††thanks: These authors contributed equally to this study.††thanks: These
authors contributed equally to this study.

# First-order transition between the plaquette valence bond solid and
antiferromagnetic phases of the Shastry-Sutherland model

Ning Xi  Department of Physics and Beijing Key Laboratory of Opto-electronic
Functional Materials and Micro-nano Devices, Renmin University of China,
Beijing 100872, China  Hongyu Chen  Department of Physics and Beijing Key
Laboratory of Opto-electronic Functional Materials and Micro-nano Devices,
Renmin University of China, Beijing 100872, China  Z. Y. Xie
<EMAIL_ADDRESS>Department of Physics and Beijing Key Laboratory of
Opto-electronic Functional Materials and Micro-nano Devices, Renmin University
of China, Beijing 100872, China  Rong Yu<EMAIL_ADDRESS>Department of
Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and
Micro-nano Devices, Renmin University of China, Beijing 100872, China

###### Abstract

We study the ground state phase diagram of the Shastry-Sutherland model by
using the variational optimization of the infinite tensor network states, and
find a weakly first-order transition between the plaquette and the
antiferromagnetic states. The full plaquette state strongly competes with the
empty plaquette ground state, with an energy difference less than $10^{-4}J$.
We show a staggered ring exchange interaction that preserves the Shastry-
Sutherland lattice symmetry can stabilize the full plaquette ground state. In
light of this, we propose the triple point where the full plaquette, empty
plaquette, and antiferromagnetic phases meet as a deconfined quantum critical
point.

Introduction. Enhanced quantum fluctuations in frustrated spin system can give
rise to exotic quantum phases, including quantum spin liquid (QSL), valence
bond solid (VBS), and spin nematicity (Diep_Book, ; Mila_Book, ;
Balents_Nature:2010, ; Savary_2016, ; ZhouYi2017, ; Yu_PRL:2015, ). The nature
of these novel quantum phases and related quantum phase transitions have been
extensively studied. Though most transitions can be described within the
standard Ginzburg-Landau-Wilson (GLW) paradigm, it has been proposed that the
transition between a VBS and an antiferromagnetic (AFM) phase is beyond the
GLW scenario, _e.g._ , via a deconfined quantum critical point (DQCP)
(Senthil2004, ). At this point, deconfined fractionalized excitations emerge,
and the enhanced symmetry allows a continuous rotation between the two
distinct order parameters. However, besides some sophisticatedly designed
models (Sandvik2007, ; Meng_EasyplaneDQCP, ; Shao_Science, ), it is still
challenging to realize a DQCP in two-dimensional (2D) frustrated spin systems.

The Shastry-Sutherland (SS) model (Shastry1981, ) is an ideal frustrated spin
model for studying the VBS-AFM transition (Koga2000, ; Chung2001, ;
Pixley2014, ; Corboz2013, ; Boos2019, ; Lee2019, ; yang2021quantum, ;
Sengupta_PRL:2013, ). It is defined on the SS lattice as sketched in Fig 1(a),
and the Hamiltonian reads

$\hat{H}_{{\rm{SS}}}=J\sum_{\langle
i,j\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+J^{\prime}\sum_{\langle\langle
i,j\rangle\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j},$ (1)

where $\mathbf{S}_{i}$ is an $S=1/2$ spin on site $i$, $J$ and $J^{\prime}$
refer to the nearest and next-nearest neighbor couplings, respectively. As
demonstrated in Fig. 1(c), the ground state is found to be a product of dimer
singlets (DSs) (Koga2000, ; Corboz2013, ) along the diagonal directions for
$J/J^{\prime}\lesssim 0.68$. For $J/J^{\prime}\gtrsim 0.68$, the ground state
first changes to a plaquette VBS, then to a Néel AFM with increasing
$J/J^{\prime}$ (Koga2000, ; Corboz2013, ; Lee2019, ). A first-order transition
between the DS and plaquette phases has been verified by various numerical
results. However, the understanding of the nature of the plaquette-AFM
transition remains controversial: A series expansion study (Koga2000, ) found
a second-order transition, while an infinite projected entangled pair state
(iPEPS) tensor network calculation (Corboz2013, ) showed it to be weakly
first-order. A recent DMRG study(Lee2019, ) proposed the transition is through
a DQCP with an emergent $O(4)$ symmetry, but another DMRG
work(yang2021quantum, ) suggested a gapless QSL settles in between the
plaquette and AFM phases.

The SS model is believed to properly describe the quantum magnetism of the
quasi-2D material $\mathrm{SrCu}_{2}(\mathrm{BO}_{3})_{2}$ (Kageyama1999, ;
PhysRevLett.82.3701, ; Koga2000, ). Evidence of evolution from the DS to a
plaquette then to an AFM state under pressure has been cumulated via inelastic
neutron scattering (INS) (Zayed2017, ), nuclear magnetic resonance (NMR)
(Waki2007, ; haravifard2016crystallization, ), Raman scattering (Bettler2020,
), and specific heat (Guo2020, ; Jimenez2021, ) measurements. Some
experimental results imply the intermediate plaquette ground state has a full
plaquette (FPL) pattern, that is, the local singlet spans on the plaquette
with a diagonal $J_{2}$ bond (Zayed2017, ; Bettler2020, ; Cui2021, ). However,
numerical calculations on the SS model (Corboz2013, ; Boos2019, ; Lee2019, ;
yang2021quantum, ) suggest an empty plaquette (EPL) ground state (Fig. 1(c)).
Though it was shown theoretically that the FPL can be stabilized as the ground
state when the SS lattice symmetry is broken (Boos2019, ), the suggested
symmetry breaking has not been observed. Therefore, it is still an open
question how to reconcile the discrepancy between theory and experiments.

In this Letter, we investigate the ground-state phase diagram of the SS model
by using a variational optimization of the infinite tensor network states with
the projected entangled simplex state (PESS) construction. Our calculation
clarifies the intermediate phase to be an EPL, and our results evidence a
weakly first-order EPL-AFM transition at $J/J^{\prime}\approx 0.79$, without a
spin liquid phase in between. Nevertheless, the FPL state is found to be
intimately competing with the EPL ground state, with an energy difference less
than $10^{-4}J$. By including a staggered ring exchange interaction in the
Hamiltonian, we show that with a perturbation that preserves the SS lattice
symmetry, the ground state changes from the EPL to FPL. In light of this
observation, we construct a global phase diagram of this generalized SS model,
and propose the triple point among the full plaquette, empty plaquette, and
antiferromagnetic phases as a deconfined quantum critical point.

Figure 1: (a) Sketch of the SS model. $J$ and $J^{\prime}$ refers to the
nearest- and next-nearest-neighbor Heisenberg couplings. (b) The 16-PESS setup
on the SS lattice. The tensors are placed at the plaquette within the
$J^{\prime}$ coupling. Each $U$ tensor (blue) consists of $4$ physical spin
indices and $4$ auxiliary ones, while an $S$ tensor contains $4$ auxiliary
indices only. (c) The ground-state phase diagram of the SS model. DS, EPL, and
AFM refer to dimer singlet, empty plaquette, and antifferromagnetic phases,
respectively. Both the DS-EPL and EPL-AFM transitions are found to be first-
order. (d) Typical configurations, corresponding spin-spin correlations and
magnetic order parameters of the DS, EPL, FPL, and AFM states in the
calculation. The EPL and FPL have distinct characters of symmetry breaking.

Model and method. We consider the ground states of the SS model defined in Eq.
(1). To investigate the stability of the EPL versus FPL, we also consider a
generalized SS model by including a staggered ring-exchange interaction
$\hat{H}_{{\rm{Q}}}$ between the nearest neighbor spin pairs of a plaquette
with $J^{\prime}$ interaction (see inset of Fig. 4(a)):

$\hat{H}_{{\rm{Q}}}=-Q\sum_{ijkl\in\boxslash}\left(\mathbf{S}_{i}\cdot\mathbf{S}_{j}\right)\left(\mathbf{S}_{k}\cdot\mathbf{S}_{l}\right)+\left(\mathbf{S}_{i}\cdot\mathbf{S}_{k}\right)\left(\mathbf{S}_{j}\cdot\mathbf{S}_{l}\right).$
(2)

The ground states are obtained by using variational optimization of the
16-PESS tensor network states SM . The PESS construction of the tensor network
states has been shown to give an excellent description to the ground state of
frustrated spin systems (Xie2014, ; Liao2017, ). On the SS lattice, the
16-PESS is constructed with $U$ and $S$ tensors as illustrated in Fig. 1(b). A
projection tensor $U$ is defined on a plaquette with the $J^{\prime}$
interaction, and carries the four spins in this plaquette. Four adjacent $U$
tensors are then connected by an entangled simplex tensor $S$. The $S$ tensor
is introduced to describe the entanglement among the spin clusters but itself
does not carry any physical spin degree of freedom. In this work, we find a
$2\times 2$ unit cell with one independent pair of $U$ and $S$ tensors is
sufficient to characterize the ground state. The calculation is performed in
an infinitely large system by employing the translational symmetry.

To determine the ground states, we adopt an advanced variational optimization
method to globally minimize the ground-state energy
${\langle\psi|\hat{H}|\psi\rangle}/{\langle\psi|\psi\rangle}$. We are inspired
by differentiable programming (Liao2019, ; ADSRG2020, ; ADPEPS2021, ), which
can be effectively combined with other well-developed techniques (Jiang2008, ;
Corboz2014, ; Phien2015, ; Corboz2016, ; wang2011cluster, ). To be specific,
the state for optimization is initialized from an arbitrary state or an
approximately converged state obtained from either simple update (Jiang2008, ;
Xie2014, ) or cluster update (wang2011cluster, ). Then we use the corner
transfer matrix renormalization group (CTMRG) method (Corboz2014, ) to
contract the infinite network and get the approximate environment of the local
tensors. After that, we use the quasi-Newton L-BFGS algorithm to minimize the
energy density, which can be effectively implemented by the Zygote package
(Zygote, ). The automatic differentiation provides a global optimization
strategy of the ground state, and is probably more reliable than local
optimization apporaches, especially for critical systems where many competing
states exist.

Figure 2: (a) The ground-state energy per site with $J/J^{\prime}$ at $D=5$.
Blue (red) dashed line is the fitted energy $E_{A}$ ($E_{P}$) in the AFM
(Plaquette) phase (see text). The insert shows the energy differences
$E-E_{A}$ and $E-E_{P}$. (b) The first-order derivative $dE/dJ$ with
$J/J^{\prime}$ shows a clear discontinuity at the plaquette-AFM transition.
(c) The staggered magnetization $m_{m}$ with $J/J^{\prime}$ for $D=5$. Closed
circles show the $m_{m}$ value of lowest energy configurations, while the open
squares are obtained with biased configurations (see maintext) and exhibit
clear hysteresis loop. (d) Same to (c) but for the plaquette order parameter
$m_{p}$. (e) Finite-$D$ analysis and the extrapolated $m_{m}$ with
$J/J^{\prime}$. (f) Extrapolation of the transition points with $1/D$. The
transition point at each finite $D$ is determined from the crossing point of
the $E-E_{A}$ and $E-E_{P}$ curves in the inset of panel (a).

