††thanks: They contribute equally to this work.††thanks: They contribute
equally to this work.

# Synthetic tensor gauge fields

Shaoliang Zhang<EMAIL_ADDRESS>School of Physics, Institute for
Quantum Science and Engineering, Huazhong University of Science and
Technology, Wuhan 430074, China  Chenwei Lv<EMAIL_ADDRESS>Department of
Physics and Astronomy, Purdue University, West Lafayette, IN, 47907, USA  Qi
Zhou<EMAIL_ADDRESS>Department of Physics and Astronomy, Purdue
University, West Lafayette, IN, 47907, USA  Purdue Quantum Science and
Engineering Institute, Purdue University, West Lafayette, IN 47907, USA

###### Abstract

Synthetic gauge fields have provided physicists with a unique tool to explore
a wide range of fundamentally important phenomena in physics. However, only
synthetic vector gauge fields are currently available in experiments. The
study of tensor gauge fields, which play a vital role in fracton phase of
matter, remains purely theoretical. Here, we propose schemes to realize
synthetic tensor gauge fields using techniques readily available in
laboratories. A lattice tilted by a strong linear potential and a weak
quadratic potential naturally yields a rank-2 electric field for a lineon
formed by a particle-hole pair. Such a rank-2 electric field leads to a new
type of Bloch oscillations, where neither a single particle nor a single hole
responds but a lineon vibrates. A synthetic vector gauge field carrying a
position-dependent phase could also be implemented to produce the same
synthetic tensor gauge field for a lineon. In higher dimensions, the interplay
between interactions and vector gauge potentials imprints a phase to the ring-
exchange interaction and thus generates synthetic tensor gauge fields for
planons. Such tensor gauge fields make it possible to realize a dipolar
Harper-Hofstadter model in laboratories.

The study of couplings between matter and gauge fields is an essential topic
in physics, telling us how our universe functions at different length scales.
At the cosmological scales, gauge fields lay the cornerstone to build our
universe Lasenby  _et al._ (1998). At microscopic scales, interesting phases
arise in quantum matter subject to gauge fields Thouless  _et al._ (1982);
Tsui  _et al._ (1982); König  _et al._ (2007). In addition to exploring gauge
fields that readily exist in nature, experimentalists have also made efforts
to realize synthetic gauge fields using ultracold atoms, photonics and other
highly controllable systems Lin  _et al._ (2011); Aidelsburger  _et al._
(2011, 2013); Miyake  _et al._ (2013); Schweizer  _et al._ (2019); Görg  _et
al._ (2019); Yang  _et al._ (2020); Li  _et al._ (2022); Zhou  _et al._
(2022); Hafezi  _et al._ (2011); Khanikaev  _et al._ (2013); Mittal  _et al._
(2014); Süsstrunk and Huber (2015); Xiao  _et al._ (2015); Lee  _et al._
(2018); Boada  _et al._ (2012); Celi  _et al._ (2014); An  _et al._ (2017);
Lustig  _et al._ (2019). Synthetic gauge fields have allowed physicists to
explore fundamental problems in high-energy physics, simulate novel
topological quantum matter in condensed matter physics and accomplish many
other significant tasks.

Whereas outstanding progress has been made in studying synthetic gauge fields
in the past decade, experimentalists have only realized synthetic vector gauge
fields as of now. Tensor gauge fields, however, bring us even richer physics.
Such gauge fields are crucial in the study of fracton phases of matter Chamon
(2005); Bravyi  _et al._ (2011); Haah (2011); Yoshida (2013); Vijay  _et al._
(2015, 2016); Pretko (2017a, b); Nandkishore and Hermele (2019); Pretko  _et
al._ (2020). A remarkable property of these phases is that a single fracton is
immobile due to the presence of the dipole conservation law in certain quantum
systems Scherg  _et al._ (2021); Lake  _et al._ (2022, 2023); Kohlert  _et
al._ (2023). Nevertheless, bound states of fractons, for instance, a dipole
formed by a particle-hole pair, could move. Some dipoles only move in the
direction parallel to the dipole moments, and thus are referred to as lineons.
In contrast, the so-called planons move in directions perpendicular to the
dipole moments.

A unique feature of lineons and planons is that they couple to tensor gauge
fields, offering physicists a rich playground to explore exotic phenomena
unattainable in traditional platforms Pretko (2017c); Pai and Pretko (2018);
Ma  _et al._ (2018); Bulmash and Barkeshli (2018). For instance, the kinetic
energy of a lineon in a bosonic system is written as,

$\hat{K}_{L}=-\sum_{m}(t_{2}e^{-iA_{xx}}\hat{b}^{\dagger
2}_{{m}}\hat{b}_{m+1}\hat{b}_{m-1}+h.c.),$ (1)

where $\hat{b}^{\dagger}_{{m}}$ ($\hat{b}_{{m}}$) is the creation
(annihilation) operator for bosons at site ${m}$ and $t_{2}$ is the strength
of the correlated tunneling. $A_{xx}$ is a rank-2 gauge potential and couples
to the second derivative of the bosonic field, since the kinetic energy
$\sim(A_{xx}-a^{2}\partial^{2}_{x}\theta_{B})^{2}$, where $a$ is the lattice
spacing, $\theta_{B}$ is the phase of the bosonic field and $x$ is the
coordinate of the lattice in the continuum limit. Whereas correlated
tunnelings similar to that in Eq.(1) have been realized in various systems
Yang  _et al._ (2020); Scherg  _et al._ (2021); Zhou  _et al._ (2022); Kohlert
_et al._ (2023), a fundamental question is how to imprint a phase to such
exotic hoppings. It is desirable to access a finite $A_{xx}$ and other tensor
gauge fields in experiments, which will allow experimentalists to test
profound theoretical predictions of fracton phases of matter and explore how
quantum matter interact with tensor gauge fields in laboratories.

