††thanks: corresponding author.

# Quantum thermodynamics of nonequilibrium processes in lattice gauge theories

Zohreh Davoudi<EMAIL_ADDRESS>Department of Physics, University of Maryland,
College Park, MD 20742 USA  Maryland Center for Fundamental Physics,
University of Maryland, College Park, MD 20742, USA  Joint Center for Quantum
Information and Computer Science, NIST and University of Maryland, College
Park, MD 20742 USA  The NSF Institute for Robust Quantum Simulation,
University of Maryland, College Park, Maryland 20742, USA  Christopher
Jarzynski<EMAIL_ADDRESS>Department of Chemistry and Biochemistry,
University of Maryland, College Park, MD 20742 USA  Institute for Physical
Science and Technology, University of Maryland, College Park, MD 20742, USA
Department of Physics, University of Maryland, College Park, MD 20742 USA  The
NSF Institute for Robust Quantum Simulation, University of Maryland, College
Park, Maryland 20742, USA  Niklas Mueller<EMAIL_ADDRESS>InQubator for
Quantum Simulation (IQuS), Department of Physics, University of Washington,
Seattle, WA 98195, USA  Greeshma Oruganti<EMAIL_ADDRESS>Institute for
Physical Science and Technology, University of Maryland, College Park, MD
20742, USA  The NSF Institute for Robust Quantum Simulation, University of
Maryland, College Park, Maryland 20742, USA  Connor Powers<EMAIL_ADDRESS>Department of Physics, University of Maryland, College Park, MD 20742 USA
Maryland Center for Fundamental Physics, University of Maryland, College Park,
MD 20742, USA  Joint Center for Quantum Information and Computer Science, NIST
and University of Maryland, College Park, MD 20742 USA  The NSF Institute for
Robust Quantum Simulation, University of Maryland, College Park, Maryland
20742, USA  Nicole Yunger Halpern<EMAIL_ADDRESS>Joint Center for Quantum
Information and Computer Science, NIST and University of Maryland, College
Park, MD 20742 USA  Institute for Physical Science and Technology, University
of Maryland, College Park, MD 20742, USA  The NSF Institute for Robust Quantum
Simulation, University of Maryland, College Park, Maryland 20742, USA

###### Abstract

A key objective in nuclear and high-energy physics is to describe
nonequilibrium dynamics of matter, e.g., in the early universe and in particle
colliders, starting from the Standard Model. Classical-computing methods, via
the framework of lattice gauge theory, have experienced limited success in
this mission. Quantum simulation of lattice gauge theories holds promise for
overcoming computational limitations. Because of local constraints (Gauss’s
laws), lattice gauge theories have an intricate Hilbert-space structure. This
structure complicates the definition of thermodynamic properties of systems
coupled to reservoirs during equilibrium and nonequilibrium processes. We show
how to define thermodynamic quantities such as work and heat using strong-
coupling thermodynamics, a framework that has recently burgeoned within the
field of quantum thermodynamics. Our definitions suit instantaneous quenches,
simple nonequilibrium processes undertaken in quantum simulators. To
illustrate our framework, we compute the work and heat exchanged during a
quench in a $Z_{2}$ lattice gauge theory coupled to matter in 1+1 dimensions.
The thermodynamic quantities, as functions of the quench parameter, evidence
an expected phase transition. For general thermal states, we derive a simple
relation between a quantum many-body system’s entanglement Hamiltonian,
measurable with quantum-information-processing tools, and the Hamiltonian of
mean force, used to define strong-coupling thermodynamic quantities.

††preprint: UMD-PP-024-05, IQuS@UW-21-075

_Introduction._ —Gauge theories are ubiquitous descriptions of fundamental
interactions in the Standard Model in particle physics [1, 2, 3, 4, 5], of
emergent low-energy phenomena in condensed-matter and synthetic quantum
systems [6, 7, 8, 9], and of diverse fault-tolerant quantum-computing codes
[10, 11, 12, 13, 14]. An overarching goal is to simulate strongly interacting
matter starting from gauge theories. Key focuses include nonequilibrium
phenomena described by quantum chromomodynamics (QCD), the theory of the
strong force. Such phenomena arise in ultrarelativistic particle collisions
[15, 16] and in the early universe [17, 18]. Theoretical studies [19, 20, 21,
22, 23] of in- and out-of-equilibrium phases of QCD, and of its thermalization
mechanisms, are often restricted to extreme parameter regimes, to facilitate
perturbation theory. Alternatively, such studies feature simpler (often lower-
dimensional) models to capture qualitative features of QCD.

Studying QCD and other strongly interacting gauge theories requires
nonperturbative tools, as are enabled by lattice gauge theory (LGT) [24, 25,
26, 27, 28]. In a LGT, quantum fields are defined on a discrete spacetime of
finite extent. Within the path-integral formulation of LGTs, Monte-Carlo
simulations are viable if Euclidean (imaginary) time replaces Minkowski (real)
time. The scheme permits parallels with statistical mechanics: Euclidean time
stands in for inverse temperature, and vacuum expectation values serve as
thermal averages. LGT has enabled thermodynamic studies of gauge theories,
including the QCD equation of state at small chemical potentials [29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 39, 40]. Nonetheless, the sampling weight in
Monte-Carlo computations can become nonreal—a sign problem can occur—demanding
infeasibly many samples to resolve the signal [41]. This problem occurs at
finite fermionic densities (prevalent in LGTs, condensed-matter, and nuclear
many-body systems) and in out-of-equilibrium processes [40, 42, 43, 44]. These
limitations do not inhibit Hamiltonian-based approaches, such as tensor-
network methods [45, 46, 47] and quantum simulation [48, 49, 50, 51]. These
approaches, hence, suit thermodynamic studies of gauge theories, in and out of
equilibrium. Still, a modern description of LGTs in the language of quantum
thermodynamics needs to be developed.

The field of quantum thermodynamics extends conventional thermodynamics to
small and quantum systems that exchange heat and work [52, 53, 54]. A typical
set-up features a subsystem of interest (called the _system_) interacting with
a _reservoir_ of inaccessible degrees of freedom (DOFs) [55, 56]. The coupling
is often weak because a system and reservoir typically interact only at their
shared boundary. This boundary is of lower dimensionality than the system,
whose volume is proportional to its internal energy. Hence, the interaction
energy is much smaller than the system’s internal energy, as well as the
reservoir’s. Yet quantum systems and reservoirs can be small while behaving
thermodynamically [57, 58, 59], and their interactions need not be negligible.
Hence the subfield of strong-coupling (quantum) thermodynamics has burgeoned
recently [60, 61, 62, 63, 64, 65, 66, 67, 68, 69]. Strong coupling blurs the
system-reservoir boundary, complicating the definition of the system’s
internal energy, work, and heat. Nonetheless, definitions have been developed
and shown to obey thermodynamic laws [70, 67, 71, 64, 72, 66].