First-order plaquette-AFM transition. As demonstrated in Fig. 1(c), we find
the ground state is the DS for $J/J^{\prime}<0.675$, consistent with previous
works (Lee2019, ; Corboz2013, ; Koga2000, ). Increasing $J/J^{\prime}$ the
ground state first changes to the EPL, then to the AFM at $J/J^{\prime}\approx
0.79$. To examine the plaquette-AFM transition, we calculate the ground-state
energy $E$ and its derivative $dE/dJ$. The results for $D=5$ are shown in Fig.
2(a) and (b), respectively. Though $E$ varies smoothly across the transition,
$dE/dJ$ shows a small discontinuity at $J/J^{\prime}\approx 0.78$, featuring a
weakly first-order transition. We further calculated the order parameters
$m_{m}$ and $m_{p}$ of the AFM and plaquette phases, respectively.

$\displaystyle m_{m}$ $\displaystyle=\sqrt{\langle
m_{m}^{x}\rangle^{2}+\langle m_{m}^{y}\rangle^{2}+\langle
m_{m}^{z}\rangle^{2}},$ (3) $\displaystyle m_{p}$
$\displaystyle=\bigg{|}\sum\limits_{\left\langle
ij\right\rangle\in\boxempty_{A}}\langle\mathbf{S}_{i}\cdot\mathbf{S}_{j}\rangle-\sum\limits_{\left\langle
ij\right\rangle\in\boxempty_{B}}\langle\mathbf{S}_{i}\cdot\mathbf{S}_{j}\rangle\bigg{|}.$
(4)

Here $\left\langle m_{m}^{\alpha}\right\rangle=\frac{1}{N}\sum_{i}\left\langle
e^{i\mathbf{Q}\cdot\mathbf{r}_{i}}S_{r_{i}}^{\alpha}\right\rangle_{r_{i}}$ for
$\alpha=x,y,z$ and $\mathbf{Q}=(\pi,\pi)$, and $\boxempty_{A/B}$ label the two
inequivalent empty plaquettes. As shown by the solid curves with closed
symbols in Fig. 2(c) and (d), $m_{m}$ and $m_{p}$ both exhibit small but
finite abrupt jumps near $J/J^{\prime}\approx 0.78$, consistent with the
$dE/dJ$ result. To further verify the first-order nature of the transition, we
first stabilize the plaquette (AFM) state from our optimization at a
$J/J^{\prime}$ ratio far from the transition, then slowly increase (decrease)
$J/J^{\prime}$. At each step, we take the converged state obtained from last
step as the initial state for optimization. We repeat this procedure until the
system is driven through the transition to the AFM (plaquette) phase. As shown
by the dashed curves with open symbols in Fig. 2(c) and (d), both $m_{m}$ and
$m_{p}$ exhibit clear hysteresis loops, indicating the existence of metastable
states, which is a prominent signature of a first-order transition.

This first-order transition is shown in Fig. 2(e) for all the finite $D$
values we studied, and the large-$D$ limit is obtained by extrapolation. It
shows clearly that in each curve the AFM order parameter $m_{m}$ experiences a
jump. Though the discontinuity $\Delta m_{m}$ at the jump reduces with
increasing $D$, it remains finite in the large-$D$ limit. This evidences a
weakly first-order transition in the thermodynamic limit.

The transition point can alternatively be determined from the calculated
energies at finite $D$. As shown in Fig. 2(a), for each $D$ we fit the
energies away from the transition ($J/J^{\prime}<0.78$ in the plaquette phase
and $J/J^{\prime}>0.79$ in the AFM phase for $D=5$) to polynomial functions SM
. Denote $E_{P}$ and $E_{A}$ to be the fitted energies in the plaquette and
AFM phases, respectively, and define $\delta E_{P(A)}=E-E_{P(A)}$. The
transition point can then be accurately determined as the crossing point of
$\delta E_{P}$ and $\delta E_{A}$, as demonstrated in the inset of Fig. 2(a).
The transition points at finite $D$ values determined in this way are plotted
in Fig. 2(f). The estimated transition points in the large-$D$ limit from two
different fits are both converged to $(J/J^{\prime})_{c}\approx 0.79$, which
is also consistent with the value obtained from $m_{m}$.

Nature of the plaquette ground state. In the parameter regime $0.68\lesssim
J/J^{\prime}\lesssim 0.79$, we are able to stabilize both two plaquette states
in the calculation (see Fig. 1(d)), and we find the energy of the EPL state is
always lower than that of the FPL. Interestingly, near the transition point,
the energies of these two states become very close, as shown in Fig. 3(a). In
the large-$D$ limit, the energy difference is less than $10^{-4}J$. Such a
small energy difference is about the same order to the one between the EPL and
AFM states near the transition (see inset of Fig. 2(a)). This implies that the
FPL state, though never favored as the ground state of the SS model, emerges
as a low-lying competing state near the plaquette-AFM transition. Actually,
besides the FPL, we can also stabilize other metastable states with
competitive energies near the transition. The emergence of these metastable
states suggests enhanced low-energy fluctuations, and is consistent with the
weakly first-order transition we found.

Figure 3: (a) Energies of EPL and FPL states versus $1/D$ at
$J/J^{\prime}=0.77$. The horizontal dashed dot lines show the plaquette
ground-state energies in an iPEPS (Corboz2013, ) and a DMRG (Lee2019, ) study
for comparison. (b) The AFM ground-state energy versus $1/D$ at
$J/J^{\prime}=0.8$. The horizontal dashed dot lines show the AFM ground-state
energies from another DMRG (yang2021quantum, ) and a series expansion
(Koga2000, ) calculation for comparison.

Global phase diagram and possible DQCP. The quasidegeneracy between the EPL
and FPL states prompts that a FPL ground state can be stabilized nearby if we
consider a global phase diagram with an extra tuning parameter. Such a global
phase diagram would help solving the paradox on the nature of the plaquette
ground state between theory and experiments. A previous theoretical work
(Boos2019, ) found a quasi-one-dimensional singlet phase adiabatically
connected to the FPL when the symmetry between the two diagonal $J^{\prime}$
bonds is broken. But the orthogonal lattice distortion accounting for this
symmetry breaking has not been verified experimentally. Here we adopt a
different strategy, namely, to tune the stability of the EPL and FPL phases
without breaking the SS lattice symmetry. For this purpose, we generalize the
SS model by including a ring-exchange-like interaction.

The model Hamiltonian reads $\hat{H}=\hat{H}_{{\rm{SS}}}+\hat{H}_{{\rm{Q}}}$,
where $\hat{H}_{{\rm{SS}}}$ and $\hat{H}_{{\rm{Q}}}$ are given in Eqs. (1) and
(2), respectively. The global phase diagram in the $J$-$Q$ plane is
illustrated in Fig. 4(a). It is shown that the EPL, FPL, and AFM states span a
broad regime, and the ring-exchange term $\hat{H}_{{\rm{Q}}}$ favors the FPL
ground state over the EPL state. We find the FPL-AFM transition is also weakly
first-order, as evidenced by the discontinuity of the ground-state energy
derivative $dE/dQ$ (Fig. 4(b) and (c)). Remarkably, going along the FPL-AFM
transition trajectory toward the triple point among the EPL, FPL, and AFM
phases, the discontinuity of the energy derivative reduces. It is fully
suppressed when extrapolating to the the triple point as shown in Fig. 4(c).
This implies that the triple point is a quantum critical point. Note that our
model preserves the SS lattice symmetry and the three phases break distinct
symmetries. We therefore propose this point as a DQCP, because a continuous
transition between any two ordered phases is prohibited within the LGW
paradigm. Again, because the EPL and FPL breaks different lattice symmetries,
their order parameters can be combined to a complex (two-component) monopole
operator (Lee2019, ). Further taking the three-component spin order parameter
of the AFM phase, this infers an enlarged $SO(5)$ symmetry at the proposed
DQCP.

Figure 4: (a) Ground-state phase diagram of the generalized SS model that
preserves the SS lattice symmetry. Besides the DS phase, EPL, FPL, and AFM
phases are stabilized as the ground states. Symbols show the transition points
and the lines are guides to eyes. The triple point of the EPL, FPL, and AFM
phases is proposed as a DQCP (see text). Inset: Sketch of the generalized SS
model with the staggered ring-exchange-like $\hat{H}_{{\rm{Q}}}$ interaction.
(b) The first-order derivative of the ground-state energy $dE/dQ$ with
$(Q-Q_{c})/J$, where $Q_{c}$ is the AFM-FPL transition point. (c) The
discontinuity of $dE/dQ$ at $Q=Q_{c}$ versus $J^{\prime}/J$. The discontinuity
vanishes when approaching to the triple point, implying it is a DQCP.

Discussions and conclusion. As mentioned in the introduction, the nature of
the transition between the plaquette and AFM phases is under active debate. To
address this issue numerically, it is important to accurately determine the
ground-state energy. In our calculations, the ground state converges quickly
with increasing $D$. Compared to other numerical methods, as shown in Fig.
3(a), the energy of the EPL state at $D=6$ in our calculation is lower than
that of a previous iPEPS work (Ref. (Corboz2013, )) for $D=7$, and close to
the $D=10$ result and a DMRG one (Lee2019, ). For the AFM state shown in Fig.
3(b), the energy at $D=6$ in our work is lower than those of DMRG
(yang2021quantum, ) and series expansion results (Koga2000, ).

The discontinuities of $dE/dJ$ and $m_{m}$ unambiguously show a first-order
EPL-AFM transition in the SS model, and rule out a continuous transition
Lee2019  or an intermediate QSL phase yang2021quantum . The extrapolated
transition points from different estimates are converged in our calculation.
This suggests a QSL regime (yang2021quantum, ), even if exists, should be very
narrow in the parameter space. Our results also shed light to the VBS-AFM
transitions in the square lattice $J_{1}$-$J_{2}$ and checkerboard (CB) models
(Gu_J1J2:2020, ; Wang_PRL:2018, ; Brenig_PRB:2002, ; Fouet_PRB:2003, ;
Berg_PRL:2003, ; Zou_arXiv:2020, ), as both SS and CB models can be realized
by depleting the $J_{2}$ bonds from the $J_{1}$-$J_{2}$ model. In the CB
model, the full and empty plaquettes have a larger energy difference. We then
expect the VBS-AFM transition there to be stronger first-order. In the
$J_{1}$-$J_{2}$ model, however, the VBS-AFM transition should be weaker, or
could even be intervened by an intermediate QSL, because the frustration is
not released by depleting the $J_{2}$ bonds.

In the shaded regime of the global phase diagram in Fig. 4, enormous competing
states with close energies appear. This prevents us from locating the triple
point precisely, and we cannot fully rule out a QSL in this regime. On the
other hand, this suggests enhanced fluctuations which usually emerge near a
QCP. Details of this phase diagram deserves future investigation.

The purpose of studying the model with the ring-exchange-like interaction in
this work is to illustrate the existence of a DQCP by adding a relevant
perturbation to the SS model. Certainly a realistic model for
$\mathrm{SrCu}_{2}(\mathrm{BO}_{3})_{2}$ may contain more complicated
interactions. Our argument for the DQCP is indeed generic: this intriguing
physics applies to any perturbation that preserves the SS lattice symmetry and
can tune the ground state between EPL and FPL. Note that a DQCP does not exist
in models breaking the SS lattice symmetry (Boos2019, ), and our proposal
obviously differs from other theoretical suggestions on realizing a DQCP in
the SS model (Lee2019, ; yang2021quantum, ).