In this work, we point out some schemes to realize synthetic tensor gauge
fields using currently available experimental techniques. We show that a
combination of a strong linear potential and a weak quadratic potential yields
a rank-2 electric field for lineons that are confined in one dimension.
Alternatively, external lasers could be implemented to induce photon-assisted
tunnelings in a linearly tilted lattice and the same tensor gauge fields are
obtained. Such electric fields lead to a new type of Bloch oscillations, in
which neither a single hole nor a single particle moves but a dipole responds
to the rank-2 electric field. In higher dimensions, interactions and vector
gauge potentials jointly supply a phase to a ring-exchange interaction and
thus deliver tensor gauge fields for planons. For instance, a planon may
experience a higher rank magnetic field that allows us to access a dipolar
Harper-Hofstadter model.

Figure 1: (a) Tensor gauge fields are absent in a linearly tilted lattice. (b)
Two pathways of second-order processes give rise to the nearest neighbor
tunneling of a dipole. (c) The long-range tunneling of a dipole is suppressed
due to a destructive interference of two pathways. (d) A quadratic potential
amounts to a finite rank-2 electric field $E_{xx}$.

We start from lineons. Since the motion of a lineon is confined in one
dimension, it is sufficient to consider a one-dimensional system. As shown in
Fig. 1(a), the Hamiltonian of one-dimensional bosons in a linearly tilted
lattice is written as
$\hat{H}_{B}=\hat{K}_{B}+\hat{V}_{B}+\sum_{m}\frac{U}{2}\hat{n}_{m}(\hat{n}_{m}-1)$,
where

$\displaystyle\hat{K}_{B}$
$\displaystyle=-\sum_{m}(t_{1}\hat{b}^{\dagger}_{{m}}\hat{b}_{m+1}+h.c.),$ (2)
$\displaystyle\hat{V}_{B}$ $\displaystyle=\sum_{m}m\Delta\hat{n}_{m},$ (3)

and $\hat{n}_{m}=\hat{b}^{\dagger}_{{m}}\hat{b}_{{m}}$. $\Delta$ is a constant
and $t_{1}$ is the single-particle tunneling strength. In the limit where
$t_{1}\ll U$, double occupancy is prohibited, and each site is filled by one
boson. When $t_{1}\ll\Delta$ is also satisfied, the tunneling of either the
particle or the hole is suppressed. Nevertheless, a dipole can hop through
some second-order processes, as shown in Fig. 1(b). These two pathways yield a
finite tunneling amplitude of the dipole,

$\hat{\tilde{K}}_{L}=-\sum_{m}(t_{2}\hat{b}^{\dagger
2}_{{m}}\hat{b}_{m+1}\hat{b}_{m-1}+h.c.),$ (4)

where $t_{2}=-t_{1}^{2}(\frac{1}{U+\Delta}-\frac{1}{\Delta})\approx
t_{1}^{2}U/\Delta^{2}$. Here, we consider the limit $U\ll\Delta$ to avoid the
irrelevant tiling-assisted resonance when $U\sim\Delta$. If we define
$\hat{D}_{m}^{\dagger}=\hat{b}^{\dagger}_{m}\hat{b}_{m-1}$,
$\hat{\tilde{K}}_{L}=-\sum_{m}(t_{2}\hat{D}_{m}^{\dagger}\hat{D}_{m+1}+h.c.)$,
describing nearest neighbor tunnelings of a dipole. Long-range tunnelings
vanish due to destructive interference as shown in Fig. 1(c).

Whereas the above perturbative approach provides us with a qualitative
picture, a more rigorous scheme is going to the rotating frame for eliminating
the linear potential $\hat{V}_{B}$. The Hamiltonian becomes time-dependent,
$\hat{\tilde{H}}_{B}=\hat{\mathcal{U}}\hat{H}\hat{\mathcal{U}}^{-1}+i\hbar(\partial_{t}\hat{\mathcal{U}})\hat{\mathcal{U}}^{-1}=\hat{\tilde{K}}_{B}(t)+\frac{U}{2}\sum_{m}\hat{n}_{m}(\hat{n}_{m}-1)$,
where $\hat{\tilde{K}}_{B}=-\sum_{m}(t_{1}e^{-i\Delta
t/\hbar}\hat{b}_{m}^{\dagger}b_{m+1}+h.c.)$, and

$\hat{\mathcal{U}}=e^{i\hat{V}_{B}t/\hbar}.$ (5)

