# Theory of Dirac spin liquids on spin-$S$ triangular lattice:
possible application to $\alpha$-CrOOH(D)

Vladimir Calvera<EMAIL_ADDRESS>Department of Physics, Stanford
University, Stanford, CA 94305  Chong Wang<EMAIL_ADDRESS>Perimeter Institute
for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

###### Abstract

Triangular lattice quantum antiferromagnet has recently emerged to be a
promising playground for realizing Dirac spin liquids (DSLs) – a class of
highly entangled quantum phases hosting emergent gauge fields and gapless
Dirac fermions. While previous theories and experiments focused mainly on
$S=1/2$ spin systems, more recently signals of a DSL were detected in an
$S=3/2$ system $\alpha$-CrOOH(D) in Ref. Liu  _et al._ (2020). In this work we
develop a theory of DSLs on triangular lattice with spin-$S$ moments. We argue
that in the most natural scenario, a spin-$S$ system realizes a $U(2S)$ DSL,
described at low energy by gapless Dirac fermions coupled with an emergent
$U(2S)$ gauge field (also known as $U(2S)$ QCD3). An appealing feature of this
scenario is that at sufficiently large $S$, the $U(2S)$ QCD becomes
intrinsically unstable toward spontaneous symmetry breaking and confinement.
The confined phase is simply the $120^{\circ}$ coplanar magnetic order, which
agrees with semiclassical (large-$S$) results on simple Heisenberg-like
models. Other scenarios are nevertheless possible, especially at small $S$
when quantum fluctuations are strong. For $S=3/2$, we argue that a $U(1)$ DSL
is also theoretically possible and phenomenologically compatible with existing
measurements. One way to distinguish the $U(3)$ DSL from the $U(1)$ DSL is to
break time-reversal symmetry, for example by adding a spin chirality term
${\bf S}_{i}\cdot({\bf S}_{j}\times{\bf S}_{k})$ in numerical simulations: the
$U(1)$ DSL becomes the standard Kalmeyer-Laughlin chiral spin liquid with
semion/anti-semion excitation; the $U(3)$ DSL, in contrast, becomes a non-
abelian chiral spin liquid described by the $SU(2)_{3}$ topological order,
with Fibonacci-like anyons.

Triangular lattice antiferromagnet was historically the first system
suggested, by AndersonAnderson (1973), to realize quantum spin liquid
statesSavary and Balents (2017); Zhou  _et al._ (2017). Although the ground
state of the nearest-neighbor Heisenberg model forms a classical $120^{\circ}$
coplanar magnetic order even for $S=1/2$Sachdev (1992); Bernu  _et al._
(1992); Capriotti  _et al._ (1999); White and Chernyshev (2007), the order can
be destroyed upon including some relatively weak further-neighbor couplings.
For example, for $S=1/2$ systems with a second-neighbor exchange $J_{2}$,
multiple numerical studies indicate the existence of a quantum spin liquid
phase in the range $0.07\leq J_{2}/J_{1}\leq 0.15$, sandwiched between the
familiar $120^{\circ}$ coplanar magnetic order at $J2/J1<0.07$ and a strip
magnetic order at $J_{2}/J_{1}>0.15$Manuel and Ceccatto (1999); Kaneko  _et
al._ (2014); Iqbal  _et al._ (2016); Li  _et al._ (2015); Hu  _et al._ (2015);
Zhu and White (2015); Saadatmand and McCulloch (2016); Wietek and Läuchli
(2017); Gong  _et al._ (2017); Bauer and Fjærestad (2017); Castells-Graells
_et al._ (2019); Ferrari and Becca (2019); Hu  _et al._ (2019) .

It has become increasingly clear recently that the quantum spin liquid state
realized this way, at least on $S=1/2$ triangular lattice, is likely a $U(1)$
Dirac spin liquid ($U(1)$ DSL). This conclusion comes from a synthesis of
significant progresses in formal theories, numerical simulations and
experimental observations: the stability of DSL as a gapless phase was
established through a series of theoretical argumentsBorokhov  _et al._
(2002); Hermele  _et al._ (2004); Song  _et al._ (2019, 2020) and lattice
simulationsKarthik and Narayanan (2016, 2016, 2019). It was also shown
theoretically that the spin-spin correlation function in the DSL has
nontrivial weight at the Brillouin zone corners (the ${\bf
K}=(2\pi/3,-2\pi/3)$ and ${\bf K}^{\prime}=-{\bf K}$ points), which comes from
monopole fluctuations in the $U(1)$ gauge field and cannot be understood from
spinon mean field theorySong  _et al._ (2019, 2020). This spectral weight was
observed in numerical simulations of the $J_{1}-J_{2}$ Heisenberg modelFerrari
and Becca (2019); Hu  _et al._ (2019), as well as in neutron scattering
experiments on powders of NaYbO2, an $S=1/2$ triangular systemDing  _et al._
(2019). Although more refined experimental confirmations are still needed,
there are good reasons to be optimistic on this front.

More recentlyLiu  _et al._ (2020) DSL-like behaviors were also observed in an
$S=3/2$ triangular system $\alpha$-CrOOH(D) (delafossites green-grey powder):
the system does not order down to $\sim 2K$ which is much lower than the Curie
temperature $\theta_{CW}\sim-211.6(5)K$; the specific heat behaves roughly as
$C\sim T^{2.2}$ which is consistent with a relativistic critical state with
$C\sim T^{2}$; furthermore the spectral weight in neutron scattering (again on
powders) accumulates around the ${\bf K},{\bf K}^{\prime}$ points which is
similar to the $S=1/2$ case. Assuming these signatures indeed come from a spin
liquid phase at low energy, the obvious question is: what kind of spin liquid
is it? Is it the same spin liquid as the $S=1/2$ $U(1)$ DSL in terms of
universal properties, or is it a distinct phase?

We can even consider a more general setting. Assuming that a $U(1)$ DSL is
indeed realized, say in the $S=1/2$ $J_{1}-J_{2}$ model in appropriate
regimes, one can ask how this spin liquid evolves as we increase $S$ (with the
form of the Hamiltonian fixed). For sufficiently large spin (beyond some
critical value $S_{c}$) the system becomes semi-classical and the spin liquid
should disappear from the phase diagram. Indeed for very large $S$ where
$1/S$-expansion becomes reliable, it is known that for small $J_{2}/J_{1}$ the
system forms the well known $120^{\circ}$ coplanar anti-ferromagnet, and when
$J_{2}/J_{1}$ exceeds a critical value $\sim 1/8$ the system goes through a
first order transition and forms a stripe antiferromagnetJolicoeur  _et al._
(1990); Chubukov and Jolicoeur (1992). The experiments on $\alpha$-CrOOH(D)
suggests that $S_{c}>3/2$, so the natural question is: what types of spin
liquids should we expect for $1/2<S<S_{c}$?

In order to develop a theory of critical Dirac spin liquids for general $S$,
we start with a parton decomposition of the spin operators (for a general
review of the parton approach see Ref. Wen (2004)):

${\bf
S}_{i,\mu}=\frac{1}{2}\sum_{a=1}^{2S}\sum_{\alpha,\beta=\uparrow,\downarrow}f^{\dagger}_{i,a,\alpha}{\sigma}_{\mu}^{\alpha\beta}f_{i,a,\beta},$
(1)

where $i$ is the lattice site index, $\mu=x,y,z$ denotes the spin components,
$\sigma_{\mu}$ is the Pauli matrix and $f_{i,a,\alpha}$ is a fermion
annihilation operator that carries spin-$1/2$ (indexed by $\alpha$) and a
color index $a=1,2...2S$. The auxiliary fermion $f$ is also known as the
spinon. This representation of the spin operator introduces some redundancies.
The physical spin Hilbert space can be recovered by imposing the constraints
of single occupancy for each color $a$
($\sum_{\alpha}f^{\dagger}_{\alpha,a}f_{\alpha,a}=1$) and total symmetrization
over all different colors. These constraints can be conveniently formulated as
a local $Sp(2S)$ gauge invariance on the spinons:

$\left(\begin{array}[]{c}f_{1,\uparrow}\\\ \vdots\\\ f_{2S,\uparrow}\\\
f^{\dagger}_{1,\downarrow}\\\ \vdots\\\ f^{\dagger}_{2S,\downarrow}\\\
\end{array}\right)\longrightarrow U\left(\begin{array}[]{c}f_{1,\uparrow}\\\
\vdots\\\ f_{2S,\uparrow}\\\ f^{\dagger}_{1,\downarrow}\\\ \vdots\\\
f^{\dagger}_{2S,\downarrow}\\\ \end{array}\right),$ (2)

where the $Sp(2S)$ matrix $U$ can vary from site to site. For $S=1/2$, this
recovers the more familiar $Sp(1)=SU(2)$ gauge symmetry.

We should now proceed with a spinon mean field ansatz. Motivated by the anzatz
for the $U(1)$ DSL in $S=1/2$ systems, we use the following mean field
Hamiltonian:

$H_{MF}^{S}=-t\sum_{a=1}^{2S}\sum_{<ij>}(-1)^{h_{ij}}(f_{i,a,\alpha}^{\dagger}f_{j,a,\alpha}+h.c.)$
(3)

where the sign factor $(-1)^{h_{ij}}$ gives a $\pi$-flux on all upward
triangles and zero flux on all downward triangles. For later convenience, we
assume $t>0$ without loss of generality. This ansatz breaks the microscopic
$Sp(2S)$ gauge symmetry down to $U(2S)$:

$f_{a,\alpha}\to U_{ab}f_{b,\alpha},$ (4)

where $U\in U(2S)$. The mean field ansatz also satisfies the average
constraint $\langle\sum_{\alpha}f^{\dagger}_{a,\alpha}f_{a,\alpha}\rangle=1$
for each color $a$. The hopping amplitude is designed so that at low energy
the spinons form gapless Dirac cones at two points in the reduced Brillouin
zone (two valleys). Including the degeneracy from the spin and color indices,
there are in total $4\times 2S$ two-component Dirac fermions at low energy,
and they couple to a $U(2S)=(U(1)\times SU(2S))/\mathbb{Z}_{2S}$ gauge field
in the fundamental representation. The resulting low energy theory is a
$U(2S)$ gauge theory with $N_{f}=4$ flavors of fundamental Dirac fermions.
This ansatz is very similar to that of the standard $U(1)$ DSL for $S=1/2$ –
the only difference is that we have $2S$ colors instead of only $1$ color for
$S=1/2$, and we shall call this state a $U(2S)$ DSL. The low energy theory,
known as $U(2S)$ QCD3 with $N_{f}=4$, is described by the following
Lagrangian:

$\mathcal{L}^{(S)}=\sum_{i=1}^{4}\bar{\psi}_{i}i\not{D}_{\bm{\mathcal{A}}}\psi_{i}+...,$
(5)

where $\bm{\mathcal{A}}$ is a $U(2S)$ gauge field and $\psi_{i}$ is a Dirac
fermion that transforms as a $U(2S)$ fundamental and a two-component spinor
under the emergent Lorentz symmetry. The $...$ represents additional terms
allowed by physical symmetries such as four-fermion interactions and monopole
tunnelings. For $S=1/2$ we recover the QED3 description of the standard $U(1)$
DSL.

