††thanks: Corresponding author<EMAIL_ADDRESS>
# Dynamic mass generation on two-dimensional electronic hyperbolic lattices

Noble Gluscevich  Equal contributors  Department of Physics, Lehigh
University, Bethlehem, Pennsylvania, 18015, USA  Abhisek Samanta  Equal
contributors  Department of Physics, The Ohio State University, Columbus, Ohio
43210, USA  Sourav Manna  Department of Condensed Matter Physics, Weizmann
Institute of Science, Rehovot 7610001, Israel  Raymond and Beverly Sackler
School of Physics and Astronomy, Tel-Aviv University, Tel Aviv 6997801, Israel
Bitan Roy  Department of Physics, Lehigh University, Bethlehem, Pennsylvania,
18015, USA

###### Abstract

Free electrons hopping on hyperbolic lattices embedded on a curved space of
negative curvature can foster (a) Dirac liquids, (b) Fermi liquids, and (c)
flat bands, respectively characterized by a vanishing, constant, and divergent
density of states near the half-filling. From numerical self-consistent
Hartree-Fock analyses, here we show that the nearest-neighbor and on-site
Coulomb repulsions respectively give rise to charge-density-wave and
antiferromagnetic orders featuring staggered patterns of average electronic
density and magnetization in all these systems, when the hyperbolic
tessellation is accomplished by periodic arrangements of even $p$-gons. Both
quantum orders dynamically open mass gaps near the charge neutrality point.
Only on hyperbolic Dirac materials these orderings take place via quantum
phase transitions (QPTs) beyond critical interactions, which however decreases
with increasing curvature, showcasing curvature induced weak coupling QPTs. We
present scaling of these masses with the corresponding interaction strengths
and display their spatial variations.

_Introduction_. From table salts to quantum materials, crystals are ubiquitous
in nature. They feature discrete translational, rotational and reflection
symmetries, manifesting periodic arrangements of regular polygons with $p$
arms ($p$-gon). Yet another integer plays a pivotal role for the geometric
classification of crystals, the number of nearest-neighbor (NN) sites $q$,
connected to each vertex of a $p$-gon. On a flat Euclidean plane, these
symmetry constraints are encoded in an equation $(p-2)(q-2)=4$, permitting
only triangular ($2p=q=6$), square ($p=q=4$), and honeycomb ($p=2q=6$)
lattices. On the other hand, hyperbolic tessellation, realized on a negatively
curved space, can be accomplished when a mathematical inequality
$(p-2)(q-2)>4$ is satisfied. Therefore, curved hyperbolic space accommodates
infinitely many periodic lattices, characterized by a pair of integers $(p,q)$
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

Naturally, hyperbolic quantum materials are expected to harbor a variety of
electronic band structures [5, 7, 8, 9, 10, 11, 12, 13, 14]. They can be
classified in terms of the density of states (DOS) at half-filling, for
example. As such, a family of hyperbolic lattices with even integer $p$
(bipartite) fosters (a) Dirac liquids, (b) Fermi liquids, and (c) flat bands
depending on $p$ and $q$, which can be captured from a free electron tight-
binding model with only the NN hopping. Respectively, they feature vanishing,
constant, and divergent DOS at zero energy [$\rho(0)$]. See Table 1.

$p$-gon | Dirac liquid | Fermi liquid | Flat band
---|---|---|---
$p/2=$ odd | $q=3$ | $q=4,5$ | $\times$
$p/2=$ even | $\times$ | $q=3$ | $q=4,5$
Table 1: Classification of bipartitie (even $p$) electronic hyperbolic
lattices, characterized by pair of integers $p$ and $q$, in terms of the
density of states at zero energy $\rho(0)$. Dirac liquid, Fermi liquid, and
flat band are respectively identified from vanishing, constant, and divergent
$\rho(0)$.

On such electronic hyperbolic lattices, we numerically investigate the role of
NN ($V$) and on-site Hubbard ($U$) repulsions at half-filling within the mean-
field or Hartree-Fock approximation. Respectively, they are shown to support
staggered patterns of electronic density and spin, two quantum phases named
charge-density-wave (CDW) and antiferromagnet (AFM). While infinitesimal $V$
and $U$ are sufficient to nucleate these phases on hyperbolic lattices hosting
Fermi liquids and flat bands, in Dirac liquids such orderings take place at
finite interactions via quantum phase transitions (QPTs), due to the linearly
vanishing DOS therein. These findings are summarized in Fig. 1 and Fig. 2.
More intriguingly, as the curvature is strengthened in Dirac materials
(realized for larger $p$), the critical couplings for these two ordering
decrease monotonically, as shown in Fig. 3, indicating curvature induced weak
coupling QPTs.