In conclusion, our numerical results unambiguously show a weakly first-order
plaquette-AFM transition in the SS model. Though the ground state favors the
EPL configuration, the FPL is energetically competitive, and can be stabilized
as the ground state under a ring-exchange-like perturbation that preserves the
SS lattice symmetry. Moreover, we provide evidence for a DQCP at the triple
point where the EPL, FPL, and AFM phases meet in the global phase diagram of
the generalized SS model.

Acknowledgments. We thank fruitful discussions with Y. Cui, W. Ding, C. Liu,
B. Normand, L. Wang, Y. Wang, W. Yu, and H. Zou. This work was supported by
the National R&D Program of China (Grants No. 2017YFA0302900 and No.
2016YFA0300500), the National Natural Science Foundation of China (Grants No.
12174441 and No. 11774420), and the Fundamental Research Funds for the Central
Universities and the Research Funds of Renmin University of China (Grants No.
18XNLG24 and No. 20XNLG19).

## References

  * (1) H. T. Diep ed., “Frustrated Spin Systems", World Scientific (2004).
  * (2) C. Lacroix, P. Mendels, and F. Mila ed., “Introduction to Frustrated Magnetism, Materials, Experiments, Theory", Springer (2010).
  * (3) L. Balents, “Spin Liquid in Frustrated Magnets", Nature 464, 199 (2010).
  * (4) L. Savary and L. Balents, “Quantum Spin Liquids: A Review", Rep. Prog. Phys. 80, 016502 (2016).
  * (5) Y. Zhou, K. Kanoda, and T.-K. Ng, “Quantum spin liquid states", Rev. Mod. Phys. 89, 025003 (2017).
  * (6) R. Yu and Q. Si, “Antiferroquadrupolar and Ising-Nematic Orders of a Frustrated Bilinear-Biquadratic Heisenberg Model and Implications for the Magnetism of FeSe", Phys. Rev. Lett. 115, 116401 (2015).
  * (7) T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, “Deconfined quantum critical points", Science 303, 1490 (2004).
  * (8) A. W. Sandvik, “Evidence for deconfined quantum criticality in a two-dimensional heisenberg model with four-spin interactions", Phys. Rev. Lett. 98, 227202 (2007).
  * (9) N. Ma et al., “Dynamical signature of fractionalization at the deconfined quantum critical point", Phys. Rev. B 98, 174421 (2018).
  * (10) H. Shao, W. Guo, and A. W. Sandvik, “Quantum criticality with two length scales", Science 352, 213 (2016).
  * (11) B. S. Shastry and B. Sutherland, “Exact ground state of a quantum mechanical antiferromagnet", Physica B+C 108, 1069 (1981).
  * (12) A. Koga and N. Kawakami, “Quantum Phase Transitions in the Shastry-Sutherland Model for ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}{)}_{2}$", Phys. Rev. Lett. 84, 4461 (2000).
  * (13) C. H. Chung, J. B. Marston, and S. Sachdev, “Quantum phases of the shastry-sutherland antiferromagnet: Application to ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}{)}_{2}$", Phys. Rev. B 64, 134407 (2001).
  * (14) J. H. Pixley, R. Yu, and Q. Si, “Quantum phases of the shastry-sutherland kondo lattice: Implications for the global phase diagram of heavy-fermion metals", Phys. Rev. Lett. 113, 176402 (2014).
  * (15) P. Corboz and F. Mila, “Tensor network study of the shastry-sutherland model in zero magnetic field", Phys. Rev. B 87, 115144 (2013).
  * (16) C. Boos, S. P. G. Crone, I. A. Niesen, P. Corboz, K. P. Schmidt, and F. Mila, “Competition between intermediate plaquette phases in ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}){}_{2}$ under pressure", Phys. Rev. B 100, 10 (2019).
  * (17) J. Y. Lee, Y. Z. You, S. Sachdev, and A. Vishwanath, “Signatures of a Deconfined Phase Transition on the Shastry-Sutherland Lattice: Applications to Quantum Critical SrCu2(BO3)2", Phys. Rev. X 9, 11 (2019).
  * (18) J. Yang, A. W. Sandvik, and L. Wang, “Quantum criticality and spin liquid phase in the shastry-sutherland model", arXiv:2104.08887 (2021).
  * (19) K. Wierschem and P. Sengupta, “Columnar antiferromagnetic order and spin supersolid phase on the extended Shastry-Sutherland lattice", Phys. Rev. Lett. 110, 207207 (2013).
  * (20) H. Kageyama, K. Yoshimura, R. Stern, N. V. Mushnikov, K. Onizuka, M. Kato, K. Kosuge, C. P. Slichter, T. Goto, and Y. Ueda, “Exact Dimer Ground State and Quantized Magnetization Plateaus in the Two-Dimensional Spin System ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}){}_{2}$", Phys. Rev. Lett. 82, 3168 (1999).
  * (21) S. Miyahara and K. Ueda, “Exact Dimer Ground State of the Two Dimensional Heisenberg Spin System ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}){}_{2}$", Phys. Rev. Lett. 82, 3701 (1999).
  * (22) M. E. Zayed et al., “4-spin plaquette singlet state in the Shastry-Sutherland compound SrCu2 (BO3)2", Nat. Phys. 13, 962 (2017).
  * (23) T. Waki, K. Arai, M. Takigawa, Y. Saiga, Y. Uwatoko, H. Kageyama, and Y. Ueda, “A novel ordered phase in SrCu2 (BO3)2 under high pressure", J. Phys. Soc. Jpn. 76, 7 (2007).
  * (24) S. Haravifard, D. Graf, A. E. Feiguin, C. D. Batista, J. C. Lang, D. M. Silevitch, G. Srajer, B. D. Gaulin, H. A. Dabkowska, and T. F. Rosenbaum, “Crystallization of spin superlattices with pressure and field in the layered magnet ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}{)}_{2}$", Nat. Commun. 7, 1 (2016).
  * (25) S. Bettler, L. Stoppel, Z. Yan, S. Gvasaliya, and A. Zheludev, “Sign switching of dimer correlations in ${\mathrm{SrCu}}_{2}{({\mathrm{BO}}_{3})}_{2}$ under hydrostatic pressure", Phys. Rev. Research 2, 012010 (2020).
  * (26) J. Guo, G. Sun, B. Zhao, L. Wang, W. Hong, V. A. Sidorov, N. Ma, Q. Wu, S. Li, Z.-Y. Meng, A. W. Sandvik, and L. Sun, “Quantum Phases of ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}{)}_{2}$ from High-Pressure Thermodynamics", Phys. Rev. Lett. 124, 206602 (2020).
  * (27) J. Larrea Jiménez, S. P. G. Crone, E. Fogh, M. E. Zayed, R. Lortz, E. Pomjakushina, K. Conder, A. M. Lëuchli, L. Weber, S. Wessel et al., “A quantum magnetic analogue to the critical point of water", Nature 592, 370 (2021).
  * (28) Y. Cui et al., “Proximate deconfined quantum critical point in the Shastry-Sutherland material", unpublished.
  * (29) See Supplemental Material [http://link…] for details about the variational optimization and fitting and extrapolation, which also includes Refs. Xie2014 ; Liao2017 ; Jiang2008 ; Phien2015 ; Corboz2016 ; wang2011cluster ; Liao2019 ; Zygote ; Frank2004 .
  * (30) Z. Y. Xie, J. Chen, J. F. Yu, X. Kong, B. Normand, and T. Xiang, “Tensor renormalization of quantum many-body systems using projected entangled simplex states", Phys. Rev. X 4 011025 (2014).
  * (31) H. J. Liao, Z. Y. Xie, J. Chen, Z. Y. Liu, H. D. Xie, R. Z. Huang, B. Normand, and T. Xiang, “Gapless spin-liquid ground state in the $s=1/2$ kagome antiferromagnet", Phys. Rev. Lett. 118, 137202 (2017).
  * (32) H.-J. Liao, J.-G. Liu, L. Wang, and T. Xiang, “Differentiable programming tensor networks", Phys. Rev. X 9, 031041 (2019).
  * (33) Bin-Bin Chen, Yuan Gao, Yi-Bin Guo, Yuzhi Liu, Hui-Hai Zhao, Hai-Jun Liao, Lei Wang, Tao Xiang, Wei Li, and Z. Y. Xie, “Automatic differentiation for second renormalization of tensor networks", Phys. Rev. B 101, 220409(R) (2020).
  * (34) Boris Ponsioen, Fakher F. Assaad, Philippe Corboz, “Automatic differentiation applied to excitations with Projected Entangled Pair States", arXiv:2107.03399
  * (35) H. C. Jiang, Z. Y. Weng, and T. Xiang, “Accurate determination of tensor network state of quantum lattice models in two dimensions", Phys. Rev. Lett. 101, 090603 (2008).
  * (36) P. Corboz, T. M. Rice, and M. Troyer, “Competing states in the t-J model: uniform d-wave state versus stripe state", Phys. Rev. Lett. 113, 046402 (2014).
  * (37) H. N. Phien, J. A. Bengua, H. D. Tuan, P. Corboz, and R. Orús, “Infinite projected entangled pair states algorithm improved: Fast full update and gauge fixing", Phys. Rev. B 92, 035142 (2015).
  * (38) P. Corboz, “Variational optimization with infinite projected entangled-pair states", Phys. Rev. B 94, 035133 (2016).
  * (39) L. Wang and F. Verstraete, “Cluster update for tensor network states", arXiv:1110.4362 (2011).
  * (40) Zygote package,  https://fluxml.ai/Zygote.jl/dev/.
  * (41) F. Verstraete, and J. I. Cirac, “Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions", arXiv:cond-mat/0407066 (2004).
  * (42) W.-Y. Liu, S.-S. Gong, Y.-B. Li, D. Poilblanc, W.-Q. Chen, and Z.-C. Gu, “Gapless quantum spin liquid and global phase diagram of the spin-1/2 $J_{1}$-$J_{2}$ square antiferromagnetic Heisenberg model", arXiv:2009.01821 (2020).
  * (43) L. Wang and A. W. Sandvik, “Critical level crossings and gapless spin liquid in the square-lattice spin-1/2 $J_{1}$-$J_{2}$ Heisenberg antiferromagnet", Phys. Rev. Lett. 121, 107202 (2018).
  * (44) W. Brenig and A. Honecker, “Planar pyrochlore: A strong-coupling analysis", Phys. Rev. B 65, 140407 (2002).
  * (45) J.-B. Fouet, M. Mambrini, P. Sindzingre, and C. Lhuillier, “Planar pyrochlore: A valence-bond crystal", Phys. Rev. B 67, 054411 (2003).
  * (46) E. Berg, E. Altman, and A. Auerbach, “Singlet Excitations in Pyrochlore: A Study of Quantum Frustration", Phys. Rev. Lett. 90, 147204 (2003).
  * (47) H. Zou, F. Yang, and W. Ku, “Nearly degenerate ground states of a checkerboard antiferromagnet and their bosonic interpretation", arXiv:2011.06520 (2020).

## SUPPLEMENTAL MATERIAL – First-order transition between the plaquette
valence bond solid and antiferromagnetic phases of the Shastry-Sutherland
model

###  S .1 Variational Optimization for the 16-PESS tensor network states

To find the tensor-network-state representation of a target state, many
optimization methods, including time evolution with simple update (SU)
[30,31,35],cluster update (CU)[39],full update(FU)[37,38], and traditional
variational optimization [38,41],have been developed. In this work, we adopt
an advanced variational optimization of the tensor network states with the
projected entangled simplex state (PESS) construction. This is inspired by the
differentiable programming [32-34], which has shown great potential in the
study of classical systems and quantum many-body systems. Here we construct
the 16-PESS tensor network states on a square lattice as shown in Fig. 1(b) of
the main text. We choose a 4-site cluster as a unit cell expressed by a $U$
tensor. Every four adjacent units are connected by an entangled simplex tensor
$S$. The simplex tensors $S$ are used to describe the entanglement of these
clusters. The $U$ and $S$ tensors are all placed at plaquettes with the
$J^{\prime}$ coupling.

