# cQASM v1.0
Towards a Common Quantum Assembly Language

N. Khammassi1, G.G. Guerreschi2, I. Ashraf1, J. W. Hogaboam2,
C. G. Almudever1, K. Bertels1

1 QuTech, Delft University of Technology, The Netherlands
2 Intel Labs, USA

###### Abstract

The quantum assembly language (QASM) is a popular intermediate representation
used in many quantum compilation and simulation tools to describe quantum
circuits. Currently, multiple different dialects of QASM are used in different
quantum computing tools. This makes the interaction between those tools
tedious and time-consuming due to the need for translators between theses
different syntaxes. Beside requiring a multitude of translators, the
translation process exposes the constant risk of loosing information due to
the potential incompatibilities between the different dialects. Moreover,
several tools introduce details of specific target hardware or qubit
technologies within the QASM syntax and prevent porting the code to other
hardwares. In this paper, we propose a common QASM syntax definition, named
cQASM, which aims to abstract away qubit technology details and guarantee the
interoperability between all the quantum compilation and simulation tools
supporting this standard. Our vision is to enable an extensive quantum
computing toolbox shared by all the quantum computing community.

##  I Introduction

Building quantum computers requires implementing multiple functional layers.
At the most abstract level, algorithm designers formulate quantum algorithms
in a high-level language that requires one or more compilation steps to
translate the algorithm description into a set of instructions that can be
executed by quantum hardware. Compilers can internally use different
intermediate representations to, for example, perform optimizations,
instruction scheduling or qubits mapping, but it is desirable, for portability
and flexibility, that the outcome of the compilation is a hardware-independent
quantum assembly code (QASM). An additional translation process, possibly
including further compilation and optimization steps, is then used to generate
the hardware-specific micro-code. The micro-code can then be executed on the
target micro-architecture which provides the hardware-based control logic
needed to execute the instructions on the target quantum processor.

The quantum assembly language (QASM) is a hardware-independent instruction set
which aims to provide a compact and expressive intermediate representation to
describe quantum circuits. This representation is intended to be used not only
by quantum compilers, but also quantum computer simulators, reversible circuit
synthesis frameworks or microcode synthesis backends and other tools that
needs to represent quantum circuits in a straightforward way. Algorithm
designers can also benefit from a comprehensive QASM during the design and
test of relatively small quantum circuits, algorithms and protocols.

QASM first appeared in 2005 in a set of software tools from MIT [1] and in
[2]. Since then, many different custom QASM dialects have been defined and
used by various tools either as input or as an intermediate quantum circuit
representation. This makes the interaction between the different tools
difficult and time-consuming since it requires translations between different
dialects with the constant risk of loosing information in the translation
process when two dialects do not offer the same features or have different
focus. With this paper, we propose the first version of a common QASM language
(cQASM) to help bringing the quantum computing community together and trigger
the effort of a standardized QASM language. Our goal is to provide a common
ground for all the tools that have been or will be developed to compile,
simulate and ultimately execute quantum circuits.

Figure 1: Compilation of Quantum Algorithms : an algorithm written in high-
level language can be optimized and compiled into a common technology-
independent cQASM code then into hardware specific executable QASM (eQASM)
code.

In the definition of the cQASM syntax, we took particular care to include the
user experiences and expectations from different parts of the quantum
community. We discussed and included ways to describe the recent variational
quantum algorithms and sampling tasks proposed for pre-error corrected
devices, while we also defined compact commands to express active feedforward
and error-correction related instructions. We expect that novel algorithmic
solutions and the continuous progress in hardware architectures will continue
to drive the development and adoption of new features.

Defining a common QASM guarantees the compatibility of the different tools and
offers access to a large toolbox that includes tools for quantum compilation
and simulation for the quantum computing community.

The work is organized as follows: in the next section we give a brief overview
of some QASM variants discussed in the relevant literature of the past few
years, while in section III we present the cQASM language. Its syntax
definition is organized by topic according to:

  * •

Qubit Definition and Addressing

  * •

Measurement Register

  * •

Quantum Gates

  * •

Quantum Circuit Definition

In section IV, we introduce special features that are optional extensions of
the cQASM syntax addressing the requirements of specialized, but important,
quantum computing tools or algorithms. Finally, Section V and IV give an
overview of future extensions and draw our conclusions.

##  II Related Works

Over the last few years, several QASM variants have been defined and used in
various quantum computing tools. The initial version of QASM was first
introduced by Nielsen and Chuang to make the required illustrations for their
book [3] and by Andrew Cross through a set of tools in [1]. QASM is now being
used in different tools as the underlying quantum assembly language layer that
translates the high level quantum algorithms in low level quantum operations
which can then be executed either on quantum simulation platforms such as QX
[4] and QHipster [5] or on actual quantum processors such as demonstrated by
QuTech and Intel on Superconducting and Si-Spin qubits [6] or by IBM on the
quantum experience platform [7]. Different QASM dialects has been used by the
different compilation and simulation tools such as the QASM defined by [4] as
input for QX or the OpenQASM defined by [8].

As visually illustrated in Fig. 1, we define cQASM as a hardware-independent
language that is produced by quantum compilers as output of the translation
process from a code written in high-level language into a quantum assembly
code. Examples of compilation tools are ScaffCC [9], ProjectQ [10],
$LIQUi\ket{}$ [11] and OpenQL [12]. These compilers can include a back-end
pass to generate the technology-independent cQASM.

The cQASM instructions can then be used as input to a quantum computer
simulator or as input to a lower-level compiler which generates hardware-
specific instructions, or executable QASM (eQASM), suitable for execution by
the target quantum processor. As a concrete example, the OpenQL compiler
developed at QuTech supports multiple backends and generates different types
of eQASM, such as the QuMis code introduced in [13] and executable on a
dedicated microarchitecture controlling superconducting qubits.

##  III QASM Definition

###  III-A Case sensitivity and Comments

The cQASM syntax is not case-sensitive, i.e. upper case letters are equivalent
to lower case ones. To make the code more readable, the quantum programmer can
add comments in his code. Comments start with “#” and can be added either in a
separate line or at the end of a line containing code as shown in Code Example
1.

⬇

1 version 1.0

2 # define a quantum register of 2 qubits

3 qubits 2

4

5 # create a Bell pair via a Hadamard rotation

6 h q[0]

7 # followed by a CNOT gate

8 # q[0]: control qubit, q[1]: target qubit

9 cnot q[0],q[1]

10

11 # measure both qubits to test correlations

12 measure q[0]

13 measure q[1]

Code Example 1: Creation of Bell state.

###  III-B Qubit Definition

####  III-B1 Qubit Register

Qubits can be grouped into a simple quantum register and addressed by their
index. A qubits register can be created by specifying the number of qubits as
shown in Code Example 1 at Line 2.

####  III-B2 Qubit Addressing

Once the number of qubits is defined, the qubits can be addressed individually
through its default identifier “q[i]” where “i” is the identifier of the
target qubit: if the quantum register contains N qubits, then
$i\in\\{0,\dots,N-1\\}$ and the qubit identifiers are q[0], q[1], …, q[$N-1$].
As an example, observe line 5 of Code Example 1 where the Hadamard gate is
applied to qubit 0.

####  III-B3 Measurement Register

By default, a measurement register (binary register) is associated to the
quantum register. It is mainly used to store the outcome of measurements:
after measuring a qubit q[0], the result of the measurement is automatically
stored into bit b[0]. The outcome b[0] can be the final result of some quantum
computation or can be used to apply binary-controlled gates to other qubits.

####  III-B4 Naming Qubits and Measurement Outcomes

In order to give a meaningful name to each qubit and make the quantum program
more readable, it is possible to name qubits using the keyword “map”. The
instruction ‘map” renames a single qubit according to its two arguments: the
first is the current qubit identifier, the second is the additional name. The
syntax is presented at line 5 of Code Example 2 in which qubit q[0] is named
“data” according to the role played in the algorithm. Once the qubit has been
renamed, “data” is equivalent to “q[0]” and can be used interchangeably.

⬇

1 version 1.0

2 # define a quantum register of 3 qubits

3 qubits 3

4

5 # rename qubits

6 map q[0],data

7 map q[1],ancilla

8 map q[2],output

9

10 # address qubits via their names

11 prep_z data

12 prep_z ancilla

13 prep_z output

14 cnot data,ancilla

15 cnot data,extra

16

17 # rename classical bit

18 map b[1],error_syndrome

19 measure ancilla

20

21 #apply binary controlled Pauli-X gate

22 c-x error_syndrome,q[2]

Code Example 2: Renaming qubits and measurement outcomes and using binary
controlled operations.

Similarly to the qubits, the measurement bits can be renamed using the “map”
instruction to improve code readability. In Code Example 2, qubit q[1] is
renamed “ancilla”. When q[1] is measured, the outcome is stored by default in
b[1]. Such bit was previously renamed as “error_syndrome” (at line 6). Its
value can be used to (classically) control a Pauli-X gate which we apply to
q[2] at the end of the circuit.

###  III-C Quantum Gates

The QASM syntax supports an overcomplete, universal set of quantum gates which
includes single-, two- and three-qubits gates, as listed in TABLE I. It
provides support to commonly adopted controlled gates such as CNOT and Toffoli
gates. The syntax is common to all gates: the QASM instruction is followed by
a space and a list of arguments separated by commas. The first arguments are
the identifiers of the qubits involved in the gate (one identifier for single-
qubit gates, two for two-qubit gates, …). Certain gates require additional
parameters: for example, single-qubit arbitrary rotations need to specify the
value in radiant of the rotation angle.

The set of gates is extended by allowing each gate to be binary-controlled. By
prepending “c-” to the gate name, the quantum operation is executed if and
only if a specific measurement outcome is equal to 1. In this case, the first
parameter after the instruction name will not be the qubit identifier, but the
identifier of the classical bit. Any gate defined in TABLE I can be extended
in the way just described.

