# Layer 2 be or Layer not 2 be: Scaling on Uniswap v3

Austin Adams
Uniswap Labs
<EMAIL_ADDRESS>We thank Maude Wilson, Gordon Liao, Ciamac Moallemi, Ed
Felten, Brad Bachu, Xin Wan, and Bridgett Frey for their comments.

###### Abstract

This paper studies the market structure impact of cheaper and faster chains on
the Uniswap v3 Protocol. The Uniswap Protocol is the largest decentralized
application on Ethereum by both gas and blockspace used, and user behaviors of
the protocol are very sensitive to fluctuations in gas prices and market
structure due to the economic factors of the Protocol. We focus on the chains
where Uniswap v3 has the most activity, giving us the best comparison to
Ethereum mainnet. Because of cheaper gas and lower block times, we find
evidence that the majority of swaps get better gas-adjusted execution on these
chains, liquidity providers are more capital efficient, and liquidity
providers have increased fee returns from more arbitrage. We also present
evidence that two second block times may be too long for optimal liquidity
provider returns, compared to first come, first served. We argue that many of
the current drawbacks with AMMs may be due to chain dynamics and are vastly
improved with cheaper and faster transactions.

##  1 Introduction

Automated market makers (AMMs) - like the Uniswap Protocol - have become the
dominant market structure for onchain markets, trading over $3 trillion
dollars to date.111Link to source The Ethereum Protocol, currently the largest
chain by onchain volume, has a limited availability for computation - or block
space - and as user demand increases, the cost has risen due to limited
supply. This is by design - block space needs to be constrained due to
security and state growth. But as markets mature and the average transaction
becomes more complex (increasing aggregate transaction costs), Ethereum’s
base-layer throughput cannot feasibly keep up. As a result, there are
comparatively fewer transactions on mainnet today than a few years ago, and
those transactions are more expensive than they used to be.

To scale Ethereum, the ecosystem has chosen a modular scaling approach through
utilizing layer 2 protocols (L2s). L2s utilize the trustless data availability
of Ethereum while outsourcing computational validation to various schemes,
such as zero-knowledge or optimistic fraud proofs. Network costs on L2s are
much lower compared to Ethereum mainnet, and that cost will reduce even
further with the forthcoming implementation in Dencun of proto dank-
sharing.222Dencun refers to the joint consensus hardfork Deneb and execution
hardfork Cancun, which goes by the portmanteau, Dencun At the current time,
prediction markets price a 60x decrease in the already cheap data availability
costs of L2s, but the benefit will likely be lower due to induced demand.

In this paper, we will study the impact of cheaper and faster transaction
costs on the Uniswap Protocol. To do this, we will contrast market structure
of Uniswap v3 on largest L2s to Ethereum mainnet. We will focus on L2s,
because the largest deployments of Uniswap v3 are on these chains and Ethereum
mainnet. However, we should note that sidechains or alternative L1s - like
Solana or Monad - have some of the same benefits.

Lower costs will have significant benefits for Uniswap Protocol users.
According to Adams et al. (2023a), 22% of the total cost of the average
Ethereum transaction on the Uniswap Interface is Ethereum gas costs. On
transactions less than $1k, gas costs can make up as much as 98% of the total
costs of trading. Lowering this cost allows users to pay less for the same
service, potentially making trading costs comparable to centralized exchanges.
Additionally, cheaper arbitrage transactions, and the related increase in fees
paid by arbitrageurs, make LPs more profitable. It is also possible that users
- both swappers and LPs - who are currently priced out of Ethereum mainnet
would take part in the ecosystem if it were cheaper.

Today, AMMs on Ethereum currently have better execution for large trades due
to higher market depth (because of higher TVL) and amortized gas costs over
the larger input amount. However, most users (and future potential users)
likely do not trade the size where it is better to trade on Ethereum. By
decreasing costs, we can show the potential of scaling beyond Ethereum, and
the practical benefits of AMMs to new users by allowing cheaper, more
efficient, and easier transactions.

Contributions. In this paper, we contribute to the blossoming field of Uniswap
v3 empirical research by focusing on the impact cheaper and faster chains have
on the largest application by blockspace on Ethereum - the Uniswap Protocol.
We show that the around 90% of Uniswap Interface swappers would have better
execution on the studied L2s. LPs on the Protocol also benefit from 50%
average higher capital efficiency, and 20% more average arbitrage fee returns.
We also talk about how these differences could alleviate many of the concerns
for wide-spread adoption of AMMs outside of the crypto ecosystem.

There is currently a lack of research on the impact of faster chains on
Uniswap v3, because the required data is cumbersome to pull, maintain, and
utilize. To support more research in this area, we developed and open-sourced
a Python package that recreates Uniswap v3 functionality natively in Python -
called v3-polars - along with this paper. It can be found here. All empirical
work in this paper is either created using this package and can be found in
this repository, or the link is provided. Therefore, the entire paper should
be replicable.

Overall, we present the first overview of the entire Uniswap v3 ecosystem on
Ethereum mainnet and L2s and compare them. We discuss the current benefits -
and the downsides - of the current L2 ecosystem for the Uniswap Protocol. As
Ethereum continues to move towards modularity, we hope this paper provides
insight into what the future of onchain decentralized markets may look like,
and how these changes could solve some of the challenges with widespread AMM
adoption today.

Related Literature. Literature on automated market makers has grown rapidly in
the past few years. Early roots can be traced back to Hanson (2007) and Othman
et al. (2013). Some recent literature about automated market makers include
Capponi and Jia (2021), Lehar and Parlour (2021), and Milionis et al. (2022).
Uniswap v3 specifically has been described in Adams et al. (2021) and
empirically studied by Adams et al. (2023a).