The optimization procedure starts from a PESS tensor network state, which can
be obtained from a random initilization or a state after time evolution
(combined with SU, CU, and FU). The variational process further optimizes the
expectation value of the energy associated with the PESS state iteratively,
until the energy is converged. The state with the lowest energy will be
eventually selected as the candidate ground state of the system for a given
auxiliary bond dimension $D$. Deep inside a phase, the nature of the ground
state and the ground-state energy is not sensitive to the way of
initialization. However, when the system is close to a phase transition, the
converged state in usual optimization process may heavily depend on the
initial states, as being trapped in local minimum is a very common problem. To
relieve this issue, careful variational treatment is very important.

Figure S1: Tensor-diagram representation of the norm of the variational state
${\langle\psi|\psi\rangle}$ and the expectation value of the local energy
$E_{i}={\langle\psi|\hat{H_{i}}|\psi\rangle}/{\langle\psi|\psi\rangle}$. The
blue and red dots refer to the $U$ and $S$ tensors in the 16-PESS
construction, respectively, and the grey dots denote the environment tensors
constructed by CTMRG. The blue rectangle sandwiched between the two layers of
network in the right panel refers to the local Hamiltonian operator $H_{i}$.

The basic idea of our variational process is to minimize the ground-state
energy density
$E_{i}={\langle\psi|\hat{H_{i}}|\psi\rangle}/{\langle\psi|\psi\rangle}$
globally. Within the 16-PESS tensor network construction, $E_{i}$ and the norm
of the variational state $\langle\psi|\psi\rangle$ can be effectively
evaluated, as illustrated in Fig. S1. The total ground-state energy $E$ is
then obtained by summing over the local energies. For the specific details of
the optimization, we use the corner transfer matrix renormalization group
(CTMRG) [38] method to contract the infinite network and get the approximate
environment of the local tensors. After that, we use the quasi-Newton L-BFGS
algorithm to search for the minimum of energy expectation value, in which an
automatic gradient method supported by Zygote [40] is applied. This automatic
differentiation program can compute $\partial E_{i}/\partial U(S)$
effectively. This process involves nonlinear optimization and it is the key
point for a global minimization.

Figure S2: Convergence of expectation values versus $\chi$ with $D=5$. The
corner mark $c$ here denotes the final (convergent) value (with the largest
$\chi$ we employed). (a): Convergence of the energy as a function of $\chi$.
(b): Convergence of the antiferromagnetic order parameter $m_{m}$ as a
function of $\chi$. (c): Convergence of the plaquette order parameters
$m_{p}$($\widetilde{m}_{P}$) as a function of $\chi$. Here $\widetilde{m}_{P}$
is the order parameter of the FPL state, defined as
$\widetilde{m}_{P}=\sum\limits_{\left\langle
ij\right\rangle\in\boxslash}\langle\mathbf{S}_{i}\cdot\mathbf{S}_{j}\rangle-\sum\limits_{\left\langle
ij\right\rangle\in\boxbslash}\langle\mathbf{S}_{i}\cdot\mathbf{S}_{j}\rangle$
and $m_{p}$ is the order parameter of the EPL state, defined in Eq. (4) of the
main text.

When the optimization is converged, we further use the CTMRG to contract the
infinite network and calculate the expectation value of energy. To obtain a
more reliable extrapolation and comparison, we push the CTMRG truncation
dimension $\chi$ to a sufficiently large value. In the practical calculation,
we use $\chi$ as large as $160$ for convergence. As demonstrated in Fig. S2,
all the quantities are converged within $10^{-7}$.

###  S .2 Fitting And Extrapolation

We can estimate the transition point from the discontinuity of the order
parameters as shown in Fig. 2 in the main text. To further confirm the result,
we can use the level-crossing method described as follows, to obtain a more
accurate result.

For each $D$, we fit the energies obtained deep in each phase to a cubic
polynomial function. Define the energy difference $\delta
E_{P(A)}=E-E_{P(A)}$, where $E$ is the calculated energy, and $E_{P(A)}$ is
the fitted energy in the PL (AFM) state. The transition point can then be
determined as the crossing point of $\delta E_{P}$ and $\delta E_{A}$ for this
transition (see also main text). The crossing points for $D=$4, 5, 6 are shown
in Fig. S3, respectively. These data are then used for extrapolating the
transition point in the large-$D$ limit, as shown in Fig. 2(f) of the main
text. The transition point for infinite $D$ value is then estimated to be
$J/J^{\prime}\approx 0.79$ from both exponential and power-law fittings.

Figure S3: The energy differences $\delta E_{A}$ and $\delta E_{P}$ versus
$J/J^{\prime}$ at $D=$ 4, 5, 6 for (a), (b), (c) respectively.