TABLE I: Supported Quantum Gates Quantum Gate |  Description |  Example
---|---|---
I |  Identity |  i q[2]
H |  Hadamard |  h q[0]
X |  Pauli-X |  x q[1]
Y |  Pauli-Y |  y q[3]
Z |  Pauli-Z |  z q[7]
Rx |  Arbitrary x-rotation |  rx q[0], 3.14
Ry |  Arbitrary y-rotation |  ry q[3], 3.14
Rz |  Arbitrary z-rotation |  rz q[2], 0.71
X90 |  X90 |  x90 q[1]
Y90 |  Y90 |  y90 q[0]
mX90 |  mX90 |  mx90 q[1]
mY90 |  mY90 |  my90 q[0]
S |  Phase |  s q[6]
Sdag |  Phase dagger |  sdag q[6]
T |  T |  t q[1]
Tdag |  T dagger |  tdag q[3]
CNOT |  CNOT |  cnot q[0],q[1]
Toffoli |  Toffoli |  toffoli q[3],q[5],q[7]
CZ |  CPHASE |  cz q[1],q[2]
SWAP |  Swap |  swap q[0],q[3]
CRK |  Controlled Phase Shift ($\pi/2^{k}$ ) |  crk q[0],q[1],k
CR |  Controlled Phase Shift (arbitrary angle) |  cr q[0],q[1],angle
c-X |  Binary-Controlled X |  c-x b[0],q[2]
c-Z |  Binary-Controlled Z |  c-z b[1],q[2]
TABLE II: Supported State Preparation and Measurement Instruction |  Description |  Example
---|---|---
prep_x |  State preparation in x basis |  prep_x q[0]
prep_y |  State preparation in y basis |  prep_y q[1]
prep_z |  State preparation in z basis |  prep_z q[1]
measure_x |  Measurement in x basis |  measure_x q[0]
measure_y |  Measurement in y basis |  measure_y q[1]
measure_z , measure |  Measurement in z basis |  measure q[2]
measure_all |  Measurement of all qubits in z basis |  measure_all
measure_parity |  Measurement of the product of Pauli matrices |  measure_parity q[0],x,q[2],z

###  III-D Measurements

####  III-D1 Partial Measurement (Single Qubit)

As shown in the previous examples, each qubit can be measured in the z-basis
individually using the keyword “measure” followed by the target qubit, for
instance “measure q[0]”. By default, the qubit is measured in the z-basis, the
outcome is binary (either +1 or -1) and the probability of each of the two
possibility is related to the probability of qubit q[0] being in state
$\ket{0}$ or $\ket{1}$ respectively. At the end of the measurement, the state
of qubit q[0] collapses into the corresponding z-basis state. Single qubit
measurement in all three basis (x-, y-, z-basis) are allowed by using the
dedicated instruction measure_x, measure_y or measure_z.

####  III-D2 Register Measurement (All Qubits)

Alternatively, one can measure the entire quantum register at once using same
keyword measure_all without specifying any target qubit.

###  III-E Feedback Support : Binary-Controlled Quantum Gates

Binary-controlled gates are quantum gates which are controlled by measurement
outcomes. The programmer can use a binary measurement outcome to control a
quantum operation. The latter will be executed only if that binary value is 1.
In the following example we put the state of the first qubit q[0] into
superposition, then we measure it and use the measurement outcome b[0] to
conditionally apply the Pauli-X gate on qubit q[1].

Multiple measurement outcomes can be used to control a quantum operation, in
this case all the control bits are placed before the qubits used in that
quantum operation.

⬇

1 h q[0]

2 measure_z q[0] # measurement outcome in b0

3 # simple binary-contolled gate

4 c-x b[0],q[1] # apply Pauli-X to q[1] if b[0]=1

5 measure_z q[2]

6 measure_z q[3]

7 measure_z q[4]

8 # multi-binary controlled gate

9 c-x b[2],b[3],b[4],q[5] # apply pauli-x to q4 if b2=1 and b3=1 and b4=1

10 # binary controlled gate using an arbitrary mask :

11 # we want to apply a Pauli-X to q[4] if b[0]=0 and b[1]=1

12 not b[0] # negate b0

13 c-x b[0],b[1],q[4] # multi-bits controlled X gate

14 not b[0] # restore the measurement register

Code Example 3: Multi-binary-controlled quantum gate.

Sometimes, the programmer might need to use an arbitrary binary mask where
some measurement outcomes are ones and others are zeros. In this case the
programmer can use the not classical operation to invert a bit before using it
to control an operation.

###  III-F Display Results

In section III-D we introduced the description of the measurement operations.
After a measurement is performed, the outcome must be accessible to the user.
This feature is implemented through the command “display b[i]” that returns
the outcome value of the latest measurement involving the i-th qubit.

In addition, when the QASM code describes a simulation and not an actual
experiment, more information about the quantum state may be readily accessible
depending on the simulation tools. For example, simulators that represent
quantum states as dense vectors store all the quantum amplitudes at each step
of the algorithm. In this situation, the command display can be used to
inspect the quantum state by printing all computational-basis amplitudes to
the screen.

Later, Section IV will introduce special features like the “measurement
averaging”. The command “display” is compatible with these extensions and
should print the average measurement outcome of a qubit (a double precision
float in [0..1]) along with the overall number of measurements (integer), the
number of +1 and -1 measurements. The later information allows for more
flexibility when post-processing the results.

##  IV Special Features

###  IV-A Parity Measurements (Pauli String Observables)

Measurements do not always involve a single qubit at a time, in fact several
important algorithms require multi-qubit measurements. This situation is
fundamentally different from cases in which each qubit is measured
independently. Here, we consider observables that are constituted by products
of single-qubit Pauli matrices on a subset of qubits. In general, any
observable can be described as a Pauli string after appropriate extra gates
are applied.

When the measurement takes place at the end of the algorithm, as is the case
for the important class of variational algorithms (see Code Example 7), two
options are available. The simplest strategy is to separately measure all
qubits involved in the Pauli string and then multiply the measurement
outcomes.

However, this strategy is not viable when the measurement must not extract
more information then the single outcome bit. This is the case when parity
measurements are performed during error correction codes or to extract
information about stabilizer states. The keyword “measure_parity” is
introduced to describe the latter scenario. It is followed by an even number
of arguments: each pair of arguments are constituted by the qubit identifier
and the specification of the corresponding measurement axis (chosen among X,
Y, and Z axis). The number of qubits involved is automatically defined by
(half) the number of arguments following the instruction. The outcome is a
single bit that will be copied in all bit registers associated with the qubits
involved in the measurement. See Code Example 4 for the explicit use.

⬇

1 version 1.0

2 # define a quantum register of 3 qubits

3 qubits 4

4

5 # apply a short sequence of gates

6 h q[0]

7 h q[1]

8 h q[2]

9 cnot q[2],q[3]

10

11 # measure the Pauli string Z0Z2

12 measure_parity q[0],z,q[2],z

13 # the outcome is stored in both b[0] and b[2]

14

15 # measure the Pauli string X1Y3

16 measure_parity q[1],x,q[3],y

17 # the outcome is stored in both b[1] and b[3]

Code Example 4: Parity-like multi-qubit measurements.

###  IV-B Demolition Measurement

In certain hardware architectures, measuring one or more qubits not only
collapses their state, but also removes the qubits from the register.
Consider, for example, quantum linear optics setup with dual rail encoding:
the qubit state is determined by a single photon being in one of two modes and
the photon detection removes it from both modes leaving them empty. While it
is important to properly describe this feature at the level of eQASM, for the
scope of the current instruction set it is sufficient to assume that an extra
qubit is added to substitute the demolished one. It is implicit in the
instruction set that this extra qubit is renamed accordingly to the demolished
one and that its state corresponds to the usual result of projective
measurements.

###  IV-C Parallel Quantum Gates

A set of quantum gates can be scheduled to start in parallel using the syntax
shown in Code Example 5. Gates between brackets and separated by “$\mid$” are
scheduled to be executed in parallel. The brackets allow the expression of
parallelism of a set of gates over multiple lines and avoid having verbose
single lines. For instance, prep_z gates on q[0], q[1] and q[2] in Code
Example 5 are parallel gates. Similarly, the measurements are scheduled to be
executed simultaneously.

⬇

1 { prep_z q[0] | prep_z q[1] | prep_z q[2] }

2 h q[0, 1, 2]

3 h q[0:2]

4 cnot q[0], q[3]

5 cnot q[1], q[3]

6 { measure q[0] | measure q[1] | measure q[2] }

7 c-x b[0:2],q[0]

Code Example 5: Parallelism Specification.

###  IV-D Single Gate Multiple-Qubits (SGMQ)

In many cases, addressing many qubits at once can be very useful when applying
a single quantum operation to a set of qubits similarly to SIMD (Single
Instruction Multiple Data) fashion in classical computing. The following
notations aim to simplify the addressing of a set of qubits:

  * •

Contiguous range of qubits: a set of qubits within a contiguous range can be
addressed as “q[i:j]”, in this case the qubits {i,i+1,$\dots$,j} are included
in the qubit set.

  * •

Arbitrary set of qubits: alternatively, an arbitrary set of qubits can be
addressed as “q[i,j,k,l]” where i, j and k are arbitrary index within a valid
range of qubits.

  * •

Arbitrary set of qubit ranges: finally the two previous addressing modes can
be combined to match a set of contiguous ranges of qubits, for instance
“q[i:j,k:l,m:n] designates a set of 3 qubit ranges forming
{i,i+1,$\dots$,j}$\cup${k,$\dots$,l}$\cup${m,$\dots$,n}.

In Code Example 5, Line 2, a Hadamard gate is applied simultaneously to qubits
0, 1, 2. In case the indices are contiguous, the list can be abbreviated with
ellipsis, e.g. h q[0:2] such as in Line 3 which is equivalent to line 2.

