# Towards Greener Data Centers via Programmable Data Plane

Garegin Grigoryan
<EMAIL_ADDRESS>
Alfred University  Minseok Kwon
<EMAIL_ADDRESS>
Rochester Institute of Technology

###### Abstract

The energy demands of data centers are increasing and are expected to grow
exponentially. Reducing the energy consumption of data centers decreases
operational expenses, as well as their carbon footprint. We design techniques
to reduce data center power consumption by leveraging Software-Defined
Networking (SDN) and programmable data plane concepts. Relying solely on in-
data plane registers, our proposed system P4Green consolidates traffic in the
least number of network switches and shifts workloads to the servers with the
available renewable energy. Unlike existing SDN-based solutions, P4Green’s
operation does not depend on a centralized controller, making the system
scalable and failure-resistant. Our proof-of-concept simulations show that
traffic consolidation can reduce data centers’ aggregation switch usage by 36%
compared to standard data center load balancing techniques, while workload
control can boost renewable energy consumption for 46% of the daily traffic.

##  I Introduction

With the rapid growth of cloud-based data aggregation and computation, the
data centers hosting clouds became the major consumers of electrical power and
produce a significant carbon footprint [26, 9]. A Data Center Network (DCN)
that includes servers and network infrastructure consumes about 40-50% of the
total energy consumed by a data center overall, and that share is likely to
grow [25, 5]. As the efficiency of data centers’ non-IT components improves
[12, 22], optimizing the energy consumption of computing and networking
components becomes critical. Only by 2030, data center user behavior is
expected to increase the energy consumption of data centers by 20% in case the
current technological development trends do not change. However, the likely
end of Moore’s law may lead to data centers’ energy consumption surge by up to
134% [19].

Energy consumption of network components is not proportional to traffic [10],
hence even switches that transmit little traffic consume considerable energy
resources. To reduce the energy consumption of network infrastructure, the
state-of-the-art approaches propose carefully choosing the architecture for a
data center (e.g., three-tier vs fat-tree data center network) to maximize the
utilization of the available resources in a data center. To reduce the energy
consumption of servers, workload scheduling systems [13], virtualization
techniques [16], and server hardware optimizations [18] were proposed. Using
renewable energy can reduce the carbon footprint of data center servers. One
challenge is the volatility of renewable energy and the high costs of its
storage [23]. To overcome this, workload migration techniques were proposed
[26, 30]. Often these approaches require expensive hardware modifications; in
the meantime, re-organizing the data center and consolidation of resources may
lead to a lack of redundancy and therefore decreased quality of service during
peak hours of a data center operation.

In this work, we present P4Green, a dynamical adaptive system aimed at
reducing the energy consumption of server and network components at a data
center. Instead of relying on dedicated hardware, we leverage the technologies
of switch programmability. Offloading green traffic engineering to
programmable switches has several benefits. First, such switches can be
programmed to arbitrarily process each data packet at the line rate; second,
they can collect and store information reflecting the state of the data center
network and its servers, including their renewable energy capacities. In
addition, programmable switches are faster and more power-efficient than
switches with fixed-function ASIC [3, 27].

Specifically, we leverage the Software-Defined Networking (SDN) via P4
programming language for the data plane and P4Runtime API for the control
plane [6, 28]. The goal of P4Green is to consolidate traffic in the least
number of switches based on the traffic volume and to schedule workloads on
servers with the available renewable energy. P4Green switches evaluate the
data center traffic per pre-defined time epochs and based on that adjust the
packet forwarding algorithms to include or exclude additional network
switches. In addition, they collect information about the availability of
renewable energy at the data center servers, make workload allocation
decisions, and forward the traffic to the most appropriate server. Unlike
existing SDN-based traffic engineering solutions, P4Green does not rely on a
controller and uses only its in-data plane registers for making forwarding
decisions. P4Green operates in the data plane at the line rate, improving the
scalability of the system and making it robust to control plane failures.