We can then derive the Floquet Hamiltonian Goldman and Dalibard (2014),

$\hat{\tilde{H}}_{L}=\sum_{m}\bigg{(}\frac{U}{2}-2t_{2}\bigg{)}\hat{n}_{m}(\hat{n}_{m}-1)+4t_{2}\hat{n}_{m+1}\hat{n}_{m}+\hat{\tilde{K}}_{L}.$
(6)

Though $\hat{\tilde{H}}_{L}$ includes the kinetic energy of a lineon,
$\hat{\tilde{K}}_{L}$, the lineon does not couple to a tensor gauge field. We
then need to add a phase to $t_{2}$ in Eq.(4), i.e., $t_{2}\rightarrow
t_{2}e^{-iA_{xx}}$. Since a constant $A_{xx}$ can be gauged away and does not
lead to any physical consequence, here, we consider the simplest means to
obtain a time-dependent $A_{xx}$. We add an extra small quadratic potential to
the Hamiltonian, $\hat{H}_{B}^{\prime}=\hat{H}_{B}+\hat{V}_{B}^{\prime}$,
where

$\hat{V}_{B}^{\prime}=\sum_{m}\frac{1}{2}m^{2}\Delta^{\prime}\hat{n}_{m},$ (7)

and $\Delta^{\prime}$ is a constant. To eliminate
$\hat{V}_{B}+\hat{V}_{B}^{\prime}$, the unitary transformation in Eq.(5) needs
to be modified,

$\hat{\mathcal{U}}^{\prime}=e^{i(\hat{V}_{B}+\hat{V}_{B}^{\prime})t/\hbar}.$
(8)

Using this $\hat{\mathcal{U}}^{\prime}$, $\hat{\tilde{K}}_{B}$ is replaced by
$\hat{\tilde{K}}^{\prime}_{B}=-\sum_{m}(t_{1}e^{-i(\Delta+\Delta^{\prime}(m+1/2))t/\hbar}\hat{b}_{m}^{\dagger}\hat{b}_{m+1}+h.c.)$.
Applying the same method, we find $\hat{\tilde{K}}_{L}$ is replaced by
$\hat{\tilde{K}}^{\prime}_{L}=-\sum_{m}(t_{2}e^{-i\Delta^{\prime}t/\hbar}\hat{b}_{m}^{{\dagger}2}\hat{b}_{m+1}\hat{b}_{m-1}+h.c.)$
in the limit $\Delta\gg\Delta^{\prime}$. Compare this expression with Eq.(1),
we find that $A_{xx}=\Delta^{\prime}t/\hbar$ is a rank-2 tensor gauge
potential that varies linearly with time.

A time-dependent $A_{xx}$ produces a rank-2 electric field acting on dipoles,

$E_{xx}=-\frac{\partial A_{xx}}{\partial t}=-\Delta^{\prime}/\hbar.$ (9)

This result can also be understood from that a quadratic potential in Eq.(7)
produces a linear potential for a dipole. From Eq.(7), we see that when a
dipole, i.e., a particle at site $m$ and a hole at site $m-1$, move by one
lattice space, the potential energy increases by a constant $\Delta^{\prime}$.
A dipole thus experiences a linear potential or equivalently, a constant
electric field given by Eq.(9).

We should mention that a combination of a linear potential and a quadratic one
is equal to a quadratic potential that merely shifts the origin, as a well-
known result. The conventional harmonic traps used in many experiments are
thus able to produce $E_{xx}$. But different from ordinary experiments, the
central part of the harmonic trap should be avoided. Due to the slow increase
of the potential there, single-particle tunnelings cannot be suppressed.
Instead, we should use the region where the gradient of the potential is
strong enough to prohibit single-particle tunnelings. This can be achieved by
first preparing the sample in a weak harmonic trap and then adding a strong
linear potential.

Figure 2:  Time-dependent density distribution $\langle\hat{n}_{m}(t)\rangle$
of a lineon. The parameters are chosen as $U/t_{1}=4$, $\Delta/t_{1}=40$, and
$\Delta^{\prime}/t_{1}=-0.01$ (a), $0.01$ (b),$0$ (c), $0.01$(d). A particle
excitation is isolated from the hole in (d). These conditions lead to Bloch
oscillations in opposite directions in (a) and (b), a symmetric expansion in
(c) and immobile dynamics in (d). (e)$x_{D}$ as a function of time. Results of
the exact diagonalization of the effective Hamiltonian of a single dipole are
shown in red dotted (a), green dashed (b) and blue dash-dotted lines (c).
Black curves are results of the full Hamiltonians using TEBD. Horizontal lines
denote $-4t_{2}/\Delta^{\prime}$.