* [FLJ∗22] Fu J., Li S., Jiang Y., Lin K.-Y., Qian C., Loy C. C., Wu W., Liu Z.:  Stylegan-human: A data-centric odyssey of human generation.  In _European Conference on Computer Vision_ (2022).
  * [FXZ∗24] Fei B., Xu J., Zhang R., Zhou Q., Yang W., He Y.:  3d gaussian as a new vision era: A survey.  _arXiv preprint arXiv:2402.07181_ (2024).
  * [GAA∗23] Gal R., Alaluf Y., Atzmon Y., Patashnik O., Bermano A. H., Chechik G., Cohen-Or D.:  An image is worth one word: Personalizing text-to-image generation using textual inversion.  In _ICLR_ (2023).
  * [GEVDM18] Graham B., Engelcke M., Van Der Maaten L.:  3d semantic segmentation with submanifold sparse convolutional networks.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ (2018), pp. 9224–9232.
  * [GII∗21] Grigorev A., Iskakov K., Ianina A., Bashirov R., Zakharkin I., Vakhitov A., Lempitsky V.:  Stylepeople: A generative model of fullbody human avatars.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2021), pp. 5151–5160.
  * [GJC∗23] Guo C., Jiang T., Chen X., Song J., Hilliges O.:  Vid2avatar: 3d avatar reconstruction from videos in the wild via self-supervised scene decomposition.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2023), pp. 12858–12868.
  * [GL23] Guédon A., Lepetit V.:  Sugar: Surface-aligned gaussian splatting for efficient 3d mesh reconstruction and high-quality mesh rendering.  _arXiv preprint arXiv:2311.12775_ (2023).
  * [GNK18] Güler R. A., Neverova N., Kokkinos I.:  Densepose: Dense human pose estimation in the wild.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2018).
  * [GPAM∗14] Goodfellow I., Pouget-Abadie J., Mirza M., Xu B., Warde-Farley D., Ozair S., Courville A., Bengio Y.:  Generative adversarial nets.  _Advances in neural information processing systems 27_ (2014).
  * [GPR∗23] Goel S., Pavlakos G., Rajasegaran J., Kanazawa* A., Malik* J.:  Humans in 4D: Reconstructing and tracking humans with transformers.  In _International Conference on Computer Vision (ICCV)_ (2023).
  * [GSW∗22] Gao J., Shen T., Wang Z., Chen W., Yin K., Li D., Litany O., Gojcic Z., Fidler S.:  Get3d: A generative model of high quality 3d textured shapes learned from images.  In _Advances in Neural Information Processing Systems_ (2022).
  * [GWH∗20] Guo Y., Wang H., Hu Q., Liu H., Liu L., Bennamoun M.:  Deep learning for 3d point clouds: A survey.  _IEEE transactions on pattern analysis and machine intelligence 43_ , 12 (2020), 4338–4364.
  * [GYK∗22] Gao X., Yang J., Kim J., Peng S., Liu Z., Tong X.:  Mps-nerf: Generalizable 3d human rendering from multiview images.  _IEEE Transactions on Pattern Analysis and Machine Intelligence_ (2022).
  * [GZH∗23] Guo X., Zheng M., Hou L., Gao Y., Deng Y., Ma C., Hu W., Zha Z., Huang H., Wan P., et al.:  I2v-adapter: A general image-to-video adapter for video diffusion models.  _arXiv preprint arXiv:2312.16693_ (2023).
  * [GZL∗24] Gong L., Zhu Y., Li W., Kang X., Wang B., Ge T., Zheng B.:  Atomovideo: High fidelity image-to-video generation.  _arXiv preprint arXiv:2403.01800_ (2024).
  * [GZX∗22] Gao X., Zhong C., Xiang J., Hong Y., Guo Y., Zhang J.:  Reconstructing personalized semantic facial nerf models from monocular video.  _ACM Transactions on Graphics (TOG) 41_ , 6 (2022), 1–12.
  * [HCH∗23] Han X., Cao Y., Han K., Zhu X., Deng J., Song Y.-Z., Xiang T., Wong K.-Y. K.:  Headsculpt: Crafting 3d head avatars with text.  _arXiv preprint arXiv:2306.03038_ (2023).
  * [HCJS20] He T., Collomosse J., Jin H., Soatto S.:  Geo-pifu: Geometry and pixel aligned implicit functions for single-view human reconstruction.  In _Advances on Neural Information Processing Systems_ (2020).
  * [HCL∗22] Hong F., Chen Z., Lan Y., Pan L., Liu Z.:  Eva3d: Compositional 3d human generation from 2d image collections.  _arXiv preprint arXiv:2210.04888_ (2022).
  * [HHL∗23] Hu J., Hui K.-H., Liu Z., Zhang H., Fu C.-W.:  Clipxplore: Coupled clip and shape spaces for 3d shape exploration.  In _SIGGRAPH Asia 2023 Conference Papers_ (2023), pp. 1–12.
  * [HHP∗23] Hu S., Hong F., Pan L., Mei H., Yang L., Liu Z.:  Sherf: Generalizable human nerf from a single image.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ (2023), pp. 9352–9364.
  * [HL23] Hu S., Liu Z.:  Gauhuman: Articulated gaussian splatting from monocular human videos.  _arXiv preprint arXiv:2312.02973_ (2023).
  * [HMZA21] Huang X., Mei G., Zhang J., Abbas R.:  A comprehensive survey on point cloud registration.  _arXiv preprint arXiv:2103.02690_ (2021).
  * [HSW∗21] Hu E. J., Shen Y., Wallis P., Allen-Zhu Z., Li Y., Wang S., Wang L., Chen W.:  Lora: Low-rank adaptation of large language models.  _arXiv preprint arXiv:2106.09685_ (2021).
  * [HSY∗23] Huang Y.-H., Sun Y.-T., Yang Z., Lyu X., Cao Y.-P., Qi X.:  Sc-gs: Sparse-controlled gaussian splatting for editable dynamic scenes.  _arXiv preprint arXiv:2312.14937_ (2023).
  * [HSZ∗23] Huang X., Shao R., Zhang Q., Zhang H., Feng Y., Liu Y., Wang Q.:  Humannorm: Learning normal diffusion model for high-quality and realistic 3d human generation.  _arXiv preprint arXiv:2310.01406_ (2023).
  * [HTC∗24] Hong F., Tang J., Cao Z., Shi M., Wu T., Chen Z., Wang T., Pan L., Lin D., Liu Z.:  3dtopia: Large text-to-3d generation model with hybrid diffusion priors.  _arXiv preprint arXiv:2403.02234_ (2024).
  * [HTE∗23] Haque A., Tancik M., Efros A. A., Holynski A., Kanazawa A.:  Instruct-nerf2nerf: Editing 3d scenes with instructions.  _arXiv preprint arXiv:2303.12789_ (2023).
  * [HWZ∗23] Huang Y., Wang J., Zeng A., Cao H., Qi X., Shi Y., Zha Z.-J., Zhang L.:  Dreamwaltz: Make a scene with complex 3d animatable avatars.  _arXiv preprint arXiv:2305.12529_ (2023).
  * [HXL∗20] Huang Z., Xu Y., Lassner C., Li H., Tung T.:  Arch: Animatable reconstruction of clothed humans.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2020).
  * [HXS∗21] He T., Xu Y., Saito S., Soatto S., Tung T.:  Arch++: Animation-ready clothed human reconstruction revisited.  In _International Conference on Computer Vision_ (2021).
  * [HY23] Huang J., Yu H.:  Point’n move: Interactive scene object manipulation on gaussian splatting radiance fields.  _arXiv preprint arXiv:2311.16737_ (2023).
  * [HYX∗23] Huang Y., Yi H., Xiu Y., Liao T., Tang J., Cai D., Thies J.:  Tech: Text-guided reconstruction of lifelike clothed humans.  _arXiv preprint arXiv:2308.08545_ (2023).
  * [HZG∗23] Hong Y., Zhang K., Gu J., Bi S., Zhou Y., Liu D., Liu F., Sunkavalli K., Bui T., Tan H.:  Lrm: Large reconstruction model for single image to 3d.  _arXiv preprint arXiv:2311.04400_ (2023).
  * [HZJ∗21] Hong Y., Zhang J., Jiang B., Guo Y., Liu L., Bao H.:  Stereopifu: Depth aware clothed human digitization via stereo vision.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2021).
  * [HZP∗22] Hong F., Zhang M., Pan L., Cai Z., Yang L., Liu Z.:  Avatarclip: zero-shot text-driven generation and animation of 3d avatars.  _ACM Transactions on Graphics (TOG)_ (2022).
  * [JBAT17] Jackson A. S., Bulat A., Argyriou V., Tzimiropoulos G.:  Large pose 3d face reconstruction from a single image via direct volumetric cnn regression.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2017).
  * [JHBZ22] Jiang B., Hong Y., Bao H., Zhang J.:  Selfrecon: Self reconstruction your digital avatar from monocular video.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2022), pp. 5605–5615.
  * [JJW∗22] Jiang S., Jiang H., Wang Z., Luo H., Chen W., Xu L.:  Humangen: Generating human radiance fields with explicit priors.  _arXiv preprint arXiv:2212.05321_ (2022).
  * [JMB∗22] Jain A., Mildenhall B., Barron J. T., Abbeel P., Poole B.:  Zero-shot text-guided object generation with dream fields.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2022).
  * [JN23] Jun H., Nichol A.:  Shap-e: Generating conditional 3d implicit functions.  _arXiv preprint arXiv:2305.02463_ (2023).
  * [JWF∗24] Jin P., Wu Y., Fan Y., Sun Z., Yang W., Yuan L.:  Act as you wish: Fine-grained control of motion diffusion model with hierarchical semantic graphs.  _Advances in Neural Information Processing Systems 36_ (2024).
  * [JWZ∗23] Jiang R., Wang C., Zhang J., Chai M., He M., Chen D., Liao J.:  Avatarcraft: Transforming text into neural human avatars with parameterized shape and pose control.  _arXiv preprint arXiv:2303.17606_ (2023).
  * [JYQ∗22] Jiang Y., Yang S., Qiu H., Wu W., Loy C. C., Liu Z.:  Text2human: Text-driven controllable human image generation.  _ACM TOG_ (2022).
  * [JYS∗22] Jiang W., Yi K. M., Samei G., Tuzel O., Ranjan A.:  Neuman: Neural human radiance field from a single video.  In _European Conference on Computer Vision_ (2022), Springer, pp. 402–418.
  * [KAL∗21] Karras T., Aittala M., Laine S., Härkönen E., Hellsten J., Lehtinen J., Aila T.:  Alias-free generative adversarial networks.  _Advances in Neural Information Processing Systems 34_ (2021), 852–863.
  * [KAZ∗23] Kolotouros N., Alldieck T., Zanfir A., Bazavan E. G., Fieraru M., Sminchisescu C.:  Dreamhuman: Animatable 3d avatars from text.  _arXiv preprint arXiv:2306.09329_ (2023).
  * [KDJ∗23] Kwak J.-g., Dong E., Jin Y., Ko H., Mahajan S., Yi K. M.:  Vivid-1-to-3: Novel view synthesis with video diffusion models.  _arXiv preprint arXiv:2312.01305_ (2023).
  * [KH13] Kazhdan M., Hoppe H.:  Screened poisson surface reconstruction.  _ACM Transactions on Graphics (ToG) 32_ , 3 (2013), 1–13.
  * [KKCF21] Kwon Y., Kim D., Ceylan D., Fuchs H.:  Neural human performer: Learning generalizable radiance fields for human performance rendering.  _Advances on Neural Information Processing Systems_ (2021).
  * [KKLD23] Kerbl B., Kopanas G., Leimkühler T., Drettakis G.:  3d gaussian splatting for real-time radiance field rendering.  _ACM Transactions on Graphics (ToG) 42_ , 4 (2023), 1–14.
  * [KLA∗20] Karras T., Laine S., Aittala M., Hellsten J., Lehtinen J., Aila T.:  Analyzing and improving the image quality of stylegan.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ (2020), pp. 8110–8119.
  * [KPD19] Kolotouros N., Pavlakos G., Daniilidis K.:  Convolutional mesh regression for single-image human shape reconstruction.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2019).
  * [KW13] Kingma D. P., Welling M.:  Auto-encoding variational bayes.  _arXiv preprint arXiv:1312.6114_ (2013).
  * [LBB∗17] Li T., Bolkart T., Black M. J., Li H., Romero J.:  Learning a model of facial shape and expression from 4d scans.  _ACM Transactions on Graphics (TOG)_ (2017).
  * [LC98] Lorensen W. E., Cline H. E.:  Marching cubes: A high resolution 3d surface construction algorithm.  In _Seminal graphics: pioneering efforts that shaped the field_. 1998.
  * [LCCT23] Li W., Chen R., Chen X., Tan P.:  Sweetdreamer: Aligning geometric priors in 2d diffusion for consistent text-to-3d.  _arXiv preprint arXiv:2310.02596_ (2023).