To summarize our contributions, we:

  * •

Designed P4Green, a system for reducing power consumption of network devices
in a data center and load balancing workloads towards servers with available
green resources;

  * •

P4Green traffic engineering and forwarding operates at the line rate, fully in
the data plane; the control plane is used only at the initialization stage;

  * •

We implemented and tested the prototype of P4Green using Mininet and bmv2
switch emulator;

  * •

Our results show a 36% reduction of data centers’ aggregation switch usage
compared to a traditional ECMP load balancing; as well as a boost of energy
consumption for 46% of the daily traffic in a distributed data center
scenario.

##  II Design

Figure 1: P4Green architecture and workflow

P4Green leverages the properties of the P4 switch architecture that allow
limited programmability in the forwarding engine and full programmability at
the control plane. The goals of P4Green are:

  * •

Traffic consolidation during light load. P4Green consolidates traffic by
dynamically changing the number of active aggregation switches based on
traffic volume, with no control plane involvement.

  * •

Green energy-driven workload control. Server workload is determined by the
resources available at the server such as renewable energy. Switches obtain
resource information from each server, choose appropriate servers for target
workloads, and forward workload traffic to the chosen servers.

Overall, P4Green helps reduce the energy consumption of a data center network
by minimizing the number of active switches; in addition, P4Green boosts
renewable energy usage by sending workloads toward servers that report the
availability of such energy. In this section, we describe how we implement the
functionality of P4Green while overcoming challenges, such as the data plane
computational limitations, the dependency on the control plane, and the TCP
session affinity requirements.

###  II-A Overview

The architecture of P4Green is shown in Figure 1 with its workflow. P4Green is
implemented with a single P4 program for the data plane and a Python program
for the SDN controller that uses P4Runtime API to set up the switches. When a
packet arrives at the switch, it first is parsed to extract information needed
for packet forwarding and future analysis. In addition to the standard fields
such as Ethernet and IP headers, the parser extracts the TCP header as well as
the TCP timestamp option. The TCP header is for load balancing across
different aggregation switches via 5-tuple hashing. The TCP timestamp option
enables TCP session affinity when distributing traffic to different servers.
After parsing, the packet is passed to the ingress pipeline in which an output
port is assigned based on the following factors:

  * •

Switch type: core, aggregation, or access;

  * •

Packet type: (a) Aggregation_in, packets that move from non-aggregation
switches towards aggregation switches; (b) Server_in, packets that move in to
servers; (c) Server_out, packets that move out from servers.

As shown in Figure 1, the ingress pipeline consists of the Traffic
Consolidation and Workload Control modules together with Longest Prefix
Matching (LPM) and the Host_info match-action tables. Next, we describe these
components in detail.

Figure 2: A simplified data center network Figure 3: Traffic Consolidation
workflow

###  II-B Traffic Consolidation

A typical data center network has three layers of switches: access switches
that connect servers to the network, core switches that forward heavy inter-
traffic, and aggregation switches that forward inter-traffic between access
and core switches as well as intra-traffic between access switches (see Figure
2). The connectivity between the access and core layers exists as long as
there is at least one active aggregation switch. The Traffic Consolidation
module helps determine the number of aggregation switches to be activated
dynamically depending on traffic volume. Specifically, it estimates traffic
arriving at core and access switches per epoch time, enables (or disables)
more aggregation switches to forward traffic if traffic volume grows (or
drops) significantly. This module runs only for ingress packets at core
switches (Aggregation_in packets), and for both ingress and server packets at
access switches (Aggregation_in and Server_out packets). The workflow of the
module is defined in more detail in Figure 3 and in the following subsections.

####  II-B1 Initialization

First, three registers are initialized:

  * •

aggr_switches: number of enabled switches (initially set to 1).

  * •

epoch_start: starting timestamp for measuring traffic within a time interval.

  * •

traffic: counter to estimate traffic volume.

These registers are set to the same values for all switches. This can be done
solely at the switches themselves with no controller involved. There are also
other registers possibly having different values based on the network topology
and the preferences of network operators. They are indeed to be initialized by
the controller:

  * •

switch_type: Core or access switches execute switch-type-specific commands
while aggregation switches simply forward packets to the destination IP
address (see Figure 1).

  * •

epoch_length: The length of an epoch is used for estimating traffic volume.