Similarly to the qubits, multiple measurement outcomes can be addressed
simultaneously using the list notation the same way as the qubits, for
instance, at the end of the previous, the measurement outcomes of qubits 0, 1
and 2 are used to control a Pauli-X quantum operation on a qubit 0. This
addressing mode reduce the verbosity of the code, improve its readability and
offers a compact way to express parallelism.

###  IV-E Sub-circuit Definition

For better readability, a QASM circuit can be split into a set of functional
sub-circuits. The list of QASM instructions forming a sub-circuit is provided,
after the name of the sub-circuit itself, starting with a dot. Code Example 6
partition the circuit in 3 parts: the initialization starting at line 5, the
single iteration of a Grover-step from line 17, and finally the measurements
after line 37.

In addition, the definition of sub-circuits allow the specification of loops
over each sub-circuit as detailed in the next paragraph and already included
in example mentioned above.

⬇

1 version 1.0

2

3 # define a quantum register of 9 qubits

4 qubits 9

5

6 map q[4],oracle

7

8 # sub-circuit for state initialization

9 .init

10 x oracle

11 { h q[0] | h q[1] | h q[2] | h q[3] | h oracle }

12

13 # core step of Grover’s algorithm

14 # loop with 3 iterations

15 .grover(3)

16

17 # search for |x> = |0100>

18

19 # oracle implementation

20 x q[2]

21 toffoli q[0],q[1],q[5]

22 toffoli q[1],q[5],q[6]

23 toffoli q[2],q[6],q[7]

24 toffoli q[3],q[7],q[8]

25 cnot q[8],oracle

26 toffoli q[3],q[7],q[8]

27 toffoli q[2],q[6],q[7]

28 toffoli q[1],q[5],q[6]

29 toffoli q[0],q[1],q[5]

30 x q[2]

31

32 # Grover diffusion operator

33 { h q[0] | h q[1] | h q[2] | h q[3] }

34 { x q[0] | x q[1] | x q[2] | x q[3] }

35 h q[3]

36 toffoli q[0],q[1],q[5]

37 toffoli q[1],q[5],q[6]

38 toffoli q[2],q[6],q[7]

39 cnot q[7],q[3]

40 toffoli q[2],q[6],q[7]

41 toffoli q[1],q[5],q[6]

42 toffoli q[0],q[1],q[5]

43 h q[3]

44 { x q[0] | x q[1] | x q[2] | x q[3] }

45 { h q[0] | h q[1] | h q[2] | h q[3] }

46 display

47

48 # final measurement

49 .measure

50 h oracle

51 measure oracle

52 display

Code Example 6: Grover Algorithm. TABLE III: Supported Code Constructs Statement |  Description |  Example
---|---|---
{gate args $\mid$ gate args} |  Apply set of gates in parallel |  {h q[0] $\mid$ h q[1] }
gate qubitlist |  Single gate multiple qubits operation |  h q[0,2,3]
.function |  Functional sub-circuit |  (see Code Example 6)
.function(iterations) |  Static loops |  (see Code Example 6)

###  IV-F Static Loops

The loop control structure is an essential component of many quantum
algorithms such as the Grover’s algorithm. For now, QASM provides support for
the FOR-loop by simply adding the number of times a sub-circuit needs to be
executed right after the name of the sub-circuit such as at line 10 of the
Grover’s algorithm example given in Figure 10.

###  IV-G Quantum Gate Scheduling

Instruction scheduling in the VLIW fashion requires the specification of
“bundles” which can be expressed using a wait instruction (example: $wait\,5$
specifies that an execution unit should wait 5 time units before executing the
next instruction). Together with the parallelism specification, the wait
instruction provides accurate time specification without changing the
structure or the syntax of the QASM language.

Beside setting the starting time of each instruction in a scheduled QASM code,
the “wait” instruction effectively sets the duration of each instruction. This
possibility allows the user to determine by hand the duration of each kind of
gate and, in simulation mode, allows to tailor the simulation to different
qubit technologies characterized by different gate durations. The gate
duration is a critical information in many cases such as noise simulation
where qubits suffer decoherence over time or quantum computer architecture
simulation where the overall execution time of a quantum program needs to be
calculated.

Should be noted that the simulators or tools which do not support the notion
of the time information can ignore the “wait” instruction, the duration of
each instruction is then a single cycle. The number of cycles is an integer
equal to or greater than 1.

###  IV-H Measurement Averaging

In many quantum algorithms, the measurement outcomes of the qubits are not
directly used as raw binary values, but collected and post-processed. One
common form of post processing consists in computing the average measurement
outcome of a given qubit. For instance, a programmer might need to evaluate
the robustness of an error correction code through characterizing the physical
vs logical error rate, for that the quantum circuit is executed several times
under quantum noise and the average measurements are collected and used to
compute the actual error rate. Another example is the execution of experiments
to observe Rabi oscillations.

The ubiquitous necessity of this feature makes it an attractive candidate for
implementation in simulation tools and in actual hardware. In such case the
measurement outcomes for each qubit are accumulated and the average ground
state measurement is stored and returned when requested. For instance, a
programmer can execute a loop with several thousands of iterations and the
measurement are collected and made available at the end of this loop. The
available information includes the number of +1 and -1 measurements for each
qubit and by extension the average measurement in the ground or excited state.
The display command described in section III-F can display this information at
any point of the quantum circuit.

In order to accommodate such use cases, one syntax element allowing the
resetting of the average measurement for each qubit when needed is required: a
command named reset_averaging can be used to reset the averaging engine
whenever needed. The syntax of the later command is as follows:
reset_averaging q[i] where q[i] is one or more qubits identifier such as
specified in the qubit addressing Section III-B.

###  IV-I Expectation Value of Observables

⬇

1 version 1.0

2 # define a quantum register of 3 qubits

3 qubits 3

4

5 # reset the counters for the average procedure

6 reset_averaging

7

8 # prepare and measure the quantum state 1000 times

9 # to accumulate a large outcome statistics

10 .average(1000)

11 # state preparation

12 prep_z q[0:3]

13 { rx q[0] 3.14 | ry q[1] 0.23 | h q[2] }

14 rx q[2] 3.14

15 cnot q[2],q[1]

16 { z q[1] | rx q[2] 1.57 }

17

18 # measure of $Z_1$

19 measure_z q[1]

20 # the corresponding average is automatically updated

21

22 # measure of $X_0 X_2$

23 measure_parity q[0],x,q[2],x

24 # the corresponding average is automatically updated

25

26 # estimate the observable A

27 .result

28 # show the average of $X_0 X_2$ together with its latest outcome

29 display b[0]

30 # show the average of $Z_1$ together with its latest outcome

31 display b[1]

32

33 # the expectation value of $\hat{A}$ follows

34 # from a straightforward postprocess

Code Example 7: Central routine of a typical variational quantum algorithm.
Here we prepare a certain quantum state and then measure the expectation value
of the observable ${\hat{A}=\alpha Z_{1}+\beta X_{0}X_{2}}$.

Recently, a novel class of quantum algorithms emerged that is particularly
relevant to near term devices due to its intrinsic robustness to systematic
noise. These algorithms are commonly called _variational quantum algorithms_
[14, 15, 16] and alternate short quantum computations with classical post-
process of the outcomes: the basic idea is that a short quantum circuit
suffices to prepare an approximate solution of the problem at hand, but the
specific form of the quantum gates is _a priori_ unclear and must be
determined by try-and-fail approach guided by a classical optimizer. To
implement this kind of hybrid algorithms, one needs to 1) compute expectation
values of several, possibly non-commuting, observables, 2) compute the quality
of the approximate solution, 3) provide this information to a classical
optimizer that suggests how to modify the quantum circuit to improve the
approximate solution, and 4) reprogram the quantum computer before start the
next iteration from point 1).

While the development of instructions for a natural execution of the quantum-
to-classical-to-quantum iteration is outside of the scope of the current work,
we believe that it is important to illustrate how one can perform the steps 1)
and 2) above using the instructions already introduced. Code Example 7 uses
static loops to prepare and measure 1000 times the relevant terms of
observable $\hat{A}$. The average is obtained through the dedicated command
introduced in the previous subsection.

##  V Future Extensions

The proposed QASM syntax specification allows the description of quantum
circuits in a relatively compact way while remaining at the quantum gate level
and abstracting away low-level hardware details. Despite being an assembly
language, several features such as SGMQ instructions allows the expression of
gate-level parallelism in a relatively compact way and allowing simulation
tools and compiler to specify the execution timing and the gate scheduling in
a straightforward way.

Despite the expressiveness that can be provided by the proposed cQASM syntax,
several limitations worth to be mentioned and addressed in future versions:

  * •

Control flow and conditional branching.

  * •

Interleaving classical and quantum instructions: this requires classical
registers and a classical instruction set.

  * •

Sub-circuit reuse: currently, the sub-circuits cannot be recalled many times,
the reuse of a sub-circuit as a recallable function can offer make algorithm
specification even more compact, promote code reuse and save instruction
space.

  * •

Along with classical instructions, a classical memory can offer a storage
space for intermediate classical computations which can be needed in some
quantum algorithms.

##  VI Conclusion

In this paper we proposed the common Quantum Assembly language (cQASM) to pave
the way toward a hardware-agnostic language that is shared by the quantum
computing community and implemented by the many tools required to achieve
large-scale quantum computation. We described the cQASM syntax and the
semantic behind it and we distinguished two main parts of the syntax, one
minimal syntax required to express basic quantum circuits, and a set of
extensions to offer different features such as parallelism expression, loop
constructions or special measurements.

The first version of cQASM allows the intuitive description of quantum
circuits, but its current syntax does not address the interaction between
quantum and classical computation in a completely satisfactory way. These
important features, like branch support, conditional loops or dynamic rotation
angle computation, will be introduced in future version where classical
instructions can be interleaved with quantum ones and interact together. We
hope for a large participation in the definition of the instruction set
regulating the interface between quantum and classical aspects of quantum
algorithms and their execution.