To explicitly show the effect of $E_{xx}$, we have numerically computed the
dynamics of a lineon using the time-evolution block-decimation (TEBD) method.
The Hamiltonian is chosen as $\hat{H}_{B}^{\prime}$ which includes both a
strong linear potential and a weak quadratic potential. The initial state
includes a single dipole, i.e., a single particle-hole excitation in a Mott
insulator. Whereas the initial state of the dipole can be an arbitrary
wavepacket, for convenience, we let the superposition state of the dipole
occupy two lattice sites, i.e.,
$\ket{\Psi(t=0)}=(\hat{D}^{\dagger}_{0}+\hat{D}^{\dagger}_{1})|\text{MI}\rangle/2$,
where $|\text{MI}\rangle=\prod_{m}\hat{b}^{\dagger}_{m}|0\rangle$ is a Mott
insulator with one particle per site. As shown in Fig. 2(a-c), when
$\Delta^{\prime}=0$, the wavepacket simply expands symmetrically towards both
directions as $t$ increases. Once $\Delta^{\prime}\neq 0$, the wavepacket
moves towards one direction as a result of a finite $E_{xx}$. Changing the
sign of $\Delta^{\prime}$ reverses the direction of the motion. In contrast,
if the separation between the particle and the hole is larger than one lattice
spacing, each of them becomes an isolated excitation. Neither the particle nor
the hole moves, as shown in Fig. 2(d).

To further quantitatively characterize the dynamics of a dipole, we define

${x}_{D}=\frac{1}{2}\langle\sum_{m=1}^{L}[m-(L{+}1)/2]^{2}\hat{n}_{m}\rangle-\frac{1}{24}L(L^{2}-1).$
(10)

In the absence of a dipole, $\langle\hat{n}_{m}\rangle=1$, $x_{D}=0$. When
there is an additional particle at site $m+1$ and a hole at site $m$,
$x_{D}=m-L/2$. We therefore read from ${x}_{D}$ the position of the dipole.
From Fig. 2(e), it is clear that $x_{D}$ oscillates as time goes by when
$\Delta^{\prime}\neq 0$. We have smoothed the curves by averaging the results
within a much smaller time scale $2\pi\hbar/\Delta$ (Supplementary Materials).
In the presence of a finite $E_{xx}=-\Delta^{\prime}/\hbar$, the energy
penalty for a dipole to hop to the nearest neighbor site is $\hbar E_{xx}$.
The effective Hamiltonian of a single dipole is written as $\hat{H}_{\rm
eff}=\hat{\tilde{K}}_{L}-\hbar
E_{xx}\sum_{m}m\hat{D}^{\dagger}_{m}\hat{D}_{m}{/2}$. The period of the Bloch
oscillation of a dipole is given by $T_{D}=2\pi/|E_{xx}|$. Using Eq.(9), we
obtain

$T_{D}=2\pi\frac{\hbar}{|\Delta^{\prime}|}.$ (11)

The results of $H_{\rm eff}$ are shown in Fig.2(e). Similarly, the oscillation
amplitude is given by $-4t_{2}/\Delta^{\prime}$ (Supplementary material). Both
expressions agree well with the numerical solution of the full Hamiltonian of
bosons $\hat{H}^{\prime}_{B}$ using TEBD.

As for vector gauge fields, though a linear external potential in the real
space is equivalent to a time-dependent vector potential, it is useful to
realize the latter directly in experiments. The same consideration applies to
tensor gauge fields. To this end, we consider another method of using external
lasers to induce photon-assisted tunnelings. Such tunnelings naturally carry
the phases of external lasers and can be made position-dependent, as shown in
Fig. 3. This scheme was used in pioneering works of realizing the Harper-
Hofstadter model Aidelsburger  _et al._ (2013); Miyake  _et al._ (2013). Here,
a key difference is that, instead of using a resonant coupling, we consider a
finite detuning, $\Delta=\Delta_{0}-\hbar\omega$, where $\Delta_{0}$ is the
energy mismatch between the nearest neighbor sites of the linearly tilted
lattice, and $\omega$ is the frequency of the driving field. If one uses Raman
lasers, $\omega=\omega_{1}-\omega_{2}$ will be the difference between the
frequencies of the two lasers. As such, there exists a residual linear
potential described by Eq.(3). The one-dimensional system is then described by
the same Hamiltonian described before. Most of the remaining discussions apply
except an important difference that $\hat{K}_{B}$ is replaced by

$\hat{K}^{\prime}_{B}=-\sum_{m}(t_{1}e^{-im\theta}\hat{b}^{\dagger}_{{m}}\hat{b}_{m+1}+h.c.).$
(12)

Correspondingly, the difference in the phase carried by the particle and the
hole leads to a finite phase in the tunneling of a dipole. The kinetic energy
of a dipole is then readily given by Eq.(1), where $A_{xx}=\theta$, and there
is no need to introduce an extra quadratic potential. To have a time-dependent
$A_{xx}$, we recall that $\theta=({\bf k}_{1}-{\bf k}_{2})\cdot\bf{a}$, where
${\bf k}_{i=1,2}$ is the wave vectors of the Raman lasers and ${\bf a}$ is the
unit vector of the lattice. The incident angles of the Raman lasers can then
be changed as time goes by.

Figure 3: (a) Raman lasers (green and red arrows) produce photon-assisted
tunnelings in a linearly tilted lattice. A finite detuning
$\Delta=\Delta_{0}-\hbar\omega$ leads to a residual linear potential. (b) The
site-dependent phase of the single-particle tunneling provides a phase in the
tunneling of a dipole (violet arrow).