  * [LCY∗23] Liu J.-W., Cao Y.-P., Yang T., Xu E. Z., Keppo J., Shan Y., Qie X., Shou M. Z.:  Hosnerf: Dynamic human-object-scene neural radiance fields from a single video.  _arXiv preprint arXiv:2304.12281_ (2023).
  * [LDZY23] Lin Y., Dai Z., Zhu S., Yao Y.:  Gaussian-flow: 4d reconstruction with dynamic 3d gaussian particle.  _arXiv preprint arXiv:2312.03431_ (2023).
  * [LGL∗23] Long X., Guo Y.-C., Lin C., Liu Y., Dou Z., Liu L., Ma Y., Zhang S.-H., Habermann M., Theobalt C., et al.:  Wonder3d: Single image to 3d using cross-domain diffusion.  _arXiv preprint arXiv:2310.15008_ (2023).
  * [LGT∗23] Lin C.-H., Gao J., Tang L., Takikawa T., Zeng X., Huang X., Kreis K., Fidler S., Liu M.-Y., Lin T.-Y.:  Magic3d: High-resolution text-to-3d content creation.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2023).
  * [LHG∗23] Lin Y., Han H., Gong C., Xu Z., Zhang Y., Li X.:  Consistent123: One image to highly consistent 3d asset using case-aware diffusion priors.  _arXiv preprint arXiv:2309.17261_ (2023).
  * [LHQ∗23] Liu Y., Huang X., Qin M., Lin Q., Wang H.:  Animatable 3d gaussian: Fast and high-quality reconstruction of multiple human avatars.  _arXiv preprint arXiv:2311.16482_ (2023).
  * [LHR∗21] Liu L., Habermann M., Rudnev V., Sarkar K., Gu J., Theobalt C.:  Neural actor: Neural free-view synthesis of human actors with pose control.  _ACM SIGGRAPH Asia_ (2021).
  * [LLL∗23] Liu Z., Li Y., Lin Y., Yu X., Peng S., Cao Y.-P., Qi X., Huang X., Liang D., Ouyang W.:  Unidream: Unifying diffusion priors for relightable text-to-3d generation.  _arXiv preprint arXiv:2312.08754_ (2023).
  * [LLQ∗16] Liu Z., Luo P., Qiu S., Wang X., Tang X.:  Deepfashion: Powering robust clothes recognition and retrieval with rich annotations.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2016).
  * [LLXH22] Li J., Li D., Xiong C., Hoi S.:  Blip: Bootstrapping language-image pre-training for unified vision-language understanding and generation.  In _ICML_ (2022).
  * [LLZ∗23] Liu Y., Lin C., Zeng Z., Long X., Liu L., Komura T., Wang W.:  Syncdreamer: Generating multiview-consistent images from a single-view image.  _arXiv preprint arXiv:2309.03453_ (2023).
  * [LMR∗15] Loper M., Mahmood N., Romero J., Pons-Moll G., Black M. J.:  SMPL: A skinned multi-person linear model.  _ACM Trans. Graphics, Asia_ (2015).
  * [LSC∗23] Liu M., Shi R., Chen L., Zhang Z., Xu C., Wei X., Chen H., Zeng C., Gu J., Su H.:  One-2-3-45++: Fast single image to 3d objects with consistent multi-view generation and 3d diffusion.  _arXiv preprint arXiv:2311.07885_ (2023).
  * [LSS∗21] Lombardi S., Simon T., Schwartz G., Zollhoefer M., Sheikh Y., Saragih J.:  Mixture of volumetric primitives for efficient neural rendering.  _ACM Transactions on Graphics (ToG) 40_ , 4 (2021), 1–13.
  * [LTV∗22] Li R., Tanke J., Vo M., Zollhöfer M., Gall J., Kanazawa A., Lassner C.:  Tava: Template-free animatable volumetric actors.  In _European Conference on Computer Vision_ (2022), Springer, pp. 419–436.
  * [LTYY23] Li M., Tao J., Yang Z., Yang Y.:  Human101: Training 100+ fps human gaussians in 100s from 1 view.  _arXiv preprint arXiv:2312.15258_ (2023).
  * [LTZ∗23] Li J., Tan H., Zhang K., Xu Z., Luan F., Xu Y., Hong Y., Sunkavalli K., Shakhnarovich G., Bi S.:  Instant3d: Fast text-to-3d with sparse-view generation and large reconstruction model.  _arXiv preprint arXiv:2311.06214_ (2023).
  * [LWC∗24] Liu F., Wang H., Chen W., Sun H., Duan Y.:  Make-your-3d: Fast and consistent subject-driven 3d content generation.  _arXiv preprint arXiv:2403.09625_ (2024).
  * [LWL∗24] Liu X., Wu C., Liu X., Liu J., Wu J., Zhao C., Feng H., Ding E., Wang J.:  Gea: Reconstructing expressive 3d gaussian avatar from monocular video.  _arXiv preprint arXiv:2402.16607_ (2024).
  * [LWVH∗23] Liu R., Wu R., Van Hoorick B., Tokmakov P., Zakharov S., Vondrick C.:  Zero-1-to-3: Zero-shot one image to 3d object.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ (2023), pp. 9298–9309.
  * [LWW∗23] Liu H., Wang X., Wan Z., Shen Y., Song Y., Liao J., Chen Q.:  Headartist: Text-conditioned 3d head generation with self score distillation.  _arXiv preprint arXiv:2312.07539_ (2023).
  * [LXJ∗23] Liu M., Xu C., Jin H., Chen L., Xu Z., Su H., et al.:  One-2-3-45: Any single image to 3d mesh in 45 seconds without per-shape optimization.  _arXiv preprint arXiv:2306.16928_ (2023).
  * [LXZ∗23] Lorraine J., Xie K., Zeng X., Lin C.-H., Takikawa T., Sharp N., Lin T.-Y., Liu M.-Y., Fidler S., Lucas J.:  Att3d: Amortized text-to-3d object synthesis.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ (2023), pp. 17946–17956.
  * [LYX∗23] Liao T., Yi H., Xiu Y., Tang J., Huang Y., Thies J., Black M. J.:  Tada! text to animatable digital avatars.  _arXiv preprint arXiv:2308.10899_ (2023).
  * [LYX∗24] Li M., Yao S., Xie Z., Chen K., Jiang Y.-G.:  Gaussianbody: Clothed human reconstruction via 3d gaussian splatting.  _arXiv preprint arXiv:2401.09720_ (2024).
  * [LZ21] Lassner C., Zollhofer M.:  Pulsar: Efficient sphere-based neural rendering.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2021), pp. 1440–1449.
  * [LZK∗24] Li X., Zhang Q., Kang D., Cheng W., Gao Y., Zhang J., Liang Z., Liao J., Cao Y.-P., Shan Y.:  Advances in 3d generation: A survey.  _arXiv preprint arXiv:2401.17807_ (2024).
  * [LZT∗23] Liu X., Zhan X., Tang J., Shan Y., Zeng G., Lin D., Liu X., Liu Z.:  Humangaussian: Text-driven 3d human generation with gaussian splatting.  _arXiv preprint arXiv:2311.17061_ (2023).
  * [LZW∗23] Li C., Zhang C., Waghwase A., Lee L.-H., Rameau F., Yang Y., Bae S.-H., Hong C. S.:  Generative ai meets 3d: A survey on text-to-3d in aigc era.  _arXiv preprint arXiv:2305.06131_ (2023).
  * [MBOL∗22] Michel O., Bar-On R., Liu R., Benaim S., Hanocka R.:  Text2mesh: Text-driven neural stylization for meshes.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2022).
  * [MHS∗21] Meng C., He Y., Song Y., Song J., Wu J., Zhu J.-Y., Ermon S.:  Sdedit: Guided image synthesis and editing with stochastic differential equations.  _arXiv preprint arXiv:2108.01073_ (2021).
  * [MKRLV23] Melas-Kyriazi L., Rupprecht C., Laina I., Vedaldi A.:  Realfusion: 360 $\\{$$\backslash$deg$\\}$ reconstruction of any object from a single image.  _arXiv preprint arXiv:2302.10663_ (2023).
  * [MKXBP22] Mohammad Khalid N., Xie T., Belilovsky E., Popa T.:  Clip-mesh: Generating textured meshes from text using pretrained image-text models.  In _SIGGRAPH Asia 2022 conference papers_ (2022), pp. 1–8.
  * [MNR∗24] Mercier A., Nakhli R., Reddy M., Yasarla R., Cai H., Porikli F., Berger G.:  Hexagen3d: Stablediffusion is just one step away from fast and diverse text-to-3d generation.  _arXiv preprint arXiv:2401.07727_ (2024).
  * [MRP∗22] Metzer G., Richardson E., Patashnik O., Giryes R., Cohen-Or D.:  Latent-nerf for shape-guided generation of 3d shapes and textures.  _arXiv preprint arXiv:2211.07600_ (2022).
  * [MST∗21] Mildenhall B., Srinivasan P. P., Tancik M., Barron J. T., Ramamoorthi R., Ng R.:  Nerf: Representing scenes as neural radiance fields for view synthesis.  _Communications of the ACM_ (2021).
  * [MSVW23] Mu J., Sang S., Vasconcelos N., Wang X.:  Actorsnerf: Animatable few-shot human rendering with generalizable nerfs.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ (2023), pp. 18391–18401.
  * [MYR∗20] Ma Q., Yang J., Ranjan A., Pujades S., Pons-Moll G., Tang S., Black M. J.:  Learning to dress 3d people in generative clothing.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2020), pp. 6469–6478.
  * [NJD∗22] Nichol A., Jun H., Dhariwal P., Mishkin P., Chen M.:  Point-e: A system for generating 3d point clouds from complex prompts.  _arXiv preprint arXiv:2212.08751_ (2022).
  * [NPLT∗19] Nguyen-Phuoc T., Li C., Theis L., Richardt C., Yang Y.-L.:  Hologan: Unsupervised learning of 3d representations from natural images.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ (2019), pp. 7588–7597.
  * [NPRM∗20] Nguyen-Phuoc T. H., Richardt C., Mai L., Yang Y., Mitra N.:  Blockgan: Learning 3d object-aware scene representations from unlabelled images.  _Advances in neural information processing systems 33_ (2020), 6767–6778.
  * [NSLH21] Noguchi A., Sun X., Lin S., Harada T.:  Neural articulated radiance field.  In _International Conference on Computer Vision_ (2021).
  * [NSLH22] Noguchi A., Sun X., Lin S., Harada T.:  Unsupervised learning of efficient geometry-aware neural articulated representations.  In _European Conference on Computer Vision_ (2022).
  * [NYD16] Newell A., Yang K., Deng J.:  Stacked hourglass networks for human pose estimation.  In _European Conference on Computer Vision_ (2016).
  * [ODM∗23] Oquab M., Darcet T., Moutakanni T., Vo H., Szafraniec M., Khalidov V., Fernandez P., Haziza D., Massa F., El-Nouby A., et al.:  Dinov2: Learning robust visual features without supervision.  _arXiv preprint arXiv:2304.07193_ (2023).
  * [OELS∗22] Or-El R., Luo X., Shan M., Shechtman E., Park J. J., Kemelmacher-Shlizerman I.:  Stylesdf: High-resolution 3d-consistent image and geometry generation.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2022).
  * [PBV23] Petrovich M., Black M. J., Varol G.:  Tmr: Text-to-motion retrieval using contrastive 3d human motion synthesis.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ (2023), pp. 9488–9497.
  * [PCG∗19] Pavlakos G., Choutas V., Ghorbani N., Bolkart T., Osman A. A. A., Tzionas D., Black M. J.:  Expressive body capture: 3d hands, face, and body from a single image.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2019).
  * [PDW∗21] Peng S., Dong J., Wang Q., Zhang S., Shuai Q., Zhou X., Bao H.:  Animatable neural radiance fields for modeling dynamic human bodies.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ (2021), pp. 14314–14323.
  * [PET∗23] Pan M. M., Elgharib M., Teotia K., Tewari A., Golyanik V., Kortylewski A., Theobalt C., et al.:  Avatarstudio: Text-driven editing of 3d dynamic human head avatars.  _arXiv preprint arXiv:2306.00547_ (2023).
  * [PFS∗19] Park J. J., Florence P., Straub J., Newcombe R., Lovegrove S.:  Deepsdf: Learning continuous signed distance functions for shape representation.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2019).
  * [PJBM22] Poole B., Jain A., Barron J. T., Mildenhall B.:  Dreamfusion: Text-to-3d using 2d diffusion.  In _International Conference on Learning Representations_ (2022).
  * [PNM∗20] Peng S., Niemeyer M., Mescheder L., Pollefeys M., Geiger A.:  Convolutional occupancy networks.  In _Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part III 16_ (2020), Springer, pp. 523–540.
  * [PT02] Pantazopoulos I., Tzafestas S.:  Occlusion culling algorithms: A comprehensive survey.  _Journal of Intelligent and Robotic Systems 35_ (2002), 123–156.