Now what is the fate of the theory Eq. (5) at low energy? It is known that
when $N_{f}\gg 2S$, the Dirac fermions and gauge fields all remain gapless at
low energy and the theory flows under renormalization group (RG) to a
conformal field theory (CFT). An emergent $SU(N_{f})=SU(4)$ symmetry is
expected among the Dirac fermions ($2$ spins and $2$ valleys) in the
CFTHermele  _et al._ (2005). For $N_{f}=4$, the conformality remains as long
as $2S<N_{c}$ for some $N_{c}$. The exact value of $N_{c}$ is currently
unknown and requires future numerical simulation. Based on the apparent
stability of $U(1)$ DSL we expect $N_{c}>1$. If $N_{c}>3$, the $U(3)$ DSL will
be a stable gapless phase that could be realized in materials like
$\alpha$-CrOOH(D).111To be more precise: the stability of $U(2S)$ DSL requires
not only the conformality of the pristine $U(2S)$ QCD3, but also the
irrelevance of operators that are allowed by microscopic symmetries (the $...$
terms in Eq. (5)). For the purpose of this paper we do not carefully
distinguish the two types of instabilities. However this will be important in
future numerical studies to decide the critical value $S_{c}$.

In $U(1)$ DSL for $S=1/2$ systems, monopole operators of the $U(1)$ gauge
field form an important class of critical fluctuationBorokhov  _et al._
(2002); Hermele  _et al._ (2004); Dyer  _et al._ (2013); Song  _et al._ (2019,
2020). This is also true for general $U(2S)$ DSLs. To introduce the monopoles
in $U(2S)$ QCD3, let us first write the $U(2S)$ gauge field as

$\bm{\mathcal{A}}=\mathcal{A}+\frac{1}{2S}aI,$ (6)

where $\mathcal{A}$ is an $SU(2S)$ gauge field while $a$ is a $U(1)$ gauge
field ($I$ being the $2S\times 2S$ identity matrix). The normalization factor
$1/2S$ above is chosen so that the minimum $U(1)$ gauge charge carried by
$SU(2S)$-invariant operators is unity. We can now define monopoles for the $a$
gauge field as operators that insert nontrivial $U(1)$ gauge flux $\int da$.
The Dirac quantization rule states that the monopole should be local with
respect to operators that are gauge-invariant under other gauge symmetries
(here being the $SU(2S)$). Therefore the minimum $U(1)$ monopole carries
$2\pi$ flux of $a$. Equivalently, the conserved flux current is

$j^{\mu}_{\rm{flux}}=\frac{1}{2\pi}\epsilon^{\mu\nu\lambda}\partial_{\nu}a_{\lambda}.$
(7)

One subtlety compared to $U(1)$ DSL is that the monopoles are not directly
local with respect to the Dirac fermions $\psi$, since they couple to $a$ as
charge $1/2S$ fields. This means that if the $U(2S)$ QCD remains gapless at
low energy and the $\psi$ fermions are deconfined, the $U(1)$ monopole must be
accompanied by an $SU(2S)$ flux, so that it can be local with respect to
$\psi$. Here it is useful to consider a $U(1)^{\otimes 2S}$ subgroup of
$U(2S)$ and denote the gauge field for each $U(1)$ subgroup as
$\mathcal{A}^{(a)}$, namely
$\bm{\mathcal{A}}=\textit{diag}(\mathcal{A}^{(a)})$. We can now label the flux
configuration by specifying the gauge flux of each $\mathcal{A}^{(a)}$:

$\int d\mathcal{A}^{(a)}=2\pi n_{a},\hskip 5.0ptn_{a}\in\mathbb{Z}.$ (8)

The topological flux of the $a$ gauge field is

$\int da=2\pi\sum_{a}n_{a},$ (9)

and the most relevant monopole (the one with the lowest scaling dimension or
energy) corresponds to

$(n_{1},n_{2}...n_{2S})=(1,0,...0),$ (10)

or other configurations obtained from it through $SU(2S)$ rotations. This has
been quantitatively discussed using a large $N_{f}$ expansion in Ref. Dyer
_et al._ (2013). For $N_{f}\gg 2S$, the scaling dimension of the fundamental
monopole is

$\Delta_{\mathcal{M}}=0.265N_{f}-0.0383-0.516(2S-1)+O(1/N_{f})$ (11)

Although we only have $N_{f}=4$, the qualitative aspects (such as symmetry
quantum numbers) of the most monopole operators are expected to remain the
same as long as the theory stays gapless.

The above minimum monopole, denoted as $\mathcal{M}_{\mathcal{A}}$,
effectively behaves as a $U(1)$ monopole seen only by one color of Dirac
fermion $\psi_{a=1}$. Since $\psi_{a=1}$ has exactly the same band structure
as the Dirac fermions in the $U(1)$ DSL, many properties of
$\mathcal{M}_{\mathcal{A}}$ will be identical to the monopoles in the $U(1)$
DSL. Each two-component Dirac fermion with color index $a=1$ contributes a
zero mode in the flux background $\chi_{i}$ ($i=1,...4$). Gauge invariance
requires filling half of the zero modes, so that a gauge-invariant monopole
operator looks likeBorokhov  _et al._ (2002):

$\chi^{\dagger}_{i}\chi^{\dagger}_{j}\mathcal{M}_{\mathcal{A}},$ (12)

where $\mathcal{M}_{\mathcal{A}}$ is a “bare” flux-insertion operator with all
the zero modes empty. There are in total $C^{4}_{2}=6$ such operators that
form a vector representation of the $SO(6)=SU(4)/\mathbb{Z}_{2}$ flavor
symmetry. Three of these six operators form a triplet under the microscopic
spin $SO(3)$ and the remaining three are singlets. The spin triplet monopole
makes the most important contribution to the spectral weight of spin-spin
correlation function. These monopoles could also carry nontrivial lattice
momenta, and the neutron spectral weight will accumulate at the momentum of
the spin triplet monopole. The monopole momentum comes from a nontrivial Berry
phase as the gauge flux moves in a lattice of gauge charges, and the pattern
of the gauge charges is determined by the topology of the underlying spinon
band structure. It was shown in Refs. Song  _et al._ (2019, 2020) that for the
triangular mean field ansatz Eq. (3), the spin triplet monopoles carry a
lattice momentum ${\bf K}=(2\pi/3,-2\pi/3)$. This ${\bf K}$ momentum makes the
monopole sharply distinct from spinon scattering operators
$\psi^{\dagger}_{i}\psi_{j}$, which can at most have a lattice momenta at the
${\bf M}$ points ($(\pi,0)$, $(0,\pi)$ and $(\pi,\pi)$) according to their
band structure. This is why observations of spectral weight at ${\bf K}$,
${\bf M}$ and symmetry related momenta are considered important evidences for
the existence of the DSL. The same analysis shows that the three spin-singlet
monopoles carry momenta ${\bf K}+{\bf M}$ – this will be important for
measurements on singlet excitations, such as X-ray scatterings.

We now ask what happens if $S>S_{c}$ and the $U(2S)$ QCD3 becomes unstable.
The leading instability of $U(2S)$ QCD is believed to be a spontaneous
breaking of the $SU(N_{f})$ flavor symmetry down to $SU(N_{f}/2)\times
SU(N_{f}/2)\times U(1)$ (recall that $N_{f}$ is even due to fermion doubling
from parity anomaly)Appelquist and Nash (1990); Komargodski and Seiberg
(2018). One can think of this “spontaneous chiral symmetry breaking” as the
formation of a Dirac mass term of the form, up to $SU(N_{f})$ flavor
rotations,

$m\bar{\psi}\sigma^{z}\psi,$ (13)

which has value $+m$ for $N_{f}/2$ flavors of Dirac fermions and $-m$ for the
other $N_{f}/2$ flavors. Below this mass scale only the $U(2S)\sim
SU(2S)\times U(1)$ gauge field remains. The $SU(2S)$ gauge field is expected
to eventually become gapped and confine the $\psi$ fermions at lower energy,
and only $SU(2S)$-invariant objects such as
$\psi_{a=1}\psi_{a=2}...\psi_{a=2S}$ remain as gapped excitations. The $U(1)$
gauge field now remains as a free photon field in the IR.222In fact this
sequence of events should be viewed as a picture to aid our thinking. In
general they will not be clearly separated in energy scale, unless $S$ happens
to be barely above $S_{c}$.  It is well known that a free $U(1)$ gauge field
in $(2+1)d$ should be viewed as a conventional symmetry breaking phase, a
“superfluid” that breaks the $U(1)$ flux conservation symmetry, with the
photons being the Goldstone bosons. The free $U(1)$ gauge theory has a unique
monopole operator, which serves as the condensed order parameter of this
superfluid.

Now back to the $U(2S)$ DSLs on triangular lattice, with fixed $N_{f}=4$. For
$S>S_{c}$, we expect the spontaneous chiral symmetry breaking $SU(4)\to
SU(2)\times SU(2)\times U(1)$ followed by the confinement of the $SU(2S)$
gauge field. Microscopically we do not have the full $SU(4)$ symmetry, so the
ordering pattern will be decided by microscopic details and cannot be
determined from the effective field theory. For reasons that will become clear
later, we consider a symmetry breaking pattern represented by the mass Eq.
(13), where $\sigma^{z}$ refers to a Pauli matrix in the spin index. This mass
term produces a quantum spin Hall insulator with a mutual Chern-Simons term

$\frac{1}{2\pi}A^{S_{z}}da,$ (14)

where $A^{S_{z}}$ is a probe $U(1)$ gauge field that couples to the conserved
$S_{z}$. This term assigns a unit $S_{z}$ spin to each flux quanta of $a$, so
that a monopole operator $\mathcal{M}$ behaves like $S_{x}+iS_{y}$. This
monopole is a linear combination of the original $SO(6)$-vector monopoles in
the QCD3 theory – the 6-fold degeneracy is now lifted due to the chiral
symmetry breaking. In particular, the $\mathcal{M}$ operator inherits the
lattice momentum of the spin-triplet monopoles of the QCD3 theory. So
$\mathcal{M}$ should carry lattice momentum ${\bf K}$. An operator
$S_{x}+iS_{y}$ with lattice momentum ${\bf K}$ is nothing but the order
parameter of the $120^{\circ}$ magnetic order, so the resulting state from
spontaneous chiral symmetry breaking and confinement is simply the familiar
$120^{\circ}$ order. Notice that when $S<S_{c}$, this instability towards
$120^{\circ}$ order does not happen spontaneously, but can nevertheless take
place as we drive the $U(2S)$ DSL through a (likely continuous) phase
transition, which is described by a QCD-Gross-Neveu field theory.