In higher dimensions, physics becomes even richer. Whereas the previously
discussed method still works for lineons, there exist planons that could move
in two directions that are perpendicular to the dipole moment. To simplify
notations, we consider two dimensions as an example. A ring exchange
interaction allows a planon to move,

$\hat{\tilde{K}}_{P}=-\sum_{\bf m}(t_{2}^{\prime}\hat{b}^{\dagger}_{\bf
m}\hat{b}^{\dagger}_{{\bf m}+\hat{\bf x}+\hat{\bf y}}\hat{b}_{{\bf m}+\hat{\bf
x}}\hat{b}_{{\bf m}+\hat{\bf y}}+h.c.),$ (13)

where ${\bf m}=(m_{x},m_{y})$ is the lattice index and $\hat{\bf x}$ and
$\hat{\bf y}$ are the unit vectors in the $x$ and $y$ directions,
respectively. If we define the creation operator for a planon as
$\hat{D}^{\dagger}_{{\bf m},{\bf\hat{n}}}=\hat{b}^{\dagger}_{\bf
m}\hat{b}_{{\bf m}-\hat{\bf n}}$, where $\hat{\bf n}=\hat{\bf x},\hat{\bf y}$,
$t_{2}^{\prime}$ corresponds to the tunneling strength of a planon. When
applying similar second-order processes of single-particle tunnelings as that
in discussions about lineons, it produces both the ring exchange interaction
and kinetic energies of lineons (Supplementary Materials).

To access purely a ring-exchange interaction with a controllable phase, we
consider an alternative scheme using particles with long-range interactions in
a lattice. As shown in Fig. 4(a), the lattice potential in the $x$ direction
is strong such that the tunneling along this direction is suppressed, i.e.,
$t_{x}=0$. In the $y$ direction, Raman lasers imprint a $x$-dependent phase in
the tunneling, $t_{y}e^{im_{x}\theta}$. The Hamiltonian is written as
$\hat{H}_{M}=\hat{K}_{M}+\hat{U}_{M}$, where

$\displaystyle\hat{K}_{M}$ $\displaystyle=-\sum_{\bf
m}(t_{y}e^{im_{x}\theta}\hat{b}^{\dagger}_{\bf m}\hat{b}_{{\bf m}+\hat{\bf
y}}+h.c.),$ (14) $\displaystyle\hat{U}_{M}$
$\displaystyle=\frac{U}{2}\sum_{\bf m}\hat{n}_{\bf m}(\hat{n}_{\bf
m}-1)+V\sum_{\langle{\bf m},{\bf n}\rangle}\hat{n}_{\bf m}\hat{n}_{\bf n}.$
(15)

$V$ denotes the nearest neighbor interaction. Here, we consider an isotropic
nearest-neighbor interaction for simplicity. Anisotropic interactions do not
change the main results.

When a dipole is aligned along the $x$ direction, the tunneling of either the
particle or the hole is suppressed in the limit where $V\gg t_{y}$, as such
tunnelings lead to an extra energy penalty, as shown in Fig. 4(a).
Nevertheless, second-order processes allow such a dipole to tunnel in the $y$
direction. The kinetic energy of this planon is written as

$\hat{K}_{P}=-\sum_{\bf m}(J_{y}e^{-iA_{xy}}\hat{D}^{\dagger}_{\bf m,\hat{\bf
x}}\hat{D}_{{\bf m}+\hat{\bf y},\hat{\bf x}}+h.c.),$ (16)

where $J_{y}=-2t_{y}^{2}/V$, and $A_{xy}=\theta$ denotes a synthetic tensor
gauge field. Similar to previous discussions, a time-dependent $A_{xy}$ gives
rise to an electric field $E_{xy}=-\partial\theta/\partial t$ acting on a
planon. This time, planon will perform Bloch oscillations in the direction
perpendicular to the dipole moment.

Figure 4:  (a) The tunneling of a planon (green thick arrow) through second
order processes of single-particle tunnelings (thin black arrow). (b) A
$z$-dependent $A_{xy}$ provides a magnetic field for the planon. (c) The side
view of (b) represents a dipolar Harper-Hofstadter model where each plaquette
is threaded with a finite flux $\phi$.

Going to three dimensions, the variance of $A_{xy}$ in the $z$ direction leads
to a magnetic field $B_{xyz}=-\partial A_{xy}/\partial z$. As shown in Fig.
4(b), such a magnetic field can be accessed by introducing a $z$-dependent
$\theta=\phi z$. To this end, we consider a three-dimensional model,
$\hat{H}^{\prime}_{M}=\hat{K}^{\prime}_{M}+\hat{U}_{M}$, where

$\hat{K}^{\prime}_{M}=-\sum_{\bf
m}(t_{y}e^{im_{x}m_{z}\phi}\hat{b}^{\dagger}_{\bf m}\hat{b}_{{\bf m}+\hat{\bf
y}}+t_{z}\hat{b}^{\dagger}_{\bf m}\hat{b}_{{\bf m}+\hat{\bf z}}+h.c.).$ (17)