Figure 1: Charge-density-wave (CDW) on hyperbolic lattices resulting from the
nearest-neighbor Coulomb repulsion ($V$). The spatial variation of the average
electronic density measured from the half-filling on (a) $(10,3)$, (b)
$(12,3)$, and (c) $(12,4)$ hyperbolic lattices for $V=0.8$. Scaling of the CDW
order parameter at the center (red) and averaged over the entire system (blue)
with $V$ in these systems are respectively shown in (d), (e), and (f). Density
of states for free fermions (black) and in the presence of the CDW order (red)
in these systems are shown in (g), (h) and (i), respectively, for $V=1.0$,
displaying the formation of a mass gap. Here, the results are presented for
third generation $(10,3)$, $(12,3)$, and $(12,4)$ hyperbolic lattices,
respectively containing $2880$, $7680$, and $13080$ sites [16]. Critical
coupling for the CDW ordering in $(10,3)$ Dirac system is estimated to be
$V_{c}\approx 0.69$ [inset of (d)], while in $(12,3)$ Fermi liquids it follows
the BCS scaling [inset of (e)].

_Free fermions_. A tight-binding model for free fermions, allowed to hop only
between the NN sites of hyperbolic lattices [16], takes the form

$H_{0}=-\sum_{\langle i,j\rangle}t_{ij}c^{\dagger}_{i}c_{j}.$ (1)

Here $c^{\dagger}_{i}$ ($c_{i}$) is the fermionic creation (annihilation)
operator on the $i$th site and $\langle\cdots\rangle$ restricts the summation
within the NN sites. The spin independent NN hopping amplitude $t_{ij}$ is
assumed to be constant $t$, which we set to be unity. For the sake of
simplicity, we suppress the spin index of the fermionic degrees of freedom.
Inclusion of spin thus only causes a mere doubling of $H_{0}$. For even
integer $p$, $H_{0}$ possesses a particle-hole symmetry, and the system
belongs to the class BDI. A half-filled system keeps all the negative
(positive) energy states occupied (empty), and the average fermionic density
at each site is $1/2$, manifesting the sublattice symmetry. Then $\rho(0)$
serves as a smoking gun probe to group quantum hyperbolic materials into three
broad classes, characterized by vanishing (Dirac liquid), constant (Fermi
liquid), and divergent (flat band) $\rho(0)$, as shown in Table 1.

Figure 2: On-site Hubbard repulsion ($U$) mediated antiferromagnet (AFM)
ordering on hyperbolic lattices. All the details are the same as in Fig. 1,
but for the AFM order [Eq. (9)]. Top row: Self-consistent solutions of
magnetization at each site measured from its value at half-filling (zero),
featuring the AFM order for $U=6.0$. Here, the results are presented for third
generation $(10,3)$, and second generation $(12,3)$ and $(12,4)$ hyperbolic
lattices, respectively containing $2880$, $972$, and $1320$ sites [16].

_NN repulsion_. We now set out to investigate the role of electronic
interactions in these three families of quantum fluids. At first, we consider
spinless fermions and the NN Coulomb repulsion ($V$), captured by the
Hamiltonian

$H_{V}=H_{0}+\frac{V}{2}\sum_{\langle i,j\rangle}n_{i}n_{j}-\mu N,$ (2)

where $n_{i}=c^{\dagger}_{i}c_{i}$ is the fermionic density on site $i$, $N$
is the total number of spinless fermions in the half-filled system and $\mu$
is the chemical potential. Performing a Hartree decomposition of the quartic
term, we arrive at the effective single-particle Hamiltonian [17, 18, 19]

$H^{\rm Har}_{V}=H_{0}+V\sum_{\langle i,j\rangle}\bigg{(}\langle
n_{B,i}\rangle n_{A,j}+\langle n_{A,i}\rangle n_{B,j}\bigg{)}-\mu N.$ (3)