These considerations raise the question _can LGTs obey weak-coupling
thermodynamics?_ Local constraints, or Gauss’s laws, constitute the defining
feature of (lattice) gauge theories. Electrodynamics—a U(1) gauge
theory—offers a familiar example: the electric charge at a site balances the
electric-field flux emanating from the site. A gauge theory’s Hilbert space is
restricted to a subspace that satisfies Gauss’s laws. This restriction
complicates the partitioning of the system into subsystems: the fields’ values
on one side of the partition depend on the values on the other side. Yet such
a partitioning is used to define quantities such as the bipartite entanglement
entropy [73, 74, 75, 76, 77, 78, 79]. The system-reservoir partitioning in
LGTs, we posit, resembles that in strong-coupling thermodynamics. Hence, we
use strong-coupling thermodynamics to define thermodynamic quantities for
LGTs.

Motivated by quantum simulators’ capabilities [80, 81], we answer the
following questions: what are the work and heat exchanged during instantaneous
quenches [82]—simple nonequilibrium processes created in quantum simulations?
Can these quantities signal different phases and phase transitions
111Relationships between work distributions and phase transitions have been
explored recently in the context of spin systems [157, 158, 159, 160, 161,
162, 163, 164].? Can such quantities be measured efficiently with quantum-
information-processing tools?

Capitalizing on our upcoming work [70], we describe how to compute changes in
internal energies. We show how to divide these changes into work and heat
consistently with the first and second laws of thermodynamics. A simple model
illustrates this framework: a $Z_{2}$ LGT coupled to hardcore bosonic matter
in 1+1 dimensions (D). We observe that thermodynamic quantities, as functions
of chemical potential, signal an expected phase transition. Furthermore, our
work bridges the fields of quantum information theory and strong-coupling
quantum thermodynamics: we show that the entanglement Hamiltonian [84, 85],
which can often be efficiently measured experimentally [86, 87, 88, 89], is
related to the Hamiltonian of mean force [64], which underpins strong-coupling
quantum thermodynamics.

Figure 1: Overview of this work’s quench protocol. The system starts in a
global Gibbs state. At $t=0$, the system Hamiltonian $H_{S}$ is quenched
instantaneously. Under the new total Hamiltonian, the system-reservoir
composite equilibrates to a global Gibbs state with the same temperature as
the initial state.

_Overview of strong-coupling quantum thermodynamics._ —Consider a system $S$
and a reservoir $R$. The composite $S\cup R$ evolves under the Hamiltonian

$H_{S\cup R}\coloneqq H_{S}+H_{R}+V_{S\cup R}\,.$ (1)

$H_{S}$ and $H_{R}$ denote system’s and reservoir’s Hamiltonians,
respectively; and $V_{S\cup R}$ denotes the interaction. During a
thermodynamic process, $S$ can absorb heat $Q$ and work $W$, quantities we aim
to compute.

In weak-coupling quantum thermodynamics, the interaction Hamiltonian,
$V_{S\cup R}$, contributes negligibly to the total internal energy, $U_{S\cup
R}\coloneqq\langle H_{S\cup R}\rangle$. Hence, the system’s internal energy is
the expectation value of the Hamiltonian $H_{S}$ that acts nontrivially on the
system alone: $U_{S}\coloneqq\langle H_{S}\rangle$ [71]. Thus, one can measure
the system’s internal energy by accessing only system DOFs. In contrast, when
$\langle V_{S\cup R}\rangle$ is comparable to $\langle H_{S}\rangle$, one must
use strong-coupling thermodynamics.

How much $V_{S\cup R}$ contributes to $U_{S}$ is ambiguous. To resolve the
ambiguity, one can define the global Gibbs state, the thermal-equilibrium
state with respect to $H_{S\cup R}$ and an inverse temperature $\beta$:
$\pi_{S\cup R}\coloneqq e^{-\beta H_{S\cup R}}/Z_{{S\cup R}}$. The partition
function is $Z_{{S\cup R}}\coloneqq{\text{Tr}}(e^{-\beta H_{S\cup R}})$.
Throughout this work, we denote Gibbs states by $\pi$ and general density
matrices by $\rho$. ${\text{Tr}}_{X}$ denotes the trace over the Hilbert space
of subsystem $X$. Consider tracing out the reservoir DOFs from the global
Gibbs state $\pi_{S\cup R}$. The system’s reduced density matrix, $\pi_{S}$,
is a Gibbs state with respect to an effective Hamiltonian $H_{S}^{*}$, the
_Hamiltonian of mean force_ [64]: $\pi_{S}\coloneqq{\text{Tr}}_{R}(\pi_{S\cup
R})\equiv e^{-\beta H_{S}^{*}}/Z_{S}^{*}$, wherein

$H_{S}^{*}\coloneqq-\frac{1}{\beta}\ln\frac{{\text{Tr}}_{R}\left(e^{-\beta
H_{S\cup R}}\right)}{Z_{R}}\,,$ (2)

$Z_{R}\coloneqq{\text{Tr}}_{R}(e^{-\beta H_{R}})$, and
$Z_{S}^{*}\coloneqq{\text{Tr}}_{S}(e^{-\beta H_{S}^{*}})=Z_{S\cup R}/Z_{R}$.
Equation (2) underlies an intuitive definition for the system’s free energy
[64, 90, 91],

$F_{S}\equiv
U_{S}-\frac{\mathcal{S}}{\beta}\coloneqq-\frac{1}{\beta}\ln(Z_{S}^{*})\,.$ (3)

$\mathcal{S}\coloneqq-{\text{Tr}}(\rho_{S}\ln\rho_{S})$ denotes the von
Neumann entropy, and $\rho_{S}$ denotes the system’s reduced state
222Boltzmann’s constant is set to one throughout..

It is natural to associate the effective Hamiltonian’s expectation value,
$\langle H_{S}^{*}\rangle$, with the system’s internal energy, $U_{S}$.
Consequently, one need not access reservoir DOFs to compute $U_{S}$. Any
change in $U_{S}$ comes from work and heat, by the first law of
thermodynamics: $\Delta U_{S}=W+Q$. By the second law, $\Delta F_{S}\leq W$,
or $Q\leq\Delta\mathcal{S}/\beta$. Here, $\Delta U_{S}$, $\Delta F_{S}$ and
$\Delta\mathcal{S}$ denote, respectively, net changes in the system’s internal
energy, free energy and entropy, during a thermodynamic process. Intuition
guides the separation of $\Delta U_{S}$ into $W$ and $Q$: work is associated
with changes in the system’s Hamiltonian; and heat, with changes in its state
[53]. To define work and heat concretely, we now specify a thermodynamic
process.