While there are relatively few papers on the empirics of Uniswap v3 outside of
Ethereum mainnet, most likely due to the difficulty of generating the
necessary data, there are a few notable contributions. Chemaya and Liu (2022)
presents findings showing that CPMMs on L2s give users better prices compared
to Ethereum, but that users do not optimally switch based on the best total
cost - showing a perceived security metric for the chain. Caparros et al.
(2023) focuses on slippage and concentration between Polygon and Ethereum,
showing similar benefits from downstream gas cost reductions. Mitchell (2022)
shows that the average full-range liquidity position does not have increased
total fee returns on L2s, which will not contradict our results - as we focus
only on arbitrage fee returns.

A technical discussion of L2s is out of the scope of this paper. For that, see
Thibault et al. (2022).

##  2 Overview of Current Uniswap v3 Deployments

In this paper, we will focus on the Ethereum mainnet, the two rollups with
significant Uniswap v3 deployments (Arbitrum and Optimism), and the sidechain,
Polygon. Some argue that Polygon is not technically an L2 due to a distinct
validator set, but we will refer to it as an L2 for brevity. While Uniswap v3
is deployed on other chains, these chains have the most volume and TVL of the
Uniswap v3 deployments.333For an exhaustive and updated list of canonical
deployments, see here The majority of volume is still found on mainnet, and
during times of volatility volume is proportionally higher. However, it is
important to keep in mind that although volume may be higher on mainnet, the
majority of transactions actually occur on L2s. Ethereum generally only has
about 25% share of transaction count, but around 60% of volume.

Figure 1A: Share of Volume by Blockchain on Uniswap v3

Source

Figure 1B: Share of Transactions by Blockchain on Uniswap v3

Source

During times of high volume, volume returns proportionally to Ethereum
mainnet, as Ethereum’s share of total volume spikes during high volume days
(which can be seen in the spikes of Ethereum market share in Figure 1A). There
are few potential reasons for this. First, when there is volatility, the
larger value locked on Ethereum mainnet has greater turnover, resulting in a
more extreme volume spike. Users (or likely arbitrageurs) may also be more
inelastic to gas costs during times of volatility, making them more likely to
swap at high gas costs due to the presence of profitable arbitrage
opportunities resulting from volatility. This negates one of the major
benefits of L2s, leading to increased volume on mainnet.

Table 1: Uniswap v3 Protocol Statistics

Note: All calculations are year to date (Jan 1st, 2023 - Jan 1st, 2024)
Note: Ethereum statistics only include Uniswap v3 (to allow direct
comparisons)
Source

Any analysis of Protocol volume must look at both the amount of trades being
executed and the size of those trades to get the complete picture. While the
majority of volume is still found on mainnet, the majority of transactions are
actually occurring on L2s. While Arbitrum has almost twice as many swaps as
mainnet, the median size of mainnet swaps is almost three times larger than
Arbitrum, leading to higher aggregate volume on Ethereum. One explanation for
this is the higher liquidity on mainnet, and the arbitrage that occurs is
larger as a result. More liquidity leads to larger arbitrage transactions, but
not more swapping opportunities, leading to a larger average swap size from
arbitrage. The other main reason is that fixed gas costs for swaps on mainnet
are larger, and people are less likely to execute small transactions with high
gas costs.

Historically, Ethereum has dominated the number of users and volume trading on
it likely due to first mover advantage and censorship resistance properties.
However, retail users likely benefit the most from L2s because of lower gas
costs and better average execution. The main market structure benefit of
Ethereum is the large capital base in pools. We expect to see price-sensitive
users begin to swap on cheaper chains as adoption continues, especially if
their drawbacks are improved.

##  3 Benefits for Uniswap v3

There are two main categories of swappers on blockchain-based markets:
arbitrageurs and retail flow. Aggregate statistics combine these two users,
although they have different reasons for trading. Arbitrageurs will execute
orders if they are profitable, and we will revisit their impact on market
structure later. Retail flow is just as important (providing the bulk of
positive returns for LPs in the $\mathsf{LVR}$ framework established in
Milionis et al. (2022)) and are a bulk of the users who benefit from cheaper
chains.

To focus on retail users, we must first segment out arbitrageurs. To help us
segment, we will take advantage of the fact that arbitrageurs do not typically
arbitrage using the Uniswap Interface. The Interface contracts were designed
to support safe onchain trading, and are typically less gas efficient and less
customizable than optimally designed MEV contracts, which are preferred by
arbitrageurs. Retail users are less gas sensitive than MEV traders, preferring
better trade outcomes on the relatively fewer swaps they do, so they are more
likely to use the Interface. Because of this, we can segment out the Interface
users as a separate cohort of predominantly retail users.444Note that while
arbitrageurs do not use the Uniswap Interface, not all retail flow uses the
Uniswap Interface. This means that we are likely dropping significant retail
flow from our sample and should not be taken as exhaustive. We will utilize
this same segmentation feature in reverse in later sections.

We can see that as a percentage of total Interface transactions, trades on
Ethereum are almost double the next highest chain (around 48% for Ethereum to
28% on Arbitrum). These retail transactions are smaller than arbitrage
transactions, but still larger than retail transactions on other chains.

Arbitrage transactions are likely lower bounded by gas costs, as arbitrageurs
will only execute if they are able to pay off their fixed costs for
transacting (gas). However, Uniswap Interface volume has no implicit lower-
bound, but even these trades on the Uniswap Interface on Ethereum are bigger.
This could indicate that users who utilize lower trade sizes must move off of
Ethereum and are effectively priced out. We will focus on costs for users to
find the optimal switching point.