The above analysis motivates us to conjecture that a $U(2S)$ DSL is realized
in the spin-$S$ $J_{1}-J_{2}$ model on triangular lattice, at some
intermediate $J_{2}/J_{1}$. The appealing feature of this conjectured scenario
is that it naturally reproduces the following nontrivial facts about the
$J_{1}-J_{2}$ model:

  1. 1.

At small $J_{2}/J_{1}$ the ground state forms a simple $120^{\circ}$ coplanar
order for any $S$.

  2. 2.

At $S=1/2$ the model appears to realize a $U(1)$ DSL phase at intermediate
$J_{2}/J_{1}$.

  3. 3.

At large $S$ the model becomes semiclassical with no spin liquid in the phase
diagram.

Figure 1: We conjecture that the two theories in this figure are dual to each
other in the IR. Left: Spin $S$ triangular lattice antiferromagnet with
nearest and next-nearest Heisenberg couplings $J_{1}$, $J_{2}$, roughly in the
regime $J_{2}/J_{1}\sim 0.1$. Right: $U(2S)$ QCD3 with $N_{f}=4$. This is a
strong-weak duality in the following sense: at large $S$ the triangular
lattice antiferromagnet is weakly fluctuating and forms a classical order,
while the $U(2S)$ QCD is strongly fluctuating and confines at low energy; at
small $S$ (say $S=1/2$) the quantum fluctuations of the lattice spins lead to
a spin liquid phase with resonating (fluctuating) singlet configurations,
while the QCD becomes stable at low energy and stays gapless and deconfined.

Our conjecture can also be phrased as an intriguing strong-weak duality
between triangular lattice antiferromagnets and $U(N)$ QCD3 with $N_{f}=4$, as
we illustrate in Fig. 1.

The above non-abelian scenario, although natural, is not the only possible
scenario, especially at small $S$ (like $S=1$ or $3/2$). It is possible that
the omitted $...$ terms in Eq. (5) are strong enough to drive the system to a
different phase, for example by spontaneously breaking (Higgsing) the $U(2S)$
gauge symmetry to a subgroup and gapping out some of the Dirac fermions. Such
“descendent” states can be described in the parton approach by adding
additional terms in the mean field Hamiltonian Eq. (3). In general a $U(2S)$
DSL can have many different possible descendent phases. In the Appendix we
discuss a number of such phases for $S=3/2$. Motivated by experiments on
$\alpha$-CrOOH(D) for $S=3/2$, we can further demand the descendent phase to
be (1) gapless with critical spin fluctuations at ${\bf K}$ momentum and (2)
in proximity to the $120^{\circ}$ order through a continuous phase transition.
It turns out that there is a unique descendent phase which satisfy these
demands: a $U(1)$ DSL with the same universal properties as realized in
$S=1/2$ systems. The corresponding parton mean field Hamiltonian is the
original Eq. (3) plus the following perturbation:

$\delta H=\Delta\sum_{\langle\langle
ij\rangle\rangle}(-1)^{s_{ij}}(\epsilon_{\alpha\beta}f^{\dagger}_{i,a=2,\alpha}f^{\dagger}_{j,a=3,\beta}+h.c.).$
(15)

The above term is a second-neighbor $p$-wave pairing between the $a=2,3$
colors, where the factor $(-1)^{s_{ij}}$ inverts sign when $i,j$ are inverted,
and the direction is chosen so that the flux on each triangle formed by one
second neighbor and two nearest neighbors is uniformly $1$, namely
$(-1)^{h_{ij}+h_{jk}+s_{ik}}=1$ when $i,j,k$ runs counterclockwise on the
triangle. This second-neighbor term gaps out the Dirac cones from the two
colors $a=2,3$ without breaking the physical symmetries, and breaks the gauge
symmetry from $U(3)$ down to $U(1)\times U(1)$. The first $U(1)$ gauge
symmetry acts as $f_{a=1}\to e^{i\theta}f_{a=1}$, while the second $U(1)$
gauge symmetry acts as $(f_{a=2},f_{a=3}^{\dagger})\to
e^{i\theta^{\prime}}(f_{a=2},f_{a=3}^{\dagger})$. The gauge field from the
second $U(1)$ gauge symmetry couples only to $a=2,3$ fermions which are now
gapped, so at low energy it should confine due to monopole proliferation. In
the Appendix we show that the proliferated monopole here does not break any
physical symmetry. This leaves behind a single $U(1)$ gauge field coupled with
the Dirac fermions from $f_{a=1}$, which is nothing but a $U(1)$ DSL. The band
structure for $f_{a=1}$ is identical to the spinons in the standard $U(1)$ DSL
for $S=1/2$ systems, so we expect the same universal properties. In
particular, the most important spin fluctuations come from the monopoles at
${\bf K}$ momentum, and the most important singlet fluctuations come from
monopoles at ${\bf K}+{\bf M}$.

The above procedure using terms like Eq. (15) can also be applied to other
$S>1/2$. For any half-integer $S$ the simplest descendent state is the $U(1)$
DSL, while for integer $S$ the simplest descendent is a symmetric, gapped
state with short-range entanglement. This scenario is similar to the
Heisenberg spin chains in one dimension.

The two plausible DSLs for $S=3/2$, one with $U(3)$ gauge symmetry and the
other with $U(1)$, are both compatible with existing measurements. They will
differ in more refined details including various critical exponents, but it is
quite nontrivial to either compute or measure such quantities. A natural
question is whether we can distinguish the two scenarios using some readily
accessible tool. We now show that a promising way to distinguish the different
scenarios is to break time-reversal symmetry and measure the resulting
topological order, which is achievable in numerical calculations, for example
using the density-matrix-renormalization-group (DMRG) approach.

Specifically, we can introduce a spin chirality term ${\bf S}_{i}\cdot({\bf
S}_{j}\times{\bf S}_{k})$ which breaks time-reversal and reflection but
preserve all other physical symmetries. For both the $U(3)$ and $U(1)$ DSLs,
this spin chirality term will induce an $SU(4)$-symmetric, gauge-invariant
mass gap for the Dirac fermions $\bar{\psi}\psi$. The gapped Dirac fermions
form nontrivial Chern bands and generate a Chern-Simons (CS) term for the
$U(3)$ or $U(1)$ gauge field. The resulting state is therefore a gapped
topological order. For $U(1)$ DSL this is the well known Kalmeyer-Laughlin
chiral spin liquid with semion/anti-semion excitationsKalmeyer and Laughlin
(1987). For $U(3)$ DSL the gapped Dirac fermions generate a CS term for the
$U(3)$ gauge field $\mathcal{A}$:

$\pm\frac{2}{4\pi}{\rm{Tr}}\left(\mathcal{A}d\mathcal{A}-\frac{2i}{3}\mathcal{A}^{3}\right),$
(16)

which is also known as $U(3)_{\pm 2}$ CS theory. By level-rank duality, this
theory with fermionic charge is dual to the $SU(2)_{\mp 3}$ CS theory with
bosonic charge. This topological order is non-abelian, with Fibonacci-like
anyons. The two topological orders can be readily distinguished numerically,
for example by examining the entanglement spectrum.

To summarize, we have discussed two possible gapless Dirac spin liquids – with
gauge symmetries $U(3)$ and $U(1)$, respectively – that are compatible with
recent experiments on the spin-$3/2$ triangular lattice system
$\alpha$-CrOOH(D). The $U(3)$ DSL appears to be theoretically more natural but
the $U(1)$ DSL is also possible. The two states can be distinguished, at least
numerically, by examining the resulting topological order upon time-reversal
breaking. Our discussion also calls for careful numerical studies of the
$U(N)$ QCD3 theory, for example using lattice gauge theory simulations to
determine the critical value $N_{c}$.

Acknowledgements: We thank Yin-Chen He, Tim Hsieh, Sung-Sik Lee and Liujun Zou
for helpful discussions. Research at Perimeter Institute is supported by the
Government of Canada through the Department of Innovation, Science and
Economic Development Canada and by the Province of Ontario through the
Ministry of Research, Innovation and Science.