  * •

traffic_threshold(s): This threshold(s) is used to enable or disable
aggregation switches by recalculating the aggr_switches register.

Note that these registers can be modified during the operation of the switch
using either the control or the data plane. As we show further in this
section, some of these registers are modified automatically by the data plane
of the switch.

####  II-B2 Operation

Core and access switches use ECMP for forwarding packets to aggregation
switches based on 5-tuple hashing. These switches use aggr_switches as the
width of the ECMP hashing, i.e., the number of possible distinct outputs,
which controls the number of aggregation switches that forward traffic. The
value of aggr_switches changes based on the traffic volume that core and
access switches process. The counter traffic is updated for each packet
arrival and reset to zero at the end of each epoch, defined by epoch_length.
All updates to traffic are done directly in the switch.

Specifically, for each incoming packet, the P4Green program reads the ingress
timestamp in the metadata. If the timestamp exceeds epoch_start by
epoch_length, then traffic contains the traffic volume the switch has
processed during epoch_length. The traffic volume is then used to recalculate
aggr_switches by comparing traffic to traffic_thresholds provided by the
control plane. Depending on the measured traffic volume, aggr_switches is
either increased or decreased, and then used as the width of the ECMP hashing
for assigning output ports. Finally, traffic is reset to zero and epoch_start
is reset to the current timestamp, and traffic load evaluation is restarted
for a new epoch.

In P4Green, the initialization is the only step that needs input from the
control plane. The control plane program uses P4Runtime API to initialize the
switch state variables. The traffic load evaluation and ECMP modification are
performed solely in the data plane. In the meantime, P4Green requires a power
control module to power off switches that do not forward any traffic. Such a
module can control the switches’ state by checking the value in aggr_switches
register of the switch. We leave the design of the power control module out of
the scope of this paper.

###  II-C Workload Control

Distributed data centers naturally have heterogeneous servers in terms of
geographical locations, renewable energy and volatile resources. In iLoad
[15], servers determine their workload, e.g., machine learning tasks or web
traffic, based on locations and resources. A programmable switch receives
those workload requests directly from servers, and makes adjustments to packet
forwarding but with little input from the controller. The controller in iLoad
regularly updates the forwarding table at the switch, using the servers’
workload requests information stored in the data plane. P4Green adopts a
similar approach, but enhances it by completely eliminating the control plane
from switch operation. We also use the TCP timestamp to provide TCP session
affinity, leveraging the approach presented in [4]. The algorithm is discussed
below with its workflow depicted in Figure 4.

####  II-C1 Initialization

Figure 4: Workload Control workflow

At the startup, registers and tables are initialized:

  * •

The servers_data register block for storing per-server resource availability
index is initially set to 0 for every server. servers_data is used to select
the most pertinent server ID for the target workload.

  * •

The virtual_ip address, set by the control plane for processing client
packets.

In addition, the switch has the Host_info table with actions to be performed
when forwarding packets based on the server’s ID (i.e., rewriting the MAC
headers and the destination IP address).

####  II-C2 Operation

First, P4Green classifies the incoming packets into Server_out (Case A) and
Server_in (Case B) categories. For each category, further classification is
performed by P4Green.

a) Info-packets: A server sends the switch an info-packet that contains a
resource availability index (e.g., renewable energy, CPU, or memory
availability). An info-packet is an IP packet with the first three octets of
the destination IP address taken from the subnet to which the switch belongs
and the last octet used to encode the payload. The protocol field has the
specific value (0x8F in our example) for the switch to identify info-packets.

Handling info-packets is illustrated on Figure 4, Case A1. The P4 program
parses info-packets to obtain the sender ID (ingress port plus source IP
address) and the resource availability index. The index is stored in
server_data and used to choose the most relevant server for client workload
requests.

b) Client requests: Each access switch is assigned with a Virtual IP address
(VIP) at the initialization time, and clients send Server_in packets using
VIPs. Assuming the data center uses the TCP protocol, upon receiving a
Server_in packet, the switch extracts the TCP SYN flag to identify if it is a
new request (Case B1 in Figure 4 when SYN=1). The switch then selects the most
relevant server to handle this client request. If no server has reported the
resource availability index greater than zero, the switch simply uses round-
robin to select the server. If there are such servers, the switch selects the
next server with available resources. The packet is then augmented with the
server ID and passed to the Host_info match-action table that in turn replaces
the VIP with the destination IP address of the selected server.