There are two main costs to users, gas costs and price impact. From Table 1,
we can directly calculate the gas cost to users. The median swap gas cost over
the last year on Uniswap v3 on Ethereum mainnet is $10.00, while the average
Arbitrum swap cost is $0.28 - a staggering 97% decrease in gas cost.

The other cost of swapping for users is price impact, which is the movement of
the “midmarket price” of the pool from execution of the trade. Uniswap v3
allows users to “concentrate liquidity” between prices, concentrating
liquidity compared to a CPMM (like Uniswap v2). This concentration of
liquidity has led to market depth dwarfing that of centralized exchange in
some cases. For example, Uniswap v3 has two times the market depth of ETH/USD
versus Binance - see Liao and Robinson (2022) for more discussion on this.

Small Swappers Benefit From Lower Gas.

First, we will examine the difference in all-in swap cost of an USDC to ETH
trade on Ethereum mainnet vs. Arbitrum. As previously stated, we will break
down the cost into two factors: price impact and gas costs. We saw in Table 1
that gas costs on the studied L2s are much smaller, however overall liquidity
is lower (and thus the price impact of a trade is higher). We utilize similar
methodology to Chemaya and Liu (2022) who calculate the optimal time for
switching between chains.

Figure 2: Breakeven size of ETH/USDC on Ethereum vs. Arbitrum

Note: The above calculation does not take into account the increase in gas
cost of a higher swap size (due to the cost of shifting ticks). Since these
costs are logarithmic to the swap size and a small portion of the overall gas
cost, they should not notably impact these values, but slightly higher than
calculated. The grey line represents the breakeven cost at each time if gas
was always the median value over the sample. We also focus only on Arbitrum,
because of non-standard gas calculations with Optimism.

In Figure 2, we examine the trade size in dollars of an USDC to ETH swap that
is needed to breakeven - that is a swap of USDC to ETH where we would get an
equal output of ETH on Arbitrum and Ethereum (including gas). In summary,
above the pink line a user receives more ETH on Ethereum and below would
receive more ETH on Arbitrum. The grey line calculates the same value, but
holds gas fixed to the median value over the time period to show the impact of
price impact.

First, let us focus on the pink line which utilizes gas costs as of the time
of the swap. Notice that the size of this swap varies drastically over the
last year. To pick a number as a baseline, we choose $25k, as breakeven is
strictly over this value. To put this into context, around 10.9% of swaps that
utilize the ETH/USDC pool from the Uniswap Interface are above $25k during
this time period.555Source Only 2.5% of swaps are greater than $125k, which is
the breakeven value at the end of the sample period. These numbers indicate
that the vast majority of users do not trade with the transaction size needed
to benefit from the larger capital base on Ethereum, meaning they are likely
better of mainly operating on cheaper chains. Note that there are more swaps
that go through these pools than just Uniswap Interface swaps, however, the
numbers do no change drastically.

As previously mentioned, there are two costs measured here - the price impact
on the pool and gas costs. The grey line in the figure above charts the
breakeven swap size if gas was always always the median cost over the period.
This shows us fluctuations due to price impact differences, removing the
impact of gas. We can see that while the differences in price impact from
Ethereum to Arbitrum are a major driver of the breakeven trade size, but we
can see other fluctuations due to differences in gas costs.

If drawbacks for L2s are addressed appropriately, then there may be no reason
for Ethereum to have significantly more TVL than an L2. With gas costs
lowering by an estimated 20x-60x the current rate on rollups due to EIP-4844
introduced in Dencun hardfork, this may further increase the breakeven swap
amount. If liquidity differences between L2s and mainnet converge, there may
not be any trade that is better on Ethereum mainnet.666Source for 20x and
source for 60x

LPs Increased Concentration leads to greater capital efficiency.

We have shown that smaller swappers get better execution on Uniswap v3 due to
lower fixed gas costs from L2s. However, perceived security of the chain and
subpar UX limit the amount of capital that is currently deployed on these
chains.

One of the major critiques of Uniswap v2 was that it created capital
inefficiencies by locking capital at prices it was unlikely to ever trade at.
Uniswap v3 changed this by allowing the concentration of liquidity around
desired price intervals. However, a more concentrated position is more likely
to require rebalancing as the price of the pool moves. Rebalancing incurs a
gas cost, meaning that when the gas costs are higher, efficient users will
rebalance less. On mainnet, this means capital is still not as concentrated as
it could be if gas had no cost. We could see more concentration on L2s, where
lower gas prices mean the fixed costs for LPing are lower.

The data supports the theory that as LPs are able to adjust their placement of
liquidity more cheaply on L2s, there is more concentration around the
midmarket price. Because of gas costs, L2s have at least 75% more
concentration than Ethereum. There are also significantly more liquidity
positions opened on L2s, showing that users are rebalancing their positions
more frequently. Not only that, we see that there are more unique liquidity
providers on L2s, meaning users may be priced out of providing liquidity on
Ethereum.

Figure 3: Median liquidity concentration around mid-market of ETH/USDC

Note: We calculate the liquidity distribution around the current mid-price
every 15 minutes from 03-01-2023 to 01-01-2024 and take the normalized
liquidity distribution around mid for the studied chains.

Normalizing liquidity at each sample period, we calculate that median
liquidity at the current mid market price is 70% of the total on Arbitrum and
80% of total on Optimism. On Ethereum, it is only 40%. This means that in the
median sample, L2s have at least 75% more concentration at the pool midpoint
than Ethereum.