_Instantaneous quenches within strong-coupling quantum thermodynamics._
—During a _quench_ , the system’s Hamiltonian changes rapidly. Quenches
generate nonequilibrium conditions and are studied in diverse quantum-
simulation experiments [93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 89].
Figure 1 depicts the quench studied here. The system-reservoir composite
begins in a Gibbs state: $\rho_{S\cup R}^{\text{i}}\coloneqq\rho_{S\cup
R}(t{=}0^{-})=\pi_{S\cup R}^{\text{i}}\coloneqq e^{-\beta H_{S\cup
R}^{\text{i}}}/{\rm{Tr}\big{(}e^{-\beta H_{S\cup R}^{\text{i}}}\big{)}}$,
wherein the initial composite Hamiltonian $H_{S\cup R}^{\text{i}}\coloneqq
H_{S\cup R}(t{=}0^{-})$. Equation (1), with a time-dependent $H_{S}(t)$,
specifies $H_{S\cup R}$. At time $t=0$, $H_{S}(t)$ is instantaneously quenched
from $H_{S}(t{=}0^{-})\coloneqq H_{S}^{\text{i}}$ to
$H_{S}(t{=}0^{+})\coloneqq H_{S}^{\text{f}}$. The system-reservoir composite
then equilibrates to a global Gibbs state under the final total Hamiltonian
$H_{S\cup R}(t{=}0^{+})\coloneqq H_{S\cup R}^{\text{f}}$. The initial and
final global Gibbs states have the same temperature, $\beta^{-1}$, by
assumption. Justifications include the possibility that $S\cup R$ couples
weakly to a larger reservoir, at a temperature $\beta^{-1}$, in the distant
past and future [70].

Practical work and heat definitions depend on system DOFs alone. We present
such definitions, for quenches of the type in Fig. 1, in Ref. [70]; see also
Refs. [90, 61, 64]. The system’s internal energy is

$U_{S}(t)\coloneqq\langle
H_{S}^{*}(t)\rangle={\text{Tr}}\bm{(}\rho_{S}(t)H_{S}^{*}(t)\bm{)}\,,$ (4)

for arbitrary states $\rho_{S}(t)$. $H_{S}^{*}$ acquires time dependence under
the replacement $\text{Tr}_{R}\bm{(}e^{-\beta H_{S\cup
R}}\bm{)}\mapsto\text{Tr}_{R}\bm{(}e^{-\beta H_{S\cup R}(t)}\bm{)}$ in Eq.
(2). During the quench, the internal energy changes by an amount $\Delta
U_{S}$ equal to the work absorbed by $S$:

$W\coloneqq{\text{Tr}}_{S}\bm{(}\rho^{\text{i}}_{S}H_{S}^{*}(t=0^{+})\bm{)}-{\text{Tr}}_{S}\bm{(}\rho^{\text{i}}_{S}H_{S}^{*}(t=0^{-})\bm{)}\,.$
(5)

During the subsequent equilibration, the internal-energy change equals the
heat absorbed by $S$:

$Q\coloneqq{\text{Tr}}_{S}\bm{(}\rho^{\text{f}}_{S}H_{S}^{*}(t=0^{+})\bm{)}-{\text{Tr}}_{S}\bm{(}\rho^{\text{i}}_{S}H_{S}^{*}(t=0^{+})\bm{)}\,.$
(6)

These definitions are intuitive and obey the first and second laws of
thermodynamics [70]. Thus, upon identifying a Hamiltonian of the form in Eq.
(1), one calculates work and heat by measuring the system’s internal energy,
$\langle H_{S}^{*}\rangle$. We show next how to measure this quantity.

_Measuring thermodynamic quantities in quantum simulations._ —We aim to derive
a relation between strong-coupling thermodynamic quantities and a quantity
called the entanglement Hamiltonian. Every density matrix $\rho$ can be
expressed as $\rho=\sum_{k}p_{k}\lvert k\rangle\\!\langle k\rvert$. The
$\lvert k\rangle$ denote eigenstates; and the $p_{k}\in[0,1]$, probabilities.
One can define $p_{k}\coloneqq e^{-\lambda_{k}}$ for some $\lambda_{k}\geq 0$:
$\rho=\sum_{k}e^{-\lambda_{k}}\lvert k\rangle\\!\langle k\rvert$. This
expansion has the form of an un-normalized thermal state, at unit temperature,
with respect to a Hamiltonian with eigenenergies $\lambda_{k}$ and eigenstates
$|k\rangle$. For $\rho=\rho_{S}\coloneqq\rm{Tr}_{R}(\rho_{S\cup R})$, this
Hamiltonian is the _(bipartite) entanglement Hamiltonian_ [84, 85],

$H^{\rm{ent}}_{S}\coloneqq-\ln(\rho_{S})\,.$ (7)

This operator contains more information than the bipartite entanglement
entropy. It has spawned numerous studies in quantum information theory and
many-body physics 333For instance, the $\lambda_{k}$ statistics imply whether
a system behaves ergodically [165, 166, 167, 168, 169]. (Ergodicity is
necessary for thermalization.) Also, a gap in the $H_{S}^{\rm ent}$ spectrum
may signal a topologically ordered phase [84, 169, 170].. Parameterized
ansatze for entanglement Hamiltonians [85], with random-measurement protocols
[105, 106, 107, 108], enable tomography of ground and nonequilibrium states
[109, 110, 86, 87, 88], including of LGTs [89, 111, 112].

Figure 2:  (a) Partitioning of DOFs into system ($S$) and reservoir ($R$)
DOFs. Partitioning of total Hamiltonian’s (b) hopping terms, (c) electric-
field terms, and (d) mass and chemical-potential terms into $H_{S}(t)$ (teal),
$H_{R}$ (light red), and $V_{S\cup R}$ (purple). Each row shows the total
system. Matter (gauge-field) DOFs reside on lattice sites (links). The last
link wraps around to indicate the lattice’s periodicity.

To leverage entanglement-Hamiltonian-tomography tools, we prove a relation
between $H^{\rm{ent}}_{S}$ and $H_{S}^{*}$ for thermal states. We rewrite the
Hamiltonian of mean force as

$\displaystyle H_{S}^{*}$
$\displaystyle=-\frac{1}{\beta}\ln\left(\frac{{\text{Tr}}_{R}(e^{-\beta
H_{S\cup R}})}{{\text{Tr}}_{R}(e^{-\beta H_{R}})}\right)$ (8a)
$\displaystyle=-\frac{1}{\beta}\ln\left(\rho_{S}\frac{Z_{S\cup
R}}{Z_{R}}\right)\,.$ (8b)

The system’s reduced state is
$\rho_{S}=\pi_{S}\coloneqq{\text{Tr}}_{R}(e^{-\beta H_{S\cup R}})/Z_{S\cup
R}$. Using Eqs. (7) and (3) yields the relation between the entanglement
Hamiltonian and the Hamiltonian of mean force:

$H_{S}^{*}=\frac{1}{\beta}H^{\rm{ent}}_{S}+F_{S}\,.$ (9)

$\mathbb{I}_{S}$ denotes the identity operator defined on the system’s Hilbert
space.

One can measure the first term in Eq. (9) using the aforementioned tomography
tools. To calculate $F_{S}$, one must calculate partition functions
(classically or via quantum simulation), which can be costly [113, 114, 115,
116], demanding access to DOFs outside $S$. To avoid computing the second term
in Eq. 9, we study the average dissipated work, $W_{\rm{diss}}\coloneqq
W-\Delta F_{S}$, a measure of entropy production [117, 118]. For our quench
protocol,

$W_{\rm{diss}}=\frac{1}{\beta}{\text{Tr}}_{S}\bm{(}\rho^{\text{i}}_{S}\left[H^{\rm{ent}}_{S}(t{=}0^{+})-H^{\rm{ent}}_{S}(t{=}0^{-})\right]\bm{)}\,.$
(10)

Hence, one can compute $W_{\rm{diss}}$ upon measuring the entanglement
Hamiltonian alone.

