyle\frac{217\zeta(5)(E_{R}+\mu)^{3/2}}{128\sqrt{2m}\pi^{5}T_{c}^{4}},$
$\displaystyle\gamma_{4-}=$
$\displaystyle\frac{\gamma_{4+}}{\sqrt{1+\mu^{\prime}}},$

and

$\displaystyle\beta_{-}=$
$\displaystyle\frac{\beta_{+}}{\sqrt{1+\mu^{\prime}}},$
$\displaystyle\beta_{1-}=$
$\displaystyle\frac{279\zeta(5)\sqrt{m(E_{R}+\mu)}}{64\sqrt{2}\pi^{5}T_{c}^{4}},$
$\displaystyle\beta_{2-}=$
$\displaystyle-\frac{279\zeta(5)\sqrt{E_{R}(E_{R}+\mu)}}{256\pi^{5}T_{c}^{4}}.$
(53)

The qualify factor for $\mu>0$ is

$\displaystyle
Q=2.63127\frac{h\sqrt{-\epsilon}}{T_{c}}\frac{1}{\sqrt{1+\mu^{\prime}}},$ (54)

and for $\mu<0$,

$\displaystyle Q=2.63127\frac{h\sqrt{-\epsilon}}{T_{c}}\sqrt{1+\mu^{\prime}},$
(55)

Notably, this model gives zero nonlinear paraconductivity.

When we apply that $1~{}T\sim 0.001$, $T_{c}\sim 0.01$, $D\sim 0.05$,
$\chi\sim 1$, $r_{t}\sim 0.1$, and $J\sim 1.5$, one can get $\gamma\sim
10^{5}~{}T^{-1}A^{-1}$ and $Q\sim 10^{-1}h\sqrt{-\epsilon}$. Additionally, we
present the normalized $\gamma$ and $Q$ as a function of $\mu^{\prime}$ in
Fig. S5.

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#  Wrong Colored Vermeer: Color-Symmetric Image Distortion

Hendrik Richter
HTWK Leipzig University of Applied Sciences
D–04251 Leipzig, Germany.
Email<EMAIL_ADDRESS>
###### Abstract

Color symmetry implies that the colors of geometrical objects are assigned
according to their symmetry properties. It is defined by associating the
elements of the symmetry group with a color permutation. I use this concept
for generative art and apply symmetry-consistent color distortions to images
of paintings by Johannes Vermeer. The color permutations are realized as
mappings of the HSV color space onto itself.

## Color Symmetry

Symmetry is an ubiquitous concept in many branches of mathematics, most
clearly visible, for instance, in geometry or group theory or mathematical
physics. Symmetry is even more frequently present in visual arts, for example
in architecture or painting or drawing. Thus, juxtaposing mathematics and art
immediately suggest dealing with symmetry. In the following, I focus on a
special kind of symmetry: color symmetry.

Suppose we have a geometrical object with some symmetry properties. This
implies that the object in invariant under certain transformations. A complete
set of these transformations yields the symmetry group $\mathcal{G}$ of the
geometrical object. Further assume that the geometrical object is (or can be)
colored. Thus, color symmetry means that the coloring of the object is
consistent with the symmetry group. More specifically, an element
$g\in\mathcal{G}$ is called a color symmetry if $g$ is associated with a color
permutation over a color set [7]. In other words, $g$ permutes the colors
consistently by a color permutation assigning which elements of $\mathcal{G}$
change and which elements of $\mathcal{G}$ preserve colors, see for instance
[1, 2, 4, 5, 6] for explicitly addressing the topic of color symmetry in
generative art. In principle, a color symmetry may change all colors or even
preserve all colors. From an artistic point of view, however, color symmetry
is more interesting if we have a permutation which mixes color-changing and
color-preserving elements.