## Acknowledgment

The authors would like to thank Prof. F. Chong, Y. Ding and A. Holmes
(University of Chicago), Dr. A. J. Abhari (IBM), Prof. M. R. Martonosi
(Princeton University) for their contributions and valuable discussions in
defining the syntax of the common quantum assembly language v1.0.

## References

  * [1] A. Cross, “QASM tools,” https://www.media.mit.edu/quanta/quanta-web/projects/qasm-tools/, published: 2005.
  * [2] T. Metodi, D. Thaker, A. Cross, F. Chong, and I. Chuang, “A quantum logic array microarchitecture: Scalable quantum data movement and computation,” in _MICRO_. IEEE Computer Society, 2005, pp. 305–318.
  * [3] M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information: 10th Anniversary Edition_ , 10th ed. New York, NY, USA: Cambridge University Press, 2011.
  * [4] N. Khammassi, I. Ashraf, X. Fu, C. Almudever, and K. Bertels, “Qx: A high-performance quantum computer simulation platform,” _IEEE 2017 Design, Automation & Test in Europe Conference & Exhibition (DATE)_, pp. 464–469, March 2017.
  * [5] M. Smelyanskiy, N. P. D. Sawaya, and A. Aspuru-Guzik, “qhipster: The quantum high performance software testing environment,” _arXiv:1601.07195_ , 2016\. [Online]. Available: http://arxiv.org/abs/1601.07195
  * [6] “Qutech and intel demonstrate full stack implementation of programmable quantum computer prototype,” Hipeac Compilation Architecture Info N 50, April, 2017.
  * [7] IBM, “IBM Quantum Experience,” https://www.research.ibm.com/ibm-q/.
  * [8] A. W. Cross, L. S. Bishop, J. A. Smolin, and J. M. Gambetta, “Open Quantum Assembly Language,” _arXiv:1707.03429_ , 2017. [Online]. Available: https://arxiv.org/abs/1707.03429
  * [9] A. J. Abhari, S. Patil, D. Kudrow, J. Heckey, A. Lvov, F. T. Chong, and M. Martonosi, “Scaffcc: Scalable compilation and analysis of quantum programs,” 2015.
  * [10] D. S. Steiger, T. Häner, and M. Troyer, “Projectq: An open source software framework for quantum computing,” 2016.
  * [11] K. M. S. Dave Wecker, “Liqui$\ket{}$: A software design architecture and domain-specific language for quantum computing,” February 2014.
  * [12] “OpenQL : A Quantum Programming Language for Quantum Accelerators (to appear in Jul. 2018).” 2018.
  * [13] X. Fu, M. A. Rol, C. C. Bultink, J. van Someren, N. Khammassi, I. Ashraf, R. F. L. Vermeulen, J. C. de Sterke, W. J. Vlothuizen, R. N. Schouten, C. G. Almudever, L. DiCarlo, and K. Bertels, “An experimental microarchitecture for a superconducting quantum processor,” 2017.
  * [14] E. Farhi, J. Goldstone, and S. Gutmann, “A quantum approximate optimization algorithm,” _arXiv:1411.4028_ , pp. 1–16, 2014. [Online]. Available: http://arxiv.org/abs/1411.4028
  * [15] A. Peruzzo, J. R. McClean, P. J. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, “A variational eigenvalue solver on a photonic quantum processor,” _Nature Communications_ , vol. 5, no. May, p. 4213, jul 2014. [Online]. Available: http://www.nature.com/doifinder/10.1038/ncomms5213
  * [16] D. Wecker, M. B. Hastings, and M. Troyer, “Towards practical quantum variational algorithms,” 2015.


# Asteroseismology reveals the near-core magnetic field strength in the
early-B star HD 43317

D. M. Bowman1,2 D. Lecoanet3,4,2 T. Van Reeth1,2 1Institute of Astronomy, KU
Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
email<EMAIL_ADDRESS>
2Kavli Institute for Theoretical Physics, University of California,
Santa Barbara, CA 93106, USA
3Department of Engineering Sciences and Applied Mathematics,
Northwestern University, Evanston, IL 60208, USA
4CIERA, Northwestern University, Evanston, IL 60201, USA

(2022)

###### Abstract

Spectropolarimetic campaigns have established that large-scale magnetic fields
are present at the surfaces of approximately 10% of massive dwarf stars.
However, there is a dearth of magnetic field measurements for their deep
interiors. Asteroseismology of gravity-mode pulsations combined with rotating
magneto-hydrodynamical calculations of the early-B main-sequence star HD 43317
constrain its magnetic field strength to be approximately $5\times 10^{5}$ G
just outside its convective core. This proof-of-concept study for magneto-
asteroseismology opens a new window into the observational characterisation of
magnetic fields inside massive stars.

###### keywords:

magnetic fields, stars: early-type, stars: oscillations, stars: rotation

††volume: 361††journal: Massive Stars Near and Far††editors: N. St-Louis, J.
S. Vink & J. Mackey, eds.

##  1 Introduction

Only about 10% of early-type (i.e. spectral types O, B and A) dwarf stars are
known to host a strong, large-scale magnetic field at their surface, which is
thanks to dedicated spectropolarimetric campaigns [Wade et al. (2014), Grunhut
et al. (2017), Shultz et al. (2019), Sikora et al. (2019), (Wade et al. 2014;
Grunhut et al. 2017; Shultz et al. 2019; Sikora et al. 2019)]. The magnetic
fields are predominantly dipolar, typically inclined with respect to the
rotation axis, and have strengths between approximately 100 G and a few tens
of kG. Large-scale magnetic fields in main-sequence stars are thought to be
formed through the star formation process [Neiner et al. (2015), (e.g. Neiner
et al. 2015)] or through binary star mergers that leave behind a strongly
magnetic and apparently single star [Schneider et al. (2019), (e.g. Schneider
et al. 2019)]. Regardless of how they arise, magnetic fields are one of the
largest uncertainties in stellar structure and evolution theory, as they
strongly impact interior rotation, mixing and angular momentum transport
[Maeder & Meynet, Keszthelyi et al. (2019), (Maeder & Meynet 2005; Keszthelyi
et al. 2019)].

However, there are few inferences of the strength and geometry of a magnetic
field below the surface of an early-type star, specifically in the deep
interior near the convective cores of main-sequence massive stars. Thus any
direct inference of interior stellar properties from asteroseismology – the
study of stellar structure from pulsations – are extremely valuable in
improving stellar structure and evolution theory [Aerts (2021), (Aerts 2021)].
To date, only a handful of massive stars have undergone forward asteroseismic
modelling to constrain their interior properties (see [Bowman (2020), Bowman
2020] for a review).

Amongst massive stars, there are two main types of pulsations that can be
excited: pressure and gravity modes, which are standing waves that are
restored by the pressure force and buoyancy, respectively. Coherent pulsation
modes (i.e. standing waves) are excited by a heat-engine mechanism operating
in the iron-nickel opacity bump at 200 000 K, and have pulsation periods of
order hours to days [Dziembowski & Pamyatnykh (1993), Dziembowski et al.
(1993), (Dziembowski & Pamyatnykh 1993; Dziembowski et al. 1993)]. Gravity
modes in particular are sensitive to the properties (e.g. mass and size) of
the convective core in main-sequence massive stars, because they probe the
Brunt-Väisälä frequency profile. Furthermore, rotation shifts pulsation mode
frequencies such that the interior rotation profiles of massive stars can be
deduced from gravity-mode period spacing patterns [Bouabid et al. (2013),
(Bouabid et al. 2013)] and rotationally split multiplets [Aerts et al. (2003),
(e.g. Aerts et al. 2003)].

In these proceedings, we summarise the methods and results of applying
magneto-asteroseismology to the magnetic, pulsating, early-B star HD 43317
published by [Lecoanet, Bowman, & Van Reeth (2022), Lecoanet, Bowman & Van
Reeth (2022)], to which the reader is referred for full details.

##  2 The Magnetic Pulsator HD 43317

HD 43317 is an early-type main-sequence star with spectral type B3.5 V. From
dedicated high-resolution spectropolarimetry, a precise rotation period of
0.897673(4) d and a large-scale dipolar surface magnetic field with strength
$B_{\rm p}=1312\pm 332$ G have been measured [Briquet et al. (2013),
Buysschaert et al. (2017), (Briquet et al. 2013; Buysschaert et al. 2017)].
Furthermore, atmospheric modelling demonstrated that HD 43317 is likely to be
a single star with solar metallicity, and has an effective temperature of
$T_{\rm eff}=17350\pm 750$ K, a surface gravity of $\log\,g=4.0\pm 0.1$ and a
projected surface rotational velocity of $v\,\sin\,i=115\pm 9$ km s-1 [Pápics
et al. (2012), (Pápics et al. 2012)].

HD 43317 was observed by the CoRoT space mission [Auvergne et al. (2009),
(Auvergne et al. 2009)], which assembled a light curve spanning 150.5 d with
an average cadence of 32 s. Frequency analysis of the CoRoT light curve
revealed dozens of significant gravity-mode frequencies [Pápics et al. (2012),
Buysschaert et al. (2018), (Pápics et al. 2012; Buysschaert et al. 2018)]. The
detection of gravity modes in a strongly magnetic star makes HD 43317 unique
among early-type stars, since strong magnetic fields are expected to suppress
their excitation (e.g. [Saio (2005), Lecoanet et al. (2017), Saio 2005;
Lecoanet et al. 2017]).