$\hat{m}=(m_{x},m_{y},m_{z})$ has been promoted to the index of a three-
dimensional lattice. For a fixed $m_{x}$, the phase carried by $t_{y}$ changes
linearly as a function of $m_{z}$. We thus just need to generalize the scheme
of realizing the Harper-Hofstadter model to a $m_{x}$ dependent one. The
simplest scheme is to use the synthetic dimension formed by internal degrees
of freedom like hyperfine spins or different atomic species, which may feel
different vector gauge potentials Aidelsburger  _et al._ (2013). As the
tunneling in the $x$ direction is suppressed, it is sufficient to consider
only two layers with $m_{x}=0$ and $m_{x}=1$, respectively. In other words,
only a two-component system is required. Applying component-selective Raman
lasers that do not affect the first component, the tunneling of this component
is unchanged. $t_{y}$ in the first layer with $m_{x}=0$ thus does not acquire
an additional phase. In contrast, the Raman lasers imprint a phase to the
tunneling of the second component. $t_{y}$ in the second layer with $m_{x}=1$
thus becomes $t_{y}e^{im_{z}\phi}$. The minimal version of Eq.(17) is then
realized. Alternatively, if one directly implements the scheme in Aidelsburger
_et al._ (2013), where these two components feel opposite vector gauge
potentials, $\phi\rightarrow 2\phi$.

Once accessing Eq.(17) in laboratories, a dipolar Harper-Hofstadter model is
realizable. Similar to the previous discussions, a dipole aligned in the $x$
direction can also tunnel in the $z$ direction with a tunneling strength
$J_{z}=-t_{z}^{2}/V$. If we view the system along the $x$-axis, the dipole
appears to be a two-dimensional particle moving in the $y-z$ plane, as shown
in Fig. 4(c). This thus realizes a dipolar Harper-Hofstadter model,

$\begin{split}\hat{H}_{\rm dHH}=&\frac{1}{2}\sum_{\bf
m}(J_{y}e^{-im_{z}\phi}\hat{D}_{\bf m,\hat{x}}\hat{D}^{\dagger}_{\bf
m+\hat{y},\hat{x}}\\\ &+J_{z}\hat{D}_{\bf m,\hat{x}}\hat{D}^{\dagger}_{\bf
m+\hat{z},\hat{x}}+h.c.).\end{split}$ (18)

where the flux included in a single plaquette becomes finite. This dipolar
Harper-Hofstadter model is expected to host many new quantum phenomena. Unlike
conventional Chern insulators, the edge of a finite strip of such a model
features a vanishing particle current but a finite dipole current that is
worth future theoretical and experimental studies.

We have shown that some simple protocols fulfill the lofty goal of creating
synthetic tensor gauge fields in laboratories. Our scheme can be generalized
to fermions and multi-component systems. It is also possible to realize even
higher rank gauge fields using related schemes. We hope that our work may
stimulate more interest to study higher rank gauge fields.

This work is supported by the U.S. Department of Energy, Office of Science
through the Quantum Science Center (QSC), a National Quantum Information
Science Research Center. S.Zhang is supported by National Natural Science
Foundation of China(Grant No.12174138).

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Supplementary Materials for

“Synthetic tensor gauge fields”

In this supplementary material, we present results of the fast oscillation due
to the micromotion of single particles, the amplitude of Bloch oscillation,
and a scheme that produces both the ring-exchange interaction and the kinetics
of lineons.

##  .1 Fast oscillations due to the micromotion of single-particles

When deriving the effective Hamiltonian for the dipole using the Floquet
theory, we have dropped off the micromotion of single-particles. In the
presence of a strong linear potential, a single particle experiences a Bloch
oscillation with a frequency $\omega_{s}=\Delta/\hbar$. This is a micromotion
in a much smaller time scale than that of the Bloch oscillation of the dipole.
Whereas numerical results clearly show such fast oscillations, we consider the
average over time in such a small scale $T_{f}=2\pi/\omega_{s}$ to smooth the
results. Here, we show similar numerical results as that in Fig. 2 (e) using
exact diagonalization with $L=7$. The main figure in Fig. S1(a) show the
result before doing the average. The fast oscillations are highlighted in the
inset. Fig. S1(b) shows the result after doing the average.

Figure S1:  $x_{D}$ as a function of the time before (a) and after (b)
averaging over a small time period $T_{f}=2\pi\hbar/\Delta$. The parameters
are chosen as $U/t_{1}=4$, $\Delta/t_{1}=40$ and
$\Delta^{\prime}/t_{1}=-0.03$, $0.03$. Results of the exact diagonalization of
the effective Hamiltonian of a single dipole are shown in red dotted
($\Delta^{\prime}/t_{1}=0.03$), green dashed ($\Delta^{\prime}/t_{1}=-0.03$)
lines. Black curves are the results of the exact diagonalization of the full
Hamiltonians. Horizontal lines denote $-4t_{2}/\Delta^{\prime}$. Insets show
the results in a small time window.

