# Intrinsic PUF Instance on Non-Volatile NAND Flash Memory

Surbhi Vasudeva
Department of Computer Engineering
San Jose State University
San Jose, CA 95192
surbhi.vasudeva.edu
Sara Tehranipoor
Department of Electrical and Computer Engineering
Santa Clara University
Santa Clara, CA 95050
<EMAIL_ADDRESS>
Nima Karimian
Department of Computer Engineering
San Jose State University
San Jose, CA 95192
<EMAIL_ADDRESS>

###### Abstract

Embedded systems or micro controller based modules have become increasingly
prevalent in our daily lives. However, the security of embedded devices as
well as the authenticity of hardware has become an increasing concern within
the growing Internet of Things (IoT) space. In this paper we setup an
experiment environment where SLC flash program disturbance is observed.We
discovered that intra-page disturbance is easier to be produced than inter-
page disturbance. We also observed that adjacent pages are paired in (2n,
2n+1) manner, and disturbance only occurs within a pair. Lastly, we found that
as page number increases from 0 to 63, it becomes more difficult to observe
the first bit flip within a page, and thus more difficult to achieve the
disturbance stable state.

_Keywords_ NAND Flash $\cdot$ Security $\cdot$ Hardware Security Primitives

##  1 Introduction and Motivation

Technology is prevalent in every field today. Lives have become so much
dependent on electronic devices. The electronic devices in use are mostly part
of a bigger network - the Internet of Things (IoT) [1, 2], in the form of
sensors or nodes. The information reaches its final destination for analysis
or application through the cloud. Within this setup, there is no guarantee of
the integrity of the data. Authentication [3, 4, 5] appears to be the solution
to this problem. In the past, when it became difficult to distinguish/identify
humans based on their physical traits, biometric authentication [6, 7] was
invented. Similarly, electronic devices, for example, RFIDs, look alike. The
only difference is the ID stored in the chip memory, which is vulnerable and
prone to attack. Also, it can be modified easily [8]. As an example, For
costlier hardware devices that are produced in bulk, there are so many people
involved in the production chain. At any point in time, people can decide to
use the product for their benefit and introduce the product in the market
while solely enjoying the profits without sharing with the parent company.
Such an act is unacceptable for a reputed company. The company’s reputation
and millions of dollars could be at stake. What if the device could have a
fingerprint just like human beings? Then it becomes easier to identify the
device uniquely without any ID embedded in it. How this is possible is due to
the inherent nature of silicon devices as the physical variations that happen
on physical devices cannot be reproduced through manufacturing. Hardware
security primitives are potential solutions [9, 10, 11, 12, 13, 14, 15, 16] to
secure consumer electronic devices [17]. Physically unclonable function (PUF)
is the technique of digital fingerprint on physical semiconductor devices.
This fingerprint remains almost unaffected due to any temperature, humidity,
or stability variations [18, 19, 20]. There has been research that
investigates PUF from volatile memories which can be found in [21, 22, 23, 24,
25, 26, 27, 28] and non-volatile memories in [29].

Most electronic devices need memory to store any information. From the list of
all the available memories, NAND Flash is the widely used one where many
different applications such as USB drives, media players, digital cameras, and
smartphones use it. That is because it has lower power consumption, has
shorter write and erase time (high access speed), and has a low bit cost. With
the introduction of PUF, IoT security especially in healthcare domain [30,
20], can be enhanced as the PUF provides a unique ID, which is unclonable.
Secondly, if data is stored with PUF encrypted on memory, the memory is
secured as there is no chance of decryption without the PUF. Third, PUF
behaves like a random function, i.e., generating random output values, and
there are numerous applications of a good and reliable random number
generator. This function is unpredictable for even an attacker with physical
access to the system. Also, It is impossible to produce a copy of the same
physical system even when the functionality is known. Physical Unclonable
Functions (PUFs) provide a unique signature [31, 32]. Where the signature is
the bits that have been stable no matter how many read and write operations
applied to that cell. An extracted signature can be used to authenticate a
chip and produce a random number that can be used in cryptography keys. The
variation takes place between blocks and between pages. That means the chip
can have multiple unique signatures in different locations, and different
methods can extract them. Some PUF models are vulnerable to attacks [33].

In this paper, we implemented PUF extraction using the Samsung NAND Flash
Memory (K9F1G08U0E) with the FMC interface of the experimental discovery board
(STM32F429ZIT6). The software used is STMCUBE32IDE with C programming
language. We researched different program disturb algorithms and implemented
them on NAND flash to extract the PUF and observe the performance of the same.

Specifically, we have the following contributions in this paper:

  * •

Observed SLC Single Page Program Disturbance.

  * •

Observed SLC Multi-Page Program Disturbance.

  * •

Observed SLC pages within a block are paired.

The remainder of the paper is organized as follows: Section II are describing
related work, reviews the basics of NAND FLASH, followed by a description of
PUF and various techniques to identify PUF in Section III. Section IV
describes the methodology and approach followed to obtain PUF. Section V
demonstrates the hardware and software setup, along with operations performed.
Section VI shows the results of our NAND Flash PUF implementations, and
finally, we close with conclusions in Section VII.

##  2 Related Works

Shijie et al. [34] introduced a PUF-based key generator for NAND flash chips.
They proposed three methods to extract raw PUF output numbers including
position map and partial programming and partial erasure to select reliable
keys. Prabhu et al. [35] proposed seven NAND flash PUFs based on program
disturb, read disturb and program operation latency. They have experimented
with fourteen devices. They have used Pearson correlation to evaluate and
measure the robustness of the signatures. Their results showed NAND flash
based on program latency and program disturb are the most useful ones. They
have observed that NFPUF based on program disturb is the best one to
distinguish between different chips but takes more time for extraction and
NFPUF based on program latency is the fastest among all PUFs. Cai et al. [36]
demonstrated the effects of two-step programming on MLC NAND flash. Their work
exposed the dangers of two-step programming and how it could be exploited. The
process of two-step programming involves programming a single cell’s LSB and
MSB at different times. Most of the existing work is limited to analyzing and
generating PUF for a single NAND flash chip either SLC or MLC. In this paper,
we will observe SLC Single Page Program disturbance and SLC pages within a
block are paired.