_Lattice gauge theories and strong-coupling quantum thermodynamics._ —We use
the Hamiltonian formulation of LGTs [26], which suits quantum simulation. As
matter and gauge fields undergo local symmetry-group transformations, the
observables remain invariant. The symmetry restricts the states to a _physical
subspace_ : for each site $n$, a _Gauss-law_ operator $G_{n}$ acts on $n$ and
commutes with the Hamiltonian. The full Hilbert space (spanned by eigenbases
of electric and matter fields) decomposes into $G_{n}$ eigenspaces, for each
$n$. The eigenspace labeled by some eigenvalue $g$, and shared by all the
$G_{n}$, is the physical subspace. In the Standard Model of particle physics,
nature sets the value of $g$. That is, for any physical state
$|\Psi_{\rm{phys}}\rangle$,

$G_{n}|\Psi_{\rm{phys}}\rangle=g|\Psi_{\rm{phys}}\rangle\,,\quad\forall n\,.$
(11)

In electrodynamics, $G_{n}=\bm{\nabla}\cdot\bm{E}_{n}-\rho_{n}$. $\bm{E}_{n}$
denotes the electric field, and $\rho_{n}$ denotes the (dynamical) electric-
charge density, both at site $n$. Gauss’s law follows from setting $g=0$
($g\neq 0$) in Eq. 11 in the absence (presence) of a background static
electric charge.

One may impose Gauss’s laws by manually removing the unphysical states from
the full Hilbert space [119, 120]. Alternatively, the Hamiltonian $H_{S\cup
R}$ may be replaced with

$H^{\prime}_{S\cup R}=H_{S\cup R}+\sum_{n}f(G_{n})\,.$ (12)

$f(G_{n})$ denotes a function of Gauss-law operators. Chosen properly, it
penalizes transitions to unphysical states [121, 122, 123, 124, 125, 126, 127,
128]. Consider partitioning a lattice into a system $S$ and a reservoir $R$.
Some Gauss-law penalty terms act on both $S$ and $R$: Gauss-law operators are
multibody operators consisting of gauge and matter fields. Such penalty terms,
thus, contribute to the $V_{S\cup R}$ in Eq. (1). Their contribution must be
large to constrain the state to the physical subspace. Therefore, one cannot
neglect the internal energy’s dependence on $V_{S\cup R}$ when computing
thermodynamic quantities; see the SM. Consequently, systems described by LGTs
call for strong-coupling quantum thermodynamics.

_Example of $Z_{2}$ LGT coupled to matter in (1+1)D._—We aim to numerically
compute the thermodynamic quantities exchanged during our quench protocol in a
LGT, using strong-coupling quantum thermodynamics. Consider $Z_{2}$ gauge
fields (spin-$\frac{1}{2}$ hardcore bosons) coupled to matter fields (chosen
to be spin-$\frac{1}{2}$ hardcore bosons). The initial state evolves under the
Hamiltonian

$\begin{split}H&=H_{\mathrm{h}}+H_{\mathrm{e}}+H_{\mathrm{m}}+H_{\mathrm{\mu}}+H_{c}\\\
&\equiv-t\sum_{n=0}^{N-1}\left(\sigma^{+}_{n}\tilde{\sigma}^{z}_{n}\sigma^{-}_{n+1}+{\rm{h.c.}}\right)-\epsilon\sum_{n=0}^{N-1}\tilde{\sigma}^{x}_{n}\\\
&\quad+m\sum_{n=0}^{N-1}(-1)^{n}\sigma^{+}_{n}\sigma^{-}_{n}-\mu\sum_{n=0}^{N-1}\sigma^{+}_{n}\sigma^{-}_{n}+c\sum_{n=0}^{N-1}\mathbb{I}_{n}\,,\end{split}$
(13)

on a one-dimensional $N$-site spatial lattice with periodic boundary
conditions ($\sigma_{N}=\sigma_{0}$). $H_{\rm h}$, $H_{\rm e}$, $H_{\rm m}$,
and $H_{\mu}$ represent the matter-hopping, electric-field, matter-mass, and
matter-chemical-potential terms, respectively 444The mass term’s sign
alternates between even-index and odd-index sites, consistently with the
staggering of fermionic matter [25]. Despite working with a bosonic theory, we
keep the staggering convention.. The constant term $H_{c}$ is added such that
$H_{S}(t)$, $H_{R}$, and $H_{S\cup R}(t)$ have non-negative eigenvalues. $t$,
$\epsilon$, $m$, and $\mu$ denote the hopping strength, electric-field
strength, matter mass, and matter chemical potential, respectively. Each Pauli
operator $\sigma_{n}$ acts on the Hilbert space of the site-$n$ matter field.
Each Pauli operator $\tilde{\sigma}_{n}$ acts on the Hilbert space of the
gauge field rightward of $n$. Specifically, $\tilde{\sigma}_{n}^{x}$
($\tilde{\sigma}_{n}^{z}$) denotes the electric-field (gauge-link) operator
555The gauge link and electric field are two conjugate operators in a gauge
theory, similarly to position and momentum in quantum mechanics..

The $\mathbb{Z}_{2}$ gauge transformation is generated by the Gauss-law
operator 666 For a pure gauge theory, which lacks matter fields, the operator
reduces to $G_{n}\coloneqq\tilde{\sigma}^{x}_{n}\tilde{\sigma}^{x}_{n-1}$.
This is the familiar generator of a local $\mathbb{Z}_{2}$ symmetry.

$G_{n}=\tilde{\sigma}^{x}_{n}\tilde{\sigma}^{x}_{n-1}\exp\left(i\pi\left[\sigma^{+}_{n}\sigma^{-}_{n}+\frac{(-1)^{n}-1}{2}\right]\right)\,.$
(14)

The physical states obey Eq. (11) with $g=1$ 777$G_{n}$ has only two
eigenvalues, $\pm 1$. Physically, when $g=1$, if no matter (or antimatter)
occupies site $n$, the electric field on the left equals that on the right. If
site $n$ contains matter (or antimatter), the electric-field eigenvalues on
the left and right have opposite signs.. One can realize the gauge-invariant
dynamics by adding $\sum_{n}f(G_{n})=\kappa\sum_{n}(1-G_{n})$ to $H$, then
taking the limit $\kappa\rightarrow\infty$ 888Numerical computations of this
work are performed by manually restricting to the physical Hilbert space..