If SYN=0 (Case B2 in Figure 4), the server that previously served the
corresponding TCP session needs to be selected for the TCP session affinity
requirement. Finding such a server is, however, challenging since the original
destination IP address is the VIP, not the server IP address. In the meantime,
storing a per-session table of connections in the switches is not scalable. To
get around this problem, we leverage TCP option fields to encode the server ID
for each flow as proposed in [4]. It uses the TCP property that hosts echo the
TCP timestamp (in ms) to each other while in session. A TCP Server_out packet
encodes the server ID into the last three bits of timestamp (see Case A2 for
Server_out packets in Figure 4). Now, the server ID can be obtained from the
last three bits of the TCP timestamp echo field for Seriver_in packets of
existing client requests (when SYN$=$0). The packet is then matched to the
Host_info table to rewrite its headers. Note that we use only the three least
significant bits of the TCP timestamp here, while the rest is not affected.
Our solution does not require modifications to the server or client TCP/IP
stack, except enabled TCP timestamp support. An alternative to leveraging TCP
timestamp is using other reliable protocols such as QUIC [20] with the
Connection ID field in its header.

##  III Evaluation

Figure 5: Traffic consolidation in P4Green

We implement P4Green as a Mininet emulator on the bmv2 [2] switch, and test
its traffic consolidation and workload control aspects as a proof-of-concept.
For testing, we use the network topology illustrated in Figure 2 that consists
of eight servers with unique IP addresses, four access switches with virtual
IP addresses, three aggregation switches, and one core switch connected to the
external network. First, traffic is sent from outside to the access switches,
and we measure how traffic volume affects packet forwarding. Moreover, we test
how traffic consolidation helps minimize network resources usage. We use
iperf3 [1] for traffic generation. After that, Hosts 1 and 2 send info-packets
to access switch 1 with their renewable energy availability index. An external
host then sends workload requests to VIP 1, and we observe which servers
respond to those requests.

Note that the Mininet emulator used in our experiments is not designed for
processing large volumes of traffic. Hence, the traffic rates in our
evaluation emulate the patterns of the daily traffic and do not represent the
realistic data center traffic volumes.

###  III-A Traffic Consolidation

At access and core switches, we set the threshold for enabling two and three
aggregation switches to 10KB and 20KB, respectively. Traffic is measured every
epoch_length which is set to 1 second. Traffic is generated with iperf3 to
simulate a typical daily load with the peak in the afternoon. The results are
displayed in Figure 5. With the low traffic (approximately less than 20KB on
Aggregation Switch 1 or $\approx$37% of the peak load), only one aggregation
switch receives traffic. As traffic grows, the second aggregation switch is
activated in ECMP and helps decrease the load on the first switch. Once the
traffic exceeds a threshold of 32KB ($\approx$ 60% of the peak load), the
traffic flows through all aggregation switches. Overall, the operation hours
of aggregation switches are reduced by more than 36%. In a data center
environment with up to 32 aggregation switches, each of them consuming about
400W*h (following the estimations made in [24]), that can save up to 4.6kW*h,
not considering the reductions in the energy needs of cooling facilities due
to reduced heat dissipation. The exact amounts of energy savings depend on the
topology of the data center network and and its traffic behaviour.

###  III-B Workload Control

Figure 6: Green load balancing in P4Green (in Server 1’s time zone) Figure 7:
Reported renewable energy availability index by Server 1 and Server 2 (in
Server 1’s time zone)

To test the workload control capabilities, we measure how traffic that servers
receive changes as each server informs the closest access switch of available
renewable energy. In the experiment, we simulate the case where Server 1 has
more renewable energy in the first half of the day, while Server 2 has more in
the second half. We assume that Server 1 is located in a time zone six hours
ahead of the time zone of Server 2.