Table 2: Uniswap v3 LP Positions Selected Statistics

Note: Everything calculated as of end of 2023 except where marked.
Source

There is also evidence that users are rebalancing more frequently, resulting
in more efficiently used concentrated liquidity. More concentration of
liquidity makes capital more efficiently used on the Uniswap Protocol, as less
capital is locked up far away from the current midpoint. When a position is
rebalanced in Uniswap v3, technically a new position is created, meaning that
more frequent rebalancing creates more unique positions. In Table 2, we see
significantly more positions created on L2s. Arbitrum has almost three times
more positions created this year than Ethereum - a similar trend in Polygon as
well. We see that users (likely sophisticated) are rebalancing more
frequently.

However, we find that it is not just that the same set of users are providing
more concentrated liquidity. In Table 2, we see that more unique wallets are
provide liquidity, as there are almost three times the unique wallets
providing liquidity on Arbitrum. This could indicate that many would-be LPs
are priced out by the expensive gas cost of liquidity provision on mainnet
Ethereum.

To zoom in on a specific pool, we will focus on arguably the most important
pool on Uniswap v3, the 5 bps ETH/USDC pool. Since, most onchain pools are
paired tokens paired against (W)ETH, this is the pool used to utilize
stablecoins in Defi via multi-hop trades through WETH/USDC.

Table 3: ETH/USDC 5 bps Selected Statistics YTD

Note: ** excludes JIT LPs/Limit Orders (tick-range = 1 tick spacing)
Source

While all the 5 bps ETH/USDC pools have similar dynamics, we will focus on
Ethereum vs. Arbitrum. One notable difference between Ethereum and Arbitrum is
the average holding time of each position. Ethereum holding time is more than
75x longer from first mint to first burn than Arbitrum, meaning that positions
are rebalanced less frequently. There are fewer liquidity providers on
Ethereum mainnet, they are holding their positions longer, and their positions
are wider.

As a result, there are ultimately fewer but larger positions on Ethereum. This
could be due to perceived security risks inherent to L2s, as even a low chance
of total loss of funds - due to a security issue - must be internalized when
making trading decisions. This low chance causes large LPs to utilize less
capital on L2s, creating smaller liquidity positions on average. We will
discuss these tradeoffs more in later sections.

Increased Fee Returns For Liquidity Providers.

We have already shown the positive effects on market structure for LPs from
lower gas - increased LP concentration, more unique liquidity providers, and
decreased holding times. A second important aspect of the studied chains are
their shorter block times. Historically, it has been difficult to calculate
the benefit for market structure changes on accured LP fees, as LPs,
arbitraguers, and swappers may respond differently to market structure
changes. LPs are especially difficult due to utilizing different portfolio
benchmarks. Liquidity providers may have arbitrary reasons for providing
liquidity, and may benchmark against holding an arbitrary portfolio of tokens
(impermanent loss), loss vs rebalancing ($\mathsf{LVR}$), or any other
arbitrary strategies. However, we will focus on the well-researched
$\mathsf{LVR}$, established in Milionis et al. (2023a), to compare between L1
and L2 LP performance. We also choose this because improving $\mathsf{LVR}$
generally improves returns on other strategies, and because the benchmark for
the portfolio is the same between chains and at arbitrary times.

In the $\mathsf{LVR}$ with fees framework, arbitrage losses increase with the
square root of block time. This is because the LP fee of the pool creates a
lower bound for price movements in which arbitrage transactions are not
profitable, described as the no-trade region. Shorter block times means there
is less time for the market price of the assets to move at a given level of
volatility and therefore a higher likelihood that arbitrage transactions on
that price movement will not be profitable (due to staying within the no-trade
region).

When using Fees - $\mathsf{LVR}$ to compare full-range positions (or any
identical LP strategy), the costs of hedging the LP position are the exactly
the same. This is because the path of the asset and the hedge occurs outside
of the decentralized exchange by assumption , meaning for like assets and like
LP strategies, the LVR cost is identical. Because the daily LVR cost is
identical between all the studied pools, we are able to simplify the
comparison to just the fee returns of the LPs, which can be found directly
from swap transactions. This allows a simple comparison of Fees -
$\mathsf{LVR}$ utilizing only change in fees.

From Mitchell (2022), we can see there is very little difference between
aggregate ETH/USDC fee returns of full-range positions on L1s and L2s. We know
that there is more retail flow transacting on Ethereum, which may offset the
decreased fee returns from arbitrage. We have also shown that retail traders
continually trade on Ethereum mainnet - even though this is not optimal,
likely because of lagged adoption of other chains. However, because of the
coordination required between LPs and swappers, liquidity providers may stay
on Ethereum because the fees from retail flow offset the increased costs from
arbitrage (even if pareto optimal for both parties to be on L2s). Retail
traders are the bulk of positive returns for liquidity providers, so liquidity
providers must stay where they are.

Additionally, retail trading fees are hard to model and their reasons for
trading are not well understood. We believe as adoption of cheaper chains
increased, users will either move to cheaper but decentralized chains, or the
choice will be abstracted away through market structure innovations like
Crosschain UniswapX. Accordingly, we want to present data that attempts to
quantify the impact of changing block times on arbitrage fees from LPs. To
focus on the impact of arbitrage of lower costs and shorter block times, we
will remove retail transactions, as arbitrage fee returns react instantly to
market structure changes.

To segment arbitrage fee returns, we utilize router addresses for all retail
flow on Arbitrum, Ethereum, and Optimism. Since Uniswap v3 requires a contract
to interact with it (and cannot be called directly by an end user’s wallet)
and contract deployments cost several magnitudes more than a swap, it feasible
to check these contracts by hand. For example, on Arbitrum, we checked around
300 contracts (68 of which are interfaces) spanning more than 95% of volume
and transactions. For context, around 1 million wallets have swapped on
Arbitrum. The transparent nature of the blockchain also makes it feasible to
see what these contracts are doing to ensure they are being used as expected.
While we likely missed a very small number of interfaces, we believe this
sample well-spans the collection of retail interfaces.777If anything, we are
biasing against L2s in this methodology, as gas optimizations are less
meaningful on L2s, meaning some searchers utilize Uniswap Interface contracts
for arbitrage on L2s. The transaction counts and volume also follow a power
law distribution when segmented by originating contract, and we went from
largest first. For replication, a list of all the tracked contracts can be
found in this repository.