##  3 Background

###  3.1 NAND Flash

The NAND Flash cell is made of a floating gate transistor, where the electrons
held by the floating gate decide the threshold voltage ($V_{th}$) of the cell.
Normally to switch on an NMOS transistor we need $V_{g}>V_{th}$. During the
program stage, if we inject electrons in the floating gate, we will require a
relatively higher voltage ($V_{g}$) to offset the negative charge of electrons
( and turn on the channel ($V_{g}-V_{\delta}>V_{th}$). If there are no
electrons captured in the floating gate a relatively small gate voltage is
sufficient to make $V_{g}>V_{th}$ and turn on the transistor. NAND Flash is
non-volatile and can retain data even when power is not supplied. When power
is detached from NAND flash memory, a metal-oxide-semiconductor provides an
extra charge to the memory cell that helps to keep the data. This metal-oxide-
semiconductor is known as the Floating gate. Regular kinds of NAND flash
storage incorporate SLC, MLC, TLC, QLC, and 3D NAND. What isolates each type
is the number of bits every cell employs. The more bits put away in every
cell, the more affordable the NAND flash would cost. NAND Flash stores data in
blocks. The basic unit of erase operation is this block, whereas the basic
unit of a write operation is a page. Each page consists of a data area and a
spare area. This data area and spare area vary, and it depends upon flash size
and manufacturer. This spare area is mainly used to store the error correction
algorithm or to store the map table-related information in a page layout.
Basic operations performed in NAND Flash are to read or program a page or
erase a block.

Figure 1: Flash Memory Organization

###  3.2 PUF within NAND Flash Memories

Flash cells are densely packed, so process variations in terms of feature
geometry can have a significant effect on the coupling between nearby cells,
or the same cell behavior. Because of these variations, some cells might be
more or less susceptible to write/read disturb/wear. Physical Unclonable
Functions (PUFs) provide a unique signature. Where the signature is the bits
that have been stable no matter how many read and write operations applied to
that cell. An extracted signature can be used to authenticate a chip and
produce a random number that can be used in cryptography keys. The variation
takes place between blocks and between pages. That means the chip can have
multiple unique signatures in different locations, and different methods can
extract them. Physical Unclonable Function helps to prevent the storage of
secret keys in the storage memory.

###  3.3 PUF Generation Techniques

There are different methods to extract raw PUF output numbers from NAND Flash
memory chips, which are Program disturb, Read disturb, and Program operation
latency. In Program disturb, a block will be erased and then repeatedly
program one page in the block. Between each Program, the adjacent pages should
be read for Program disturb-induced errors. For each bit on the adjacent page,
we have recorded the number of programs required to cause the first-bit error.
Those values comprise the signature. Extracting a signature requires one erase
operation and many program operations, and these programs don’t affect
reliability. The disadvantage of program disturb is that it causes irreparable
damage to the page, and it is quite slow.

The second method is to read disturb. Within this technique, we first erase
the entire block, then program it with random data. Next, we read from each
page several million times to induce a read-disturb. After every 1000 read
iterations, we checked each page in the block for errors. If we observed an
error on a page, we recorded the bit, the page, and the cycle of the error.
This process is repeated 10 million times. Thus, we use the read cycle counts
for all the bits in the block as the signature. The Read disturbs are less
destructive, and it causes less noise disturbance in signature than the
Program disturbance technique, yet it is slow.

Program operation latency is another method that is used to extract
signatures. Within this technique, we program one bit at a time on a page and
record the latency for each operation. There are two types of operation
latency we can introduce, which are a single page and multiple pages. The
advantage of this method is that it is fast, and it dost not cause wearing the
Flash device. On the other hand, the information content of the signature is
low and is easier to forge.

##  4 Methodology and Approach

###  4.1 Algorithm

After looking into different methods to extract the signature. We decided to
use the program disturb in our project. We used this method because we can
notice the distribution happening more frequently in the cells than using read
disturb. That is because program disturb can appear in earlier iterations, and
it needs approximately ten thousand iterations to be able to see it. Also, it
could appear earlier than that. On the other hand, with read disturb, we can
not see a lot of distribution happening within ten thousand times. it needs to
be run for longer iterations to observe the read disturb. The general idea to
achieve the program disturbance is by first erasing the targeted block. Then,
choose a page within that block and program it repeatedly for approximately
ten thousand cycles. In each cycle, we will read the adjacent page for any
errors. If there is any error i.e a bit is flipped. We will record the cycle
number corresponding to that bit in our signature. We will continue to monitor
for other bits and then for the number of cycles shows the pseudo-code that is
used to achieve this approach.

###  4.2 Procedure

We found stable bits in the NAND Flash memory in different ways, on the same
page, on the adjacent page, and on all pages within the block. To start with,
the first approach, which is finding the stable bits on the same page. We
erased block zero and program page one with value AA in hexadecimal. After
that, we monitored changes in bits that were occurring on the same page. We
did that by comparing the values of the bits on the same page with AA. If the
value changes, that means that the bit is not stable. Otherwise, they are
stable bits, and we stored them in an array. After that, we repeated the
program operation and the steps after it for ten thousand iterations. Then, by
the end of the ten thousand iterations, we had stable bits that generate a
PUF.

The second approach is similar to the first one, but this time we are reading
the adjacent pages. In this, we erased block zero, and then programmed page
two with value AA in hexadecimal. After that, we read the adjacent pages,
which are page one and page three. In these adjacent pages, we compared the
bits inside them with the value of FF (FF is the value generated due to the
erase operation; it set all bits to ones). If it is the same value, that means
it is stable. If it changed, that means it is not stable. After that, we
stored the stable bits in an array and keep comparing them at each iteration
ten thousand times. Where in each iteration, we write to the same page, which
is page 2, and read bits from page 0 and page 3.

In the last approach, erased block zero and programmed all pages with AA. Then
choose page one and program it again with AA ten thousand times. Then, read
the adjacent pages, which are page zero and page two—compare the bits on them
with the value of AA. If it changed, it is not stable; otherwise, it is the
stable bit and stores them in an array. After that, I moved to the next page,
which is page two, and programs it with AA ten thousand times. Next, read the
adjacent page of it, which is page one and page three, and compare the value
it has with AA. If it did not change, we would put them in an array. Applied
the same steps to all pages in the block, and there are 64 pages. ECC needs to
be disabled. In case ECC is enabled, then it will be corrected automatically.

##  5 Hardware Setup and Implementation

###  5.1 Hardware Connection

The hardware connections between the STM32F429I and the NAND Flash are made
using female jumper wires. Connections to a total of 16 pins on NAND Flash is
required. Among the 16, there are 8 data pins, 6 control pins, Vcc, and GND.
The same 8 data pins on the NAND Flash would be used to pass command, address,
and data. The 6 control pins on the NAND Flash are CLE(Command Latch Enable),
ALE(Address Latch Enable), R/B(Ready/Busy Output), WE(Write Enable), RE(Read
Enable), CE(Chip Enable). Power is supplied to NAND Flash through a
microcontroller by connecting its Vcc 3V to the Vcc of the NAND Flash and GND
of the microcontroller to GND of NAND Flash. Corresponding FMC(Flexible Memory
Controller) pins from the microcontroller are connected to NAND Flash control
and data pins. STM32CubeIDE is used to develop the software code in the C
programming language. STM32CubeIDE has the STM32CubeMX graphic tool to
visualize the board, select pins and configure them as well. The STM32CubeIDE
provides the facility to generate definitions and library files with the
configured pins from the graphic tool.