##  3 Forward Asteroseismic Modelling

The gravity-mode pulsation mode frequencies of HD 43317 extracted from the
CoRoT light curve allowed [Buysschaert et al. (2018), Buysschaert et al.
(2018)] to perform forward asteroseismic modelling to ascertain the best-
fitting interior structure. As a first step, the effective temperature and
surface gravity from spectroscopy were used to delimit the parameter space in
the Kiel diagram (see Fig. 1). Within this parameter space, a grid of non-
rotating stellar structure models was calculated using the open-source MESA
evolution code [Paxton et al. (2011), Paxton et al. (2013), Paxton et al.
(2015), (r8118; Paxton et al. 2011, 2013, 2015)]. The free parameters were the
mass, $M$, the central core hydrogen content, $X_{\rm c}$, as a proxy for age
on the main sequence, and the amount of convective boundary mixing (CBM) of
the convective core, $f_{\rm CBM}$, assuming a diffusive exponential
prescription. The metallicity was fixed at solar as indicated by the
spectroscopic abundance analysis (i.e. $Z=0.014$), and a fixed amount of
mixing was set in the radiative envelope, specifically $D_{\rm env}=10$ cm2
s-1. Models were calculated from the zero-age main sequence until core
hydrogen depletion (i.e. $X_{\rm c}=0$).

Figure 1: Kiel diagram showing the 20 best-fitting asteroseismic structure
models as black crosses for HD 43317. The 1- and 2-$\sigma$ spectroscopic
confidence intervals are denoted by the two black rectangles. Each calculated
stellar structure model is shown as a circle, which have been colour-coded by
their resulting asteroseismic $\chi^{2}$ value in the forward asteroseismic
modelling methodology. Figure adapted from [Buysschaert et al. (2018),
Buysschaert et al. (2018)], their figure 7.

For every structure model, the adiabatic pulsation mode frequencies for dipole
(i.e. $\ell=1$) and quadrupole (i.e. $\ell=2$) gravity modes and all possible
azimuthal orders (i.e. $|m|\leq\ell$) were calculated using the GYRE stellar
pulsation code [Townsend & Teitler, Townsend et al. (2018), (Townsend &
Teitler 2013; Townsend et al. 2018)]. A uniform (i.e. rigid) interior rotation
profile was assumed applying the measured surface rotation period of
0.897673(4) d and the traditional approximation for rotation (TAR) was used to
include the effects of the Coriolis force on the pulsation mode frequencies.
However, such an approach neglects the horizontal component and the structural
deformation due to the centrifugal force, because [Buysschaert et al. (2018),
Buysschaert et al. (2018)] used non-rotating structure models.

Forward asteroseismic modelling is the quantitative comparison of observed
pulsation mode frequencies to those theoretically predicted from a grid of
stellar structure models by use of a merit function [Aerts (2021), (Aerts
2021)]. After eliminating the unlikely mode geometries, [Buysschaert et al.
(2018), Buysschaert et al. (2018)] identified the best-fitting structure model
and then recalculated a finer grid of MESA models to oversample the parameter
space identified by the global $\chi^{2}$ minimum and repeated the forward
asteroseismic modelling exercise. The 20 best-fitting structure models that
satisfy the $2\sigma$ confidence interval from forward asteroseismic modelling
are shown in Fig. 1. The gravity mode frequencies of HD 43317 were identified
to have radial orders between $-15\leq n_{\rm pg}\leq-1$, and the best overall
structure model had a mass of $M=5.8^{+0.1}_{-0.2}$ M⊙, a core hydrogen
content of $X_{\rm c}=0.54^{+0.01}_{-0.02}$ and a parameterisation of CBM to
be $f_{\rm CBM}=0.004^{+0.014}_{-0.002}$ [Buysschaert et al. (2018),
(Buysschaert et al. 2018)].

The results of the forward asteroseismic modelling make HD 43317 particularly
interesting for two reasons, both of which support the hypothesis that a
strong magnetic field must exist in the deep interior. Firstly, the low amount
of measured CBM is plausibly a consequence of a magnetic field suppressing
convective motions from ‘overshooting’ the Schwarzschild boundary of the
convective core into the overlying radiative envelope. This is observed in
magneto-hydrodynamical (MHD) simulations of magnetised core convection
[Browning et al. (2004), Featherstone et al. (2009), Augustson et al. (2016),
(e.g. Browning et al. 2004; Featherstone et al. 2009; Augustson et al. 2016)].
Secondly, the gravity-mode frequencies identified in HD 43317 are particularly
high frequency and have atypical radial orders for gravity modes in stars with
masses between $3\lesssim M\lesssim 9$ M⊙ (see [Pedersen et al. (2021),
Pedersen et al. 2021]). This lends further support to the presence of a strong
interior magnetic field, because, if present, it would interact with and
suppress low-frequency gravity modes leaving only high-frequency gravity modes
unaffected.

##  4 Interior Magnetic Field Inference

To test the hypothesis of a strong magnetic field in the deep interior of HD
43317, [Lecoanet, Bowman, & Van Reeth (2022), Lecoanet, Bowman & Van Reeth
(2022)] performed a magneto-asteroseismic analysis to infer the critical
magnetic field strength needed to suppress non-observed gravity modes.
Previous work has found that gravity waves can strongly interact with magnetic
fields in stellar interiors [Rogers & MacGregor (2010), Fuller2015, Lecoanet
et al. (2017), (e.g. Rogers & MacGregor 2010; Fuller et al. 2015; Lecoanet et
al. 2017)]. If a gravity wave of frequency $f$ enters a region where

$f\lesssim f_{\rm B}=\frac{1}{2\pi}\sqrt{\frac{B_{\rm
r}}{\pi\rho}\frac{N\Lambda}{r}}$ (1)

where $B_{\rm r}$ is the radial magnetic field, $\rho$ is the density,
$\Lambda=\sqrt{\ell(\ell+1)}$ with $\ell$ the spherical harmonic degree of the
gravity wave, and $r$ is the local radius, it converts into a magnetic (i.e.
Alfvén) wave, which in turn prevents the formation of a gravity mode (i.e.
standing gravity wave). The best fitting structure model as found by
[Buysschaert et al. (2018), Buysschaert et al. (2018)], as well as the
resultant magnetic interaction frequency ($f_{\rm B}$; cf. Eqn. 1) for HD
43317 are shown in the left panel of Fig. 2. The lowest observed gravity mode
frequency of HD 43317 is close to the value of $f_{\rm B}$ in the near-core
region at $r=0.18R_{\star}$. The critical magnetic field strength required to
suppress all of the non-observed low-frequency gravity modes given the
structure model of HD 43317 is inferred to be $B_{\rm r}\simeq 4.7\times
10^{5}$ G at $r=0.18R_{\star}$, as shown in the right panel of Fig. 2
[Lecoanet, Bowman, & Van Reeth (2022), (Lecoanet, Bowman & Van Reeth 2022)].

Figure 2: Left: The best fitting structure model from forward asteroseismic
modelling by [Buysschaert et al. (2018), Buysschaert et al. (2018)], showing
the Lamb ($S_{\rm l}$) and Brunt-Väisälä ($N$) frequency profiles, the lowest
observed frequency gravity mode ($f_{\rm g}$) and the magnetic interaction
frequency ($f_{\rm B}$; cf. Eqn. 1). Right: Gravity mode frequencies as a
function of critical magnetic field strength needed to suppress their
propagation, in which observed gravity modes are shown as thick symbols. The
horizontal line denotes the maximum permitted magnetic field strength without
suppressing the lowest frequency observed gravity mode. Figures adapted from
[Lecoanet, Bowman, & Van Reeth (2022), Lecoanet, Bowman & Van Reeth et al.
(2022)], their figures 1 and 3.

To accurately quantify the magnetic field strength in the near-core of HD
43317, [Lecoanet, Bowman, & Van Reeth (2022), Lecoanet, Bowman & Van Reeth
(2022)] calculated the linear waves for a rotating and magnetised stellar
model taking the best model in Fig. 2 as input and using the Wentzel-Kramers-
Brillouin-Jeffreys (WKBJ) approximation with the dedalus code [Lecoanet et al.
(2019), Vasil et al. (2019), Burns et al. (2020), (Lecoanet et al. 2019; Vasil
et al. 2019; Burns et al. 2020)]. Spherical geometry, a predominately dipolar
magnetic field geometry and uniform interior rotation at the rate used in the
prior asteroseismic analysis were assumed. A graphical representation of the
results of these dedalus calculations showing the azimuthal velocity structure
of gravity modes above and below the critical magnetic field threshold is
shown in the left panel of Fig. 3. The important conclusion is that
frequencies below the lowest observed $n_{\rm pg}=-15$ gravity mode strongly
interact with the interior magnetic field. This places a strong constraint on
the maximum magnetic field strength inside HD 43317 corroborating the observed
distribution of gravity modes. To check the robustness of their results,
[Lecoanet, Bowman, & Van Reeth (2022), Lecoanet, Bowman & Van Reeth (2022)]
propagated the uncertainties on the best-fitting structure model into the
magnetic field inference and find they have only a small impact, as shown in
the right panel of Fig. 3. Thus the magnetic inference is robust to the
asteroseismic uncertainties.

Figure 3: Left: dedalus calculation showing the azimuthal velocity structure
for the $(\ell,m)=(2,-1)$ gravity wave with $f=0.872$ d-1 and $n_{\rm pg}=-15$
in the left quadrants and $f=0.840$ d-1 and $n_{\rm pg}=-16$ in the right
quadrants. The top and bottom quadrants show the incoming and outgoing wave
structure, respectively, which in the case of the right quadrants show that
the gravity wave is converted into an Alfvén wave, hence not observed. Right:
Inferred magnetic field strengths in the near-core region taking the 20 best-
fitting structure modelling from [Buysschaert et al. (2018), Buysschaert et
al. (2018)] as input. The upper and lower limits are assuming radial orders of
$n_{\rm pg}=-15$ and $-16$ for the lowest frequency observed gravity mode,
respectively. Figures adapted from [Lecoanet, Bowman, & Van Reeth (2022),
Lecoanet, Bowman & Van Reeth et al. (2022)], their figures 4 and 5.