The lattice can be partitioned into a system $S$ and a reservoir $R$, as in
Fig. 2(a). Also, the Hamiltonian (13) decomposes as in Eq. (1). Some hopping
terms act only on sites in $S$ ($R$) and so belong in $H_{S}$ ($H_{R}$). Other
terms describe hopping between a site in $S$ and a site in $R$. These
interaction terms belong in $V_{S\cup R}$; see Fig. 2(b). The electric-field
Hamiltonian can be partitioned as follows; see Figure 2(c). Terms acting on
links in $S$ belong in $H_{S}(t)$. Terms acting on links in $R$, but touching
the $S$-$R$ boundary, belong in $V_{S\cup R}$. Terms acting on links elsewhere
in $R$ belong in $H_{R}$. The interaction terms are consistent with the
interrelation, by Gauss’s law, of system DOFs with the link DOFs on the
reservoir side of a boundary. Terms in $H_{\text{m}}$ and $H_{\mu}$ belong in
$H_{S}(t)$ and $H_{R}$, depending on whether they act on a site in $S$ or $R$;
see Fig. 2(d). Finally, if penalty terms enforcing Gauss’s law act at the
boundary, they belong in $V_{S\cup R}$. Otherwise, they belong in $H_{S}$ or
$H_{R}$.

We consider quenching the system’s chemical potential from $\mu_{\rm{i}}=0$ to
$\mu_{\rm{f}}\neq 0$. We further define the system’s _chiral condensate_ 999
$\Sigma$ has the form that the chiral condensate would have if the LGT were
coupled to staggered fermions (a Jordan-Wigner transformation maps the fermion
operators to Pauli operators). Then, the chiral condensate serves as an order
parameter for chiral phase transitions. $\Sigma$ is not technically a chiral
condensate in our model, however. The reason is that the gauge fields couple
directly to spin-$\frac{1}{2}$ DOFs, not to fermions. Nonetheless, inspired by
theories with staggered fermions, we call $\Sigma$ the chiral condensate.,

$\Sigma\coloneqq\frac{1}{N_{S}}\sum_{n=0}^{N_{S}-1}(-1)^{n}\langle\sigma^{+}_{n}\sigma^{-}_{n}\rangle\,.$
(15)

$N_{S}$ denotes the system size. The expectation value is computed in the
final thermal state, whose $\mu=\mu_{\text{f}}$. If
$\mu_{\text{f}}\gg\epsilon,m$, the system’s thermal state is dominated by the
matter fields’ all-spin-up state, yielding $\Sigma=0$. At other
$\mu_{\text{f}}$ values, $\Sigma$ could be nonvanishing. In fact, $\Sigma$
suddenly changes from finite to vanishing values as $\mu_{\text{f}}$
increases, indicating a phase transition, as Fig. 3(a) shows. This phase
transition does not look perfectly sharp, due to finite-temperature (see the
SM) and finite-size effects. We regard the phase transition as occurring at
the critical value $\mu_{\text{f}}=\mu_{\text{f}}^{\rm c}$, where
$\Sigma^{\prime}\coloneqq d\Sigma/d\mu_{\text{f}}$ is maximized.

Figure 3 displays thermodynamic quantities, calculated as functions of
$\mu_{\text{f}}$: $W_{\rm diss}$, $W$, $Q$, $\Delta F_{S}$, and
$\Delta\mathcal{S}$. As $\mu_{\text{f}}$ grows, $W_{\rm{diss}}\coloneqq
W-\Delta F_{S}$ remains near zero until around the $\mu_{\mathrm{f}}$ value
where the phase transition occurs. Afterward, $W_{\rm diss}$ increases. Hence
$W_{\rm diss}$ indicates the phase transition clearly. Similarly, $W$ ($\beta
Q$) begins deviating from $\Delta F_{S}$ ($\Delta\mathcal{S}$) around the
phase transition. Nonetheless, to calculate work or heat, one must compute
partition-function ratios dependent on all the DOFs in $S\cup R$; see Eq.
(8b). In contrast, determining the dissipated work requires only measurements
of the system’s entanglement Hamiltonian; see Eq. (10). Measuring the
dissipated work, using quantum simulators, is therefore more feasible.

Figure 3: Thermodynamic quantities as functions of the final chemical
potential, $\mu_{\text{f}}$, during the instantaneous quench from
$\mu_{\text{i}}=0$: (a) Dissipated work $W_{\rm{diss}}$ (solid, black) and
chiral condensate $\Sigma$ (dot-dashed, teal). (b) Work $W$ (solid, black) and
change $\Delta F_{S}$ in the system’s free energy (dotted, black). (c)
Normalized heat $\beta Q$ (solid, black) and change $\Delta\mathcal{S}$ in the
system’s entropy (dotted, black). In all subfigures, we plot
$\mu_{\text{f}}^{\rm c}$, the $\mu_{\text{f}}$ value where $\Sigma^{\prime}$
is maximized (dashed, gray). For these plots, $N=6$, $N_{S}=4$, $t=-1/2$,
$\epsilon=1/2$, $m=1/2$, $\mu_{\text{i}}=0$, and $\beta=10$.

In the SM, we apply weak-coupling quantum-thermodynamics definitions, despite
the couplings being strong. As a result, we obtain different work and heat
values, which behave qualitatively differently.

_Summary and outlook._ —This work shows how strong-coupling quantum
thermodynamics applies to lattice gauge theories in and out of equilibrium.
The work further shows how entanglement Hamiltonians can be leveraged to
measure certain thermodynamic quantities in quantum simulations. Our framework
motivates explorations of further questions in the quantum thermodynamics of
gauge theories. Examples include whether thermodynamic quantities signal
topological [8, 10, 9, 84, 135, 136, 137, 138, 139] or dynamical [140] phase
transitions. Non-Abelian and higher-dimensional gauge theories merit studying
in our framework, too. While quench protocols in LGTs have been commonly
implemented in experiments [99, 100, 101, 89, 141, 142, 143, 144, 145, 146,
103, 147, 122, 148, 149, 150], a longer-term vision is simulating particle
collisions relevant to nuclear and high-energy physics. Work has begun on
quantum simulations of scattering processes [151, 152, 153, 154, 155, 156],
and their quantum-thermodynamic properties merit studying. Generally,
developing quantum-simulation protocols for studying gauge-theory
thermodynamics is an active frontier, and this work furthers this goal.

_Acknowledgments._ —Z. D., C. J., G. O., C. P., and N. Y. H. were supported by
the National Science Foundation (NSF) Quantum Leap Challenge Institutes (QLCI)
(award no. OMA-2120757). Z.D. further acknowledges funding by the Department
of Energy (DOE), Office of Science, Early Career Award (award no.
DESC0020271), and by the Department of Physics, Maryland Center for
Fundamental Physics, and the College of Computer, Mathematical, and Natural
Sciences at the University of Maryland, College Park. She is grateful for the
hospitality of Perimeter Institute where part of this work was carried out.
Research at Perimeter Institute is supported in part by the Government of
Canada through the Department of Innovation, Science, and Economic
Development, and by the Province of Ontario through the Ministry of Colleges
and Universities. Z.D. was also supported in part by the Simons Foundation
through the Simons Foundation Emmy Noether Fellows Program at Perimeter
Institute. C. J. and N. Y. H. further acknowledge support from John Templeton
Foundation (award no. 62422). N. M. acknowledges funding by the DOE, Office of
Science, Office of Nuclear Physics, Inqubator for Quantum Simulation (award
no. DE-SC0020970). C. P. is grateful for discussions on thermal gauge theories
with Robert Pisarski, and for the hospitality of Brookhaven National
Laboratory which hosted C. P. as part of an Office of Science Graduate Student
Research Fellowship. G. O. further acknowledges support from the American
Association of University Women through an International Fellowship. Finally,
N. M., G. O., C. P., and N. Y. H. thank the participants of the InQubator for
Quantum Simulation (IQuS) workshop “Thermalization, from Cold Atoms to Hot
Quantum Chromodynamics” at the University of Washington in September 2023 for
many valuable discussions.