Here $\langle n_{A}\rangle$ ($\langle n_{B}\rangle$) corresponds to the site
dependent self-consistent average electronic density on the sublattice $A$
($B$). We measure them relative to the uniform density at half-filling
according to $\langle n_{A,i}\rangle=1/2+\delta_{A,i}$ and $\langle
n_{B,i}\rangle=1/2-\delta_{B,i}$. To maintain the system at the half-filling,
throughout we choose $\mu=V/2$ and ensure that
$\sum_{i}\left(\delta_{A,i}-\delta_{B,i}\right)=0$. The positive definite
quantities $\delta_{A}$ and $\delta_{B}$ yield the local CDW order parameter,
defined as

$~{}\delta_{\rm CDW}=\frac{1}{2}\left(\delta_{A}+\delta_{B}\right),$ (4)

where for a given $A$ site, the $B$ site is one of the two NN sites belonging
to the same generation of the hyperbolic lattice [16]. We numerically compute
$\delta_{A}$ and $\delta_{B}$, and subsequently $\delta_{\rm CDW}$ in the
entire system with open boundary condition for a wide range of $V$. See Fig.
1.

When $\delta_{A}$ and $\delta_{B}$, thus consequently $\delta_{\rm CDW}$ are
finite in the entire system (not necessarily uniform), it becomes an insulator
at half-filling by spontaneously breaking the sublattice symmetry. To
appreciate this outcome, we define a two-component superspinor
$\Psi^{\top}=(c_{A},c_{B})$, where $c_{A}$ and $c_{B}$ are both
$N$-dimensional spinors constituted by the annihilation operators on the sites
belonging to the $A$ and $B$ sublattices, respectively. In this basis, the
tight-binding Hamiltonian and the Hamiltonian with the CDW order parameter
respectively takes the form

$~{}\hat{h}_{0}=\left(\begin{array}[]{cc}{\bm{0}}&{\bf t}\\\ {\bf
t}^{\top}&{\bf 0}\end{array}\right)\>\>\text{and}\>\>\hat{h}_{\rm
CDW}=\left(\begin{array}[]{cc}{\bm{\Delta}}&{\bm{0}}\\\
{\bm{0}}&-{\bm{\Delta}}\end{array}\right)\equiv\hat{h}_{\Delta},$ (5)

where ${\bm{0}}$ is an $N$-dimensional null matrix, ${\bf t}$ is the
intersublattice hopping matrix, and $\pm{\bm{\Delta}}$ are $N$-dimensional
diagonal matrices, whose entries are the self-consistent solutions of
$\delta_{A}$ and $\delta_{B}$ at various sites of the system, respectively,
and $\top$ denotes transposition. As $\hat{h}_{0}$ and $\hat{h}_{\rm CDW}$
mutually _anticommute_ with each other ($\\{\hat{h}_{0},\hat{h}_{\rm
CDW}\\}=0$), the CDW order acts like a mass for gapless fermions and its
spontaneous nucleation causes insulation near the charge neutrality point by
opening a mass gap in the electronic spectra with vanishing [Fig. 1(g)],
constant [Fig. 1(h)], and divergent [Fig. 1(i)] DOS at zero energy.

As $\rho(E)\sim|E|$ near the zero energy, the condensation of the CDW order on
hyperbolic Dirac systems occurs beyond a critical NN repulsion ($V_{c}$),
slightly away from which $\delta_{\rm CDW}\sim\sqrt{V-V_{c}}$ for $V>V_{c}$,
in turn allowing us to estimate $V_{c}$ by minimizing the finite size effects
[inset of Fig. 1(d)]. By contrast, when $\rho(0)$ is finite or divergent, the
CDW order sets in for infinitesimal $V$. In the former system, the CDW order
parameter follows the BCS scaling $\delta_{\rm CDW}\sim\exp(-\kappa/V)$ due to
a constant $\rho(0)$, which is further substantiated from the exponential fit
between $\delta_{\rm CDW}$ and $1/V$ [inset of Fig. 1(e)]. On a $(12,3)$
hyperbolic lattice we find the fitting parameter $\kappa=2.36$. In the
Supplemental Material (SM) we justify these scaling behaviors from the
solutions of a gap equation [19, 20, 21]. By contrast, when $\rho(0)$ is
divergent, $\delta_{\rm CDW}$ grows considerably faster for weak $V$. Similar
scaling holds on hyperbolic lattices with other combinations of $p$ and $q$
[21]. Notice that with open boundary conditions, $\delta_{\rm CDW}$ is not
uniform in the system, rather displays spatial modulations [Fig. 1(a)-(c)].
However, a uniform CDW order with $\delta_{A}=\delta_{B}=0.5$ develops in the
entire system when $V\gg 1$. In this limit, spinless fermions occupy one of
the sublattices of bipartite hyperbolic lattices, while the other one is left
completely empty.