Take as a very simple example a 2D rectangular object as in Figure 1. The
symmetry group of the rectangle is the dihedral group $D_{2}$ of order $4$. A
rotation $\alpha$ turns the rectangle counter-clockwise 180∘, the reflections
$\beta_{h}$ and $\beta_{v}$ mirror the rectangle in the horizontal and
vertical axis. The composed symmetries are from left to right, as for instance
$\beta_{h}\beta_{v}=\alpha$ is a horizontal reflection followed by a vertical
reflection, which equals a rotation. With the identity
$e=\alpha^{2}=\beta_{h}^{2}=\beta_{v}^{2}$ and
$\beta_{h}\beta_{v}=\beta_{v}\beta_{h}=\alpha$, we obtain the symmetry group
$\mathcal{G}=(e,\alpha,\beta_{h},\beta_{v})$. The reflection axes divide the
image into 4 sections, north-east (ne), north-west (nw), south-east (se) and
south-west (sw). Each section can be mapped into any other section by the
elements of the symmetry group. A rotation maps (nw,ne) $\leftrightarrow$
(se,sw), while for a horizontal and a vertical reflection the mapping is
(nw,ne) $\leftrightarrow$ (sw,se) and (nw,sw) $\leftrightarrow$ (ne,se),
respectively. The sections in Figure 1 are further subdivided into
subsections, which can be individually colored. Thus, the rectangles show
simple color symmetries. For instance, in the left image in Figure 1 we have
triangular subsections and a rotation $\alpha$ is associated with preserving
the colors yellow $\mathbf{(y)}$ and purple $\mathbf{(p)}$ and changing blue
$\mathbf{(b)}$ to orange $\mathbf{(o)}$ and orange to blue. Using Cauchy’s
two-line notation, we may describe the color permutation $p_{\alpha}$
associated with the rotation $\alpha$ by
$p_{\alpha}=\left(\begin{smallmatrix}\mathbf{o}&\mathbf{b}&\mathbf{y}&\mathbf{p}\\\
\mathbf{b}&\mathbf{o}&\mathbf{y}&\mathbf{p}\end{smallmatrix}\right)$. Both
reflections change all colors: blue $\leftrightarrow$ yellow, orange
$\leftrightarrow$ purple, purple $\leftrightarrow$ blue and yellow
$\leftrightarrow$ orange. The color permutations $p_{\beta_{h}}$ and
$p_{\beta_{v}}$ associated with the reflections (horizontal and vertical) can
be expressed similarly by a two-line notation.

Color symmetry may likewise apply to other schemes of subdividing the sections
of a rectangle, for instance bubble-like or chessboard-like subsections. The
middle and the right image of Figure 1 shows further examples of color
symmetry where the horizontal (middle image) and vertical (right image)
reflection preserve some colors, while the rotation changes all colors. The
schemes of subdividing the sections may be different in the geometrical form
(triangle, circle, rectangle, etc.) or in the granularity. The granularity is
expressed by the number of subsection $\Lambda$. For instance, we have a
granularity of $\Lambda=\\{8,12,16\\}$ with the 8, 12 and 16 subsections in
the images in Figure 1 (from left to right). In the following I propose to
take the granularity to the level of the pixels composing the image. In other
words, each section can have as many subsections as there are pixels in it.
Thus, for any image composed of pixels color symmetries can be designed, for
instance to pre-existing source images such as paintings.

Figure 1: Simple color symmetry.

Let us consider images composed of pixels where each pixel $p_{k}$ has a color
$c_{k}$ in a HSV color space: $c_{k}=(h_{k},s_{k},v_{k})$. We start with
focusing on the hue $h_{k}$, assuming that the saturation $s_{k}$ and the
value (brightness) $v_{k}$ remain constant for now. The method is equally
applicable to the other elements of the HSV color space and in the visual
results also such mappings are used. We define a color permutation associated
with the symmetry group by a map $f(h_{k})$, which is a mapping of the hue
component of the HSV color space onto itself. A similar mapping of the color
space has recently been used to visualize number-theoretic functions with
portraits of mathematicians [8]. Here, I use general algebraic functions
solely to achieve certain color distortion effects. Thus, by using such a map
$f(h_{k})$ and for rectangular source images, a color symmetry can be imposed
by a color permutation as follows. For each subsection $\ell$,
$\ell=1,2,\ldots,\Lambda$, we set a map $f(h_{k},\ell)$, which maps the hue
component of the $k$-th pixel in subsection $\ell$, and thus changes the color
of the pixel. If the color symmetry stipulates that the color is preserved,
the map is the identity $f(h_{k},\ell)=h_{k}$. For instance, a color
permutation associated with the rotation $\alpha$ can be expressed by a two-
line notation as