##  5 Conclusions

The recent results of [Lecoanet, Bowman, & Van Reeth (2022), Lecoanet, Bowman
& Van Reeth (2022)] place a strong constraint on the maximum magnetic field
strength of $B_{\rm r}\simeq 4.7\times 10^{5}$ G at $r=0.18R_{\star}$ for the
magnetic pulsating early-B star HD 43317. The dominant probing region of the
gravity modes is just outside the convective core (see Fig. 2). Since HD 43317
has evolved beyond the zero-age main-sequence and its convective core has
receded, the current gravity-mode cavity was previously within the convective
core. The inferred near-core magnetic field strength is comparable to
predictions from MHD simulations containing a fossil field and convective core
dynamo in a main sequence massive star [Browning et al. (2004), Featherstone
et al. (2009), Augustson et al. (2016), (e.g. Browning et al. 2004;
Featherstone et al. 2009; Augustson et al. 2016)].

HD 43317 is currently the only magnetic star pulsating in gravity modes to
have undergone forward asteroseismic modelling [Buysschaert et al. (2018),
(Buysschaert et al. 2018)], but serves as a valuable proof-of-concept for
magneto-asteroseismology. The ongoing TESS space mission is providing
excellent time series data [Ricker et al. (2015), (Ricker et al. 2015)], which
have revealed a high pulsator fraction among massive stars (e.g. [Bowman et
al. (2019), Burssens et al. (2020), Bowman et al. 2019; Burssens et al.
2020]). With future spectropolarimetric and spectroscopic campaigns, such as
the CubeSpec mission [Bowman et al. (2022), (Bowman et al. 2022)], the future
of massive star asteroseismology is bright.

## Acknowledgements

DMB and TVR gratefully acknowledge funding from the Research Foundation
Flanders (FWO) by means of senior and junior postdoctoral fellowships with
grant agreements 1286521N and 12ZB620N, respectively, and FWO long stay travel
grants V411621N and V414021N, respectively. DL is supported in part by the US
National Aeronautics and Space Administration (NASA) grant 80NSSC20K1280.
Computations were conducted with support by the NASA High End Computing
Program through the NASA Advanced Supercomputing (NAS) Division at Ames
Research Center on Pleiades with allocation GIDs s2276. This research was
supported in part by the National Science Foundation under grant no. NSF
PHY-1748958.

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# Energy-scale cascade and correspondence between
the Mott and Kondo lattice physics

Danqing Hu  Beijing National Laboratory for Condensed Matter Physics and
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
University of Chinese Academy of Sciences, Beijing 100049, China  Ning-Hua
Tong  Department of Physics, Renmin University of China, Beijing 100872, China
Yi-feng Yang<EMAIL_ADDRESS>Beijing National Laboratory for Condensed
Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  University of Chinese Academy of Sciences, Beijing 100049,
China  Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China

###### Abstract

We propose an energy-scale correspondence between the Mott physics and the
Kondo lattice physics and construct a tentative phase diagram of their
correlated electrons with two characteristic energy scales $\omega^{*}$ and
$\Omega$ marking the upper boundary of a low-energy regime with well-developed
long-range coherence and the lower boundary of localized spectral weight in
the energy space, respectively. In between, there exists a crossover region
with emergent but damped quasiparticle excitations. We argue that the presence
of two separate energy scales is a generic property of correlated electrons on
a lattice and reflects an intrinsic two-stage process of the quasiparticle
dynamics to build up the lattice coherence. For the Hubbard model, they
correspond to the kink and waterfall structures on the dispersion, while for
the periodic Anderson model, they are associated with the indirect and direct
hybridization gaps. Our work reveals a deep connection between the Mott and
Kondo lattice physics and provides a basic ingredient for the study of many-
body systems.

In correlated electron systems, a “kink” denotes an abrupt slope change of the
dispersive band and has often been regarded as a fingerprint of the coupling
between electronic quasiparticles and collective bosonic excitations. It has
been intensively studied in cuprates Lanzara2001Nature ; He2001PRL ;
Shen2002PMB ; Damascelli2003RMP ; Meevasana2007 ; Armitage2010RMP , where its
appearance at an energy scale of around 30-90 meV has been attributed to spin
fluctuations or phonons, which may potentially participate in the
superconducting pairing. Different from these “bosonic kinks” which represent
the characteristic energy scales of the bosonic excitations, it has been shown
later that kink may also arise purely due to electronic mechanism without
involving bosonic excitations Byczuk2007NatPhys . This “electronic kink” was
first seen in the Hubbard model (HM) within the framework of the dynamical
mean-field theory (DMFT) Nekrasov2006  and then extended to other more
complicated models Macridin2007PRL ; Grete2011PRB ; Kainz2012PRB ;
Greger2013PRL ; Toschi2009 ; Toschi2010 . It might explain the so-called high
energy kink observed over a wide energy range from 40 to 800 meV in cuprates
and some other transition-metal oxides Yang2005PRL ; Iwasawa2005 ; Yoshida2005
; Ingle2005PRB ; Valla2007 ; Graf2007 ; Xie2007PRL ; Aizaki2012PRL .

Theoretically, the “electronic kink” has been argued to mark an important
energy scale of the Mott physics, namely the boundary of its Fermi liquid
ground state Byczuk2007NatPhys . Roughly speaking, the kink energy
$\omega^{*}$ constrains a low-energy region of well-defined Landau
quasiparticles and poses a threshold for the application of the Fermi liquid
theory. A higher energy scale $\Omega$ has also been proposed as the boundary
separating the many-body resonant state from the high-energy Hubbard bands.
$\omega^{*}$ and $\Omega$ together define a cascade of key energy scales in
the HM. Their discovery has since stimulated many debates concerning their
true physical origin. Some suggested that the kink should still be of bosonic
origin due to emergent collective spin fluctuations Rass2009 , while some
associated it with an effective Kondo energy scale Held2013PRL . A better
understanding of its origin will deepen our knowledge of the many-body
physics, but a consensus has yet to be reached. In particular, its fundamental
importance and potential implications have not been fully recognized.

Here we point out their deep connection with the general physics of
quasiparticle emergence and coherence and report a close correspondence
between the energy scales of the Mott and Kondo lattice physics. While the two
phenomena have mostly been studied separately in different families of
correlated systems based on either the HM or the periodic Anderson model
(PAM), there are increasing evidences supporting an intimate underlying
connection Held2000PRL ; Logan2016 . The Kondo lattice physics has been argued
to be a special case of the (orbital-selective) Mott transition Medici2005 ;
Pepin2007 ; Pepin2008 ; Vojta2010 , and the Mott physics in cuprates is itself
a low-energy projection of the Kondo-Heisenberg model Zhang1988 . In the
lately-discovered nickelate superconductors, both seem to exist and determine
the properties of the ground state and low-energy excitations Zhang2020 .

Our proposed correspondence is based on the simple observation that the two
energy scales of the HM have the simple relationship $\omega^{*}\propto Z$ and
$\Omega\propto{Z^{1/2}}$ with the renormalization factor Z Byczuk2007NatPhys ;
Bulla1999PB . In comparison, the PAM also has two well-known energy scales,
namely the indirect and direct hybridization gaps Coleman1984PRB ;
Auerbach1986PRL ; Benlagra2011 ; Coleman2015 , which, in the mean-field
approximation, are related to an effective hybridization $\tilde{V}$ through
$\Delta_{\rm ind}\propto\tilde{V}^{2}$ and $\Delta_{\rm dir}\propto\tilde{V}$.
This is reminiscent of $\omega^{*}$ and $\Omega$ in the HM. Since the
hybridization gaps have well-defined physical meanings, it is natural to ask
if this similarity can provide some insight on the physics of the HM or reveal
certain generic features of both models. In this work, we will establish such
a correspondence through systematic numerical studies and explore their origin
and potential implications on the physics of correlated electrons. We will
show that they represent two distinct groups of related energy scales and
provide a good characterization of the energy boundaries separating the fully
coherent and localized spectral weights with an itinerant but damped crossover
region in between. This allows us to construct a tentative phase diagram in
the energy space. We argue that the separation of the two energy scales
reveals an instrinsic two-stage process of the quasiparticle dynamics for
building up the long-range spatial coherence and represents a generic property
of a typical class of correlated electron physics on a lattice represented by
the HM and PAM. We note that the present work only concerns the paramagnetic
Fermi liquid side of the HM phase diagram but for the whole spectrum beyond
the Landau quasiparticle regime in the energy space. There is a difference
between the energy domain and temperature domain. While the Fermi liquid is
usually realized at low temperatures, the two energy scales should still be
observed in the energy spectrum, reflecting the critical charge or spin
dynamics of low-lying quasiparticle excitations.

Figure 1: Illustration of different energy scales identified from the $f$
electron DOS $\rho(\omega)=-{{\rm Im}}G(\omega)/\pi$, the real part of the $f$
electron self-energy ${\rm Re}\Sigma(\omega)$ and its slope ${\rm
Re}\Sigma^{\prime}(\omega)$, and the real part of the Green function ${\rm
Re}G(\omega)$ for (a) the HM with $U=3$ and (b) the PAM with $U=5.6$ and
$V^{2}=0.4$. $\omega^{*}$ and $\omega_{\rm m}$ are estimated from the kink in
${\rm Re}\Sigma(\omega)$ and the maximum of ${\rm Re}G(\omega)$, respectively.
For the HM, $\Omega$ is defined from the maximum of ${\rm
Re}\Sigma^{\prime}(\omega)$. For the PAM, $\Delta_{\rm ind}$, $\Delta_{\rm
dir}$, and $\Omega$ are obtained from the dispersion in Fig. 3. The insets are
the enlarged plots for the low energy scales: $\omega^{*}$, $\omega_{\rm m}$,
and $\Delta_{\rm ind}$.