##  .2 The amplitude of Bloch oscillation

Within the effective theory, the dipole dynamics is captured by a tight-
binding Hamiltonian with a linear potential,

$\hat{H}_{\rm
eff}=\sum_{m}-2t_{2}(\ket{m}\bra{m+1}+h.c.)+\Delta^{\prime}m\ket{m}\bra{m},$
(S1)

where $\ket{m}=\hat{D}^{\dagger}_{m}\ket{\rm MI}/\sqrt{2}$ denotes the
normalized single dipole state. The eigenstates of the effective Hamiltonian
$\hat{H}_{\rm eff}$ are the Wannier Stark states. The time evolution operator
is written as Hartmann  _et al._ (2004)

$U(t)=\sum_{m,m^{\prime}}J_{m-m^{\prime}}(8t_{2}\sin(\Delta^{\prime}t/2\hbar)/\Delta^{\prime})e^{i(m-m^{\prime})(\pi-\Delta^{\prime}t/\hbar)/2-im^{\prime}\Delta^{\prime}t/\hbar}\ket{m}\bra{m^{\prime}},$
(S2)

where $J_{m}(x)$ is the Bessel function of the first kind. $U(t)$ has a period
$T=2\pi\hbar/\Delta^{\prime}$. Without loss of generality, we consider an
initial state $\ket{\psi(0)}=(\ket{0}+\ket{1})/\sqrt{2}$. The quantum state
reaches its maximal extension at half of the oscillation period. The averaged
dipole displacement corresponding to the oscillation amplitude is written as

$\bra{\psi(T/2)}m\ket{\psi(T/2)}-\bra{\psi(0)}m\ket{\psi(0)}=\sum_{m}\frac{1}{2}\bigg{(}J_{m}(8t_{2}/\Delta^{\prime})-J_{m-1}(8t_{2}/\Delta^{\prime})\bigg{)}^{2}m-\frac{1}{2}=-\frac{4t_{2}}{\Delta^{\prime}}.$
(S3)

##  .3 Producing both the ring-exchange interaction and kinetics of lineons

We consider a two-dimensional Hamiltonian that is written as
$\hat{H}_{B}=\hat{K}_{B}+\hat{V}_{B}+\hat{U}_{B}$ where

$\displaystyle\begin{aligned} \hat{K}_{B}=&-\sum_{{\bf m},\hat{\bf
n}}(t_{1}\hat{b}_{\bf m}\hat{b}_{{\bf m}+\hat{\bf n}}+h.c.),\\\
\hat{V}_{B}=&\sum_{\bf m}(m_{x}\Delta_{x}+m_{y}\Delta_{y})\hat{n}_{\bf m},\\\
\hat{U}_{B}=&\frac{U}{2}\sum_{\bf m}\hat{n}_{\bf m}(\hat{n}_{\bf
m}-1)+V\sum_{\bf m}\hat{n}_{\bf m}(\hat{n}_{\bf m+\hat{x}}+\hat{n}_{\bf
m+\hat{y}}).\end{aligned}$ (S4)

$\Delta_{x}$ and $\Delta_{y}$ are energy matches between nearest-neighbor
sites along the $x$ and $y$ directions, respectively. In 2D, the finite
nearest-neighbor interaction $V$ is necessary for realizing correlated pair
tunnelings through second order processes of single-particle tunnelings. Such
$V$ can be realized by using polar molecules or Rydberg atoms arrays. As shown
by Fig.S2, when $V\neq 0$, four pathways yield a finite amplitude of the ring
exchange interaction, allowing the a planon to tunnel along the perpendicular
direction of the dipole moment,

$\hat{\tilde{K}}_{P}=-\sum_{\bf m}(t_{2}^{\prime}\hat{b}^{\dagger}_{\bf
m}\hat{b}^{\dagger}_{{\bf m}+\hat{\bf x}+\hat{\bf y}}\hat{b}_{{\bf m}+\hat{\bf
x}}\hat{b}_{{\bf m}+\hat{\bf y}}+h.c.),$ (S5)

where $t^{\prime}_{2}=t^{\prime}_{2x}+t^{\prime}_{2y}$ and
$t^{\prime}_{2x}=\frac{2V}{V^{2}-\Delta^{2}_{x}}t^{2}_{1}$,
$t^{\prime}_{2y}=\frac{2V}{V^{2}-(U+\Delta_{y})^{2}}t^{2}_{1}$.

Figure S2: The ring-exchange interaction. (a) and (b) correspond to the
second-order processes of single-particle tunnelings along the $x$ and the $y$
directions, respectively.

Some other tunnelings of a dipole can also be produced by second order
processes, as shown in Fig.S3. For example, the tunneling of a lineon exists,

$\hat{\tilde{K}}_{L}=-\sum_{\bf m}(t_{2}\hat{b}^{{\dagger}2}_{\bf
m}\hat{b}_{\bf m+\hat{y}}\hat{b}_{\bf
m-\hat{y}}+\tilde{t}_{2}\hat{b}^{\dagger}_{\bf
m-\hat{y}}\hat{b}^{\dagger}_{\bf m}\hat{b}_{\bf m+\hat{y}}\hat{b}_{\bf
m+2\hat{y}}+h.c.),$ (S6)

where $t_{2}=\frac{(2V-U)}{(V+\Delta_{y})(V-U-\Delta_{y})}t^{2}_{1}$ and
$\tilde{t}_{2}=\frac{V}{(U-2V+\Delta_{y})(U-V+\Delta_{y})}t^{2}_{1}$
correspond to the amplitude of the nearest-neighbor and the next-nearest-
neighbor tunneling of a lineon, respectively. Furthermore, the dipole can
tunnel along the diagonal direction,