Next, we remove all swap transactions that originate with one of these
interface routers (to remove retail flow). Lastly, we take all the remaining
arbitrage transactions left, re-calculate their impact on state, and generate
the full-range fee returns from all of these swaps. This is utilizes the
feeGrowthGlobal values found in Uniswap v3 state. For proofs and instructions
on replicating this methodology, see Adams and Liao (2022).

Figure 4A: Distribution of daily fee returns by token on Arbitrum

Figure 4B: Distribution of daily fee returns by token on Optimism

Note: Data calculated from March 1st, 2023 to December 1st, 2023

Calculations show that full-range LPs on L2s make an average of 20% more
returns from arbitrage compared to the same day on Ethereum mainnet.

As previously stated, the increase in fee-returns is from gas cost differences
and shorter block times. However, during this sample period (done
purposefully), Optimism and Arbitrum had nearly exact gas costs, while block
times were the main difference. To zoom in on like-samples, we focused on
March 1st, 2023 to December 1st, 2023. Over this period, we find that both
Arbitrum and Optimism outperformed Ethereum by a significant margin - around
20% on average. Finally, by focusing only on arbitrage transactions (which
will execute at any depth if profitable), we can remove the impact of
liquidity differences of these pools between chains, meaning that equalizing
the liquidity on L1s and L2s will not impact these dynamics. There is an
interaction between full-range fee returns and increased concentration of
other LPs, but it is minimal and due to gas costs. A deep dive into intra-pool
LP dynamics is out of the scope of this paper - for that, see Milionis et al.
(2023b).

Since September 2021, Ethereum mainnet returns for ETH/USDC would increase by
about $20m, surpassing some calculated losses to LPs during that time -
meaning that Ethereum LPs on ETH/USDC could be in profit from only this change
if they mark their returns with 5 minute $\mathsf{LVR}$.888See here for query
and here for data sources and Canidio and Fritsch (2023)here This assumes that
Ethereum adopts the same block production mechanisms as L2s today (or that all
capital and flow is perfectly migrated to L2s). For example, the FM-AMM from
Canidio and Fritsch (2023) and associated CowSwap implementation on Ethereum
mainnet claim increasing fees by 5-7% fixes $\mathsf{LVR}$, while simply using
faster and cheaper chains increases it by an estimated 10-15%.

Importantly, the numbers for Ethereum include fee returns from sandwich
transactions - which accounts for around 10% of ETH/USDC returns according to
Holloway (2023) \- meaning the difference is probably larger than 20%. Because
sandwich bots utilize contracts which are generally also arbitrage bots, our
methodology will include them as arbitrage transactions. Because of the
current nature of sequencers, there are fewer sandwiches to no sandwiches on
L2s, as sandwich trades without strict control of block ordering run the risk
of being unbundled.999For MEV number, see here and for a description on
unbundling Accordingly, this removes sandwich transactions on the studied L2s,
but also their associated LP returns.

In our sample period, Arbitrum had larger arbitrage fee returns than Optimism
while gas fees were similar. For LPs, the practical difference between the
chains is block times. As said previously, Arbitrum built blocks on demand,
allowing for multiple arbitrage transactions to happen in a 2 second window -
possibly hinting that 2 seconds is still too large of a block time for optimal
level of fee returns. Finding the optimal block time is likely dependent on
volatility and is not possible in our current model, thus it is out of the
scope of this paper.

##  4 Drawbacks for Uniswap v3

We have talked extensively about the benefits of faster and cheaper for users,
but current L2s also have drawbacks.

First, the sequencing of transactions is currently chosen by privileged
actors, meaning that sequencers for these rollups are currently
centralized.101010We are using the term rollup to denote that Polygon does not
have this issue Currently, rollups are designing ways to decentralize block
ordering to a distributed committee of sequencers, but these solutions are not
in production at the moment. By having one centralized sequencer, sequencers
could take advantage of their status in the system to generate MEV profits by
re-ordering transactions - similar to how PoW miners or PoS validators
extracted value. However, in practice, sequencers to date haven’t done this
and prefer known/fair block ordering methods like priority gas auctions or
first come, first served. As previously stated, the current nature of
sequencers means less sandwiches from non-privileged parties, due to the risk
of being unbundled and shielded mempools. The MEV on L2s is largely currently
confined to non-adversarial MEV (arbitrage, back-running, liquidation) due to
this. MEV extraction has also led to centralization and instability in block
production, which might not preferable for L2s.111111To see issues with MEV
extraction, see here and here

Second, the distinct ecosystems of L2s create liquidity fragmentation. In
their current design, L2s cannot trustlessly speak to each other natively in
real-time. This means that liquidity on different L2s are not composable. Put
another way, you cannot natively trade on one L2 and use the liquidity
immediately on the other. Currently, each chain must bootstrap distinct
liquidity sources for their own rollup, which duplicates capital locked up to
facilitate trading, adds additional costs for LPs due to arbitrage, and
increases price impact for users. However, there are a few mechanisms which
could facilitate interaction between L2s, such as bridging systems like
Wormhole or intent-based trading systems like Cross-chain UniswapX.121212For a
description of UniswapX, see here