## References

  * Liu  _et al._ (2020) Jiabin Liu, Benqiong Liu, Long Yuan, Boqiang Li, Lei Xie, Xiping Chen, Hongxia Zhang, Daye Xu, Wei Tong, Jinchen Wang, and Yuesheng Li, “Evidence for a gapless Dirac spin-liquid ground state in a spin-3/2 triangular-lattice antiferromagnet,” arXiv e-prints , arXiv:2010.02001 (2020), arXiv:2010.02001 [cond-mat.str-el] .
  * Anderson (1973) P.W. Anderson, “Resonating valence bonds: A new kind of insulator?” Materials Research Bulletin 8, 153 – 160 (1973).
  * Savary and Balents (2017) Lucile Savary and Leon Balents, “Quantum spin liquids: a review,” Reports on Progress in Physics 80, 016502 (2017).
  * Zhou  _et al._ (2017) Y. Zhou, K. Kanoda, and T.-K. Ng, “Quantum spin liquid states,” Reviews of Modern Physics 89, 025003 (2017), arXiv:1607.03228 [cond-mat.str-el] .
  * Sachdev (1992) Subir Sachdev, “Kagome´- and triangular-lattice heisenberg antiferromagnets: Ordering from quantum fluctuations and quantum-disordered ground states with unconfined bosonic spinons,” Phys. Rev. B 45, 12377–12396 (1992).
  * Bernu  _et al._ (1992) B. Bernu, C. Lhuillier, and L. Pierre, “Signature of néel order in exact spectra of quantum antiferromagnets on finite lattices,” Phys. Rev. Lett. 69, 2590–2593 (1992).
  * Capriotti  _et al._ (1999) Luca Capriotti, Adolfo E. Trumper, and Sandro Sorella, “Long-range néel order in the triangular heisenberg model,” Phys. Rev. Lett. 82, 3899–3902 (1999).
  * White and Chernyshev (2007) Steven R. White and A. L. Chernyshev, “Neél order in square and triangular lattice heisenberg models,” Phys. Rev. Lett. 99, 127004 (2007).
  * Manuel and Ceccatto (1999) L. O. Manuel and H. A. Ceccatto, “Magnetic and quantum disordered phases in triangular-lattice heisenberg antiferromagnets,” Phys. Rev. B 60, 9489–9493 (1999).
  * Kaneko  _et al._ (2014) Ryui Kaneko, Satoshi Morita, and Masatoshi Imada, “Gapless spin-liquid phase in an extended spin 1/2 triangular heisenberg model,” Journal of the Physical Society of Japan 83, 093707 (2014), https://doi.org/10.7566/JPSJ.83.093707 .
  * Iqbal  _et al._ (2016) Yasir Iqbal, Wen-Jun Hu, Ronny Thomale, Didier Poilblanc, and Federico Becca, “Spin liquid nature in the heisenberg ${J}_{1}-{J}_{2}$ triangular antiferromagnet,” Phys. Rev. B 93, 144411 (2016).
  * Li  _et al._ (2015) P. H. Y. Li, R. F. Bishop, and C. E. Campbell, “Quasiclassical magnetic order and its loss in a spin-$\frac{1}{2}$ heisenberg antiferromagnet on a triangular lattice with competing bonds,” Phys. Rev. B 91, 014426 (2015).
  * Hu  _et al._ (2015) Wen-Jun Hu, Shou-Shu Gong, Wei Zhu, and D. N. Sheng, “Competing spin-liquid states in the spin-$\frac{1}{2}$ heisenberg model on the triangular lattice,” Phys. Rev. B 92, 140403 (2015).
  * Zhu and White (2015) Zhenyue Zhu and Steven R. White, “Spin liquid phase of the $s=\frac{1}{2}\phantom{\rule{4.0pt}{0.0pt}}{J}_{1}-{J}_{2}$ heisenberg model on the triangular lattice,” Phys. Rev. B 92, 041105 (2015).
  * Saadatmand and McCulloch (2016) S. N. Saadatmand and I. P. McCulloch, “Symmetry fractionalization in the topological phase of the spin-$\frac{1}{2}$ ${J}_{1}\text{$-$}{J}_{2}$ triangular heisenberg model,” Phys. Rev. B 94, 121111 (2016).
  * Wietek and Läuchli (2017) Alexander Wietek and Andreas M. Läuchli, “Chiral spin liquid and quantum criticality in extended $s=\frac{1}{2}$ heisenberg models on the triangular lattice,” Phys. Rev. B 95, 035141 (2017).
  * Gong  _et al._ (2017) Shou-Shu Gong, W. Zhu, J.-X. Zhu, D. N. Sheng, and Kun Yang, “Global phase diagram and quantum spin liquids in a spin-$\frac{1}{2}$ triangular antiferromagnet,” Phys. Rev. B 96, 075116 (2017).
  * Bauer and Fjærestad (2017) Dag-Vidar Bauer and J. O. Fjærestad, “Schwinger-boson mean-field study of the ${J}_{1}\text{$-$}{J}_{2}$ heisenberg quantum antiferromagnet on the triangular lattice,” Phys. Rev. B 96, 165141 (2017).
  * Castells-Graells  _et al._ (2019) D. Castells-Graells, A. Yuste, and A. Sanpera, “Frustrated quantum spin systems in small triangular lattices studied with a numerical method,” Phys. Rev. B 100, 155119 (2019), arXiv:1806.06542 [cond-mat.str-el] .
  * Ferrari and Becca (2019) Francesco Ferrari and Federico Becca, “Dynamical structure factor of the ${J}_{1}-{J}_{2}$ heisenberg model on the triangular lattice: Magnons, spinons, and gauge fields,” Phys. Rev. X 9, 031026 (2019).
  * Hu  _et al._ (2019) Shijie Hu, W. Zhu, Sebastian Eggert, and Yin-Chen He, “Dirac Spin Liquid on the Spin-1 /2 Triangular Heisenberg Antiferromagnet,” Phys. Rev. Lett. 123, 207203 (2019), arXiv:1905.09837 [cond-mat.str-el] .
  * Borokhov  _et al._ (2002) Vadim Borokhov, Anton Kapustin, and Xinkai Wu, “Topological disorder operators in three-dimensional conformal field theory,” Journal of High Energy Physics 2002, 049 (2002).
  * Hermele  _et al._ (2004) Michael Hermele, T. Senthil, Matthew P. A. Fisher, Patrick A. Lee, Naoto Nagaosa, and Xiao-Gang Wen, “Stability of $u(1)$ spin liquids in two dimensions,” Phys. Rev. B 70, 214437 (2004).
  * Song  _et al._ (2019) Xue-Yang Song, Chong Wang, Ashvin Vishwanath, and Yin-Chen He, “Unifying description of competing orders in two-dimensional quantum magnets,” Nature Communications 10, 4254 (2019), arXiv:1811.11186 [cond-mat.str-el] .
  * Song  _et al._ (2020) Xue-Yang Song, Yin-Chen He, Ashvin Vishwanath, and Chong Wang, “From spinon band topology to the symmetry quantum numbers of monopoles in dirac spin liquids,” Phys. Rev. X 10, 011033 (2020).
  * Karthik and Narayanan (2016) Nikhil Karthik and Rajamani Narayanan, “No evidence for bilinear condensate in parity-invariant three-dimensional QED with massless fermions,” Phys. Rev. D 93, 045020 (2016), arXiv:1512.02993 [hep-lat] .
  * Karthik and Narayanan (2016) Nikhil Karthik and Rajamani Narayanan, “Scale invariance of parity-invariant three-dimensional qed,” Phys. Rev. D 94, 065026 (2016).
  * Karthik and Narayanan (2019) Nikhil Karthik and Rajamani Narayanan, “Numerical determination of monopole scaling dimension in parity-invariant three-dimensional noncompact QED,” Phys. Rev. D 100, 054514 (2019), arXiv:1908.05500 [hep-lat] .
  * Ding  _et al._ (2019) Lei Ding, Pascal Manuel, Sebastian Bachus, Franziska Grußler, Philipp Gegenwart, John Singleton, Roger D. Johnson, Helen C. Walker, Devashibhai T. Adroja, Adrian D. Hillier, and Alexand er A. Tsirlin, “Gapless spin-liquid state in the structurally disorder-free triangular antiferromagnet NaYbO2,” Phys. Rev. B 100, 144432 (2019), arXiv:1901.07810 [cond-mat.str-el] .
  * Jolicoeur  _et al._ (1990) Th. Jolicoeur, E. Dagotto, E. Gagliano, and S. Bacci, “Ground-state properties of the s=1/2 heisenberg antiferromagnet on a triangular lattice,” Phys. Rev. B 42, 4800–4803 (1990).
  * Chubukov and Jolicoeur (1992) Andrey V. Chubukov and Th. Jolicoeur, “Order-from-disorder phenomena in heisenberg antiferromagnets on a triangular lattice,” Phys. Rev. B 46, 11137–11140 (1992).
  * Wen (2004) Xiao-Gang Wen,  _Quantum Field Theory of Many-body Systems : From the Origin of Sound to an Origin of Light and Electrons._, Oxford Graduate Texts (OUP Premium, 2004).
  * Hermele  _et al._ (2005) Michael Hermele, T. Senthil, and Matthew P. A. Fisher, “Algebraic spin liquid as the mother of many competing orders,” Phys. Rev. B 72, 104404 (2005).
  * Dyer  _et al._ (2013) E. Dyer, M. Mezei, and S. S. Pufu, “Monopole Taxonomy in Three-Dimensional Conformal Field Theories,” ArXiv e-prints (2013), arXiv:1309.1160 [hep-th] .
  * Borokhov  _et al._ (2002) V. Borokhov, A. Kapustin, and X. Wu, “Topological Disorder Operators in Three-Dimensional Conformal Field Theory,” Journal of High Energy Physics 11, 049 (2002), hep-th/0206054 .
  * Appelquist and Nash (1990) Thomas Appelquist and Daniel Nash, “Critical behavior in (2+1)-dimensional qcd,” Phys. Rev. Lett. 64, 721–724 (1990).
  * Komargodski and Seiberg (2018) Zohar Komargodski and Nathan Seiberg, “A symmetry breaking scenario for QCD3,” Journal of High Energy Physics 2018, 109 (2018), arXiv:1706.08755 [hep-th] .
  * Kalmeyer and Laughlin (1987) V. Kalmeyer and R. B. Laughlin, “Equivalence of the resonating-valence-bond and fractional quantum hall states,” Phys. Rev. Lett. 59, 2095–2098 (1987).
  * Zheng  _et al._ (2015) Wayne Zheng, Jia-Wei Mei, and Yang Qi, “Classification and Monte Carlo study of symmetric $Z_{2}$ spin liquids on the triangular lattice,” arXiv e-prints , arXiv:1505.05351 (2015), arXiv:1505.05351 [cond-mat.str-el] .
  * Lu (2016) Yuan-Ming Lu, “Symmetric Z2 spin liquids and their neighboring phases on triangular lattice,” Phys. Rev. B 93, 165113 (2016), arXiv:1505.06495 [cond-mat.str-el] .

##  Appendix A Alternative DSLs on $S=3/2$ triangular lattice

We start with the parton decomposition used in the main text for the $S=3/2$
operators:

${\bf
S}_{i,\mu}=\frac{1}{2}\sum_{a=1}^{3}\sum_{\alpha,\beta=\uparrow,\downarrow}f^{\dagger}_{i,a,\alpha}{\sigma}_{\mu}^{\alpha\beta}f_{i,a,\beta},$
(17)

which comes with a microscopic $Sp(3)$ gauge symmetry. Below we describe a
sequence of mean field ansatz, all with gapless Dirac fermions at low energy,
starting from the simplest $U(3)$ Dirac ansatz used in the main text:

  1. 1.

$U(3)$: consider nearest-neighbour hopping Hamiltonian with a staggered
$\pi$-flux

$H_{1}=-t\sum_{ij}(-1)^{h_{ij}}f^{\dagger}_{i,a,\alpha}f_{j,a,\alpha},$ (18)

where the sign factor $(-1)^{h_{ij}}$ gives a $\pi$-flux on all upward
triangles and zero flux on all downward triangles. This ansatz leads to the
$U(3)$ Dirac spin liquid discussed in the main text.

  2. 2.

$U(2)\times\mathbb{Z}_{2}$: starting from the $U(3)$ theory, we can introduce
a pairing term for a single color (say $a=3$) in the mean field theory to
break the $U(3)$ gauge symmetry down to $U(2)\times\mathbb{Z}_{2}$:

$H_{3}=\sum_{i}\Delta f_{i,a=3,\uparrow}f_{i,a=3,\downarrow}+h.c.,$ (19)

which also gaps out the $f_{a=3}$ Dirac fermions. This leaves behind a $U(2)$
gauge theory with $N_{f}=4$ at low energy, together with a gapped sector with
$\mathbb{Z}_{2}$ topological order. As discussed in Refs Zheng  _et al._
(2015); Lu (2016) there are several different versions of $\mathbb{Z}_{2}$
topological orders depending on the exact form of pairing (in modern language
they are topological orders enriched by spin and lattice symmetries in
different ways). However as the gapped sector do not affect universal
behaviours of physical correlation functions, we will not distinguish
different (symmetry-enriched) topological orders here.

  3. 3.

$U(1)\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$: starting from the
$U(2)\times\mathbb{Z}_{2}$ state, we can further pair and gap out the $a=2$
fermions. This leaves behind a QED3 with two gapped $Z_{2}$ topological
ordered sectors.

  4. 4.