  * [PYG∗23] Po R., Yifan W., Golyanik V., Aberman K., Barron J. T., Bermano A. H., Chan E. R., Dekel T., Holynski A., Kanazawa A., et al.:  State of the art on diffusion models for visual computing.  _arXiv preprint arXiv:2310.07204_ (2023).
  * [PZX∗21] Peng S., Zhang Y., Xu Y., Wang Q., Shuai Q., Bao H., Zhou X.:  Neural body: Implicit neural representations with structured latent codes for novel view synthesis of dynamic humans.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2021).
  * [QMH∗23] Qian G., Mai J., Hamdi A., Ren J., Siarohin A., Li B., Lee H.-Y., Skorokhodov I., Wonka P., Tulyakov S., et al.:  Magic123: One image to high-quality 3d object generation using both 2d and 3d diffusion priors.  _arXiv preprint arXiv:2306.17843_ (2023).
  * [QSMG17] Qi C. R., Su H., Mo K., Guibas L. J.:  Pointnet: Deep learning on point sets for 3d classification and segmentation.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2017).
  * [QWM∗23] Qian Z., Wang S., Mihajlovic M., Geiger A., Tang S.:  3dgs-avatar: Animatable avatars via deformable 3d gaussian splatting.  _arXiv preprint arXiv:2312.09228_ (2023).
  * [QYSG17] Qi C. R., Yi L., Su H., Guibas L. J.:  Pointnet++: Deep hierarchical feature learning on point sets in a metric space.  _arXiv preprint arXiv:1706.02413_ (2017).
  * [Ren] https://renderpeople.com.
  * [RH01] Ramamoorthi R., Hanrahan P.:  An efficient representation for irradiance environment maps.  In _Proceedings of the 28th annual conference on Computer graphics and interactive techniques_ (2001), pp. 497–500.
  * [RKH∗21] Radford A., Kim J. W., Hallacy C., Ramesh A., Goh G., Agarwal S., Sastry G., Askell A., Mishkin P., Clark J., et al.:  Learning transferable visual models from natural language supervision.  In _International Conference on Machine Learning_ (2021).
  * [RMA∗23] Richardson E., Metzer G., Alaluf Y., Giryes R., Cohen-Or D.:  Texture: Text-guided texturing of 3d shapes.  _arXiv preprint arXiv:2302.01721_ (2023).
  * [RPT∗23] Ren J., Pan L., Tang J., Zhang C., Cao A., Zeng G., Liu Z.:  Dreamgaussian4d: Generative 4d gaussian splatting.  _arXiv preprint arXiv:2312.17142_ (2023).
  * [RRN∗20] Ravi N., Reizenstein J., Novotny D., Gordon T., Lo W.-Y., Johnson J., Gkioxari G.:  Accelerating 3d deep learning with pytorch3d.  _arXiv preprint arXiv:2007.08501_ (2020).
  * [SA07] Sorkine O., Alexa M.:  As-rigid-as-possible surface modeling.  In _Symposium on Geometry processing_ (2007), vol. 4, pp. 109–116.
  * [SCS∗22] Saharia C., Chan W., Saxena S., Li L., Whang J., Denton E. L., Ghasemipour K., Gontijo Lopes R., Karagol Ayan B., Salimans T., et al.:  Photorealistic text-to-image diffusion models with deep language understanding.  _Advances in neural information processing systems 35_ (2022), 36479–36494.
  * [SCZ∗23] Shi R., Chen H., Zhang Z., Liu M., Xu C., Wei X., Chen L., Zeng C., Su H.:  Zero123++: a single image to consistent multi-view diffusion base model.  _arXiv preprint arXiv:2310.15110_ (2023).
  * [SFHAE23] Sella E., Fiebelman G., Hedman P., Averbuch-Elor H.:  Vox-e: Text-guided voxel editing of 3d objects.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ (2023), pp. 430–440.
  * [SGY∗21] Shen T., Gao J., Yin K., Liu M.-Y., Fidler S.:  Deep marching tetrahedra: a hybrid representation for high-resolution 3d shape synthesis.  _Advances in Neural Information Processing Systems 34_ (2021), 6087–6101.
  * [SHN∗19] Saito S., Huang Z., Natsume R., Morishima S., Kanazawa A., Li H.:  Pifu: Pixel-aligned implicit function for high-resolution clothed human digitization.  In _International Conference on Computer Vision_ (2019).
  * [SJK∗23] Seo J., Jang W., Kwak M.-S., Kim H., Ko J., Kim J., Kim J.-H., Lee J., Kim S.:  Let 2d diffusion model know 3d-consistency for robust text-to-3d generation.  _arXiv preprint arXiv:2303.07937_ (2023).
  * [SJL∗21] Suo X., Jiang Y., Lin P., Zhang Y., Wu M., Guo K., Xu L.:  Neuralhumanfvv: Real-time neural volumetric human performance rendering using rgb cameras.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ (2021), pp. 6226–6237.
  * [SMH∗23] Shen T., Munkberg J., Hasselgren J., Yin K., Wang Z., Chen W., Gojcic Z., Fidler S., Sharp N., Gao J.:  Flexible isosurface extraction for gradient-based mesh optimization.  _ACM Transactions on Graphics (TOG) 42_ , 4 (2023), 1–16.
  * [SRV23] Szymanowicz S., Rupprecht C., Vedaldi A.:  Splatter image: Ultra-fast single-view 3d reconstruction.  _arXiv preprint arXiv:2312.13150_ (2023).
  * [SSC22] Sun C., Sun M., Chen H.-T.:  Direct voxel grid optimization: Super-fast convergence for radiance fields reconstruction.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2022), pp. 5459–5469.
  * [SSP07] Sumner R. W., Schmid J., Pauly M.:  Embedded deformation for shape manipulation.  In _ACM siggraph 2007 papers_. 2007, pp. 80–es.
  * [SSP∗23] Shao R., Sun J., Peng C., Zheng Z., Zhou B., Zhang H., Liu Y.:  Control4d: Dynamic portrait editing by learning 4d gan from 2d diffusion-based editor.  _arXiv preprint arXiv:2305.20082_ (2023).
  * [SSS06] Snavely N., Seitz S. M., Szeliski R.:  Photo tourism: exploring photo collections in 3d.  In _ACM siggraph 2006 papers_. 2006, pp. 835–846.
  * [SSSJ20] Saito S., Simon T., Saragih J., Joo H.:  Pifuhd: Multi-level pixel-aligned implicit function for high-resolution 3d human digitization.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2020).
  * [Sta22] Stability.AI:  Stable diffusion.  https://stability.ai/blog/stable-diffusion-public-release, 2022.
  * [SWC∗23] Shi Y., Wang J., Cao H., Tang B., Qi X., Yang T., Huang Y., Liu S., Zhang L., Shum H.-Y.:  Toss: High-quality text-guided novel view synthesis from a single image.  _arXiv preprint arXiv:2310.10644_ (2023).
  * [SWW∗23] Sun J., Wang X., Wang L., Li X., Zhang Y., Zhang H., Liu Y.:  Next3d: Generative neural texture rasterization for 3d-aware head avatars.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2023).
  * [SWY∗23] Shi Y., Wang P., Ye J., Mai L., Li K., Yang X.:  Mvdream: Multi-view diffusion for 3d generation.  _arXiv:2308.16512_ (2023).
  * [SZT∗23] Shao R., Zheng Z., Tu H., Liu B., Zhang H., Liu Y.:  Tensor4d: Efficient neural 4d decomposition for high-fidelity dynamic reconstruction and rendering.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2023), pp. 16632–16642.
  * [SZZ∗22a] Shao R., Zhang H., Zhang H., Chen M., Cao Y.-P., Yu T., Liu Y.:  Doublefield: Bridging the neural surface and radiance fields for high-fidelity human reconstruction and rendering.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2022).
  * [SZZ∗22b] Shao R., Zheng Z., Zhang H., Sun J., Liu Y.:  Diffustereo: High quality human reconstruction via diffusion-based stereo using sparse cameras.  In _European Conference on Computer Vision_ (2022).
  * [TCC∗24] Tang J., Chen Z., Chen X., Wang T., Zeng G., Liu Z.:  Lgm: Large multi-view gaussian model for high-resolution 3d content creation.  _arXiv preprint arXiv:2402.05054_ (2024).
  * [TCW∗24] Tang S., Chen J., Wang D., Tang C., Zhang F., Fan Y., Chandra V., Furukawa Y., Ranjan R.:  Mvdiffusion++: A dense high-resolution multi-view diffusion model for single or sparse-view 3d object reconstruction.  _arXiv preprint arXiv:2402.12712_ (2024).
  * [TFT∗20] Tewari A., Fried O., Thies J., Sitzmann V., Lombardi S., Sunkavalli K., Martin-Brualla R., Simon T., Saragih J., Nießner M., et al.:  State of the art on neural rendering.  In _Computer Graphics Forum_ (2020), vol. 39, Wiley Online Library, pp. 701–727.
  * [TGH∗22] Tevet G., Gordon B., Hertz A., Bermano A. H., Cohen-Or D.:  Motionclip: Exposing human motion generation to clip space.  In _European Conference on Computer Vision_ (2022), Springer, pp. 358–374.
  * [TPL∗24] Tochilkin D., Pankratz D., Liu Z., Huang Z., Letts A., Li Y., Liang D., Laforte C., Jampani V., Cao Y.-P.:  Triposr: Fast 3d object reconstruction from a single image.  _arXiv preprint arXiv:2403.02151_ (2024).
  * [TRG∗22] Tevet G., Raab S., Gordon B., Shafir Y., Cohen-Or D., Bermano A. H.:  Human motion diffusion model.  _arXiv preprint arXiv:2209.14916_ (2022).
  * [TRZ∗23] Tang J., Ren J., Zhou H., Liu Z., Zeng G.:  Dreamgaussian: Generative gaussian splatting for efficient 3d content creation.  _arXiv preprint arXiv:2309.16653_ (2023).
  * [TTM∗22] Tewari A., Thies J., Mildenhall B., Srinivasan P., Tretschk E., Yifan W., Lassner C., Sitzmann V., Martin-Brualla R., Lombardi S., et al.:  Advances in neural rendering.  In _Computer Graphics Forum_ (2022), vol. 41, Wiley Online Library, pp. 703–735.
  * [TZLW23] Tian Y., Zhang H., Liu Y., Wang L.:  Recovering 3d human mesh from monocular images: A survey.  _IEEE transactions on pattern analysis and machine intelligence_ (2023).
  * [VPY∗23] Vo K., Pham T.-T., Yamazaki K., Tran M., Le N.:  Dna: Deformable neural articulations network for template-free dynamic 3d human reconstruction from monocular rgb-d video.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2023), pp. 3675–3684.
  * [VSP∗17] Vaswani A., Shazeer N., Parmar N., Uszkoreit J., Jones L., Gomez A. N., Kaiser Ł., Polosukhin I.:  Attention is all you need.  _Advances in neural information processing systems 30_ (2017).
  * [WBL∗24] Wu J., Bian J.-W., Li X., Wang G., Reid I., Torr P., Prisacariu V. A.:  Gaussctrl: Multi-view consistent text-driven 3d gaussian splatting editing.  _arXiv preprint arXiv:2403.08733_ (2024).
  * [WCMB∗22] Watson D., Chan W., Martin-Brualla R., Ho J., Tagliasacchi A., Norouzi M.:  Novel view synthesis with diffusion models.  _arXiv preprint arXiv:2210.04628_ (2022).
  * [WCS∗22] Weng C.-Y., Curless B., Srinivasan P. P., Barron J. T., Kemelmacher-Shlizerman I.:  Humannerf: Free-viewpoint rendering of moving people from monocular video.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2022).
  * [WFX∗23] Wang P., Fan Z., Xu D., Wang D., Mohan S., Iandola F., Ranjan R., Li Y., Liu Q., Wang Z., et al.:  Steindreamer: Variance reduction for text-to-3d score distillation via stein identity.  _arXiv preprint arXiv:2401.00604_ (2023).
  * [WLL∗21] Wang P., Liu L., Liu Y., Theobalt C., Komura T., Wang W.:  Neus: Learning neural implicit surfaces by volume rendering for multi-view reconstruction.  _Advances on Neural Information Processing Systems_ (2021).
  * [WLT∗24] Weng Z., Liu J., Tan H., Xu Z., Zhou Y., Yeung-Levy S., Yang J.:  Single-view 3d human digitalization with large reconstruction models.  _arXiv preprint arXiv:2401.12175_ (2024).
  * [WLW∗23] Wang Z., Lu C., Wang Y., Bao F., Li C., Su H., Zhu J.:  Prolificdreamer: High-fidelity and diverse text-to-3d generation with variational score distillation.  _arXiv preprint arXiv:2305.16213_ (2023).