Figure 2: STM32Board connection with machine and flash memory.

###  5.2 Software Setup

STM32CubeIDE is used to develop the software code in the C programming
language. STM32CubeIDE has the STM32CubeMX graphic tool to visualize the
board, select pins and configure them as well. The STM32CubeIDE provides the
facility to generate definitions and library files with the configured pins
from the graphic tool.

###  5.3 Configuration

As part of the configuration, first, the Hardware Abstraction Layer (HAL) has
been initialized, followed by System clock configuration. Further other
peripherals like GPIO, FMC, and USART1 are initialized. A NAND Controller
instance is created, and the required setup values are configured. These
include SetupTime, WaitTime, NandBank, ECCcomputation, BlockSize, PageSize,
and others.

###  5.4 ERASE and READ Operations

Erase operation is a must before programming the page. Erase Operation takes
place at block level. Erase operation is initiated by issuing Erase setup
command 60h, followed by row address. The Erase confirm command D0h initiates
internal Block Erase operation. R/B pin is set to low. The status is checked
by issuing command 70h. Value 0 in Write Status bit (I/O 0) indicates that
Erase is completed whereas value 1 indicates an error in Erase operation. A
Read operation is initiated by writing the opcode 0x00. The CLE (command latch
enable) pin must be asserted HIGH while simultaneously asserting the CE (chip
enable) pin LOW. This command alerts the flash device to prepare the data
buffer registers to read from an incoming address. The row address and column
address is then sent after the 0x00 opcode and the address is written on
consecutive rising edges of the WE (write enable) signals. It requires 4 bytes
to locate which page to read from. 2 bytes for the row address and 2 bytes
column address. This yields a total of 64k addressable row pages and 64k
addressable columns. However the last 4 bits of the column address are not
used because there are a total 2112 bytes in the column space (2096 bytes per
page + 64 bytes of spare area for ECC). This means that 12 bits are required
to address the columns. The completion of writing the address is indicated by
sending the opcode 0x30. Now the device now knows which bytes to read from its
flash array. The data from the address is loaded onto the page register and
sent through each of the 8 I/O pins to be read by the STM32F4 micro-
controller. Each byte of data is sent on consecutive falling edges of the
RE(read enable) signal.

###  5.5 PROGRAM Operation

###  5.6 PROGRAM Operation

NAND Flash is programmed at page level. Program operation is initiated by
issuing Serial data input command 80h, followed by column and row addresses.
Then, Transmitter buffer is set and is passed in the next cycle. The
transmitter buffer consists of the data to be written on the page. The NAND
instance is initialized with frequency, setup timing, hold timing, etc. The
address consists of a column address and row address. Read operation is
performed to verify the data written to the NAND Flash. A command input of
value 00h is passed to perform read page operation. Column address and row
address are passed in the next 4 cycles. Data is read in the next cycle. Read
Page output is in the receiver buffer. The NAND instance is initialized with
frequency, setup timing, hold timing, etc. The receiver buffer holds the data
read from the page. This initiates programming process. R/B pin is set to low.
The status of write operation is checked by issuing command 70h. Value 0 in
Write status bit (I/O 0) indicates that a program is completed whereas value 1
indicates an error in program operation. A command input of value 80h is
passed, which determines it is a write operation. Column address and row
address are passed in the next 4 cycles.

1) PUF observing the same page: Program disturb technique on the same page is
observed by programming and simultaneously reading the same page for 10,000
iterations. Bit flips are observed after 4000 iterations. The continuous
programming to the same page causes disturbs, which can be seen in reading
operation after around 4000 iterations. After block erase operation, the
transmitter buffer is loaded with 0XAA for page size, which is 2048 bytes. The
program and read operations are performed in a loop for 10,000 iterations. The
program operations are invoked with the NAND controller handle, address setup
for page 2, transmitter buffer loaded with 0XAA, and number of pages to write
as 1. The read operation is invoked with the same NAND controller handle,
address, receiver buffer, and the number of pages to read as 1. The data in
the receiver buffer is then printed to the serial output using the printf
statement. At the end of the for loop, the receiver buffer holds the data read
from page 2 in last iteration. The receiver buffer data is then compared to
0XAA which is the value programmed. The indexes of the bytes whose value
matches to 0XAA are then identified to be stable bytes.

2) PUF observing the adjacent pages: As per our research, we found out that we
can obtain PUF within flash memory using page disturb technique for adjacent
pages. The initial algorithm we decided on for this technique is shown as
follows: As per this algorithm, we identified block number 0 to be our target
block and page number 3 to be our target page. Following this, we identified
page number 2 and page number 4 to be the two adjacent pages to observe any
bit flips. First of all, the target block 0 was erased, then we write pages
number 2, 3, and 4 with the same data (0xAA) once. Page number 3 is written to
for 10,000 times and observed for bit flips by comparing the RxBuffer data
byte with the data initially written to it. If the data within the byte was
equal to 0xAA, we marked that bit as stable and noted the cycle number for
that particular bit location in our stableBits array. Finally, the stableBits
array was populated after checking for every byte for any flips. In this
algorithm, the adjacent page read for bit flips is not verified bit-wise. With
one bit flip in a byte, the whole byte is disregarded. Hence, individual
stable bits are not being captured. As a solution to this, We utilized the XOR
property - if two variables having the same value are XORed together, the
output will be false. The output will be true only if the two variables are
not the same. The second logical property we utilized is masking provided by
AND operation. The change in a bit is observed by masking each bit. The
location property of a bit within a byte is used for each iteration. After
running this code, we were able to obtain the number of stable bits based on
the iteration number where the other bits were flipped.

3) PUF observing a number of pages: In this part of the experiment, the search
for stable bits was extended to all the 64 pages of the block. To create a
program disturb, the number of write iterations was set to 5000. The new
challenge when dealing with multiple pages was data gathering. Coming up with
a unique fingerprint requires complex processing. Doing all this calculation
within the limits of the Discovery board was not only challenging but also
time-consuming. Thus, the data was gathered in a raw format. This raw data was
later processed using a python script to come up with the PUF instance. The
code snippet shows the algorithm to extract the stable bits. The post-
processing of the gathered data involved the following steps:

  1. 1.

Identify stable bits: For each page in the block, the stable bits were
gathered across multiple passes.

  2. 2.

Find Correlation: For each page in the block, the stable bits were gathered
across multiple passes.

  3. 3.

Generate PUF Instance with the highest co-relation: For each page in the
block, the stable bits were gathered across multiple passes.