$U(1)_{A}$: we now start from the $U(3)$ ansatz, and add the following term on
lattice:

$\displaystyle H_{4}=$ $\displaystyle t^{\prime}\sum_{\langle\langle
ij\rangle\rangle}(-1)^{s_{ij}}(f^{\dagger}_{i,a=2,\alpha}f_{j,a=3,\alpha}+h.c.)$
(20)
$\displaystyle+h\sum_{i}(f^{\dagger}_{i,a=1,\alpha}f_{i,a=2,\alpha}+h.c.).$

The first term is a second-neighbor $p$-wave hybridyzation between the $a=2,3$
colors, where the factor $(-1)^{s_{ij}}$ inverts sign when $i,j$ are inverted,
and the direction is chosen so that the flux on each triangle formed by one
second neighbor and two nearest neighbors is uniformly $1$, namely
$(-1)^{h_{ij}+h_{jk}+s_{ik}}=1$ when $i,j,k$ runs counterclockwise on the
triangle. This second-neighbor term gaps out the Dirac cones formed by the two
colors $a=2,3$, and breaks the gauge symmetry from $U(3)$ down to $U(1)\times
U(1)\times U(1)$, but breaks no physical symmetry. The second on-site
hybridization term further breaks the gauge symmetry down to a single $U(1)$.
The low energy theory is therefore a single QED3 with $N_{f}=4$ gapless Dirac
fermions formed by $f_{a=1,\alpha}$. We will see later that this state is in
fact different from the standard $U(1)$ DSL in $S=1/2$ systems in a crucial
way: the spin-triplet monopole in this $U(1)_{A}$ theory carries a trivial
lattice momentum (${\bf\Gamma}$) instead of ${\bf K}$ for the standard $U(1)$
DSL.

  5. 5.

$U(1)_{B}$: we can also start from the $U(2)\times U(1)$ ansatz, and add the
following alternative terms:

$\displaystyle H_{5}=$ $\displaystyle\Delta_{p}\sum_{\langle\langle
ij\rangle\rangle}(-1)^{s_{ij}}(\epsilon_{\alpha\beta}f^{\dagger}_{i,a=2,\alpha}f^{\dagger}_{j,a=3,\beta}+h.c.)$
(21)
$\displaystyle+h\sum_{i}(f^{\dagger}_{i,a=1,\alpha}f_{i,a=2,\alpha}+h.c.).$

This is almost the same with the term for the $U(1)_{A}$ state, except that
$f_{3,\alpha}$ participates in a particle-hole conjugated way
($f_{3,\alpha}\to\epsilon_{\alpha\beta}f^{\dagger}_{3,\beta}$). This term also
breaks the gauge symmetry down to a single $U(1)$, and gaps out the Dirac
cones formed by the two colors $a=2,3$. The low energy theory is therefore
again a QED3 with $N_{f}=4$. We will see later that this state is identical to
the standard $U(1)$ DSL state for $S=1/2$ in terms of universal properties. In
particular, the spin-triplet monopoles do carry ${\bf K}$ momentum in the
$U(1)_{B}$ theory.

As discussed in the main text, all these DSLs have spin-triplet monopoles
which contribute to neutron scattering weight. We now analyze the lattice
momentum of these triplet monopoles in each theory. Following the discussions
in the main text, we conclude that the triplet monopoles in $U(3)$,
$U(2)\times\mathbb{Z}_{2}$, and $U(1)\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
states all have momentum ${\bf K}$, which agrees with the experiment. The
$U(1)_{A}$ and $U(1)_{B}$ phases require more analysis. This can be done by
first considering an intermediate state with $U(1)^{(1)}\times
U(1)^{(2)}\times U(1)^{(3)}$ gauge field, where the gauge field $A^{(a)}$ of
each $U(1)^{(a)}$ only couples to $f_{a}$. This is simply three copies of
QED3, each with triplet monopoles at momentum ${\bf K}$. Now $U(1)_{A}$ and
$U(1)_{B}$ phases are obtained by Higgsing the $U(1)^{\otimes 3}$ gauge
symmetry down to a single $U(1)$, with gauge field $A$. The difference is that
for $U(1)_{A}$ we have $A=A^{(1)}=A^{(2)}=A^{(3)}$ while for $U(1)_{B}$ we
have $A=A^{(1)}=A^{(2)}=-A^{(3)}$. For the monopole operators this means that
the monopole operator in $U(1)_{A}$ phase transforms under various symmetries
as $\mathcal{M}_{1}\mathcal{M}_{2}\mathcal{M}_{3}$, where $\mathcal{M}_{a}$ is
the monopole of $A^{(a)}$; while in $U(1)_{B}$ phase it transforms as
$\mathcal{M}_{1}\mathcal{M}_{2}\mathcal{M}_{3}^{\dagger}$. Therefore the
monopole lattice momentum is ${\bf K}+{\bf K}+{\bf K}=0$ for $U(1)_{A}$, and
${\bf K}+{\bf K}-{\bf K}={\bf K}$ for $U(1)_{B}$.

We therefore conclude that $U(3)$, $U(2)\times\mathbb{Z}_{2}$,
$U(1)\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ and $U(1)_{B}$ phases all
contain spin triplet monopoles at ${\bf K}$ that can contribute to neutron
scattering spectral weight, consistent with experimental observationLiu  _et
al._ (2020). We can further demand that the DSL should have a direct,
continuous transition to the $120^{\circ}$ magnetic order. This rules out
$U(2)\times\mathbb{Z}_{2}$ and $U(1)\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
states, since a Gross-Neveu type of transition will not affect the gapped
$\mathbb{Z}_{2}$ topological orders and the resulting states will not be
classical (short-range-entangled) magnetic orders. This leaves the $U(3)$ and
$U(1)_{B}$ phases to be the main contenders.

In fact to produce the same low energy theory as the $U(1)_{B}$ state, we do
not even need to have the on-site hybridization term (the $h$-term in Eq.
(21)). This will lead to a $U(1)\times U(1)$ gauge symmetry, but the second
$U(1)$ is a pure gauge theory without gapless matter, and will confine due to
monopole proliferation. From our previous monopole analysis, the monopole of
the second $U(1)$ carries lattice momentum ${\bf K}-{\bf K}=0$, so the
confinement will not break any physical symmetry. This leaves behind a single
$U(1)$ gauge theory with $N_{f}=4$ gapless Dirac fermions. This is the
candidate $U(1)$ DSL for $S=3/2$ discussed in the main text.

Finally we comment on some other mean field ansatz that naively also give DSL-
like behaviors. One can start from the $U(3)$ ansatz and make the hopping
amplitute $t$ slightly different for each flavor: $t_{1}\neq t_{2}\neq t_{3}$.
This breaks the gauge symmetry down to $U(1)\times U(1)\times U(1)$ so the low
energy theory becomes three copies of the QED3, each with $N_{f}=4$ Dirac
fermions. This theory, however, is unstable: they physical symmetries allow a
monopole-monopole coupling of form $\mathcal{M}_{1}^{\dagger}\mathcal{M}_{2}$.
The scaling dimension of each monopole is estimated in Monte Carlo
simulationKarthik and Narayanan (2019) to be $\sim 1.1$ (which is only
slightly larger than the large-$N_{f}$ estimationDyer  _et al._ (2013) $\sim
1$). This monopole-monopole coupling is therefore relevant and leads to
instability. Since the monopole scaling dimension is expected to decrease as
the color number increases, the same instability will happen if we make
$t_{1}=t_{2}\neq t_{3}$, which results in a $U(2)\times U(1)$ QCD theory with
monopole-monopole coupling. It is interesting to ask what is the effect of
this relevant monopole-monopole coupling. One answer is that it may drive the
system through a spontaneous chiral symmetry breaking followed by confinement.
A more interesting possibility is that it may drive the system back to the
$U(3)$ QCD theory. One motivation for considering this scenario is that a
strong monopole-monopole coupling will essentially identify different
monopoles and results in a theory with only one type of monopole, but this is
exactly what happens in the $U(3)$ QCD theory.


#  Quantum Correlations in Dimers

$N.Doustimotlagh$111Corresponding author.
_E-mail address_<EMAIL_ADDRESS>Phone: +86 1324 1699 374, Fax:
+0086(10)62772692,  Tsinghua University, Department of Engineering Physics,
_Beijing 100084_ , P.R.China

###### Abstract

It has been proven that the quantum discord is a more general tool to capture
non-classical correlation than quantum entanglement, because there is a non-
zero quantum discord in several mixed states that could not be measured by
quantum entanglement. But because of optimization part in formulating quantum
discord, it is very difficult and nearly impossible to find quantum discord
for some quantum states. So people proposed geometric quantum discord, which
in a bipartite state could be describe as the distance of the states from the
closest zero-discord state. To understand better geometric quantum discord,
this paper is devoted to compare it with quantum discord and measurement
induced non-locality that is in some sense dual to the geometric quantum
discord. As studying the quantum correlation experimentally is one of the
important thing in quantum information processing, so as our example for
illustrating the difference of geometric quantum discord, quantum discord and
measurement induced non-locality, our quantum system is a dimer (two spin-1/2
particles) in multiple quantum nuclear magnetic resonance.

###### keywords:

Geometric quantum discord; Measurement-induced non-locality; Quantum discord;
Dimer.

††journal: —-

##  1 Introduction

Assume a quantum system that its contents are interacting with each other, one
can describe this system by using the correlations amongst the different
constituents of it [1, 2]. Generally speaking, correlations have both quantum
and classical components. One of the most important type of quantum
correlation measures is the quantum entanglement that describes as the basic
and essential source in many quantum information processing tasks such as
teleportation [3, 4], quantum communication [5], quantum computation [1] and
dense coding [4]. It is important to known that quantum mutual information can
measure the total correlation in a bipartite quantum system [6, 7]. Quantum
mutual information in turn can be divided into quantum and classical parts and
the quantum part is named as quantum discord (QD) that first was described and
formulated by the Zurek Group [8] and Verdal group [9] independently. It has
proven that the QD is a more general tool to capture non-classical correlation
than quantum entanglement, because there is a non-zero quantum discord in
several mixed states that could not be measured by quantum entanglement[10].
The formulation of QD is according to the very difficult procedure named
numerical maximization and this procedure can not guarantee to calculate exact
results and there are analytical expressions for some special cases only [8,
9, 11]. To solve this problem, some physicists proposed another simpler
measure called geometric quantum discord (GQD)which in a bipartite state could
be describe as the distance of the states from the closest zero-discord state
[12, 13, 14]. To understand better GQD, this paper is devoted to compare it
with QD and measurement induced non-locality (MIN) [15] that is in some sense
dual to the GQD. As studying the quantum correlation experimentally is one of
the important thing in quantum information processing, so we use dimer (two
spin-1/2 particles) in multiple quantum nuclear magnetic resonance (MQ
NMR)[16, 17, 18, 19, 20] to study GQD, QD [20] and MIN numerically.