Third, some of the current optimistic rollups lack fraud proofs, which are
infrastructure to trustlessly recover from sequencer errors or incorrect
transactions.131313This is not broadly true. Arbitrum has working fraud proofs
for whitelisted actors Because optimistic rollups assume by default that
transactions are valid, sophisticated actors need to utilize fraud proofs to
show that posted transactions/blocks are invalid. After seven days, current
state is assumed to be correct - meaning transactions like withdrawals are
processed. If an invalid withdrawal is created and not disputed within seven
days, then invalid transactions will be canonical and the withdrawal is
processed. On Optimism, fraud proofs allow the trustless and retroactive
dispute of a transaction from the canonical state roots, effectively removing
it from the chain trustlessly if the fraud proof is valid. On Arbitrum, if
there are conflicting state root proposals, validators stake on histories -
effectively utilizing the the fraud proofs to solve the fork choice. For
rollups, fraud proofs are a powerful unlock for decentralization of the chain
as a whole but have implementation challenges. However, significant work is
going into productionalizing fraud proofs.

Lastly, bridging between ecosystems can be costly and time consuming. To use
the native bridges between chains, users must return to mainnet and then enter
the canonical bridge for the other ecosystem. On optimistic rollups, it takes
seven days to bridge funds back to mainnet, and depositing into a rollup costs
around 8$ at average gas prices. This adds both time and costs to transactions
that are generally time sensitive. As previously said, there are methods to
circumvent this like Cross-chain UniswapX.

There is evidence that users internalize these implicit costs when interacting
with L2s. Chemaya and Liu (2022) estimated that L2 users internalize a 3.29%
probability of total loss when trading CPMMs on Optimism and 0.68% on Polygon.
They utilize the actual revealed preference of users switching between mainnet
and the L2s to calculate this value. However, over time this number is
trending towards 0, as the security tradeoffs of these systems lessen through
the aforementioned improvements. The authors do note that other factors could
impact this parameter, such as community, user experience, and overall
integrations. We find similar results, as smaller users continue to execute
trades on Ethereum mainnet despite the relatively better execution on L2s.

On the other hand, Cong et al. (2023) finds L2s may lead to more secure
markets, because fixed costs for interacting with markets are lower. Looking
at the accuracy of Chainlink oracle feeds, Cong et al. estimate that costs for
operators dropped by 76% while each round had 79% more operators, leading to a
more secure, yet cheaper, feed. We find similar results with more unique
liquidity providers providing liquidity on Uniswap v3 and smaller average
Uniswap Interface trade size - signaling that users may be priced out of
interacting with the mainnet ecosystems due to high gas costs.

##  5 Conclusion

Overall, we find that faster and cheaper chains positively impact market
structure on decentralized markets compared to Ethereum mainnet. In sum,
swappers benefit from lower gas cost, more concentrated liquidity, and prices
that update more frequently. LPs benefit from cheaper position update costs in
concentrated liquidity, more arbitrage flows resulting in higher profit, and
lower cost of pool deployment.

As L2 costs continue to plummet for all users, the benefits of decentralized
markets will continue to shine, while removing negative barriers. The design
of current decentralized markets may also markedly change, as many of the
jarring constraints for these markets are due to high costs for end users.
Increasing a trade’s cost by double when the underlying cost is 1c is
proportionally less economically impactful than doubling the cost of a $10
translation. For decentralized markets to fulfill their full potential,
aggregate trading costs must continue to decline and user experience must
continue to improve.

We expect that users will continue their crosschain migration as perceived and
realized security of rollups continues to increase, decreasing the main
benefits of continued swapping on Ethereum mainnet. This is in the best
interest of both Ethereum and L2s, as this will dramatically decrease the cost
and increase throughput of the entire Ethereum network.

In this paper, we only studied a subset of the growing number of chains.
Because we mainly studied the most active chains today, we make no attempt to
show trading benefits of newer technologies like ZK-proofs, other alternative
chains, and the future benefits for the studied L2s due to smaller Uniswap v3
integrations.

Uniswap v3 was also optimized for Ethereum mainnet gas costs, meaning that the
trading experience for the forthcoming Uniswap v4, described in Adams et al.
(2023b), may benefit greatly from cheaper chains. Uniswap v4 allows for
customization on top of the Protocol layer, which is up to the developer.
Integrators on cheaper chains can utilize more complicated hooks, because of
lower gas costs, to improve user experience. We have seen this already with
hooks utilizing zero-knowledge proofs and Flashbots’ SUAVE, which were
deployed on L2s and are likely not cost effective on mainnet.

However, we believe that the studied generalized L2s still have many benefits
that users can utilize today, and any future improvements will only continue
to benefit the trading experience.

## References

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  * Adams et al. (2023b) Hayden Adams, Moody Salem, Noah Zinsmeister, Sara Reynolds, Austin Adams, Will Pote, Mark Toda, Alice Henshaw, Emily Williams, and Dan Robinson. 2023b.  _Uniswap v4 Core [Draft]_.  https://github.com/Uniswap/v4-core/blob/main/docs/whitepaper-v4.pdf
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  * Milionis et al. (2023a) Jason Milionis, Ciamac C Moallemi, and Tim Roughgarden. 2023a.  Automated Market Making and Arbitrage Profits in the Presence of Fees.  _arXiv preprint arXiv:2305.14604_ (2023).
  * Milionis et al. (2022) Jason Milionis, Ciamac C. Moallemi, Tim Roughgarden, and Anthony Lee Zhang. 2022.  Automated Market Making and Loss-Versus-Rebalancing.  https://doi.org/10.48550/ARXIV.2208.06046
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# Spin Nematic Liquids of the S=1 Spin Ladder in Magnetic Field

Tôru Sakai1,2 and Kiyomi Okamoto1 1Graduate School of Material Science,
University of Hyogo, Hyogo 678-1297, Japan
2National Institutes for Quantum and Radiological Science and Technology
(QST), SPring-8, Hyogo 679-5148, Japan

(Received September 1, 2019)

###### Abstract

The magnetization process of the $S=1$ spin ladder system is investigated
using the numerical exact diagonalization of finite-size clusters. The field-
induced spin nematic liquid phase was predicted to appear by our previous
work. Several ground-state phase diagrams in the plane of the single-ion
anisotropy and the external magnetic field are obtained in the present study.