In both models, the Hamiltonians include two parts: $H=H_{K}+H_{U}$. The
potential energy has the form,
$H_{U}=U\sum_{i}\left(n_{i\uparrow}-\frac{1}{2}\right)\left(n_{i\downarrow}-\frac{1}{2}\right)$,
where $U$ is the onsite Coulomb interaction and $n_{i\sigma}$ is the density
operator of the local orbital. There are two general ways to delocalize the
$f$ electrons. In the HM, one introduces a direct hopping between neighoring
lattice sites, giving rise to a kinetic energy,
$H_{K}=\sum_{k\sigma}\epsilon_{k}f_{k\sigma}^{{\dagger}}f_{k\sigma}$, where
$f^{\dagger}_{k\sigma}(f_{k\sigma})$ are the creation (annihilation) operator
of the $f$ electrons. In the PAM, the $f$ orbitals remain local and the
delocalization is achieved by hybridizing with an additional conduction band
such that
$H_{K}=\sum_{k\sigma}\epsilon_{k}c_{k\sigma}^{{\dagger}}c_{k\sigma}+V\sum_{i\sigma}(f_{i\sigma}^{{\dagger}}c_{i\sigma}+{\rm
H.c.})$, where $c^{\dagger}_{k\sigma}$($c_{k\sigma}$) are the creation
(annihilation) operator of the conduction electrons and $V$ is the bare
hybridization parameter. For simplicity, we focus on the paramagnetic state in
the half-filled one-band case with particle-hole symmetry and use the
numerical renormalization group (NRG) Wilson1975 ; Bulla2008  as the impurity
solver within the DMFT framework Metzner1989 ; Georges1992PRB ; Georges1996RMP
; Kotliar2006RMP . The local self-energy of the $f$ electrons is calculated
using $\Sigma(\omega)=UF(\omega)/G(\omega)$ with $G(\omega)=\langle\langle
f_{i\sigma};f_{i\sigma}^{{\dagger}}\rangle\rangle_{\omega}$ and
$F(\omega)=\langle\langle
f_{i\sigma}(n_{i\overline{\sigma}}-1/2);f_{i\sigma}^{{\dagger}}\rangle\rangle_{\omega}$
Bulla1998JPCM . We then solve the DMFT self-consistent equation
$G(\omega)=\int d\epsilon\rho_{0}(\epsilon)/[\omega-\epsilon-\Sigma(\omega)]$
for the HM and $G(\omega)=\int
d\epsilon\rho_{0}(\epsilon)/[\omega-V^{2}/(\omega-\epsilon+i\eta)-\Sigma(\omega)]$
for the PAM Zhang1993PRL ; Pruschke2000PRB , where
$\rho_{0}(\epsilon)=e^{-(\epsilon/t)^{2}}/(\sqrt{\pi}t)$ is the noninteracting
density of states for a hypercubic lattice with the dimensionality
$d\rightarrow\infty$. We set $t=1$ as the energy unit and choose the
logarithmic discretization parameter $\Lambda=2$ and a total number of stored
states $N_{\rm s}=800$ for the NRG calculations. Our calculations are limited
in the paramagnetic phase concerning the emergence and development of
quasiparticle coherence. Magnetic instabilities are additional issues beyond
the current work.

Figure 2: The scaling relation of different energy scales with respect to the
renormalization factor $Z$ for (a) the HM and (b) the PAM with tuning Coulomb
interaction $U$. For the PAM, the hybridization strength is set to
$V^{2}=0.4$.

Figure 1 illustrates how the different energy scales can be identified from
the $f$ electron density of states (DOS) $\rho(\omega)=-{\rm
Im}G(\omega)/\pi$, the real part of the self-energy ${\rm Re}\Sigma(\omega)$
and its $\omega$-derivative ${\rm Re}\Sigma^{\prime}(\omega)$, and the real
part of the Green function, ${\rm Re}G(\omega)$. For the HM, the DOS has a
three-peak structure including two Hubbard peaks at higher energy and a
quasiparticle peak around zero energy. The larger scale $\Omega$ is defined
here from the maximum of ${\rm Re}\Sigma^{\prime}(\omega)$. In Ref.
Byczuk2007NatPhys , it was defined from the minimum of ${\rm
Re}\Sigma(\omega)$, which, however, only follows the proposed scaling near the
Mott transition with a small $Z$. Nonetheless, both reflect the same physics
and separate roughly the quasiparticle peak and the Hubbard peaks in the DOS.
The lower scale $\omega^{*}$ can be defined from an additional slope change or
the kink in ${\rm Re}\Sigma(\omega)$ that constrains a low-energy region for
quasiparticle excitations. A different scale $\omega_{\rm m}$ may be
introduced from the maximum of ${\rm Re}G(\omega)$. $\omega^{*}$ and
$\omega_{\rm m}$ are of the same origin and related through
$\omega^{*}\approx(\sqrt{2}-1)\omega_{\rm m}$ for the HM Byczuk2007NatPhys . A
third definition may be found from the maximum ($\omega_{s}$) of the local
spin susceptibility as discussed later in Fig. 4. The relationship of these
scales can be established if we tune the local Coulomb interaction $U$ and
plot their variations with the renormalization factor, $Z^{-1}=1-{\rm
Re}\Sigma^{\prime}(0)$. The results are summarized in Fig. 2(a). We see that
$\omega^{*}\propto\omega_{m}\propto Z$ and $\Omega\propto Z^{1/2}$ over a wide
range of $Z$. Thus, $\omega^{*}$ and $\Omega$ represent two distinct groups of
related energy scales. The different definitions originate from the crossover
nature of the underlying physics.

For comparison, Fig. 1(b) shows the energy scales in the PAM. The indirect
hybridization gap $\Delta_{\rm ind}$ is determined by the DOS gap and
coincides with the additional minimum of ${\rm Re}G(\omega)$, while the direct
hybridization gap $\Delta_{\rm dir}$ cannot be clearly discerned in these
quantities and is best seen in Fig. 3 from the spectral function,
$A_{k}(\omega)=-\pi^{-1}{\rm
Im}[\omega-\Sigma(\omega)-\frac{V^{2}}{\omega-\epsilon_{k}+i\eta}]^{-1}$. The
plot implies two separated hybridization bands for the PAM. The indirect gap
$\Delta_{\rm ind}$ is the energy difference between the bottom of the upper
band and the top of the lower band and marks the boundary for the low-energy
gap. Although the PAM is in a Kondo insulating state, the self-energy below
$\Delta_{\rm ind}$ behaves as that of a Fermi liquid Pruschke2000PRB ,
reflecting the establishment of full coherence on the Kondo lattice Hu2019 ;
Qi2020 . By contrast, the direct hybridization gap $\Delta_{\rm dir}$ measures
the minimal energy difference between the two hybridization bands, below which
the spectra are governed by the itinerant $f$ character. The energy scales
$\omega^{*}$ and $\omega_{\rm m}$ can still be defined similarly as in the HM
and contain the flat part of the hybridization bands. We obtain
$\omega^{*}\propto\omega_{\rm m}\propto\Delta_{\rm ind}$. On the other hand,
the maximum of ${\rm Re}\Sigma^{\prime}(\omega)$ changes slightly with varying
$U$ and is no longer a meaningful quantity separating the boundary of the
itinerant and localized regions. Rather, we find it is better to define the
larger energy $\Omega$ from the deviation of the hybridization bands from the
original conduction bands. In Fig. 3, it is seen that $\Omega$ sets roughly
the outmost boundary of the itinerant $f$ electron spectral weight. Both
$\Omega$ and $\Delta_{\rm dir}$ reflect the separation of the hybridized
(itinerant) and unhybridized (localized) spectral regions of the $f$ electrons
in the energy space.

The scaling results for the PAM are summarized in Fig. 2(b). Similarly, we
have $\omega^{*}\propto\omega_{\rm m}\propto\Delta_{\rm ind}\propto Z$ and
$\Omega\propto\Delta_{\rm dir}\propto Z^{1/2}$ over a wide range of $Z$. The
obtained scaling relations link $\omega^{*}$ and $\Omega$ to $\Delta_{\rm
ind}$ and $\Delta_{\rm dir}$ and suggest that they reflect similar physics in
the PAM. However, unlike $\omega^{*}$ and $\Omega$ which are in reality hard
to measure, the indirect and direct hybridization gaps have unambiguous
physical meaning and can be directly probed in experiment. Their scaling
relations in the PAM can be derived from the pole properties of the Green
function,
$G_{k}(\omega)=[\omega-\Sigma(\omega)-\frac{V^{2}}{\omega-\epsilon_{k}+i\eta}]^{-1}$.
If we make the lowest order approximation, ${\rm
Re}\Sigma(\omega)\approx(1-Z^{-1})\omega$, the poles are determined by
$Z^{-1}\omega-\frac{V^{2}}{\omega-\epsilon_{k}}=0$. The direct gap is defined
as the energy differences of the two poles at $\epsilon_{k}=0$, while the
indirect gap is the energy difference at the band edges ($\epsilon_{k}=\pm
D$). We have $\Delta_{\rm dir}=Z^{1/2}V$ and $\Delta_{\rm ind}=ZV^{2}/D$,
confirming their relationship with $Z$. The relative deviation of the
hybridization bands from the conduction band may be estimated to be
$\delta=(\omega-\epsilon_{k})/\omega=ZV^{2}/\omega^{2}$. We have thus
$\Omega\approx Z^{1/2}V/\delta^{1/2}$ for a chosen small cutoff $\delta$, at
which the $f$ electron spectral weight on the dispersion is also reduced to
the order of $Z\delta$. Since all these energy scales diminish at $V=0$, their
presence reflects the delocalization of the $f$ electrons on the lattice
driven by the hybridization with conduction electrons. In this respect, the
larger scale $\Omega$ constrains the major spectral region of delocalized $f$
electrons, while $\omega^{*}$ puts a further restriction for well-established
long-range coherence.

Figure 3: Different energy scales on the intensity plot of the resolved
spectral functions for the HM and PAM. The background colors reflect the
magnitude of the $f$ electron spectral weight $A_{\rm k}(\omega)$. The blue
dashed lines denote the dispersion $Z\epsilon_{k}$ for the HM and
$\epsilon_{k}$ for the PAM. The parameters are the same as in Fig. 1. For the
PAM, the localized flat $f$ bands are outside the plotted energy window.