$p_{\alpha}=\left(\begin{smallmatrix}f_{i}(h_{k},\ell_{p})\\\
f_{j}(h_{k},\ell_{q})\end{smallmatrix}\right)\quad\forall\\{\ell_{p},\ell_{q}\\}\in\Lambda_{\alpha},\quad\ell_{p}\neq\ell_{q},$
(1)

where $\Lambda_{\alpha}$ is the set of rotation-symmetric subsections. A
rotation maps (nw,ne) $\leftrightarrow$ (se,sw). Thus, the color permutation
(1) implies that the hue $h_{k}$ of all pixels $p_{k}$ in the $\ell_{p}$-th
subsection in either nw or ne take the colors specified by $f_{i}(h_{k})$
while all pixels $p_{k}$ in the rotation-symmetric $\ell_{q}$-th subsection in
either se or sw take the colors specified by $f_{j}(h_{k})$, and vice versa.
As this means that for any pair of colors $f_{i}(h_{k})$ and $h_{j}(h_{k})$, a
rotation is taking the color $f_{i}(h_{k})$ to the color $h_{j}(h_{k})$, such
a color symmetry is even a transitive coloring [7]. If, as a simplification,
the upper map in (1) is the identity $f_{i}(h_{k},\ell_{p})=h_{k}$, then the
color of the $\ell_{p}$-th subsection are identical to the source image, while
the color of the symmetric subsection changes to $f_{j}(h_{k},\ell_{q})$. For
the reflections, the color permutations can be defined likewise. Such a
definition of a color permutation provides an algorithmic template for
symmetry-consistent color distortions of source images.

Figure 2: Vermeer’s Girl with a Pearl Earring, 1665 (upper left) and The Glass
of Wine, 1661 (middle left) and Girl with a Red Hat, 1666 (lower left) with
three color distortions each.

## Visual Results

In this section, the color permutations associated with the rectangular
symmetry group are applied and visual results are presented. As source images
three paintings by Johannes Vermeer (1632-1675) are taken. Apart from reasons
of artistic taste these paintings are particularly suitable for the color
symmetric image distortions I intended to study. Many Vermeer paintings
feature fine-scaled and strong contrasts in color, as for instance
analytically shown for the Girl with a Pearl Earring [3, 9]. The maps varying
the hue consequently create a substantial variety of color shifts in the
sections affected by color symmetry, particularly for the color maps involving
trigonometric functions. Thus, fine-scaled and strong contrasts in color lead
to substantial color distortions. Moreover, if at least the pixels’ saturation
or brightness are preserved, the “color symmetric” sections of the image
resembles the original image, but we also have a kind of alienation or
distancing effect in terms of color. Results of the color changing distortions
are shown in Figure 2. As color maps the functions
$f_{1}(h_{k})=0.45|\sin{(\sqrt{2}\cdot 20\pi h_{k}})|+0.55|\sin{(20\pi
h_{k})}|$, $f_{2}(h_{k})=0.5(1+\sin{(40\pi h_{k})})$ and
$h_{3}(h_{k})=4h_{k}(1-h_{k})$, $f_{4}(h_{k})=4h_{k}(h_{k}-1)+1$,
$f_{5}(h_{k})=0.15(1+\cos{(40\pi h_{k})})+0.5h_{k}$ and
$f_{n}(h_{k})=nh_{k}\mod 1$ are used. In addition, there are some instances
where similar functions are employed to map value or saturation, for instance
to change the dark background in the upper images of Figure 2. The images
obtained still resemble the Vermeer paintings, but also leave behind the
impression of somehow wrong colors.

Figure 3: Wrong colored Vermeer: Color-symmetric image distortions using the
images in Fig. 2.

Figure 3 presents color symmetric image distortions using the Vermeer
paintings. In each image, either a rotation, or a horizontal or a vertical
reflection is realized. Subsections as in Figure 1 are used to create
symmetric color effects, namely triangular (Girl with a Pearl Earring),
chessboard-line (The Glass of Wine) and bubble-like (Girl with a Red Hat)
subsection. Thus, works of generative art emerge in which can be seen as
quotations, homages and reinterpretations of Vermeer’s famous works.

## References

  * [1] V. Adanova, S. Tari. “Extracting Unit Cells from Tilings with Color Symmetries: Case of Counterchange Patterns.” Bridges Conference Proceedings, Jyväskylä, Finland, Aug. 9–13, 2016, pp. 453–456. http://archive.bridgesmathart.org/2016/bridges2016-453.html
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