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Supplemental Material:
“Quantum thermodynamics of nonequilibrium processes in lattice gauge theories”

##  Appendix A Supplemental Note I:
Convergence of thermodynamic quantities with $\beta$
in the lattice-gauge-theory example

Matrix exponentiations of the form $e^{-\beta H}$ can have very large or very
small eigenvalue magnitudes, especially at large $\beta$. This can lead to
instabilities in the numerical evaluation of $e^{-\beta H}$. Numerical
instabilities can arise also when a matrix logarithm is calculated. However,
absolute errors in floating-point representations increase with the magnitude
of the represented value [171]. So, we add constant shifts to the Hamiltonians
$H_{S\cup R}(t=0^{-})$ and $H_{S\cup R}(t=0^{+})$ in the form of uniform on-
site energy densities; see Eq. (13). The magnitude of the energy density,
given by $c$ in Eq. (13), is chosen such that $H_{S}(t)$, $H_{R}$, and
$H_{S\cup R}(t)$ have non-negative eigenvalues for the system sizes studied.
The energy density terms are partitioned between the sub-Hamiltonians
according to Eq. (1) in the same manner as other on-site terms shown in Fig.
2. Such constant shifts to the Hamiltonians do not change physical
observables’ qualitative behaviors.

Numerical errors may still occur when Hamiltonians have large positive
eigenvalues. Thus, we aim to test the numerical stability of the results shown
in Fig. 3. To this end, we investigate whether thermodynamic quantities, as
well as the chiral condensate, behave smoothly as functions of $\beta$.
Figures 4 and 5 display these quantities. They exhibit smooth behaviors as
$\beta$ varies, indicating the absence of instabilities. Evidence of the phase
transition becomes sharper at smaller temperatures (larger $\beta$ values).
The main-text plots correspond to $\beta=10$.

Figure 4: Convergence of thermodynamic quantities with increasing $\beta$ for
the quench of $\mu_{\rm i}=0$ to $\mu_{\rm f}>0$ as in the main text.  Figure
5: Convergence of $\Sigma$ with increasing $\beta$ for the quench of $\mu_{\rm
i}=0$ to $\mu_{\rm f}>0$, as in the main text.

##  Appendix B Supplemental Note II:
Weak-coupling quantum thermodynamics applied to
the lattice-gauge-theory example

Strong-coupling quantum thermodynamics becomes relevant when the system-
reservoir interaction contributes non-negligibly to the system’s internal
energy. The penalty terms in LGTs, introduced in Eq. 12 to impose Gauss’s
laws, act at all sites. Therefore, the boundary penalty terms’ contribution
may not appear significant, compared to the contribution of the terms acting
on the system’s bulk, particularly if the number of system DOFs exceeds the
number of DOFs at the boundary. A deeper perspective, however, is that local
constraints imposed at all sites lead to highly nonlocal structure in the
states. Hence, if the reservoir’s size is comparable to or larger than the
system’s, the system’s internal energy cannot be defined separately from the
reservoir.

We now confirm that the example in the main text is in the strong-coupling
regime. The example is a $Z_{2}$ LGT coupled to spin-$\tfrac{1}{2}$ bosonic
matter in (1+1)D with $N=6$ sites in $S\cup R$ and $N_{S}=2$ sites in $S$. We
evaluate $|\langle V_{S\cup R}\rangle/\langle H_{S}(t)\rangle|$ in our quench
protocol’s initial and final (equilibrium) states. In both cases, the ratio is
non-negligible ($\gtrsim 0.1$) at all the $\mu_{f}$ values used in our
numerical study.

Which values do thermodynamic quantities assume in this LGT if one
nevertheless applies weak-coupling thermodynamics? We plot in Fig. 6 the same
quantities as in Fig. 3 of the main text, but computed with weak-coupling
quantum-thermodynamics definitions: the system’s internal energy is equated
with the expectation value of its Hamiltonian, $H_{S}$, as opposed to the
Hamiltonian of mean force, $H_{S}^{*}$. Furthermore, the system states
$\rho_{S}=e^{-\beta H_{S}^{*}}/{\text{Tr}}(e^{-\beta H_{S}^{*}})$ become
$\rho_{S}\approx e^{-\beta H_{S}}/{\text{Tr}}(e^{-\beta H_{S}})$. This
approximation affects computations of not only expectation values, but also
equilibrium properties such as free energy and entropy.

Figure 6 shows that thermodynamic quantities’ values differ from their strong-
coupling counterparts. The weak-coupling quantities’ behaviors near the
expected phase transition are rather smooth, exhibiting diminished abilities
to signal the transition. The chiral condensate (dashed green) is calculated
from the system’s true reduced state, as in the main text.

In summary, our observations motivate the use of the strong-coupling framework
to describe the open-quantum-system thermodynamics of LGTs.

Figure 6: Computing thermodynamic quantities using weak-coupling assumptions:
For illustrative purposes, we make the approximations (argued to be incorrect
here) $e^{-\beta H_{S}^{*}(t)}/{\text{Tr}}(e^{-\beta H_{S}^{*}(t)})\approx
e^{-\beta H_{S}(t)}/{\text{Tr}}(e^{-\beta H_{S}(t)})$ and
$U_{S}(t)\approx\langle H_{S}(t)\rangle$. Thermodynamic quantities are plotted
as functions of the final chemical potential, $\mu_{\text{f}}$, during the
instantaneous quench from $\mu_{\text{i}}=0$: (a) Dissipated work
$W_{\rm{diss}}$ (solid, black) and chiral condensate $\Sigma$ (dot-dashed,
teal). (b) Work $W$ (solid, black) and change $\Delta F_{S}$ in the system’s
free energy (dotted, black). (c) Normalized heat $\beta Q$ (solid, black) and
change $\Delta\mathcal{S}$ in the system’s entropy (dotted, black). In all
subfigures, we plot $\mu_{\text{f}}^{\rm c}$, the $\mu_{\text{f}}$ value where
$\Sigma^{\prime}$ is maximized (dashed, gray). For these plots, $N=6$,
$N_{S}=4$, $t=-1/2$, $\epsilon=1/2$, $m=1/2$, $\mu_{\text{i}}=0$, and
$\beta=10$, as in Fig. 3.


# RECENT RESULTS FROM T2K

L. BERNS for the T2K collaboration  Department of Physics, Tokyo Institute of
Technology,
2–12–1 Ookayama, Meguro-ku, 152–8551 Tokyo, Japan

###### Abstract

We report on latest results from the Tokai-to-Kamioka experiment studying the
disappearance of $\nu_{\mu}$ and appearance of $\nu_{e}$ from neutrino
oscillation with accelerator generated neutrinos. Thanks to increase in
statistics and numerous upgrades to each part of the oscillation analysis, we
set the most sensitive constraints on some of the oscillation parameters,
including in particular the CP-violation angle $\delta_{\mathrm{CP}}$. Future
prospects are also briefly discussed.