For the results, Figure 6 shows the traffic processed on Servers 1 and 2
during the day in the timezone of Server 1. The reported energy values follow
the regular solar energy pattern on a sunny day (see Figure 7). As we simulate
a distributed data center in different time zones, we assume the same overall
traffic load throughout the day with all the hours presented in Server 1’s
time zone. Figure 6 shows that when both servers report zero renewable energy
between 00h:00m and 05h:00m, the standard round-robin distributes the client
workload evenly between the two servers. When Server 1 reports energy and
Server 2 does not, the modified round robin is used to send more workload to
Server 1. The figure shows that Server 1 processes a majority of the workload
between 05h:00m and 13h:00m ($\approx$ 69% of the total load). After the
energy peak hours for Server 1 before 22h:00m, $\approx$ 68% of the load is
forwarded to Server 2 with more renewable energy during that time interval.
Finally, at the end of the day (after 22h:00m), the load is distributed
evenly, since both Server 1 and Server 2 are past their peak energy hours. In
summary, P4Green distributes 46% of the total traffic towards the servers that
report more renewable energy. These results demonstrate that P4Green is
successful in load balancing and traffic engineering with no help from control
plane during the operational stage. Our implementation is publicly
available111Link to the repository: https://github.com/gareging/p4green.

##  IV Related work

ElasticTree [17] aggregates flows in the least number of networking devices
thus reducing energy consumption. The ElasticTree controller periodically
polls information from the data plane and updates the forwarding tables of the
data plane. Li et al. in [21] propose a different approach, allocating
networking resources to flows exclusively, rather than letting them share the
same links and switches. Such a design requires the control plane to analyze
the packets of unseen flows which creates a risk of a control plane
bottleneck. GRASP [14] is designed with an SDN controller that subscribes to
energy reports from the distributed data center servers and then re-actively
installs forwarding rules for new flows into the data plane. As with [21],
such an approach may not be scalable for a data center with large volumes of
traffic. Chuang et al. in [7] take into consideration a task execution length
and use an OpenFlow-based controller to monitor and reschedule the data center
jobs. iLoad [15] is an in-network green load balancing solution; however, it
still requires the control plane to analyze and update the data plane
registers during switch operations. Xu et al. in [29] design a green scheduler
for data center flows with deadlines. Work such as [8, 11] aimed at predicting
green energy generation at data center servers and proactively scheduling
workloads to the most active data centers. The authors in [13] propose a green
energy-aware algorithm for scheduling tasks in a data center.

##  V Conclusion

We leverage the emerging technologies of programmable switches to design a
data-plane system called P4Green that can allocate and de-allocate IT
resources based on traffic load and renewable energy available at servers.
P4Green eliminates the control plane from its operation and is shielded from
control plane delays and failures. Our proof-of-concept implementation shows
that it significantly reduces network switch usage and effectively allocates
workloads to servers that report the availability of volatile renewable energy
resources.

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# Fate of localization in coupled free chain and disordered chain

Xiaoshui Lin  CAS Key Laboratory of Quantum Information, University of Science
and Technology of China, Hefei, 230026, China  Ming Gong<EMAIL_ADDRESS>CAS Key Laboratory of Quantum Information, University of Science and
Technology of China, Hefei, 230026, China  Synergetic Innovation Center of
Quantum Information and Quantum Physics, University of Science and Technology
of China, Hefei, Anhui 230026, China  Hefei National Laboratory, University of
Science and Technology of China, Hefei 230088, China

###### Abstract

It has been widely believed that almost all states in one-dimensional (1d)
disordered systems with short-range hopping and uncorrelated random potential
are localized. Here, we consider the fate of these localized states by
coupling between a disordered chain (with localized states) and a free chain
(with extended states), showing that states in the overlapped and un-
overlapped regimes exhibit totally different localization behaviors, which is
not a phase transition process. In particular, while states in the overlapped
regime are localized by resonant coupling, in the un-overlapped regime of the
free chain, significant suppression of the localization with a prefactor of
$\xi^{-1}\propto t_{v}^{4}/\Delta^{4}$ appeared, where $t_{v}$ is the inter-
chain coupling strength and $\Delta$ is the energy shift between them. This
system may exhibit localization lengths that are comparable with the system
size even when the potential in the disordered chain is strong. We confirm
these results using the transfer matrix method and sparse matrix method for
systems $L\sim 10^{6}-10^{9}$. These findings extend our understanding of
localization in low-dimensional disordered systems and provide a concrete
example, which may call for much more advanced numerical methods in high-
dimensional models.