##  I Introduction

The spin nematic phase is one of interesting topics in the field of the
strongly correlated electron systems. For example, the high-magnetic field
measurement of the quasi-one-dimensional compound LiCuVO4 detected itnawa . In
addition the spin-liquid-like behavior of the $S=1$ triangular-lattice
compound NiGa2S4 was theoretically explained by the spin nematic
phasenakatsuji . The spin nematic order is the quadrapole order of the quantum
spins in two- or three-dimensional systems. On the other hand it appears as
the gapless Tomonaga-Luttinger liquid phase of the two-magnon bound state in
one-dimensional systems. In our previous workyamanokuchi  using the numerical
exact diagonalization of finite-clusters the spin nematic Tomonaga-Luttinger
liquid (TLL) phase was revealed to occur in the $S=1$ spin ladder system under
the external magnetic field in the presence of sufficiently large negative
single-ion anisotropy. In addition several ground-state phase diagrams in the
plane of the anisotropy and the magnetization were presented. However, in
order to propose some experiments to detect the spin nematic TLL phase in real
materials, the phase diagram in the plane of the anisotropy and the external
magnetic field would be much more useful. In this paper, we investigate the
$S=1$ spin ladder system with the numerical exact diagonalization of finite-
size clusters and obtain the phase diagrams in the anisotropy and field plane.

##  II Model

The $S=1$ spin ladder with the single-ion anisotropy $D$ is described by the
Hamiltonian

$\displaystyle{\cal H}=$ $\displaystyle
J_{1}\sum_{i=1,2}\sum_{j=1}^{L}\vec{S}_{i,j}\cdot\vec{S}_{i,j+1}+J_{\rm
r}\sum_{j=1}^{L}\vec{S}_{1,j}\cdot\vec{S}_{2,j}$ (1)
$\displaystyle+D\sum_{i=1,2}\sum_{j=1}^{L}(S^{z}_{i,j})^{2}$
$\displaystyle-H\sum_{i=1,2}\sum_{j=1}^{L}S^{z}_{i,j},$

where $\vec{S}_{i,j}=(S_{i,j}^{x},S_{i,j}^{y},S_{i,j}^{z})$ denotes the spin-1
operator acting on the spin at the $j$th rung and the $i$th chain. The
quantity $J_{1}$ denotes the nearest neighbor leg interaction constant,
$J_{\rm r}$ the rung interaction constant, and $H$ the strength of the
external magnetic field along the $z$ direction. We investigate the ground
state of this model using the numerical exact diagonalization of finite-size
cluster up to $L=8$. Throughout this paper we consider the negative $D$ only,
namely the easy-axis anisotropy, and fix $J_{1}=1.0$.

##  III Ground state under $H=0$

In the absence of the external magnetic field, for $D=0$, the system is in the
plaquette singlet state which is non-degenerate and has the spin gap todo2001
. On the other hand, for sufficiently large negative $D$, the system is in the
Néel state along the $z$-direction which is doubly degenerate and also has the
spin gap. The critical point $D_{\rm c}$ can be estimated by the
phenomenological renormalization group method. Namely, the size-dependent
critical point $D_{{\rm c},L}$ is determined from the equation for the scaled
gaps

$\displaystyle(L+2)\Delta_{L+2}(D_{{\rm c},L})=L\Delta_{L}(D_{{\rm c},L}),$
(2)

where $\Delta_{L}(D)$ is the lowest energy gap between the $k=0$ ground state
and the $k=\pi$ subspace in the leg direction. The scaled gap $L\Delta_{L}(D)$
is plotted versus $D$ for $J_{1}=J_{\rm r}=2.0$ in Fig. 1. Since the size
dependence of $D_{{\rm c},L}$ is quite small, we use $D_{{\rm c},6}=-0.20$ as
the best estimation of the critical point $D_{\rm c}$.

Figure 1: Scaled gap $L\Delta_{L}(D)$ plotted versus $D$ for $J_{1}=J_{\rm
r}=$2.0. Since the scaled gaps for $L=$4, 6 and 8 cross to each other almost
at the same point, we use $D_{{\rm c},6}=-0.20$ as the best estimation of the
quantum critical point $D_{\rm c}$.

##  IV Tomonaga-Luttinger liquid phases for $H>0$

Since the ground state at $H=0$ is in the plaquette singlet phase for
$0>D>D_{\rm c}$, a phase transition occurs at the critical field $H_{{\rm
c}1}$ and the gapless TLL phase is realized for $H>H_{{\rm c}1}$sakai1991a ;
sakai1991b . On the other hand, when the ground state is the Néel ordered
state for $D<D_{\rm c}$, the magnetization process is expected to be similar
to that of the $S=1/2$ Ising-like $XXZ$ ladder. In this case the TLL phase is
also realized above the critical field $H_{{\rm c}1}$. The quasiparticle
excitation, however, is different between these two TLL phases. Each
elementary magnon excitation should occur by $\delta S^{z}=2$, because the
$S^{z}=0$ state cannot occur for sufficiently large negative $D$ ($D<D_{\rm
c}$), while $\delta S^{z}=1$ for $D>D_{\rm c}$. The former TLL phase is called
the spin nematic TLL phase, to distinguish from the latter one, namely, the
conventional TLL phase. These two TLL phases can be distinguished by whether
the gapless excitation is $\delta S^{z}=1$ or 2.