##  1 Introduction

The Tokai-to-Kamioka (T2K) experiment [1] is a long baseline neutrino
experiment in Japan. A powerful beam of neutrinos produced at the J-PARC
accelerator in Tokai is sent to the Super-Kamiokande (SuperK) detector in
Kamioka $295\leavevmode\nobreak\ \mathrm{km}$ away. Because for neutrinos the
interaction basis is different from the Hamiltonian basis, the neutrino flavor
oscillates over propagation distance $L$ with frequency controlled by squared
mass splittings over energy $\Delta m^{2}/E$, and amplitude by the mixing
matrix $U$ parametrized by $\theta_{12},\theta_{13},\theta_{23},$ and
$\delta_{\mathrm{CP}}$. One of the interesting open questions in high energy
physics is the possibility of CP violation in leptonic sector via non-zero
value of $\sin\delta_{\mathrm{CP}}$. In T2K, a beam composed mostly of
$\nu_{\mu}$ (neutrino mode) or $\overline{\nu}_{\mu}$ (antineutrino mode) is
selectively generated, and $\sin\delta_{\mathrm{CP}}$ is then measured by
comparing the difference of the $P(\nu_{\mu}\to\nu_{e})$ and
$P(\overline{\nu}_{\mu}\to\overline{\nu}_{e})$ appearance probabilities. Other
questions include the sign of $\Delta m^{2}_{32}$, referred to as normal
($\Delta m^{2}_{32}>0$, NO) or inverted ordering ($\Delta m^{2}_{32}<0$, IO),
and whether $\theta_{23}$ is maximal ($=\pi/4$), or in the upper or lower
octant. These measurements have furthermore important consequences in the
context of baryon asymmetry of the universe,[2] neutrinoless double beta decay
searches,[3] and flavor symmetries.[4]

##  2 The experiment

To generate the neutrino beam, a $30\,\mathrm{GeV}$ proton beam is smashed
into a $90\,\mathrm{cm}$ graphite target, which produces by hadronic
interactions about 6 charged $\pi$ and $K$ mesons per proton. These are then
focused by means of an optical system with three magnetic horns into a
$96\,\mathrm{m}$ long decay volume, where they subsequently decay in flight to
produce neutrinos. By choosing the polarity of the horns, either positive
$\pi^{+},K^{+}$ or negative $\pi^{-},K^{-}$ particles are selected, which
mostly produce $\nu_{\mu}$ or $\overline{\nu}_{\mu}$, respectively. The far
detector is intentionally placed $2.5\,\mathrm{degrees}$ off the neutrino beam
axis to obtain a narrowband beam peaked at the $650\,\mathrm{MeV}$ oscillation
maximum.

A complex of multiple near detectors measures the neutrinos before
oscillation. INGRID is an iron-scintillator sandwich detector that monitors
the beam intensity and direction on-axis. ND280 precisely measures the
neutrino flux and interactions at the same $2.5{}^{\circ}$ off-axis angle as
SuperK. It is a composite detector with active scintillator and passive water
targets, with tracking in time projection chambers, and placed inside a magnet
for charge and momentum measurement. A recent addition is the WAGASCI+BabyMIND
[5] detector system placed at an intermediate $1.5{}^{\circ}$ off-axis angle.
While not used for the oscillation analysis presented here, first neutrino
cross-section measurements have been published recently.[6]

The far detector, SuperK,[7] is a $50\,\mathrm{kt}$ water Cherenkov detector.
The detector walls are lined with 11,146 photo-multiplier tubes (PMTs) that
detect the Cherenkov light emitted from charged particles created in the
neutrino interactions on water. It has very good $e/\mu$ PID from sharpness of
the Cherenkov rings (mis-ID rate about 0.1%), an ideal feature for a neutrino
oscillation experiment measuring the appearance rate of $\nu_{e}$ in a
$\nu_{\mu}$ beam. In T2K the incoming neutrino direction is known, such that
by assuming quasi-elastic scattering the neutrino energy can be reconstructed
from the charged lepton direction and momentum. Not being magnetized, the
$\nu/\overline{\nu}$ separation relies on the beam composition, which is
mostly $\nu_{\mu}$ or $\overline{\nu}_{\mu}$ in each operation mode. The
wrong-sign background is constrained by ND280. Recently, Gadolinium loading
into SuperK has started,[8] which will allow for some $\nu/\overline{\nu}$
separation by tagging recoil neutrons captured on Gadolinium, most interesting
for the search of supernova relic neutrinos.

##  3 Oscillation analysis flow

###  3.1 Dataset

In early 2020, we published constraints on $\delta_{\mathrm{CP}}$ that
excluded a wide range of its values at $3\sigma$ confidence level [9] for the
first time. The results presented in this work (also shown at Neutrino 2020
[10]) include additional data collected between 2019 and early 2020,
corresponding to a 33% increase in neutrino mode data. Stable operation at
$515\,\mathrm{kW}$ beam power was achieved in this most recent run, which is
the highest so far. The total amount of analyzed far detector data corresponds
to $3.6\times 10^{21}$ protons on target (POT), with about 6:5 ratio of $\nu$
: $\overline{\nu}$-mode POT.

###  3.2 Flux prediction

Various information on primary protons from beam monitors is combined with
dedicated hadron production data to predict the neutrino flux and
uncertainties. A significant reduction of the hadron production uncertainty
has been achieved in this analysis thanks to a change of the tuning method.
Instead of tuning every single interaction with thin target measurements, we
now use measurements taken with a replica of the T2K target [11] and apply a
single tuning weight based on the exiting particle’s kinematics. Further
reduction of the uncertainties are expected for the next analysis.

###  3.3 ND280 constraints

ND280 measurements are used to constrain the unoscillated flux $\times$ cross-
section. The interaction model has been updated significantly in this
analysis, including the initial state nuclear model for the dominant charged
current quasi-elastic (CCQE) scattering process, the treatment of removal
energy, a new uncertainty model on the energy dependence of CC multi-nucleon
knockout (2p2h) process, final state interaction constraints from the near
detector, and improvements to deep inelastic scattering at higher energies.

To constrain this model, a total of 18 near detector samples are used,
separated by neutrino/antineutrino mode, $\pi$ multiplicity (0, 1,
${\geq}\,2$) for interaction mode separation, lepton charge for constraining
wrong-sign background (in antineutrino mode only), and interaction target for
constraining nuclear effects (C from active scintillator, or C+O from active
scintillator with inactive water layers). The amount of ND280 data included in
the analysis has been doubled to $1.15\times 10^{21}$ ($0.834\times 10^{21}$)
POT in neutrino (antineutrino) mode. Sample selections and systematics
treatment have also been improved. Two methods are used to propagate the near
detector constraint to the far detector. One is to fit just the near detector
samples and propagate a covariance matrix of correlated flux and cross section
parameters. The other is to perform a joint fit of the near and far detector
samples and thus avoid the gaussian approximation. Both methods give
consistent results.