  * [WS23] Wang P., Shi Y.:  Imagedream: Image-prompt multi-view diffusion for 3d generation.  _arXiv preprint arXiv:2312.02201_ (2023).
  * [WSCKS23] Weng C.-Y., Srinivasan P. P., Curless B., Kemelmacher-Shlizerman I.:  Personnerf: Personalized reconstruction from photo collections.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2023), pp. 524–533.
  * [WSGT22] Wang S., Schwarz K., Geiger A., Tang S.:  Arah: Animatable volume rendering of articulated human sdfs.  In _European conference on computer vision_ (2022), Springer, pp. 1–19.
  * [WTB∗23] Wang P., Tan H., Bi S., Xu Y., Luan F., Sunkavalli K., Wang W., Xu Z., Zhang K.:  Pf-lrm: Pose-free large reconstruction model for joint pose and shape prediction.  _arXiv preprint arXiv:2311.12024_ (2023).
  * [WTZ∗21] Wang J., Tan S., Zhen X., Xu S., Zheng F., He Z., Shao L.:  Deep 3d human pose estimation: A review.  _Computer Vision and Image Understanding 210_ (2021), 103225.
  * [WWC∗24] Wang Z., Wang Y., Chen Y., Xiang C., Chen S., Yu D., Li C., Su H., Zhu J.:  Crm: Single image to 3d textured mesh with convolutional reconstruction model.  _arXiv preprint arXiv:2403.05034_ (2024).
  * [WWY23] Weng Z., Wang Z., Yeung S.:  Zeroavatar: Zero-shot 3d avatar generation from a single image.  _arXiv preprint arXiv:2305.16411_ (2023).
  * [WYW∗24] Wang Y., Yi X., Wu Z., Zhao N., Chen L., Zhang H.:  View-consistent 3d editing with gaussian splatting.  _arXiv preprint arXiv:2403.11868_ (2024).
  * [WYZ∗24] Wu T., Yuan Y.-J., Zhang L.-X., Yang J., Cao Y.-P., Yan L.-Q., Gao L.:  Recent advances in 3d gaussian splatting.  _arXiv preprint arXiv:2403.11134_ (2024).
  * [WZF∗23] Wu T., Zhang J., Fu X., Wang Y., Ren J., Pan L., Wu W., Yang L., Wang J., Qian C., et al.:  Omniobject3d: Large-vocabulary 3d object dataset for realistic perception, reconstruction and generation.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2023), pp. 803–814.
  * [WZH∗23] Wu M., Zhu H., Huang L., Zhuang Y., Lu Y., Cao X.:  High-fidelity 3d face generation from natural language descriptions.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2023), pp. 4521–4530.
  * [WZY∗24] Wu Z., Zhou P., Yi X., Yuan X., Zhang H.:  Consistent3d: Towards consistent high-fidelity text-to-3d generation with deterministic sampling prior.  _arXiv preprint arXiv:2401.09050_ (2024).
  * [WZZ∗23] Wang T., Zhang B., Zhang T., Gu S., Bao J., Baltrusaitis T., Shen J., Chen D., Wen F., Chen Q., et al.:  Rodin: A generative model for sculpting 3d digital avatars using diffusion.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ (2023), pp. 4563–4573.
  * [XCL∗23] Xu Y., Chen B., Li Z., Zhang H., Wang L., Zheng Z., Liu Y.:  Gaussian head avatar: Ultra high-fidelity head avatar via dynamic gaussians.  _arXiv preprint arXiv:2312.03029_ (2023).
  * [XFC∗23] Xing Z., Feng Q., Chen H., Dai Q., Hu H., Xu H., Wu Z., Jiang Y.-G.:  A survey on video diffusion models.  _arXiv preprint arXiv:2310.10647_ (2023).
  * [XFM22] Xu T., Fujita Y., Matsumoto E.:  Surface-aligned neural radiance fields for controllable 3d human synthesis.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2022), pp. 15883–15892.
  * [XHL∗24] Xu T.-X., Hu W., Lai Y.-K., Shan Y., Zhang S.-H.:  Texture-gs: Disentangling the geometry and texture for 3d gaussian splatting editing.  _arXiv preprint arXiv:2403.10050_ (2024).
  * [XTL∗23] Xu Y., Tan H., Luan F., Bi S., Wang P., Li J., Shi Z., Sunkavalli K., Wetzstein G., Xu Z., et al.:  Dmv3d: Denoising multi-view diffusion using 3d large reconstruction model.  _arXiv preprint arXiv:2311.09217_ (2023).
  * [XWC∗22] Xu J., Wang X., Cheng W., Cao Y.-P., Shan Y., Qie X., Gao S.:  Dream3d: Zero-shot text-to-3d synthesis using 3d shape prior and text-to-image diffusion models.  _arXiv preprint arXiv:2212.14704_ (2022).
  * [XWY∗21] Xie E., Wang W., Yu Z., Anandkumar A., Alvarez J. M., Luo P.:  Segformer: Simple and efficient design for semantic segmentation with transformers.  _Advances in Neural Information Processing Systems 34_ (2021), 12077–12090.
  * [XYC∗23] Xiu Y., Yang J., Cao X., Tzionas D., Black M. J.:  ECON: Explicit Clothed humans Obtained from Normals.  In _IEEELearning Transferable Visual Models From Natural Language Supervision Conference on Computer Vision and Pattern Recognition_ (2023).
  * [XYM∗24] Xu D., Yuan Y., Mardani M., Liu S., Song J., Wang Z., Vahdat A.:  Agg: Amortized generative 3d gaussians for single image to 3d.  _arXiv preprint arXiv:2401.04099_ (2024).
  * [XYTB22] Xiu Y., Yang J., Tzionas D., Black M. J.:  ICON: Implicit Clothed humans Obtained from Normals.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2022).
  * [YCL∗23] Yu Z., Cheng W., Liu X., Wu W., Lin K.-Y.:  Monohuman: Animatable human neural field from monocular video.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2023), pp. 16943–16953.
  * [YFW∗23] Yi T., Fang J., Wu G., Xie L., Zhang X., Liu W., Tian Q., Wang X.:  Gaussiandreamer: Fast generation from text to 3d gaussian splatting with point cloud priors.  _arXiv preprint arXiv:2310.08529_ (2023).
  * [YPN∗23] Yariv L., Puny O., Neverova N., Gafni O., Lipman Y.:  Mosaic-sdf for 3d generative models.  _arXiv preprint arXiv:2312.09222_ (2023).
  * [YSI∗23] Yuan Y., Song J., Iqbal U., Vahdat A., Kautz J.:  Physdiff: Physics-guided human motion diffusion model.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ (2023), pp. 16010–16021.
  * [YWL∗23] Ye J., Wang P., Li K., Shi Y., Wang H.:  Consistent-1-to-3: Consistent image to 3d view synthesis via geometry-aware diffusion models.  _arXiv preprint arXiv:2310.03020_ (2023).
  * [YWRR23] Yang G., Wang C., Reddy N. D., Ramanan D.:  Reconstructing animatable categories from videos.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2023), pp. 16995–17005.
  * [YYTK21] Yu A., Ye V., Tancik M., Kanazawa A.:  pixelnerf: Neural radiance fields from one or few images.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2021), pp. 4578–4587.
  * [YYZ∗23] Yang G., Yang S., Zhang J. Z., Manchester Z., Ramanan D.:  Ppr: Physically plausible reconstruction from monocular videos.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ (2023), pp. 3914–3924.
  * [YZG∗21] Yu T., Zheng Z., Guo K., Liu P., Dai Q., Liu Y.:  Function4d: Real-time human volumetric capture from very sparse consumer rgbd sensors.  In _IEEE Conference on Computer Vision and Pattern Recognition_ (2021).
  * [YZS∗23] Yang L., Zhang Z., Song Y., Hong S., Xu R., Zhao Y., Zhang W., Cui B., Yang M.-H.:  Diffusion models: A comprehensive survey of methods and applications.  _ACM Computing Surveys 56_ , 4 (2023), 1–39.
  * [YZW∗24] Yinghao X., Zifan S., Wang Y., Hansheng C., Ceyuan Y., Sida P., Yujun S., Gordon W.:  Grm: Large gaussian reconstruction model for efficient 3d reconstruction and generation, 2024.  arXiv:2403.14621.
  * [ZA23] Zhang L., Agrawala M.:  Adding conditional control to text-to-image diffusion models.  _arXiv preprint arXiv:2302.05543_ (2023).
  * [ZAB∗22] Zheng Y., Abrevaya V. F., Bühler M. C., Chen X., Black M. J., Hilliges O.:  Im avatar: Implicit morphable head avatars from videos.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2022), pp. 13545–13555.
  * [ZBS∗23] Zielonka W., Bagautdinov T., Saito S., Zollhöfer M., Thies J., Romero J.:  Drivable 3d gaussian avatars.  _arXiv preprint arXiv:2311.08581_ (2023).
  * [ZCP∗24] Zhang M., Cai Z., Pan L., Hong F., Guo X., Yang L., Liu Z.:  Motiondiffuse: Text-driven human motion generation with diffusion model.  _IEEE Transactions on Pattern Analysis and Machine Intelligence_ (2024).
  * [ZFK∗23] Zhang H., Feng Y., Kulits P., Wen Y., Thies J., Black M. J.:  Text-guided generation and editing of compositional 3d avatars.  _arXiv preprint arXiv:2309.07125_ (2023).
  * [ZGP∗23] Zhang M., Guo X., Pan L., Cai Z., Hong F., Li H., Yang L., Liu Z.:  Remodiffuse: Retrieval-augmented motion diffusion model.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ (2023), pp. 364–373.
  * [ZHZX23] Zhong C., Hu L., Zhang Z., Xia S.:  Attt2m: Text-driven human motion generation with multi-perspective attention mechanism.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ (2023), pp. 509–519.
  * [ZJY∗22] Zhang J., Jiang Z., Yang D., Xu H., Shi Y., Song G., Xu Z., Wang X., Feng J.:  Avatargen: A 3d generative model for animatable human avatars.  In _Arxiv_ (2022).
  * [ZKC∗24] Zhuang J., Kang D., Cao Y.-P., Li G., Lin L., Shan Y.:  Tip-editor: An accurate 3d editor following both text-prompts and image-prompts.  _arXiv preprint arXiv:2401.14828_ (2024).
  * [ZLJ∗23] Zeng Y., Lu Y., Ji X., Yao Y., Zhu H., Cao X.:  Avatarbooth: High-quality and customizable 3d human avatar generation.  _arXiv preprint arXiv:2306.09864_ (2023).
  * [ZMFY24] Zhou Z., Ma F., Fan H., Yang Y.:  Headstudio: Text to animatable head avatars with 3d gaussian splatting.  _arXiv preprint arXiv:2402.06149_ (2024).
  * [ZMR∗23] Zhu W., Ma X., Ro D., Ci H., Zhang J., Shi J., Gao F., Tian Q., Wang Y.:  Human motion generation: A survey.  _IEEE Transactions on Pattern Analysis and Machine Intelligence_ (2023).
  * [ZRX∗24] Zhou X., Ran X., Xiong Y., He J., Lin Z., Wang Y., Sun D., Yang M.-H.:  Gala3d: Towards text-to-3d complex scene generation via layout-guided generative gaussian splatting.  _arXiv preprint arXiv:2402.07207_ (2024).
  * [ZSZ∗21] Zheng Y., Shao R., Zhang Y., Yu T., Zheng Z., Dai Q., Liu Y.:  Deepmulticap: Performance capture of multiple characters using sparse multiview cameras.  _arXiv preprint arXiv:2105.00261_ (2021).
  * [ZT23] Zhou Z., Tulsiani S.:  Sparsefusion: Distilling view-conditioned diffusion for 3d reconstruction.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2023), pp. 12588–12597.
  * [ZWC∗23] Zheng C., Wu W., Chen C., Yang T., Zhu S., Shen J., Kehtarnavaz N., Shah M.:  Deep learning-based human pose estimation: A survey.  _ACM Computing Surveys 56_ , 1 (2023), 1–37.
  * [ZYG∗23] Zou Z.-X., Yu Z., Guo Y.-C., Li Y., Liang D., Cao Y.-P., Zhang S.-H.:  Triplane meets gaussian splatting: Fast and generalizable single-view 3d reconstruction with transformers.  _arXiv preprint arXiv:2312.09147_ (2023).
  * [ZYLD21] Zheng Z., Yu T., Liu Y., Dai Q.:  Pamir: Parametric model-conditioned implicit representation for image-based human reconstruction.  _IEEE Transactions on Pattern Analysis and Machine Intelligence_ (2021).
  * [ZZC∗23] Zhang J., Zhang Y., Cun X., Huang S., Zhang Y., Zhao H., Lu H., Shen X.:  T2m-gpt: Generating human motion from textual descriptions with discrete representations.  _arXiv preprint arXiv:2301.06052_ (2023).
  * [ZZS∗23] Zheng S., Zhou B., Shao R., Liu B., Zhang S., Nie L., Liu Y.:  Gps-gaussian: Generalizable pixel-wise 3d gaussian splatting for real-time human novel view synthesis.  _arXiv preprint arXiv:2312.02155_ (2023).