##  6 Results

###  6.1 PUF Identification

1\. PUF observing the same page: Algorithm 1 is programmed into the discovery
board and will demonstrate single page program disturb. The same page program
disturbs technique experimented on two different NAND Flash devices. The
number of iterations and the page programmed are also kept the same for both
the devices. The number of iterations experimented on both devices is 10,000.
The page programmed and observed for program disturb is page 2. The stable
indexes bytes identified in both the NAND Flash devices are shown in Figure.
The number of stable bytes identified in each device is different, and also
the indexes of the stable bytes varied. From the experiment, we found after
about 306 iterations the bits start to flip and become much worse afterward.
The explanation to the above observation is as following: Flash can only be
write from 1 to 0, once it is 0, it can only be toggled to 1 by erasing.
Initially each flash cell is programmed as 1s, and then 0xA5 is repeatedly
written to each byte. Ideally, the read output should be 0xA5 consistently.
But after some iteration, some bits flipped from 1 to 0 and can never get
back, as we are not involving erase operation for each iteration.
Consequently, the flipped bits will accumulating to a more and more chaotic
status.

2\. PUF observing the adjacent page: After performing 10,000 writes and reads,
we were able to find the number of stable Bytes. We performed a similar
operation on two devices to observe if the output obtained is unique or not.
We observed that for both the devices, all the bytes on page number 2 were
stable, whereas a limited number of stable were found on page number 4. The
image shows the number of stable bytes on page number 4, where continuous
writes were performed on page number 3.

Figure 3: Speculated Physical Layer of Block on Samsung SLC. Figure 4:
Estimating Bit Flip Pattern. Figure 5: Page 1 signature after 100k multi-page
program writes. Figure 6: Page 2 signature after 100k multi-page program
writes.

3\. PUF observing several pages: Algorithm 2 is used to observe intra-page and
adjacent-page disturbance on pages 1 through 64. However, in this study, we
only evaluated 7 pages. Table I displays the iteration at which Page 0 gets
affected from Page 1 which is less than 1,000 write iterations. However, Page
2 is undisturbed which leads to the conclusion that only pages that undergo
disturbance happen in pairs. This is dependent on how Samsung’s SLC NAND flash
memory is layered physically and the spacing between two out of three pages
are within reachable distance for multi page disturbance seen in Fig. 3 where
there is a page and pairing distance. The page distance is close enough such
that bit flipping occurs while pairing distance is far enough that one pair
does not disturb the other. Pages become more resilient to bit-flipping as
they stray away from page 0. Without knowing the physical layer of how Samsung
built it’s SLC, we speculate that page 0 is the closest to the voltage source
(or plane) of the block since we know that pairs are stacked vertically. The
driving factor for how fast a program disturbance occurs aside from how
physically close the pages are is from heat (of this voltage source). As we
climb the pages vertically, heat dissipates thus causing program disturbances
to increase exponentially. This causes the extraction of signatures to become
much more tedious in following manuscript and will be discussed as we compare
signature extraction of pages 1 and 2. After recording data into log files, we
develop a Python script to analyze and plot our observations. Our analysis
proved that the pages are physically programmed in pairs which follows the
equation (2n, 2n + 1), where n is the pair number. Programming to page 0 only
causes disturbances on page 1, programming on page 2 only affects page 3 and
so on. Further analysis shows that observations for program disturbs require
more iterations as the pages increase from 0. This is seen from plotting Table
I in Fig. 4. An exponential curve fit is applied to both inter- and intra-
page disturbances which allows for estimating when a page and it’s pair will
encounter first bit flips.

The black colored entries in the heatmap show bytes that have completely
flipped. Meaning after erasing the page, the all bits in a byte have changed
from 0xFF to 0x00. The white entries show erased bytes that remained 0xFF
after page erase. Using the non black colored entries, we can use these
markings to show the bytes and bits of a page that have not flipped from
program disturbs. After 83k iterations, the signature became stable. The
colored entries represent bit flips within a byte. In Page 2’s signature it is
quite evident that all the bits in each of the 2048 bytes do not completely
flip. Only a few of the 8 bits flip. We can use the colored entries to show
the bytes. Similar to page 1’s heatmap where the black colored entries
represent the flipped bytes and white entries show stable bytes, we observe
that 100k iterations is not enough program writes to observe a stable
signature specifically for page 2. Examples can be seen in Fig. 5 and Fig. 6.
The higher numbers are represented by warmer colors, which are indicative of
bits that were highly resistant to disturbance and only flipped after running
the experiment for high number of cycles.

Program on Page n | Page 1 | Page 2 | Page 3 | Page 4 | Page 5 | Page 6 | Page 7
---|---|---|---|---|---|---|---
Page (n-1)/k | <1 | No bit flip | 4 | No bit flip | 28 | No bit flip | 64
Page n/k | <1 | <1 | 1 | 3 | 5 | 7 | 15
Page (n+1)/k | No bit flip | 2 | No bit flip | 13 | No bit flip | 39 | No bit flip
Table 1: First bit flip for disturbed pages 1 to 7. Column 1 is showing number
of iteration for the First bit flip.

##  7 Conclusion and Future Work

In conclusion, this project helps understand the PUF and its implementation.
The program disturb technique is used to extract PUF. As Read disturb
technique needs more than a million reads to observe bit flips, program
disturb is performed. Read or Program Disturb on SLC NAND flash memory causes
lesser bit flips as compared to MLC NAND flash memory device. In this project,
unique signatures in different NAND flash memory are extracted. PUF is
captured on the same page, in the adjacent pages, and all pages. The result is
compared with two different NAND flash memory devices to compare the
signatures. PUF is identified to be unique for each NAND flash memory. Thus,
PUF extracted, could be used as a key to secure systems. As observed, upon
performing bit-wise operations on the data read from a page, the number of
stable bits is huge even after 50000 operations. An algorithm is designed to
tackle this scenario wherein it could run the outer loop for 1,00,000 writes
and find the number of stable bits after every 10,000 writes.

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Topology of Born-Infeld AdS black holes in 4D novel Einstein-Gauss-Bonnet
gravity

Pavan Kumar Yerra<EMAIL_ADDRESS>and Chandrasekhar
<EMAIL_ADDRESS>

1 Institute of Physics, Sachivalaya Marg,

Bhubaneswar, Odisha, 751005, India

2 School of Basic Sciences

Indian Institute of Technology Bhubaneswar

Bhubaneswar, Odisha, 752050, India

Abstract

The topological classification of critical points of black holes in 4D
Einstein-Gauss-Bonnet gravity coupled to Born-Infeld theory is investigated.
Considered independently, Born-infeld corrections to the Einstein action alter
the topological charge of critical points of the charged AdS black hole
system, whereas the Gauss-Bonnet corrections do not. For the combined system
though, the total topological charge of the Einstein-Gauss-Bonnet theory is
unaltered in the presence of Born-Infeld coupling.