$\hat{\tilde{K}}_{\mathrm{diag}}=-\sum_{\bf
m}(\tilde{t}_{\hat{x}+\hat{y}}\hat{b}^{\dagger}_{\bf m}\hat{b}_{\bf
m+\hat{y}}\hat{b}_{\bf m+\hat{x}+\hat{y}}\hat{b}^{\dagger}_{\bf
m+\hat{x}+2\hat{y}}+\tilde{t}_{\hat{x}-\hat{y}}\hat{b}^{\dagger}_{\bf
m}\hat{b}_{\bf m+\hat{y}}\hat{b}_{\bf m+\hat{x}-\hat{y}}\hat{b}^{\dagger}_{\bf
m+\hat{x}}+h.c.),$ (S7)

where $\tilde{t}_{\hat{x}+\hat{y}}=\tilde{t}_{\hat{x}+\hat{y}}=\tilde{t}_{2}$.

Figure S3:  (a) and (b) depict the nearest-neighbor and next-nearest-neighbor
hopping of a lineon, respectively. (c) and (d) depict the tunnelings of a
dipole along the diagonal directions.

We define $\hat{D}^{\dagger}_{\bf m,\hat{\bf y}}=\hat{b}^{\dagger}_{\bf
m}\hat{b}_{\bf m-\hat{y}}$, the creation operator of a dipole along the $y$
direction. The kinetic energy of the dipole can be written as

$\displaystyle\begin{aligned} \hat{\tilde{K}}_{D}&=-\sum_{\bf
m}\big{\\{}t^{\prime}_{2}\hat{D}^{\dagger}_{{\bf m+\hat{x}},\hat{\bf
y}}\hat{D}_{{\bf m},\hat{\bf y}}+t_{2}\hat{D}^{\dagger}_{{\bf
m+\hat{y}},\hat{\bf y}}\hat{D}_{{\bf m},\hat{\bf y}}\\\
&+\tilde{t}_{2}(\hat{D}^{\dagger}_{{\bf m+2\hat{x}},\hat{\bf
y}}+\hat{D}^{\dagger}_{{\bf m+\hat{x}+\hat{y}},\hat{\bf
y}}+\hat{D}^{\dagger}_{{\bf m+\hat{x}-\hat{y}},\hat{\bf y}})\hat{D}_{{\bf
m},\hat{\bf y}}+h.c.\big{\\}}\end{aligned}$ (S8)

We note that, in the limit where $U$ is much larger than any other energy
scales, $t^{\prime}_{2}\rightarrow\frac{2V}{V^{2}-\Delta^{2}_{x}}t_{1}^{2}$,
$t_{2}\rightarrow\frac{1}{V+\Delta_{y}}t_{1}^{2}$, and
$\tilde{t}_{2}\rightarrow 0$. In this limit, only the nearest neighbor
tunnelings exist but a dipole can tunnel in both directions, being a
superposition of a lineon and a planon.

To create a tensor gauge field, similar to discussions in the main text, an
extra small quadratic potential in the $x-y$ plane can be added,
$\hat{H}^{\prime}_{B}=\hat{H}_{B}+\hat{V}_{L}$ where

$\hat{V}^{\prime}_{B}=\frac{1}{2}\sum_{\bf m}(\alpha m_{x}+\beta
m_{y})^{2}\Delta^{\prime}\hat{n}_{\bf m},$ (S9)

where $\alpha$ and $\beta$ are both dimensionless coefficient to control the
direction of quadratic potential which satisfying $\alpha^{2}+\beta^{2}=1$.
Using a unitary transformation to eliminate
$\hat{V}_{B}+\hat{V}^{\prime}_{B}$, the single-particle tunneling acquires an
additional time-dependent phase,

$\displaystyle\begin{aligned} \hat{\tilde{K}}_{B}=&-\sum_{\bf
m}(t_{1}e^{-i\Delta_{x}t-i(m_{x}+1/2)\alpha^{2}\Delta^{\prime}t-im_{y}\alpha\beta\Delta^{\prime}t}\hat{b}^{\dagger}_{\bf
m}\hat{b}_{\bf m+\hat{x}}\\\
&+t_{1}e^{-i\Delta_{y}t-im_{x}\alpha\beta\Delta^{\prime}t-i(m_{y}+1/2)\beta^{2}\Delta^{\prime}t}\hat{b}^{\dagger}_{\bf
m}\hat{b}_{\bf m+\hat{y}}+h.c.)\end{aligned}$ (S10)

In the limit $\Delta^{\prime}\ll\Delta$, the Floquet theory gives rise to a
ring-exchange interaction,

$\hat{\tilde{K}}_{P}=-\sum_{\bf
m}(t_{2}^{\prime}e^{-i\alpha\beta\Delta^{\prime}t}\hat{b}^{\dagger}_{\bf
m}\hat{b}^{\dagger}_{{\bf m}+\hat{\bf x}+\hat{\bf y}}\hat{b}_{{\bf m}+\hat{\bf
x}}\hat{b}_{{\bf m}+\hat{\bf y}}+h.c.).$ (S11)

Similar discussions apply to other tunnelings.

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