##  2 Theory

###  2.1 Description of State of Dimer in NMR Formalism

The MQ NMR experiment has four different periods of time that are preparation,
evolution, mixing, and detection [16, 17, 18]. Our system is two spin-1/2
particles with the dynamical decoupling interaction on the preparation period
of the MQ NMR experiment which an strong external magnetic field applied on
it. We use rotating reference frame [16, 17, 18] to describe the dynamics of
the system. The MQ NMR Hamiltonian $H_{MQ}$ of the system can be writen as
[16, 17, 18, 19, 20]

$H_{MQ}=\frac{\eta}{2}(J^{+}_{1}J^{+}_{2}+J^{-}_{1}J^{-}_{2}).$ (1)

Where $R_{12}$ is the distance between two spins, $\alpha_{12}$ is the angle
between the external magnetic field $H_{0}$ and the vector $R_{12}$.
$\eta=(\gamma\hbar/R^{3}_{12})(1-3cos^{2}\alpha_{12})$ is the coupling
constant between two spins and $\gamma$ is the gyromagnetic ratio. $J^{-}_{i}$
and $J^{+}_{i}$ $(i=1,2)$ are the lowering and raising operators of spin $i$.
The thermodynamic equilibrium density matrix of the system is

$\rho_{0}=\frac{e^{\beta J_{z}}}{Tr(e^{\beta J_{z}})}.$ (2)

Where $\omega=\gamma|H_{0}|$, $\beta=\frac{\hbar\omega_{0}}{kT}$ is the
inverse temperature and $J_{z}=J_{1z}+J_{2z}$ describe the z-projection of the
total spin angular momentum. Now we can obtain the density matrix,
$\rho(\tau)$ [18, 19, 20]

$\rho(\tau)=\left(\begin{array}[]{cccc}a&0&0&e\\\ 0&b&0&0\\\ 0&0&c&0\\\
-e&0&0&f\\\ \end{array}\right).$ (3)

where

$\displaystyle a=\frac{1}{z}(\cosh\beta+\cos(\eta t)\sinh\beta),$
$\displaystyle e=\frac{1}{z}(i\sin(\eta t)\sinh\beta),$ $\displaystyle
b=c=\frac{1}{z},$ $\displaystyle d=\frac{1}{z}(\cosh\beta-\cos(\eta
t)\sinh\beta).$ (4)

Where $z=2(1+\cosh\beta)$.

###  2.2 Quantum Discord in Dimer

One can find that $\rho(\tau)$ is a particular case of $X-matrix$ which the QD
of such states has been analytically studied [21]. The QD is the difference of
total mutual and information $I(\rho)$ and the classical correlations
$C(\rho)$ [8, 9, 20, 21]:

$QD(\rho)=I(\rho)-C(\rho).$ (5)

Where $I(\rho)$ is the total mutual information and it given by [20]

$I(\rho)=S(\rho^{M})+S(\rho^{N})+\sum^{4}_{m=1}{\lambda_{m}\log_{2}{\lambda_{m}}}.$
(6)

where $N$ and $M$ denote the second and the first spins respectively.
$\rho^{M}$ and $\rho^{N}$ are the reduced density matrices and can be obtain
as:

$\displaystyle\rho^{M}=Tr_{N}\rho=diag(a+b,c+d),$
$\displaystyle\rho^{N}=Tr_{M}\rho=diag(a+c,b+d),$
$\displaystyle\rho^{M}=\rho^{N}.$

By using $(S(\rho)=Tr(\rho log_{2}\rho))$ the the appropriate entropies [20]
$S(\rho^{M})$ and $S(\rho^{N})$ can be written as:

$S(\rho^{M})=S(\rho^{N})=\frac{1}{2ln_{2}}\ln[{\frac{(z/2)^{2}-\epsilon^{2}\sinh^{2}\beta}{z^{2}}}]-\frac{\epsilon\sinh\beta}{z\ln_{2}}\ln[\frac{(z/2)+\epsilon\sinh\beta}{(z/2)-\epsilon\sinh\beta}].$
(8)

where $\epsilon=|cos(\eta\tau)|$ introduces the time-dependent parameter and
$0\leqslant\epsilon\leqslant 1$. $\lambda_{m}$s are the eigenvalues of the
density matrix and they are independent of $\epsilon$ [21]:

$\lambda_{1}=\frac{e^{\beta}}{z},\lambda_{2}=\frac{e^{-\beta}}{z},\lambda_{3}=\lambda_{4}=\frac{1}{z}.$
(9)

. We suppose that the projective measurements are applied on the subsystem
$N$. The classical correlations $C(\rho)$ [19, 20, 21] can be written as:

$C(\rho)=S(\rho^{M})-\min_{\kappa=0,1}\Xi(\kappa,\beta,\epsilon).$ (10)

Where

$\Xi(\kappa,\beta,\epsilon)=p_{0}f_{0}-p_{1}f_{1}.$ (11)

So one can write[19, 20]

$f_{i}=-\frac{1-\phi_{i}}{2}\log_{2}\frac{1-\phi_{i}}{2}-\frac{1+\phi_{i}}{2}\log_{2}\frac{1+\phi_{i}}{2}.$
(12) $p_{i}=\frac{1}{2}(1+(-1)^{i}\kappa(2(a+c)-1)).$ (13)
$\phi_{i}=\frac{1}{p}_{i}[(1-\kappa^{2})|e|^{2}+\frac{1}{4}(2(a+b)-1+(-1)^{i}\kappa(1-2(b+c)))^{2}]^{\frac{1}{2}}.$
(14)

Where $\kappa$ is an arbitrary parameter, $0\leqslant\kappa\leqslant 1$. One
can obtain that the minimum of $\Xi(\kappa,\beta,\epsilon)$ in Eq. (10) is
independent of $\epsilon$ and will be correspond to $\kappa=0$ and after some
mathematics calculation one can write:

$\Xi(\kappa=0,\beta)=f_{0}=\frac{1}{2\ln
2}[\ln(\frac{z}{4})+\phi_{0}\ln(\frac{1-\phi_{0}}{1+\phi_{0}})]+1.$ (15)

where by using Eq. (14) one can obtain
$\phi_{0}=[(\cosh(\beta)-1)/(\cosh(\beta)+1)]^{\frac{1}{2}}$. Now we are able
to write a single expression for the QD. By combining Eqs. (11), (12) and (14)
and using $\sum_{m=1}^{4}\log_{2}\lambda_{m}=-2\Xi(0,\beta)$ , one can obtain
the following expression for the QD:

$\displaystyle
QD(\rho(\tau))=\frac{1}{2ln_{2}}ln[{\frac{(z/2)^{2}-\epsilon^{2}\sinh^{2}\beta}{z^{2}}}]-\frac{\epsilon\sinh\beta}{zln_{2}}ln[\frac{(z/2)+\epsilon\sinh\beta}{(z/2)-\epsilon\sinh\beta}]$
$\displaystyle+\frac{1}{2\ln
2}[\ln(\frac{z}{4})+\phi_{0}\ln(\frac{1-\phi_{0}}{1+\phi_{0}})]+1.$

###  2.3 Geometric Quantum Discord and Measurement Induced Non-locality in
Dimer

By using the density matrix $\rho(\tau)$ we can write GQD for this density
matrix as [12, 14]

$D^{g}_{A}(\rho(\tau))=\frac{1}{4}[8(|e|^{2}+\xi_{3}-\max(\xi_{1},\xi_{2},\xi_{3})].$
(17)

where $\xi_{1}=\xi_{2}=4(|e|^{2})$ and $\xi_{3}=2[(a-c)^{2}+(b-d)^{2}]$.

Locally invariant measurements cause some global effects on quantum systems
which one can find quantum correlation in such system by measuring this
effects. Luo and Fu [15] first proposed a measure to capture these global
effect by using a geometric perspective based on the local Von Neumann
measurements wherein one of the reduced states is left invariant and called
this measure MIN [15] that is in some sense dual to the GQD [12]. Assume that
$\rho$ is the density matrix of a bipartite state that act on the associated
state space $H^{C}\otimes H^{C}$ and $\dim(H^{C}\otimes H^{C})<\infty$. The
MIN of $\rho$ can be defined as [15]

$MIN(\rho)=\max_{\prod^{C}}\|\rho-\prod^{C}(\rho)\|^{2}.$ (18)

Where $\|F^{2}\|=Tr(F^{+}F)$ is the square of the Hilbert-Schmidt norm and the
optimization should be taken over all local Von Neumann measurements
$\prod^{C}=\\{\prod^{C}_{k}\\}$ which should satisfy

$\displaystyle\sum_{k}\prod^{C}_{k}\rho^{C}\prod^{C}_{k}=\rho^{C},$
$\displaystyle\prod^{C}(\rho)=\sum_{k}(\prod^{C}_{k}\otimes
I_{D})\rho(\prod^{C}_{k}\otimes I_{D}).$

By using the density matrix $\rho(\tau)$ and equations $(5-18)$ we can write
MIN in dimer as:

$MIN((\rho(\tau)))=\frac{1}{4}[\Gamma+8|e|^{2}-4\min(\frac{\Gamma}{4},|e|^{2})].$
(20)

where $\Gamma=(a-b-c+d)^{2}$.

##  3 Numerical Results and Discussion

To understand better geometric quantum discord, this chapter was devoted to
compare it with quantum discord and measurement induced non-locality as a
measure of quantum correlation in dimer (two spin-1/2 particles).

We analyse the GQD, MIN and QD as a function of $\beta$ (reverse of the
temperature) for different values of $\epsilon$ (time dependence factor) that
one can see our result in Figs. 1 wherein (a), (b) and (c) represent GQD, MIN
and QD respectively. One can see that for bigger $\beta$ the quantum
correlation is bigger, it is natural because $\beta$ is relative to inverse of
temperature. As another result one can see that for bigger $\epsilon$ the
quantum correlations decrease. It is natural because as the time elapses,
interaction of the quantum system with environment will be longer and the
quantum correlation must decrease, and to see this result better we draw the
GQD, MIN and QD as a function of $\epsilon$ for different values of $\beta$
that one can see our result in Figs. 2 wherein (a), (b) and (c) represent GQD,
MIN and QD respectively. For understanding the effect of time and temperature
simultaneously, we draw the GQD, MIN and QD as a function of $\beta$ and
$\epsilon$ in 3-dimensions that one can see our result in Figs. 1 wherein (a),
(b) and (c) represent GQD, MIN and QD respectively. One cane see that the
behaviour of GQD, MIN and QD are nearly same.

Figure 1: GQD, QD and MIN versus $\beta$ for $\epsilon=0.1$ (black and solid),
$\epsilon=0.2$ (red and dash), $\epsilon=0.3$ (green and dot), $\epsilon=0.5$
(blue and dash-dot), $\epsilon=0.7$ (magenta and dash-dot-dot) and
$\epsilon=0.9$ (yellow and short-dash). Figure 2: GQD, QD and MIN versus
$\epsilon$ for $\beta=1$ (black and solid), $\beta=1.5$ (red and dash),
$\beta=2$ (green and dot), $\beta=3$ (blue and dash-dot), $\beta=5$ (magenta
and dash-dot-dot) and $\beta=7$ (yellow and short-dash). Figure 3: (a) GQD,
(b) MIN and (c) QD versus $\epsilon$ and $\beta$ in 3-dimensions

##  4 Summary

Quantum entanglement as a measure of non-classical correlation plays a central
role in the quantum communication and information, however, it could not
capture all of the non-classical properties of quantum phenomena. It has
proven that the quantum discord is a more general tool to capture non-
classical correlation than quantum entanglement, because there is a non-zero
quantum discord in several mixed states that could not be measured by quantum
entanglement. But because of optimization part in formulating of QD, it is
very difficult to formulate quantum discord for some quantum states. So
geometric quantum discord, which in a bipartite state could be describe as the
distance of the state from the closest zero-discord state was proposed.