Aside from the oscillation analysis, many cross-section measurements have been
published, including measurements of pionless $\nu_{\mu}$ charged-current
interactions on C vs. O targets,[12] $\nu$ vs. $\overline{\nu}$
differences,[13] $\nu_{e}/\overline{\nu}_{e}$ cross section [14] (the first
measurement of $\overline{\nu}_{e}$ in 43 years), as well as single-pion
interactions including proton tracks [15] for understanding nuclear effects.

Figure 1: _Left:_ Constraints on atmospheric oscillation parameters and
comparison with NO$\nu$A,[17] SuperK,[18] and IceCube.[19] _Right:_
Constraints from an alternative fit of $\mu$-like samples where the
atmospheric parameters for neutrinos $(\sin^{2}\theta_{23},\Delta m^{2}_{32})$
and antineutrinos
$(\sin^{2}\overline{\theta}_{23},\Delta\overline{m}^{2}_{32})$ are decoupled
from each other. The joint $\nu_{e}+\nu_{\mu}$ analysis contours from the left
figure are also shown for comparison.

###  3.4 Far detector samples

We use five single-ring samples separated by the beam mode and whether one
sees an $e$-like or $\mu$-like ring. For neutrino mode, we also have a sample
with single $e$-like ring and charged pion below Cherenkov threshold (detected
as delayed event from decay electron) for increasing the $e$-like statistics
and as control for CC1$\pi$ interactions. For this analysis, we extend the
analysis binning of $\mu$-like samples by the muon scattering angle, which
improves the separation of neutral current interaction backgrounds. All data
have been reprocessed for improved PMT gain correction. In the upcoming
analysis we will further add multi-ring samples, which will provide up to 40%
increase of the $\mu$-like events (mostly at energies higher than oscillation
maximum).

For $\mu$-like events, the disappearance rate at oscillation maximum is
sensitive to $\sin^{2}2\theta_{23}$, whereas the energy at the dip is
sensitive to $\Delta m^{2}_{32}$. For $e$-like samples, a change of event rate
correlated in $\nu/\overline{\nu}$-mode is sensitive to $\theta_{13}$ and
$\theta_{23}$-octant, whereas a change anti-correlated in
$\nu/\overline{\nu}$-mode is sensitive to $\sin\delta_{\mathrm{CP}}$ and mass
ordering. T2K’s $\theta_{13}$-constraint from $\nu_{\mu}\to\nu_{e}$
appearance, which is $\sin^{2}\theta_{13}=2.63_{-0.42}^{+0.52}\times 10^{-2}$
(NO) in this analysis, is consistent with the much stronger constraint from
reactor experiments measuring $\overline{\nu}_{e}\to\overline{\nu}_{e}$
disappearance. All following results therefore report T2K+reactor constraints
on the other oscillation parameters $(\delta_{\mathrm{CP}},\theta_{23},\Delta
m^{2}_{32})$, which improves the sensitivity especially for
$\delta_{\mathrm{CP}}$ and $\theta_{23}$-octant. This reactor-constraint on
$\theta_{13}$ has been updated since the last analysis to
$\sin^{2}\theta_{13}=(2.18\pm 0.07)\times 10^{-2}$ from Ref. [16].

##  4 Results

Fig. 1 (left) shows our latest constraints on the atmospheric mixing
parameters $(\theta_{23},\Delta m^{2}_{32})$. Improving upon our previous ones
with increased statistics and systematics model, these are the tightest
constraints on these parameters to date. We observe a slight preference of
non-maximal $\sin^{2}\theta_{23}\neq 0.5$ in upper octant with Bayes factor
$P_{\mathrm{upper}}/P_{\mathrm{lower}}=3.4$. A fit with separate atmospheric
mixing parameters for $\nu_{\mu}/\overline{\nu}_{\mu}$ (Fig. 1 right) shows
consistent results as expected from CPT and standard neutrino interactions.

Figure 2: _Left:_ The ellipses show expected event rates for each value of
$\delta_{\mathrm{CP}}$, with data-point close to maximally CP-violating value
of $-\pi/2$. The solid (dashed) ellipses show the prediction for normal
(inverted) ordering, with our data showing weak preference for normal
ordering. The colors show the prediction for various values of
$\sin^{2}\theta_{23}$, with data showing a weak preference for upper octant.
_Right:_ $\Delta\chi^{2}$ distribution for $\delta_{\mathrm{CP}}$ in each mass
ordering. The colored regions show the confidence intervals constructed with
Feldman-Cousins method.[20]

For $\nu_{e}$ vs. $\overline{\nu}_{e}$ appearance the constraint is still
rate-dominated, such that a bi-event plot is illustrative to understand the
origin of various constraints (Fig. 2 left). Upon statistical analysis, a
large region of the $\delta_{\mathrm{CP}}$ parameter space is excluded at
$3\sigma$ (Fig. 2 right), and CP-conserving values
$\delta_{\mathrm{CP}}=0,\pi$ are excluded at 90% confidence level. To check
the effects of possible biases caused by cross section model choices, we
repeat the analysis with various theory or data-driven simulated data sets.
These cause a shift of the left (right) 90% confidence interval edges on
$\delta_{\mathrm{CP}}$ by at most $0.073\,\mathrm{rad}$
($0.080\,\mathrm{rad}$). A weak preference of normal ordering is also observed
with Bayes factor $P_{\mathrm{NH}}/P_{\mathrm{IH}}=4.2$. Comparing with toy
experiments generated at maximally CP-violating value of
$\delta_{\mathrm{CP}}=-\pi/2$, our data-constraints are compatible with
sensitivity at around $1\sigma$ level. In the previous analysis, the agreement
used to be slightly below $2\sigma$ range, and was yielding by statistical
fluctuation a stronger constraint than sensitivity. The new data-constraint is
now _weaker_ due to the improvements to the analysis, including the updated
interaction model and reactor-constraint on $\theta_{13}$, reprocessing of the
far detector data, and with largest contribution the additional data. The
sensitivity however increased, resulting in improved compatibility.

##  5 Conclusion and future outlook

We presented latest neutrino oscillation results from T2K using $3.6\times
10^{21}$ protons on target with many improvements at each level of the
analysis. CP conserving values of $\delta_{\mathrm{CP}}$ are excluded at 90%
and a large $\delta_{\mathrm{CP}}$ range is excluded at $3\sigma$ confidence
level. A weak preference for normal ordering and upper octant is seen.
Additional samples are being prepared both at near and far detectors for more
statistics and systematics control. Joint fits between NO$\nu$A+T2K and SK+T2K
collaborations are also ongoing, with the aim to obtain improved oscillation
parameter constraints due to resolved degeneracies, and to understand
potentially non-trivial systematic correlations. Many upgrades are being
performed toward the future, including a large upgrade of the ND280 near
detector system for improved cross section model constraints,[21] and
$2.6\times$ stronger beam for significant increases in statistics.[22]

## References

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