##  V Phase diagrams on the $D$-$H$ plane

The purpose of this paper is to obtain the ground-state phase diagram on the
$D$-$H$ plane. We define $H_{\rm c1}$ as the critical field where the non-zero
magnetization appears for the first time with increasing $H$. At first, we
should consider the possibility of the magnetization jump at $H_{\rm c1}$. The
field $H_{\rm jump}(M)$ is defined as

$\displaystyle H_{\rm jump}(M)=[E(M)-E(0)]/M,$ (3)

where $E(M)$ is the lowest energy for $\sum_{j}S^{z}_{j}=M$. When the
magnetization jump occurs to $M$ at $H_{\rm c1}$, $H_{\rm jump}(M)<H_{\rm
jump}(M^{\prime})$ is satisfied for every $M^{\prime}$ ($M^{\prime}<M$) and
$H_{\rm c1}$ corresponds to $H_{\rm jump}(M)$. The present numerical
diagonalization for $L=8$ indicates that the smallest $H_{\rm jump}(M)$ is
given for $M\geq 3$ in the whole region $D<D_{c}$. Thus the $M=2$ state is
skipped and the magnetization jump occurs at $H_{\rm c1}$.

In order to estimate the phase boundary between the two TLL phase in the
finite magnetization phase, the cross points between the $\delta S^{z}=1$
excitation gap and the $2k_{\rm F}$ excitation gap of the two magnon bound
state in our previous work. In this paper, however, we use the cross points
between $H_{1}(M)$ and $H_{2}(M)$ defined as

$\displaystyle H_{1}(M)$ $\displaystyle=$ $\displaystyle E(M+1)-E(M)$ (4)
$\displaystyle H_{2}(M)$ $\displaystyle=$ $\displaystyle[E(M+2)-E(M)]/2,$ (5)

for $L=8$, because more points can be obtained than the previous method.

The saturation field $H_{\rm sat}$ is obtained as $E(2L)-E(2L-1)$ for larger
$D$, while $[E(2L)-E(2L-2)]/2$ for smaller $D$.

Using the numerical diagonalization for $L=8$, we obtain the $H$-$D$ phase
diagrams for $J_{r}=$0.5, 1.0 and 2.0 shown in Figs. 2, 3 and 4, respectively.
In our previous work the system size dependence of these phase boundaries is
quite small. In these phase diagrams the Haldane (H), the Néel ordered (NEEL),
the conventional Tomonaga-Luttinger liquid (CTLL), the nematic Tomonaga-
Luttinger liquid (NTLL), and the ferromagnetic (F) phases appear. Dashed
curves are the critical magnetic field with the magnetization jump.

Figure 2: Phase diagram on the $D$-$H$ plane for $J_{1}=1.0$ and $J_{\rm
r}=0.5$. Here H, NEEL, CTLL, NTLL and F correspond to the Haldane, the Néel,
the conventional Tomonaga-Luttinger liquid, the nematic Tomonaga-Luttinger
liquid, and the ferromagnetic phases, respectively. Dashed curve is the
critical magnetic field with the magnetization jump.

Figure 3: Phase diagram on the $D$-$H$ plane for $J_{1}=J_{\rm r}=1.0$.

Figure 4: Phase diagram on the $D$-$H$ plane for $J_{1}=1.0$ and $J_{\rm
r}=2.0$.

Figure 5: Ground state magnetization curves obtained from the numerical
diagonalization for $L=8$ in the case of $J_{1}=J_{\rm r}=1.0$. Black, green,
red and blue symbols correspond to $D=$0, $-1.0$, $-2.2$ and $-3.0$,
respectively. Lines and curves are guides for the eye. Solid and dashed curves
correspond to the conventional and the nematic TLL phases, and dotted lines
mean the magnetization jump.

##  VI Magnetization curve

In order to consider the experiment to detect the nematic TLL phase, it would
be useful to give the theoretical magnetization curve based on the numerical
diagonalization for $L=8$. If the magnetization $M$ is realized for
$H_{-}(M)<H<H_{+}(M)$ in the ground state of the finite-size system, the
averaged field $H_{\rm av}(M)=[H_{-}(M)+H_{+}(M)]/2$ is used to obtain the
magnetization curve in the present work. Namely, we neglect the finite-size
correction proportional to $1/L^{2}$ here. In the case of $J_{1}=J_{\rm
r}=1.0$, the ground state magnetization curves are shown in Fig. 5, where
black, green, red and blue symbols correspond to $D=$0, $-1.0$, $-2.2$ and
$-3.0$, respectively. Lines and curves are guides for the eye. Solid and
dashed curves correspond to the conventional and the nematic TLL phases, and
dotted lines mean the magnetization jump. The critical points between the two
TLL phases are not so precise, because they still include some finite-size
effects. It suggests that the reentrant quantum phase transition can occur for
$D=-2.2$.

##  VII Summary

The $S=1$ spin ladder with the easy-axis single-ion anisotropy under the
magnetic field is investigated using the numerical exact diagonalization of
finite-size clusters. We obtain the ground-state phase diagrams in the $D$-$H$
plane including the conventional and nematic TLL phases and the magnetization
jump. Some magnetization curves are also obtained.

## Acknowledgment

This work was partly supported by JSPS KAKENHI, Grant Numbers 16K05419,
16H01080 (J-Physics) and 18H04330 (J-Physics). A part of the computations was
performed using facilities of the Supercomputer Center, Institute for Solid
State Physics, University of Tokyo, and the Computer Room, Yukawa Institute
for Theoretical Physics, Kyoto University.

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