To understand better geometric quantum discord, this chapter was devoted to
compare it with quantum discord and measurement induced non-locality as a
measure of quantum correlation in dimer (two spin-1/2 particles). As studding
the quantum correlation experimentally is one of the important thing in
quantum information processing, in this chapter an scheme for measuring
quantum correlation in multi quantum nuclear magnetic resonance was described,
wherein Our quantum system is a dimer on the preparation period of this
experiment and the measures for quantum correlation is quantum discord,
geometric quantum discord and measurement induced non-locality. We analyse the
quantum discord, geometric quantum discord and measurement induced non-
locality as a function of $\beta$ (reverse of the temperature) for different
values of $\epsilon$ (time dependence factor) and we found that fore bigger
$\beta$ the quantum correlation is bigger, it is natural because $\beta$ is
relative to inverse of temperature. As another result, we showed that for
bigger $\epsilon$ the quantum correlations decreased. It is natural because as
the time elapsed, the interaction of the quantum system with environment will
be longer and the quantum correlation must decrease.

## Acknowledgements

I am so grateful to Professor G. L. Long for his kind help with this work.

## References

  * [1] M. A. Nielson and I. L. Chuang: Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2000).
  * [2] N. Doustimotlagh, S. Wang, C. You, and G.-L. Long. EPL (Europhysics Letters), 106, 60003 (2014).
  * [3] S. G. Hofer, W.f Wieczorek, M. Aspelmeyer, and K. Hammerer: Phys. Rev. A 84, 052327 (2011).
  * [4] Y. Yeo and W. Kang Chua:Phys. Rev. Lett. 96, 060502 (2006).
  * [5] R. Ursin, at all: Nature Physics 3, 481 (2007).
  * [6] B. Schumacher and M. D. Westmoreland: Phys. Rev. A 74, 042305 (2006).
  * [7] P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell: Phys. Rev. Lett. 108, 143603 (2012).
  * [8] H. Ollivier and W. Zurek: Phys. Rev. Lett. 88, 017901 (2001).
  * [9] L. Henderson and V. Vedral. Journal of Physics A: Mathematical and General, 34, 6899 (2001).
  * [10] C. s. Yu, Y. Zhang, and H. Zhao. Quantum Information Processing, 13(6):1437 (2014).
  * [11] H. Li, Y. S. Li, S. H. Wang, and G. L. Long. Communications in Theoretical Physics,61, 273 (2014).
  * [12] S. Luo and S. Fu. Physical Review A, 82, 034302 (2010).
  * [13] F. Paula, T. R. de Oliveira, and M. Sarandy. Physical Review A, 87,064101 (2013).
  * [14] N. Doustimotlagh, J. L. Guo, and S. Wang. Quantum correlations in qutrit-qutrit systems under local quantum noise channels. International Journal of Theoretical Physics, published on line,.
  * [15] S. Luo and S. Fu. Physical Review letters, 106, 120401 (2011).
  * [16] G. Drobny, A. Pines, S. Sinton, D. P. Weitekamp, and D. Wemmer. Royal Society of Chemistry,13, 49 (1978).
  * [17] W. Richter, S. Lee, W. S. Warren, and Q. He. Science, 267, 5198 (1995).
  * [18] E.B. Fel’dman, A.N. Pyrkov, JETP Lett. 88, 398 (2008).
  * [19] E. B. Fel’dman and A.I.Zenchuk. JETP Lett. 93, 459 (2011).
  * [20] E. Kuznetsova and A. Zenchuk. Physics Letters A, 376, 1029 (2012).
  * [21] M. Ali, A. Rau, and G. Alber. Physical Review A, 81, 042105(2010).


# Three Experiments to Analyze the Nature of the Heat Spreader

Seema Sethia1, Shouri Chatterjee2, Sunil Kale3, Amit Gupta4, Smruti R.
Sarangi5
1,2 Department of Electrical Engineering, IIT Delhi
3,4 Department of Mechanical Engineering, IIT Delhi
5 Department of Computer Science and Engineering, IIT Delhi
<EMAIL_ADDRESS>{shouri@ee, srk@mech, agupta@mech,
<EMAIL_ADDRESS>
###### Abstract

In this paper, we describe ongoing work to investigate the properties of the
heat spreader, and its implication on architecture research. In specific, we
conduct two experiments to quantify the heat distribution across the surface
of a spreader during normal operation. The first experiment uses T-type
thermocouples, to find the temperature difference across different points on
the spreader. We observe about 6∘C difference on average. In the second
experiment, we try to capture the temperature gradients using an infrared
camera. However, this experiment was inconclusive because of some practical
constraints such as the low emissivity of the spreader. We conclude that to
properly model the spreader, it is necessary to conduct detailed finite
element simulations. We describe a method to accurately measure the thermal
conductivity of the heat spreader such that it can be used to compute the
steady state temperature distribution across the spreader.

##  I Introduction

An oft-ignored aspect of architecture level thermal modeling is the heat
spreader. The heat spreader is typically a nickel coated copper plate placed
between the die and the heat sink (see Figure 1). Its main role is to
uniformly dissipate the heat generated by the die, and transmit the heat to
the heat sink. The heat sink is a large fin based heat exchanger that is used
to effectively dissipate the heat to the surrounding air. The heat spreader
effectively “spreads out” the heat and reduces the severity and incidence of
thermal hot spots.

Given the fact that the heat spreader is nothing more than a metal plate, and
does not have a lot of inherent complexity, it has not received a lot of
attention by the architecture community. Some prior work such as [1, 2], have
treated it as an isotherm (equal temperature at all points). We experimentally
disprove this hypothesis in this paper. Skadron et. al. [3] treat the spreader
as a mesh of points, where each point is a heat source and two adjacent points
are connected by a thermal resistance in their widely available thermal
modeling tool, HotSpot. However, there has been some recent criticism of the
equivalent thermal circuit based approach adopted by HotSpot in [4, 5]. These
works have reported a mean error of about 10% in HotSpot.

Figure 1:  The chip package

We are currently working on developing a new temperature estimation tool.
During the course of this work, we wish to look at the spreader from an
experimental viewpoint. There are several reasons for our belief that the
spreader warrants a more thorough study. (1) The conductivity of silicon is
roughly 100 W/m-K [5], whereas the conductivity of the spreader is about 400
W/m-K. Consequently, the spreader is a far more efficient lateral conductor of
heat than silicon especially at distances of the order of the dimensions of
the die. This has important implications for floor planning, thermal
management, and task allocation in multicores. For example, it is possible for
a set of active cores to heat up a set of relatively quiescent cores by
passing heat through the spreader. This will hurt the performance of the
relatively inactive cores, as well as long term lifetime reliability. (2) We
can use the spreader temperature data that we collect to calibrate temperature
simulators. (3) We can measure some thermal properties of the spreader and use
it to quantify the degradation of the material over time. We can use this
empirical data to perform more accurate FEM simulations. Lastly, for Intel
based chips that integrate the spreader with the die, it is not possible to
study the temperature profile of the die independently. We need to infer its
thermal profile by analyzing the temperature gradients on the spreader.

We performed a simple thought experiment as follows. We consider a die with a
large number of cores (128), and assumed that there is lateral heat conduction
just through the spreader. We use typical parameters from the HotSpot tool
(version 5.0) [3], and simulated a scenario in which each core dissipates
enough power to increase the die temperature measured at the center by 20∘C .
We now turn off a set of cores, and measure the effect that the active cores
have on the inactive cores. We plot the average temperature rise of the
inactive cores in Figure 3 as a fraction of the number of active cores. Figure
3 shows the normalized decrease in the MTTF (Mean Time to Failure) for three
major failure mechanisms: Electro-migration, Thermal Cycling, and Stress
Migration (see Srinivasan et. al. [6]). We observe that lateral heat
conduction can have a significant impact on inactive cores. It can heat them
up by 10 to 20∘C , and can decrease their MTTF by upto 10X.

Figure 2: Mean temp. of the inactive cores |  Figure 3: MTTF of the inactive cores
---|---

In this paper, we describe two approaches to measure the temperature
distribution across the spreader during normal operation. We can use these
numbers to calibrate temperature estimation tools. Lastly, we describe an
approach to compute the thermal conductivity of the spreader such that it can
be used to seed FEM simulations.

##  II Experiments

For all our experiments we use a 775 pin, 90 nm, Intel Pentium 4(Prescott)
chip mounted on a Dell 00M075 Dimension 4300 Motherboard. It has a nominal
frequency of 3.06GHz, 1MB L2 cache, and has two voltage steppings – 1.25 and
1.388V. We perform three experiments:

  1. 1.

Measure the temperature distribution on the surface of the heat spreader using
thermocouples.

  2. 2.

Capture the temperature distribution with an infrared camera.

  3. 3.

Measure the thermal conductivity of the heat spreader using thermocouples for
accurate steady state FEM simulation.

###  II-A Design of Thermocouples

A thermocouple consists of two wires made of different metals/alloys. At the
point of contact with the target material an EMF(voltage differential) is
generated between the two wires because of their differing thermal properties.
This difference in voltage is typically proportional to the temperature of the
target, and can be detected with a simple electronic circuit. Due to their
simplicity and accuracy, they are commonly used to perform accurate
temperature measurements. In our experiments we use 300 $\mu$m T-type
thermocouples made of copper and constantan wires. They operate best in a
temperature range between $\pm 200^{\circ}$C. To eliminate effects of
corrosion and ageing, we used brand new wires. Secondly, to ensure good
contact between the wires, we heat the tip of both wires such that they fuse
together to make a strong junction.

We connected the thermocouples to an Expert EX9018P data acquisition module.
This is a sophisticated analog to digital converter that converts the
thermocouple voltages to digital signals. It can process upto 8 thermocouple
inputs. It has an internal multiplexer that chooses one of them. The final
output is in the RS 485 serial bus format. We subsequently use an Advantech
ADAM 4520 converter to convert the RS 485 signals to RS 232 signals that can
directly be fed to the serial port of a standard PC. The ADAM 4520 chip also
helps to isolate the PC from ground loops and destructive voltage spikes. We
calibrated the thermocouples with distilled boiling water and ice. New Delhi
is 216 meters above sea level. The boiling point at this altitude is 99.304∘C
for a typical atmospheric pressure of 987.56 millibars.