Similar analyses may help to understand $\omega^{*}$ and $\Omega$ in the HM.
Figure 3 also plots the dispersion $A_{k}(\omega)=-\pi^{-1}{\rm
Im}[\omega-\epsilon_{k}-\Sigma(\omega)]^{-1}$ of the HM. Indeed, we see that
$\omega^{*}$ marks the boundary of well-defined Landau quasiparticles, above
which the dispersion has a different renormalization factor, while $\Omega$
separates the low-energy many-body states and the high energy incoherent
region, beyond which the $f$ electron weights are localized. The presence of
both $\omega^{*}$ and $\Omega$ is a reflection of the special cascade
structure of ${\rm Re}\Sigma^{\prime}(\omega)$, which, as shown in the middle
panel of Fig. 1(a), has a maximum at $\Omega$ but, with lowering energy,
decreases first to a plateau (or local minimum) on the hillside before it
eventually reaches the valley floor below $\omega^{*}$ and $\omega_{\rm m}$.
Interestingly, $\Omega$ lies roughly at the waterfall structure connecting the
quasiparticle and Hubbard bands Weber2008 ; Kohno2012 . This poses a question
concerning the relation of the two properties. For this, we note that the
waterfall is an abrupt change of the pole of the Green function with
increasing $\epsilon_{k}$. It can only occur around the inflection point of
$\omega-{\rm Re}\Sigma(\omega)$ if one wants to tentatively avoid multiple
poles. This immediately implies ${\rm Re}\Sigma^{\prime\prime}(\Omega)\approx
0$ or a maximum (minimum) in ${\rm Re}\Sigma^{\prime}(\Omega)$. Thus, $\Omega$
defined here is indeed associated with the waterfall structure in the HM. We
conclude that the two energy scales correspond to two features of the
dispersion: $\omega^{*}$ for the kink and $\Omega$ for the waterfall, which
also reflect the boundaries for the low-energy fully coherent region and the
high-energy localized region. The fact that $\Omega$ connects two regions with
opposite energy dependence in the self-energy implies
$Z^{-1}\Omega\sim\Omega^{-1}$ or $\Omega\propto Z^{1/2}$, similar to that in
the PAM.

Figure 4: The imaginary part of the local spin susceptibility, showing a
maximum at $\omega_{s}$ for both models. The parameters are the same as in
Fig. 1. The inset plots $\omega_{s}$ as functions of $Z$ with $V^{2}=0.4$
(PAM) and varying $U$.

Further insight on $\omega^{*}$ may be obtained if we consider that
$\Delta_{\rm ind}$ in the PAM is related to the spin screening,
$\omega^{*}\propto\Delta_{\rm ind}\propto T_{K}$ where $T_{K}$ is the lattice
Kondo temperature Jarrell1995PRB ; Chen2016 . The effect of spin screening
might be best seen from the local susceptibility,
$\chi_{s}(\omega)=\langle\langle S_{i}^{z};S_{i}^{z}\rangle\rangle_{\omega}$
with $S_{i}^{z}=\frac{1}{2}(n_{i\uparrow}-n_{i\downarrow})$ Shimizu2000JPSJ .
Figure 4 plots the imaginary part of the local spin susceptibilities for the
HM and PAM using the same parameters as in Fig. 1. Both exhibit a maximum at
$\omega_{s}$, which, as shown in the inset, varies linearly with $Z$ as those
of $\omega^{*}$ and $\Delta_{\rm ind}$. In fact, we find
$\omega_{s}\approx\omega^{*}$ in both models (Fig. 3). Away from half filling,
this kink or hybridization energy scale $\omega^{*}$ still exists and retains
the same relation with $\omega_{s}$ Grete2011PRB ; Kainz2012PRB . Below
$\omega_{s}$ the local susceptibility is suppressed due to the spin screening.
It is therefore natural to associate the kink with an effective spin screening
scale as proposed in Ref. Held2013PRL . We should emphasize, however, that the
presence of both $\omega^{*}$ and $\Omega$ is a property of the lattice but
not in the usual single-impurity Kondo model where only the Kondo scale
exists. This implies that the lattice feedback is crucial for separating
$\omega^{*}$ and $\Omega$ and causing the cascade structure of ${\rm
Re}\Sigma^{\prime}(\omega)$. Although DMFT does not contain explicit spatial
correlations, it necessarily includes important lattice information through
the self-consistent iterative procedure.

Thus, all energy scales can be classified into two categories, a lower one
$\omega^{*}\propto Z$ and a higher one $\Omega\propto Z^{1/2}$, which separate
the $f$ electron spectral weight into different regions of distinct
properties. If we start from localized $f$ electrons, we may conclude that
their spin screening and consequential delocalization is an energy-dependent
process marked by the two scales. While $\Omega$ sets the outmost boundary for
the many-body resonant states and covers the full energy range with
delocalized $f$ electron spectral weight, $\omega^{*}$ is a lower boundary for
quasiparticle excitations with well-established long-range coherence. This is
common for both PAM and HM and may be best seen if we construct a tentative
phase diagram in Fig. 5 in the energy space based on the properties of the $f$
electron spectra. For $Z>0$, the $\omega^{*}\propto Z$ line indicates a
screening scale and marks the upper boundary of a well-developed coherent
region irrespective of the Fermi liquid or Kondo insulating state, while
$\Omega\propto{Z^{1/2}}$ marks the lower boundary of an incoherent region with
localized $f$ spectral weight. In between, there exists a crossover region
with itinerant but strongly damped excitations of the $f$ character. With
decreasing $Z$ to zero, the system enters a Mott insulating state for the HM
or an orbital-selective Mott state for the PAM, where the many-body resonance
is suppressed and all $f$ electrons turn into fully localized magnetic
moments. Since $Z$ is dimensionless, the scaling relations suggest constant
prefactors roughly given by $D$ or $V^{2}/D$. Major correlation effects in
both properties are encapsulated in the renormalization factor $Z$ (as a
function of $U/D$) over a wide parameter region.

Figure 5: A tentative phase diagram of the HM and PAM in the energy space as a
function of the renormalization factor $Z$, showing different regions of
correlated electron spectral weights. In the shadow region, the $f$ electrons
become completely localized so that the HM is in a Mott state and the PAM is
in an orbital-selective Mott state.

We remark again that the presence of the two energy scales represents two
related but distinct features of the correlated electron physics. It is not
just a matter of theory but has real implications and observable consequences
in experiment. To see this, we may consider the gradual delocalization of the
$f$ electrons with lowering temperature. The two energy scales then turn into
two temperature scales, as observed in recent pump-probe experiment on the
heavy fermion compound CeCoIn5 Qi2020 . Determinate Quantum Monte Carlo
simulations (DQMC) taking into account spatial degree of freedom for the PAM
have confirmed two distinct stages of hybridization dynamics, namely, the
onset of precursor hybridization fluctuations and the eventual establishment
of a long-range coherent (Kondo insulating) state Hu2019 . In the HM, the two
energy scales are also expected to represent two succeeding stages of the
(nonequilibrium) quasiparticle dynamics, namely, a Fermi liquid phase with
well-established Landau quasiparticles and a higher crossover region with
itinerant but damped quasiparticles (bad metal) Pruschke1995 ; Mravlje2011PRL
; Deng2013PRL . It might also be reflected in the temperature or time
evolution of the doublon-holon excitations. Exact lattice simulations on the
HM may help clarify this issue but are often limited by the numerical accuracy
due to small lattice size. At even higher temperatures, the localized spectral
weight might be thermally excited and contribute to physical properties, but
whether or not (and how) the highly damped electrons can become Anderson
localized requires more rigorous study Antipov2016 .

Along this line of thought, we anticipate that the existence and separation of
two energy scales reflect a generic two-stage development of the electronic
coherence and is an intrinsic property of correlated electrons. Before the
long-range coherence is eventually established, there exists a precursor stage
where the electrons become partially delocalized with damped quasiparticle
excitations. On the other hand, the usual Mott transition also exists in other
models such as the Falicov-Kimball model, the Ising-Kondo model, and the
Hatsugai-Kohmoto model, which do not seem to possess the two energy scales. A
closer inspection suggests that the Mott transitions in these models are
different from that in the HM. In the Falicov-Kimball model and the Ising-
Kondo model, the ground state near the transition is not a Fermi liquid due to
“disorder scattering” Freericks2003 ; Yang2020 , while the Hatsugai-Kohmoto
model lacks the so-called dynamical spectral weight transfer which is a key
feature of the HM Yeo2019 . Thus the presence of two energy scales should not
be viewed as a generic property of the Mott transition. Rather, they are
associated with the emergence and establishment of coherent quasiparticles on
the lattice, possibly induced by the spectral weight transfer from the high-
energy localized part to the low-energy itinerant part. More elaborate studies
are required to establish this important distinction.

To summarize, we report systematic analyses of the energy-scale cascade in the
HM and the PAM and reveal a deep connection between the Mott and Kondo lattice
physics. This allows us to construct a tentative phase diagram for correlated
electrons based on two energy scales marking the upper boundary of the fully
coherent low-energy states and the lower boundary of the incoherent regime
with localized spectral weight in the energy space. For the HM, these
correspond to the kink and waterfall structures on the dispersion. For the
PAM, they are associated with the indirect and direct hybridization gaps. The
separation of two energy scales is an intrinsic property of the lattice models
and reflects a two-stage dynamical process to build up the lattice coherence
of itinerant quasiparticles. Our work clarifies the origin of these energy
scales and reveals a potentially basic and generic property for understanding
the key of correlated electrons on a lattice.

This work was supported by the National Natural Science Foundation of China
(NSFC Grant No. 11974397, No. 11774401, No. 11974420), the National Key
Research and Development Program of MOST of China (Grant No. 2017YFA0303103),
and the Strategic Priority Research Program of the Chinese Academy of Sciences
(Grant No. XDB33010100).

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