##  1 Introduction

Black holes offer a unique window to investigate the effects of strong
gravitational phenomena [1], and an understanding of their thermodynamics [2,
3, 4] is perhaps the key to insights in to the nature of degrees of freedom of
gravity [5]. Phase transitions of black holes, in particular the Hawking-Page
transition [6], evoke compelling interest, in part due to possible
interpretation in the context of gauge/gravity duality [7, 8, 9]. For charged
black holes in anti de Sitter (AdS) spacetimes, a first-order phase
transitions takes place between small and large black holes, very similar to
the liquid-gas transitions which occur in van der Waals (VdW) fluid [10].
These investigations become more invigorating in the black hole chemistry
program [11, 12, 13, 14, 15, 16], where the cosmological constant $\Lambda$ is
treated as a variable, leading to the concept of a pressure $P=-\Lambda/8\pi$,
with a thermodynamic volume $V$ acting as its conjugate. In this novel set up,
the first law $dM=TdS$ assumes the form $dM=TdS+VdP\,$, where $M$ is now
reinterpreted as enthalpy $H$ [12]. The phase transitions acquire an
appropriate meaning and analogy with liquid-gas transitions is now complete,
including the existence of a critical region where the first order phase
transitions end in a second order transition [10, 11, 17].

Criticality in phase transitions in statistical physics and especially in
black holes is an interesting topic on its own right due to emergence of
several universal phenomena and scaling. In addition to the small to large
black hole phase transition, there are now new examples of transitions such as
reentrant transitions, solid-liquid-gas type transitions with multiple
critical points in higher dimensions [18]. For many of these transitions to
exist, one typically requires the presence of both gauge and gravity higher
curvature corrections such as Born-infeld and Gauss-Bonnet terms added to the
Einstein action or involving Kerr-AdS black holes with multiple rotation
parameters. In fact, the later system can also exhibit tricritical points
though in a restricted parameter range, including the existence of
superfluidity [19, 20, 21, 22, 23, 18, 24, 25, 26, 27, 28, 29, 30].
Thermodynamics of black holes in the presence of higher derivative curvature
terms of gauge and gravity is important from holographic view point too, as
these appear naturally in discussions involving semiclassical quantum gravity
and also in the low energy limit of certain superstring theories[31, 22]. In
the context of AdS/CFT correspondence, Gauss-Bonnet terms for example are
useful as they lead to corrections in the large $N$ expansion of gauge
theories on the boundary, in the strong coupling limit. Such terms have in
fact given important contributions to the viscosity to entropy ratio, apart
from leading to novel bounds [32]. A convenient advantage involving Gauss-
Bonnet and Lovelock terms in general is that the field equations in these
theories do contain more than two derivatives of the metric in five and higher
dimensions, are topological in four dimensions, and vanish for dimension less
than three. The general Lovelock terms have similar characteristics in their
critical dimension and are useful due to the absence of ghost-like modes.

It is then interesting that in a novel limit obtained by rescaling the Gauss-
Bonnet coupling as $\alpha\to\alpha/(D-4)$ in four dimensions [33, 34], non-
trivial theory could be obtained, giving the new 4D Einstein Gauss-Bonnet
theory [35]. There have been a lot of developments with black hole solutions
obtained in a with a wide variety of non-trivial coupling to matter and gauge
terms in this novel gravity theory [36, 37, 38, 39, 40, 41, 42, 43, 44]. In
particular, Born-Infeld coupled Einstein gravity is important to study as a
possible regular theory over the Maxwell electrodynamics, as the later usually
includes infinite self energy due to a singularity at the position of
charge[45, 46]. Further, the Born-Infeld terms are also appear from the low
energy effective action of open string theories. In the past, black hole
solutions in these theories have garnered wide attention[47, 48], which has
become more interesting in the extended thermodynamic framework, as these
theories exhibit novel reentrant phase transitions in four dimensional
Einstein Born-infeld gravity, apart from the existence of the small and large
black hole transition, similar to the vdW system[49, 50]. In addition to the
traditional treatments of phase transitions in black holes which giving useful
information, it is important to look for novel methods to classify the
critical points in general black hole systems.

One new idea involves a topological approach to classification of critical
points of phase transitions in general theories of gravity, proposed in [51],
where each critical point can be bestowed with a topological charge. For
instance, critical points which carry a topological charge $Q_{t}=-1$ are
termed conventional, where as the ones with opposite topological charge are
called novel. For consistency in situations where there are multiple critical
points, a more appropriate classification is to think of the former (later)
type as coming from disappearance (appearance) of new phases [52], which seems
to hold in the case of isolated critical points as well [53]. An unexpected
result is that, the Born-Infeld corrections to the action change the
topological class of critical points of black holes [51], where as the Gauss-
Bonnet corrections do not [52]. To get deeper insight into this intriguing
topological behavior, it is desirable to study nature of critical points in a
more general theory of gravity, possibly by exploring black holes in a system
consisting of both types of terms. The main objective of this letter is to
report progress in this regard, by investigating the topological nature of
critical points in a black hole solution in 4D Einstein Gauss-Bonnet gravity
coupled to Born-Infeld Electrodynamics[45, 46, 47, 48, 49, 50] found in [54].
The thermodynamics and phase transitions in the extended thermodynamic
framework has been explored earlier in [55].

The rest of the paper is organised as follows. In section-(2), we closely
follow [54, 55, 42, 56, 57] and collect salient aspects of thermodynamics of
4D Einstein Gauss-Bonnet gravity coupled to Born-Infeld AdS black holes in
four dimensions and briefly present the topological approach required for
analysis in section(3). Our main calculations and results are presented in
section-(3) and the discussion of the topological nature of critical points is
given in subsection-(3.5). Remarks and conclusions are given in section-(4).

##  2 Thermodynamic phase structure and Topology

We consider the thermodynamics of Born-Infeld AdS black holes in four
dimensional Einstein-Gauss-Bonnet gravity in extended phase space, where we
can have pressure $P$, via $P=-\Lambda/8\pi$, and its conjugate thermodynamic
volume $V$ [12]. These black holes are characterized by the thermodynamic
quantities, temperature $T$, entropy $S$, mass $M$, and Gibbs free energy $G$,
given by [54, 55]:

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{r_{+}^{2}-\alpha+8\pi
Pr_{+}^{4}+2r_{+}^{4}\beta^{2}\Big{(}1-\sqrt{1+\frac{Q^{2}}{r_{+}^{4}\beta^{2}}}\Big{)}}{4\pi
r_{+}(r_{+}^{2}+2\alpha)},$ (2.1) $\displaystyle S$ $\displaystyle=$
$\displaystyle\pi r_{+}^{2}+4\pi\alpha\text{ln}\frac{r_{+}}{\alpha},$ (2.2)
$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{r_{+}}{2}\bigg{[}1+\frac{8\pi
Pr_{+}^{2}}{3}+\frac{\alpha}{r_{+}^{2}}+\frac{2\beta^{2}r_{+}^{2}}{3}\bigg{(}1-\sqrt{1+\frac{Q^{2}}{\beta^{2}r_{+}^{4}}}\bigg{)}+\frac{4Q^{2}}{3r_{+}^{2}}\text{}_{2}F_{1}\Big{[}\frac{1}{4},\frac{1}{2},\frac{5}{4},\frac{-Q^{2}}{\beta^{2}r_{+}^{4}}\Big{]}\bigg{]},\,\,$
(2.3) $\displaystyle G$ $\displaystyle=$ $\displaystyle M-TS,$ (2.4)

here, $r_{+}$ is the horizon radius, $\text{}_{2}F_{1}$ is the hypergeometric
function, $Q$ is the charge of the black hole, $\alpha$ is the Gauss-Bonnet
coupling parameter, and $\beta$ is Born-Infeld parameter. We note here that,
in the limit of $\alpha\rightarrow 0$, the above thermodynamic quantities
correspond to Born-Infeld AdS black holes [49], whereas, in the limit of
$\beta\rightarrow\infty$, they correspond to charged AdS black holes in four
dimensional novel Einstein-Gauss-Bonnet gravity [39, 36].

###  2.1 Phase Structure

Figure 1: The available parameter region of $(\alpha,\beta)$, for the
existence of small/intermediate/large black hole transitions and triple
points, shown with gray colored region (where there can be three critical
points). The blue colored region shows the existence of van der Waals type
small/large black hole transitions (here there is only one critical point).
Plot is displayed for $Q=1$.

Charged AdS black holes with Gauss-Bonnet coupling $\alpha$, in novel four
dimensional Einstein-Gauss-Bonnet gravity, are known to exhibit the phase
structure of van der Waals type small/large black hole first order
transitions, terminating in a second order critical point [39]. On the other
hand, with Born-Infeld coupling $\beta$, charged AdS black holes in four
dimensions exhibit a phase structure containing reentrant phase transitions,
along with the usual van der Waals type small/large black hole transitions
[49]. However, when both the couplings $(\alpha,\beta)$ are present, the phase
structure of charged AdS black holes gets an interesting modification. In
certain parameter region of $(\alpha,\beta)$, the phase structure includes a
small/intermediate/large black hole transitions and also the triple points
(where, three phases can coexist), besides the van der Waals type small/large
black hole transitions [55]. The available parameter region of
$(\alpha,\beta)$, for the existence of small/intermediate/large black hole
transitions and triple points 111All the critical points may not appear in the
phase diagram. Only those appear that globally minimize the Gibbs free
energy., is shown as gray colored region in Fig. 1. The van der Waals type
small/large black hole transitions are shown as blue colored region in Fig. 1,
where there is only one critical point. The detailed discussion of phase
structures related to these critical and triple points, can be found in [55].
Our aim in this paper, is to study the topological properties associated with
these critical points 222We do not consider the case of triple points here.
that belong to different parameter regions of $(\alpha,\beta)$ shown in the
Fig. 1.

###  2.2 Topological charge of critical points

Let us outline the proposal in [51] for assigning topological charges to
critical points. The temperature $T=T(S,P,x^{i})$ is typically a function of
entropy $S$, pressure $P$ and other thermodynamic variables $x^{i}$, which can
be extremised to get the critical points from

$(\partial_{S}T)_{P,x^{i}}=0,\ \ (\partial_{S,S}T)_{P,x^{i}}=0\,.$ (2.5)

Above relation implies the possibility of constructing a Duan’s potential [51]

$\Phi=\frac{1}{\text{sin}\theta}\,T(S,x^{i})\,,$ (2.6)

which in principle can be found upon eliminating one of the thermodynamic
variables through eqn. (2.5), with a factor of $1/\sin\theta$ added for
convenience. The set up of Duan’s $\phi$-mapping theory can now be applied by
constructing a vector field $\phi^{a}=(\phi^{S},\phi^{\theta})$, where
$\phi^{S}=(\partial_{S}\Phi)_{\theta,x^{i}}$ and
$\phi^{\theta}=(\partial_{\theta}\Phi)_{S,x^{i}}$. The existence of a
topological current $j^{\mu}$ satisfying $\partial_{\mu}\,j^{\mu}=0$ is a key
feature, which gets non-zero contribution only from the points where the
vector field $\phi^{a}$ vanishes, i.e., $\phi^{a}(x^{i})=0$. This ensures the
existence of topological charge [57, 56, 51]

$Q_{t}=\int_{\Sigma}j^{0}d^{2}x=\Sigma_{i=1}w_{i}\,,$ (2.7)

contained with in a region $\Sigma$ and $w_{i}$ is the winding number for the
$i$-th point where $\phi$ is zero. Since, $Q_{t}$ can be positive or negative,
the critical points were proposed to be further divided into two different
topological classes, i.e., the conventional (where $Q_{t}=-1$) and the novel
(where $Q_{t}=+1$) [51]. If one allows $\Sigma$ to span the full parameter
space of the thermodynamic system, the critical points of different
thermodynamic systems can be classified into different topological classes.

##  3 Topology of critical points in Born-Infeld AdS black holes in 4D
Einstein-Gauss-Bonnet gravity

First we obtain the general expression for topological charge which can be
used to analyze the critical points of Born-Infeld AdS black holes in four
dimensional Einstein-Gauss-Bonnet gravity.

###  3.1 Topological Charge

Following the discussion in subsection-(2.2), we start by computing the
thermodynamic function $\Phi$, using the eqn (2.1) for temperature 333where
for a given value of $\alpha$, entropy $S=S(r_{+})$ from eqn. (2.2). $T$,
given by

$\Phi=\frac{1}{\text{sin}\theta}T(r_{+},Q,\alpha,\beta)=\frac{1}{\text{sin}\theta}\frac{\bigg{(}r_{+}^{2}-2\alpha-\frac{2Q^{2}}{\sqrt{1+\frac{Q^{2}}{r_{+}^{4}\beta^{2}}}}\bigg{)}}{2\pi
r_{+}(r_{+}^{2}+6\alpha)}.\\\ $ (3.1)

Then, the vector field $\phi=(\phi^{r},\phi^{\theta})$ is obtained to be:

$\displaystyle\phi^{r}$ $\displaystyle=$ $\displaystyle\partial_{r_{+}}\Phi$
$\displaystyle=$
$\displaystyle\frac{\text{csc}\theta}{2\pi(r_{+}^{2}+6\alpha)^{2}}\bigg{\\{}(r_{+}^{2}+6\alpha)\Big{[}2-\frac{4Q^{4}}{r_{+}^{6}\beta^{2}\Big{(}1+\frac{Q^{2}}{r_{+}^{4}\beta^{2}}\Big{)}^{\frac{3}{2}}}\Big{]}-(3+\frac{6\alpha}{r_{+}^{2}})\Big{[}r_{+}^{2}-2\alpha-\frac{2Q^{2}}{\sqrt{1+\frac{Q^{2}}{r_{+}^{4}\beta^{2}}}}\Big{]}\bigg{\\}},$
$\displaystyle\phi^{\theta}$ $\displaystyle=$
$\displaystyle\partial_{\theta}\Phi=-\frac{\text{cot}\theta\text{csc}\theta}{2\pi
r_{+}(r_{+}^{2}+6\alpha)}\bigg{(}r_{+}^{2}-2\alpha-\frac{2Q^{2}}{\sqrt{1+\frac{Q^{2}}{r_{+}^{4}\beta^{2}}}}\bigg{)}.$
(3.2)