Figure 4: Experimental setup |  Figure 5: Position of thermocouples on the spreader |  Figure 6: Two attached thermocouples
---|---|---

###  II-B Experiment I - Thermocouple based Measurement

In this experiment we place thermocouples at different ends of the integrated
heat spreader. We mostly follow the reference procedure as described in the
Intel Thermal Design Guidelines Document [7] (Appendix D). However, instead of
applying the Kapton adhesive, we apply Halnziye HY 610 thermal paste to
achieve the dual purpose of making the thermocouples stick to the spreader and
provide good thermal conductivity for accurate measurement. This thermal paste
has mild adhesive properties. Figure 6 shows our setup. Secondly, it was not
necessary to drill holes through the heat sink since we do not connect the
thermocouples at the center. We attach them to the middle of the four sides as
shown in Figure 6. We define four positions on the spreader – TCT, TCL, TCB,
and TCR. We allow the setup to reach steady state by having a gap of at least
10 minutes between different measurements. To further minimize the error, it
is necessary to repeat each experiment by interchanging the thermocouples.
This cancels out all sources of linear error. Each such experiment set is
repeated 10 times. We report the mean values. Because of mechanical
constraints, we were not able to attach more than two thermocouples at the
same time (see Figure 6).

We also report the CPU power as measured by the Windows CPUID utility. This is
the temperature at the center of the die [7]. As a benchmark, we use a simple
script that repeatedly performs calculations using the standard Windows
calculator application.

| Left-Bottom (TCB and TCL) | Right-Top(TCR and TCT) | Top-Bottom(TCT and TCB) | Left-Right(TCL and TCR)
---|---|---|---|---
Operation | CPU | TCB | (TCB | CPU | TCT | (TCT | CPU | TCT | (TCT | CPU | TCR | (TCR
| (∘C) | (∘C) | -TCL)(∘C) | (∘C) | (∘C) | -TCR)(∘C) | (∘C) | (∘C) | -TCB)(∘C) | (∘C) | (∘C) | -TCL)(∘C)
Power Off |  | 23.90 | 0.15 |  | 20.90 | 0.50 |  | 24.75 | 0.25 |  | 25.80 | 0.40
10 mins later | 57.5 | 39.85 | 4.05 | 52.0 | 35.95 | 3.10 | 52.5 | 39.65 | 7 | 43.5 | 33.50 | 1.35
10 mins after | 89.0 | 58.50 | 6.80 | 90.0 | 55.80 | 6.35 | 88.0 | 62.55 | 14.35 | 74.5 | 56.05 | -1.15
calculator on |  |  |  |  |  |  |  |  |  |  |  |
10 mins after | 58.5 | 41.25 | 4.20 | 51.5 | 37.85 | 4.45 | 53.5 | 41.40 | 9.85 | 43.5 | 33.45 | 2.00
calculator off |  |  |  |  |  |  |  |  |  |  |  |
TABLE I: Temperatures of Points on the Surface of the Heat Spreader

Table I shows the collected data at four time instants – power off, 10 minutes
later, 10 minutes after starting the benchmark, and 10 minutes after shutting
it down. We report four sets of readings.

The die temperature varies from 51∘C to 89∘C . The spreader temperature at the
hottest point (TCT) varies from 21∘C to 62.55∘C . As a sanity check we
correlate the temperature values with the layout of the Pentium 4 processor
[8]. TCT is close to the scheduler and trace-cache. In comparison TCB is the
coolest because it abuts the L2 cache. TCL and TCR are closest to the
fetch/decode logic, and floating point units respectively. They show a
moderate amount of activity for our benchmark.

The main take-away point in this experiment is that a large temperature
variation exists across the surface of the spreader. For example, the
difference between TCT and TCB reaches 14.35∘C . The values at TCT and TCL
differ by about 7∘C , and both TCR and TCL are warmer than TCB by about 6∘C .
This experiment gives us an indication of the degree of the temperature
gradients on the surface of the spreader. However, to get a more exact
picture, we need to do a more intrusive experiment.

###  II-C Experiment II - IR Camera based Measurement

To get accurate and extensive temperature profiles, we decided to use an IR
camera that can produce a detailed temperature profile of the surface of the
spreader. A similar approach has been used by Martinez et. al. [9] to capture
the temperature profile of a die. The authors in this paper collect their data
by removing the spreader and heat sink. They use an IR transparent oil based
heat sink instead. Note that it is necessary to use some heat removal
mechanism. Otherwise, the temperature of the die will increase to unacceptable
levels, and the processor will shut itself down. We are planning to create
such kind of a setup in the future. However, we observe that in such a setup
we will not get an accurate picture of the temperature dissipation of a die
and the thermal profile of the spreader because the nature of heat transfer is
different. The latest version of the popular thermal modeling tool HotSpot 5
takes this into cognizance. Additionally, secondary heat transfer paths,
especially through the ball grid arrays, become important in this case.

We unsuccessfully try another approach. Our intuition was to remove the heat
sink during regular operation and quickly take an IR photograph of the die.
There will be an intermittent delay of less than a few seconds. However, we
hoped to possibly compensate for the error by trying to back calculate the
original temperature profile using standard results for radiative and
convective heat transfer. We use a Testo 875, 9 Hz IR camera for this purpose.
We initially underestimate the temperature of the spreader greatly. This is
possibly because of the low emissivity of the heat spreader. Consequently, to
increase the emissivity of the spreader, we coat it with Halnziye HY 610
thermal paste that has an emissivity of 0.95. The temperature values increase
by about 15∘C . The average temperature difference is about 10∘C across the
die. However, our original aim of getting a detailed temperature profile was
still not served because the thermal image was heavily dependent on the
uniformity of the thermal paste. As shown in Figure 7 some of the
hottest(darkest) regions are towards TCB (L2 cache). This is not expected to
be the case.

Figure 7: IR Photograph  Figure 8: Measurement of thermal conductance |  Figure 9: The full measurement setup
---|---

###  II-D Experiment III - Measuring Thermal Conductivity

We observe that Experiment II was inconclusive. It only reaffirmed the fact
that a temperature differential exists across the spreader. To get a better
picture, we are in the process of setting up a detailed FEM simulation
framework that will be seeded by parameters obtained from our experiments. We
describe a method to calculate the thermal conductivity of the spreader
material (nickel coated copper plate). The thermal conductivity of a material
is defined as the power that flows across a temperature gradient of 1∘C in an
object that has unit length and unit cross-sectional area. Conceptually, it is
similar to electrical conductivity, and can be used to find the steady state
distribution of temperature.

We use the comparative method. This method proposes to place the unknown
sample (sliver of the spreader material) between two samples (copper wires)
with known thermal conductivity. Both the ends of this ensemble are set to
constant temperatures by dipping them in ice and boiling water respectively.
The thermal conductivities are related by the following equation.

$\frac{\kappa_{c}\Delta T_{w1}A_{w1}}{L_{w1}}=\frac{\kappa_{sp}\Delta
T_{sp}A_{sp}}{L_{sp}}=\frac{\kappa_{c}\Delta T_{w2}A_{w2}}{L_{w2}}$ (1)

Here, $\kappa_{c}$ is the thermal conductivity of copper (400 W/m-K), and
$\kappa_{sp}$ is the unknown thermal conductivity of the spreader. $w_{1}$
refers to copper wire 1, $w_{2}$ refers to copper wire 2, $sp$ refers to the
spreader sample, $A$ represents the cross-sectional area, and $L$ represents
the length of the wire. The temperature gradient, $\Delta T$, is measured
using thermocouples. The intuition behind this equation is that there is a
constant amount of heat flow in the assembly of wires. Note that Equation 1 is
over-constrained. We create two sets of equations – (1) between the spreader
and wire 1, and (2) between the spreader and wire 2. We solve them separately
and report the mean value of thermal conductivity.

Figure 9 shows the measurement setup with the two copper wires, spreader
sample, and six thermocouples. Each adjacent pair of thermocouples measures
the temperature difference across a homogeneous section of material. Note that
Equation 1 can be used only when there is exclusively conductive heat transfer
through the ensemble. We need to reduce convective and radiative heat transfer
to the maximum extent possible. Consequently, we covered the setup with
thermally insulating glass wool. Lastly, we set the reference temperature at
both ends using boiling water and ice respectively. The entire setup is shown
in Figure 9. We allowed upto 4-5 hours for the readings to stabilize. The
experimental procedure (10 repetitions, thermocouple interchange) is the same
as that mentioned in Section II-B.

We obtain a thermal conductivity value of 369 W/m-K ($\pm 0.5$∘C ). For
reference, the thermal conductivity of copper is 400 W/m-K, and the thermal
conductivity of nickel is 90.9 W/m-K.

##  III Conclusion and Future Work

In the course of this work, we get an in-vivo estimate of the nature of
temperature gradients on the surface of the heat spreader. We would like to
extend our work to make more detailed and elaborate measurements. In specific,
we would like to drill very fine holes (diameter less than 100 nm) on the heat
sink and measure the temperatures at the center of the spreader also. Lastly,
using empirical data collected from our studies, we wish to correlate our
measurements with FEM based simulations. The final goal is four fold – (1)
Create a corpus of empirically measured temperature data, (2) Propose accurate
temperature simulation methodologies for semi-conductor packages, and (3)
Design new packaging technologies that are more thermally efficient, and
lastly (4) Come up with new architectures that can leverage these advances in
novel packaging technologies.

## References

  * [1] S. R. Sarangi, B. Greskamp, A. Tiwari, and J. Torrellas, “Eval: Utilizing processors with variation-induced timing errors,” in _MICRO_ , 2008, pp. 423–434.
  * [2] J. Srinivasan and S. Adve, “The importance of heat-sink modeling for dtm and a correction to predictive dtm for multimedia applications,” in _Proceedings of the Fourth Annual Workshop on Duplicating, Deconstructing, and Debunking (WDDD) at ISCA-05_ , 2005.
  * [3] K. Skadron, M. R. Stan, W. Huang, S. Velusamy, K. Sankaranarayanan, and D. Tarjan, “Temperature-aware microarchitecture,” in _ISCA_ , 2003, pp. 2–13.
  * [4] V. Heriz, J.-H. Park, T. Kemper, S.-M. Kang, and A. Shakouri, “Method of images for the fast calculation of temperature distributions in packaged vlsi chips,” in _Thermal Investigation of ICs and Systems, 2007. THERMINIC 2007\. 13th International Workshop on_ , sept. 2007, pp. 18 –25.
  * [5] A. Ziabari, E. Ardestani, J. Renau, and A. Shakouri, “Fast thermal simulators for architecture level integrated circuit design,” in _SemiTherm_ , 2011\.
  * [6] J. Srinivasan, S. V. Adve, P. Bose, and J. A. Rivers, “The case for lifetime reliability-aware microprocessors,” in _ISCA_ , 2004, pp. 276–287.
  * [7] “Intel pentium 4 processor on 90nm process in the 775-land lga package: Thermal and mechanical design guidelines,” Intel, Tech. Rep. 302553-004, Nov 2005\.
  * [8] H. de Vries, “Looking at Intel Prescott die, Part II,” http://chip-architect.com/news/2003_04_20_Looking_at_Intels_Prescott_part2.%html, accessed on September 17th, 2012.
  * [9] F. J. Mesa-Martinez, J. Nayfach-Battilana, and J. Renau, “Power model validation through thermal measurements,” in _Proceedings of the 34th annual international symposium on Computer architecture_ , ser. ISCA ’07. New York, NY, USA: ACM, 2007, pp. 302–311.