Figure 3: Scaling of the critical nearest-neighbor ($V_{c}$) and on-site
($U_{c}$) interactions for the charge-density-wave and antiferromagnet
ordeings, respectively, on Euclidean ($p=6$) and hyperbolic ($p=10,14,18,22$)
Dirac materials, displaying their systematic reduction with increasing $p$ or
equivalently curvature, indicating quantum phase transitions at weaker
coupling on a curved space. In all these systems $q=3$. Inset: The ratio of
$U_{c}$ to $V_{c}$ on a number of Dirac materials, expected to be locked at
$3$ (dashed line) in the mean-field limit (see text). Euclidean Dirac system
corresponds to graphene.

_On-site Hubbard repulsion_. We now consider fermions with spin degrees of
freedom (electrons), interacting via an on-site repulsion ($U$), captured by
the Hamiltonian

$H_{U}=U\sum_{i}\left(n_{i,\uparrow}-\frac{1}{2}\right)\;\left(n_{i,\downarrow}-\frac{1}{2}\right)-\mu
N,$ (6)

where $n_{i,\uparrow/\downarrow}$ is the electronic density at site $i$ with
spin projection $\uparrow/\downarrow$ in the $z$ direction. The Hartree
decomposition of $H_{U}$ leads to the single-particle Hamiltonian

$\displaystyle H^{\rm Har}_{U}$ $\displaystyle=$
$\displaystyle\sum_{x=A,B}\bigg{\\{}\left(\langle
n_{x,\uparrow}\rangle-\frac{1}{2}\right)\left(n_{x,\downarrow}-\frac{1}{2}\right)$
(7) $\displaystyle+$ $\displaystyle\left(\langle
n_{x,\downarrow}\rangle-\frac{1}{2}\right)\left(n_{x,\uparrow}-\frac{1}{2}\right)\bigg{\\}}-\mu
N,$

with $\langle n_{A,\sigma}\rangle=1/2+\sigma\delta_{A,\sigma}(\bm{r})$ and
$\langle n_{B,\sigma}\rangle=1/2-\sigma\delta_{B,\sigma}(\bm{r})$ [22, 19].
Here $\sigma=\pm$ respectively correspond to spin projections
$\uparrow/\downarrow$, and $\bm{r}$ measures the position of a site. The
overall charge neutrality of the system is maintained with the constraints
$\mu=0$ and

$\sum_{\sigma=\pm}\;\sum_{\bm{r}}\;\sigma\;\left[\delta_{A,\sigma}(\bm{r})-\delta_{B,\sigma}(\bm{r})\right]=0,$
(8)

which also keeps the average electronic density at each site equal to one.
With this ansatz, an AFM ground state is characterized by
$\delta_{A/B,\uparrow/\downarrow}>0$ and the corresponding AFM order parameter
reads as

$~{}\delta_{\rm
AFM}=\frac{1}{2}\left(\delta_{A,\uparrow}+\delta_{A,\downarrow}+\delta_{B,\uparrow}+\delta_{B,\downarrow}\right),$
(9)

where the $B$ site is one of the two NN sites in the same generation of the
hyperbolic lattice for a given site belonging to the $A$ sublattice [16]. In
noninteracting half-filled systems $\delta_{\rm AFM}=0$. Next, we follow the
same numerical procedure to self-consistently compute $\delta_{\rm AFM}$
everywhere in the system, outlined in details previously for the CDW order.
The results are summarized in Fig. 2.