The normalised vector field can be computed from
$n=(\frac{\phi^{r}}{||\phi||},\frac{\phi^{\theta}}{||\phi||})$. The
measurement of winding number $w_{i}$ enclosed by a given contour reveals the
topological charge. If a critical point is contained inside a given contour,
then its topological charge (i.e., the winding number $w_{i}$) would be non-
zero, otherwise the contour gives zero topological charge [51, 58, 59, 60, 61,
62]. We consider a contour $C$, which is piece-wise smooth and positively
oriented in the orthogonal $\theta-r$ plane, parameterized by the angle
$\vartheta\in(0,2\pi)$, as [51, 60]:

$\displaystyle\left\\{\begin{aligned} r&=a\cos\vartheta+r_{0},\\\
\theta&=b\sin\vartheta+\frac{\pi}{2}.\end{aligned}\right.$ (3.3)

Then, we compute the deflection angle $\Omega({\vartheta})$ of the vector
field $\phi$, along the contour $C$, given by

$\Omega(\vartheta)=\int_{0}^{\vartheta}\epsilon_{ab}n^{a}\partial_{\vartheta}n^{b}d\vartheta,$
(3.4)

from which, one can obtain the topological charge $Q_{t}$, as

$Q_{t}=w_{i}=\frac{1}{2\pi}\Omega(2\pi).$ (3.5)

In the following subsections, we compute the topological charges corresponding
to the critical points present in different parameter regions, obtained for
various values of $(\alpha,\beta)$.

###  3.2 Case 1: One critical point

This is the case where we have only one critical point (CP), associated with
the van der Waals type small/large black holes transitions that occur in the
blue colored region of Fig. 1. For this case, the vector field $n$, plotted in
the Fig. 2a, shows the presence of the critical point (critical values are
given in the Table 2).

Case | Critical Point (CP)
---|---
| $\beta$ |  | $\text{CP}_{1}$ | $\text{CP}_{2}$ | $\text{CP}_{3}$
Case-1 |  | $r_{c}$ | 0.4459 | - | -
0.26 | $P_{c}$ | 0.0374 | - | -
| $T_{c}$ | 0.1042 | - | -
Case-2 |  | $r_{c}$ | 0.604 | 2.0845 | 1.0714
0.38 | $P_{c}$ | 0.00413 | 0.00373 | 0.00206
| $T_{c}$ | 0.0435 | 0.0453 | 0.0404
Table 1: Critical values at $Q=1$, $\alpha=0.01$.

Case | $C_{1}$ | $C_{2}$ | $C_{3}$ | $C_{4}$
---|---|---|---|---
Case-1 | a | 0.07 | 0.07 | - | -
b | 0.4 | 0.4 | - | -
$r_{0}$ | 0.45 | 0.8 | - | -
Case-2 | a | 0.15 | 0.15 | 0.15 | 1.1
b | 0.4 | 0.4 | 0.4 | 0.7
$r_{0}$ | 0.6 | 2.08 | 1.07 | 1.4
Table 2: Parametric coefficients of contours.

We construct two contours $C_{1}$ and $C_{2}$ for the vector field $n$
(parametric coefficients of the contours are given in the Table 2). The
contour $C_{1}$ contains the critical point, which gives non-zero contribution
to the topological charge of the critical point, whereas, the contour $C_{2}$
does not contain the critical point, hence it gives zero topological charge.
This can be seen from the behavior of the deflection angle
$\Omega(\vartheta)$, along the contours $C_{1}$ and $C_{2}$, plotted in the
Fig. 2b. The angle $\Omega(\vartheta)$, along the contour $C_{1}$, decreases
to reach $-2\pi$ at $\vartheta=2\pi$, whereas, along the contour $C_{2}$, it
first decreases, then increases and finally vanishes at $\vartheta=2\pi$.

(a)

(b)

Figure 2: For case-1: (a) The vector field $n$ in $\theta-r$ plane, shows the
presence of critical point $\text{CP}_{1}$ (black dot) at
$(r,\theta)=(0.446,\frac{\pi}{2})$. The contour $C_{1}$ hides the critical
point, while, the contour $C_{2}$ does not. (b) $\Omega$ vs $\vartheta$ for
contours $C_{1}$ (red curve), and $C_{2}$ (green curve).

Thus, in this case, the topological charge of the critical point
$\text{CP}_{1}$ would be (from eqn. 3.5) $Q_{t}|_{\text{CP}_{1}}=-1$, which is
the total topological charge of the system as well (since, there is only one
critical point).

###  3.3 Case 2: Three critical points

In this case, we have three critical points, associated with the
small/intermediate/large black hole transitions that occur in the gray colored
region of Fig. 1. The presence of these three critical points (critical values
are given in the Table 2) can be seen clearly from the vector field plot shown
in Fig. 3a.

(a)

(b)

Figure 3: For case-2: (a) The vector field $n$ in the $\theta-r$ plane, shows
the presence of three critical points $\text{CP}_{1}$, $\text{CP}_{2}$, and
$\text{CP}_{3}$ (black dots) at
$(r,\theta)=(0.604,\frac{\pi}{2}),(2.085,\frac{\pi}{2}),\,\text{and}\,(1.07,\frac{\pi}{2})$,
which are contained inside the contours $C_{1}$, $C_{2}$ and $C_{3}$,
respectively. The contour $C_{4}$ hides all the three critical points. (b)
$\Omega$ vs $\vartheta$ for contours $C_{1}$ (red curve), $C_{2}$ (yellow
curve), $C_{3}$ (green curve), and $C_{4}$ (gray curve).

For this case, we then consider four contours $C_{1}$, $C_{2}$, $C_{3}$, and
$C_{4}$ (parametric coefficients of the contours are given in the Table 2).
The contours $C_{1}$, $C_{2}$, and $C_{3}$, contain the critical points
$\text{CP}_{1}$, $\text{CP}_{2}$, and $\text{CP}_{3}$, respectively, while the
contour $C_{4}$ contains all the three critical points. The behavior of the
deflection angle $\Omega(\vartheta)$ for these contours is as shown in Fig 3b,
from which we find that $\Omega(2\pi)=-2\pi,-2\pi,+2\pi,\,\text{and}\,-2\pi$,
for the contours $C_{1}$, $C_{2}$, $C_{3}$, and $C_{4}$, respectively. This
results in the topological charges of the critical points coming out to be
$Q_{t}|_{\text{CP}_{1}}=-1,\,Q_{t}|_{\text{CP}_{2}}=-1$, and
$Q_{t}|_{\text{CP}_{3}}=+1$. Therefore, in this case, the total topological
charge of the system (which is additive) would be again $-1$, given by the
contour $C_{4}$.

###  3.4 Limiting cases:

As mentioned earlier, in the limit of $\beta\rightarrow\infty$, we recover the
charged AdS black holes in four dimensional Einstein-Gauss-Bonnet gravity,
where we have only one critical point corresponding to the van der Waals type
small/large black hole transitions. In this case, one can check that the
topological charge of the critical point would be $-1$. On the other hand, in
the limit of $\alpha\rightarrow 0$, we recover the Born-Infeld AdS black
holes. In this case, we have two critical points corresponding to reentrant
phase transitions for the range:
$\frac{1}{Q\sqrt{8}}<\beta<\frac{\sqrt{3+2\sqrt{3}}}{6Q}$, otherwise, there is
only one critical point related to van der Waals type small/large black hole
transitions [49]. The topological charges of these critical points have been
computed in [51], where it has shown that the total topological charge of the
system, in the case of two critical points, vanishes.

###  3.5 Nature of the Critical points

From the view of topology, critical points have been classified into
conventional ($Q_{t}=-1$) or novel ($Q_{t}=+1$), based on the topological
charge they carry [51]. A further proposal of [52] is that, in the phase
diagram as the pressure increases, conventional critical point acts as the
phase annihilation point, where the phases disappear, whereas, the novel
critical point acts as the phase creation point, where the new (stable or
unstable) phases appear. This classification scheme continues to hold for
isolated critical points [53] and in the current case too, as we summarise
below.

(a)

(b)

Figure 4: $T$ as a function of $r_{+}$ for different pressures, showing the
appearance of new (stable or unstable) phases near the novel critical point
(green dot), and disappearance of phases near the conventional critical point
(red dot) as the pressure increases. Dashed curves denote unstable black hole
branches and solid curves denote stable black hole branches. Pressure of the
isobars increases from bottom to top. (a) for case-1, (b) for case-2 (Inset:
for $P>P_{c2}$).

For case-1 discussed earlier, there was one critical point $\text{CP}_{1}$,
for which the topological charge is $Q_{t}=-1$ (conventional). The phase
structure around this critical point is as shown in the Fig.4a, where the
number of phases in an isobar decreases as the pressure increases, making it a
phase annihilation point. For case-2, there were two conventional critical
points $(\text{CP}_{1}\,\text{and}\,\text{CP}_{2})$, and one novel critical
point $(\text{CP}_{3})$. As the pressure increases, the disappearance of
phases at the conventional critical points and the appearance of new (stable
or unstable) phases at novel critical point, can be seen from the Fig. 4b. The
behaviour of Gibbs free energy $G$ around these critical points, plotted in
Fig. 5, clearly shows the corresponding phase creation and annihilation
behaviours.

(a)  (b)

(c)  (d)

Figure 5: For case-2: Behaviour of the Gibbs free energy as a function $T$ for
different pressures, showing the appearance of new (stable or unstable) phases
near the novel critical point $\text{CP}_{3}$ (with pressure $P_{c3}$), and
the disappearance of phases near the conventional critical points
$\text{CP}_{2}$ ( with pressure $P_{c2}$), and $\text{CP}_{1}$ (with pressure
$P_{c1}$), on increase of pressure. (a) $P=0.0013$, (b) $P=0.0028$, (c)
$P=P_{c2}=0.0037$, (d) $P=\\{0.0038,P_{c1}=0.0041,0.0045\\}$.

##  4 Conclusions

We investigated the topological classification of critical points in the Born-
Infeld AdS black holes in four dimensional novel Einstein-Gauss-Bonnet gravity
in the extended phase space. With only the Born-Infeld corrections to AdS
black holes, there are just reentrant phase transitions, whereas with the
addition of Gauss-Bonnet coupling $\alpha$, the system exhibits
small/intermediate/large black hole transitions and triple points too.
Depending on the range of Gauss-Bonnet and Born-Infeld couplings
$(\alpha,\beta)$, the phase structure of these black holes can admit one or
three critical points, related to usual van der Waals type small/large black
hole transitions or an interesting small/intermediate/large black hole
transitions, respectively. We computed the topological charges associated with
these critical points and the results are summarized in Table 3.

Case | Number of | Topological | Total Topological
---|---|---|---
| Critical Points | Charge $Q_{t}$ | Charge
Case 1: |  |  |
1 | $Q_{t}|_{\text{CP}_{1}}=-1$ | $-1$
Case 2: |  | $Q_{t}|_{\text{CP}_{1}}=-1$ |
3 | $Q_{t}|_{\text{CP}_{2}}=-1$ | $-1$
| $Q_{t}|_{\text{CP}_{3}}=+1$ |
Limiting Case: $\alpha\rightarrow 0$ [51] |  | $Q_{t}|_{\text{CP}_{1}}=+1$ |
2 | $Q_{t}|_{\text{CP}_{2}}=-1$ | 0
Limiting Case: $\beta\rightarrow\infty$ |  |  |
1 | $Q_{t}|_{\text{CP}_{1}}=-1$ | $-1$
Table 3: Summary of results

It was noted earlier that the Born-infeld corrections to the Einstein action
alter the topological charge of the charged AdS black hole system 51, whereas
the Gauss-Bonnet and more generally the Lovelock type corrections do not 52.
Here, we showed that when topology of critical points is studied with both the
above Gauss-Bonnet and Born-Infeld corrections side by side (in four
dimensions), the topological nature is unaltered, possibly indicating the
dominance of former set of terms. Further, our results also obey the parity
conjecture of critical points proposed in [52], which says that, “for odd
(even) number of critical points, the total topological charge is an odd
(even) number”.

Relying on the set of examples studied thus far it would be nice to extend our
conclusions for situations where there are general higher derivative
gauge/gravity corrections to Einstein action. An important point to check is
whether the nature of higher derivative gravity corrections to Einstein action
is to preserve the topological class of black hole critical points, whereas
that of gauge corrections is to alter it. The competing nature for changing
the topological charge of critical points, between the Gauss-Bonnet and Born-
Infeld corrections in higher dimensions and with the possible inclusion of
other types of non-linear gauge/gravity terms with multiple critical points
should analysed too [63, 44, 63, 64, 65, 66, 67].

## Acknowledgements

One of us (C.B.) thanks the DST (SERB), Government of India, for financial
support through the Mathematical Research Impact Centric Support (MATRICS)
grant no. MTR/2020/000135.

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