Inside the AFM phase, each site develops a finite magnetization, which points
in the opposite directions on two complementary sublattices. Consequently, the
net magnetization in the system is zero. Notice that AFM order also develops a
mass gap near the half-filling, which can be appreciated in the following way.
For spinful fermions, the tight-binding Hamiltonian and AFM order are
respectively captured by $\hat{h}^{\rm spin}_{0}=\sigma_{0}\otimes\hat{h}_{0}$
and $\hat{h}_{\rm AFM}=\sigma_{3}\otimes\hat{h}_{\Delta}$ [Eq. (5)]. Here the
two-component Pauli matrices $\\{\sigma_{\mu}\\}$ operate on the spin indices.
As $\\{\hat{h}^{\rm spin}_{0},\hat{h}_{\rm AFM}\\}=0$, the AFM order also acts
as a mass for gapless fermions and its nucleation causes insulation at the
charge neutrality point. Otherwise, the scaling of the AFM order and its
spatial variations are qualitatively similar to the ones previously discussed
for the CDW. For instance, the critical on-site repulsion for AFM order is
$U_{c}=1.68$ on $(10,3)$ and the BCS fitting parameter $\kappa=3.54$ on
$(12,3)$ lattices, respectively. And on $(12,4)$ lattice the AFM develops
rapidly even for weak $U$.

_Weak coupling QPT_. The critical couplings for the CDW and AFM orders on
hyperbolic Dirac materials decrease monotonically with increasing $p$ (with a
fixed $q=3$) or equivalently the curvature, in comparison to those on the
honeycomb lattice, harboring Dirac fermions on Euclidean flat land [Fig. 3],
where $V_{c}\approx 0.75$ [23, 17] and $U_{c}\approx 2.22$ [24]. It strongly
indicates curvature induced QPTs at weaker coupling for Dirac fermions on
curved space. Within the mean-field approximation the ratio $U_{c}/V_{c}$ is
expected to be $3$, as the NN repulsion is operative between the fermionic
densities on three pairs of NN sites, while the Hubbard repulsion is on-site.
On hyperbolic lattices with open boundary condition although this ratio
remains almost constant $2.4$ (approximately), it is smaller than $3$ [Fig. 3
(inset)], possibly due to a rapidly increasing large number of sites located
at the edge of the system with only two NN sites, which increases $V_{c}$.

_Summary and discussion_. To summarize, from numerical Hartree-Fock self-
consistent analyses here we show that NN and on-site electronic repulsions
respectively support CDW and AFM orders on half-filled bipartite hyperbolic
lattices. While in Dirac systems such orderings take place through a QPT at
finite interactions, they nucleate even for infinitesimal $V$ and $U$ when the
DOS at half-filling is either finite or divergent. Furthermore, on hyperbolic
Dirac materials the requisite critical interactions for these two orders
decrease with increasing curvature or $p$. These predictions can be further
tested from quantum Monte Carlo simulations which also take into account the
quantum fluctuations, as they were previously executed for Euclidean Dirac
systems, such as graphene with on-site [25, 26] and NN [27, 28] repulsions
without the sign problem. In future, we will search for other exotic
correlated quantum phases on hyperbolic lattices, among which topological Mott
insulators and superconductors are the most fascinating ones.

Designer electronic materials [29, 30, 31, 32, 33] and cold atomic setups are
two promising platforms where our predicted quantum phases can be observed
experimentally. A curved designer material can be engineered by growing its
substrate (typically Cu) on another material with a different thermal
expansion coefficient. When such a heterostructure is cooled down, the Cu
substrate gets curved, which then can be decorated by the sites of desired
hyperbolic lattices. With tunable hopping amplitudes $t$, the ratios $V/t$
and/or $U/t$ can be adjusted on hyperbolic designer quantum materials to
trigger nucleation of various quantum orders. In cold atomic setups, desired
hyperbolic tessellations can be achieved by suitable arrangements of laser
traps. In such systems, at least the on-site Hubbard repulsion can be tuned to
demonstrate the nucleation of AFM order, as recently demonstrated on various
Euclidean lattices [34, 35, 36, 37].

_Acknowledgments_. N.G. and B.R. were supported by a Startup grant of B.R.
from Lehigh University. S.M. was supported by Weizmann Institute of Science,
Israel Deans fellowship through Feinberg Graduate School and Raymond and
Beverly Sackler Center for Computational Molecular and Material Science at Tel
Aviv University.

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