Anderson localization (AL), which describes the phenomenon that the disorder
totally suppresses the diffusion of the system, has attracted a great deal of
attention for many decades (Anderson, 1958; Lee and Ramakrishnan, 1985; Evers
and Mirlin, 2008; Lagendijk  _et al._ , 2009; Roati  _et al._ , 2008; Segev
_et al._ , 2013). It has been found that the spatial dimension plays an
essential role in AL (Mott and Twose, 1961; Abrahams  _et al._ , 1979; Ishii,
1973; Thouless, 1972, 1973). In the one-dimensional (1d) tight-binding model
with random potential, the localization length is given by (Thouless and
Kirkpatrick, 1981; Heinrichs, 2002)

$\xi_{0}^{-1}(E)={v^{2}\over 8t^{2}-2E^{2}}={V^{2}\over 96t^{2}-24E^{2}},$ (1)

where $v^{2}=\langle v_{i}^{2}\rangle$ is the variance of the potential
$v_{i}\in U[-V/2,V/2]$, with $V$ being the disorder strength, $t$ is the
hopping strength between the neighboring sites and $E$ is the eigenvalue (see
Eq. 14 in Ref. (Thouless and Kirkpatrick, 1981) for more details ). When the
system size $L$ is much larger than $\xi_{0}$, $L\gg\xi_{0}$, localization of
wave functions can be observed and the system is in the localized phase
without conductance. In the presence of weak disorder, $V\ll t$, we have
$|E|<2t$, thus all states should be localized with $\xi_{0}^{-1}>0$. For
example, when $V\sim 0.1t$, $\xi_{0}\sim 10^{4}$, which can be easily
confirmed by numerical simulation. The fate of the states in 1d systems will
be changed fundamentally with incommensurate potentials (Biddle  _et al._ ,
2009; Biddle and Das Sarma, 2010; Ganeshan  _et al._ , 2015; Wang  _et al._ ,
2020), long-range correlated disorders (Izrailev and Krokhin, 1999; de Moura
and Lyra, 1998; Dietz  _et al._ , 2011; Delande and Orso, 2014; Johnston and
Kramer, 1986) and many-body interactions (Bordia  _et al._ , 2016; Thiery  _et
al._ , 2018; Chiew  _et al._ , 2023; Léonard  _et al._ , 2023; Ha  _et al._ ,
2023).

Figure 1: The realization of the coupled disordered model. The free chain does
not have random potential, thus all states are extended; yet in the disordered
chain with random potential, all states are localized. The coupling between
them is the major concern of this work.

While the physics of disordered models have been widely discussed (Ishii,
1973; Brouwer  _et al._ , 1998; Heinrichs, 2002; Evers and Mirlin, 2008), the
fate of localization by the coupling of extended and localized states are much
lesser investigated. We are interested in this issue for the reason of the
dilemma that (I) the random uncorrelated potential induces localization for
the extended states in 1d systems (Abrahams  _et al._ , 1979); and (II) the
hybridization between localized and extended states may lead to delocalization
(Mott, 1987). The interplay between these two mechanisms may lead to different
physics. In this work, we propose a coupled disordered model (Fig. 1) to
address this problem. Our model is constructed from one free chain (with all
states extended) and one disordered chain (with all states localized). Two
major conclusions have been established: (1) While all states exhibit
localization in the presence of inter-chain coupling, their localization
length exhibits a distinct difference in the overlapped spectra and un-
overlapped spectra, which is not a phase transition process; (2) The
localization length for states in the un-overlapped regime of the free chain
is significantly suppressed given by

$\xi^{-1}(E)\simeq{t_{\text{v}}^{4}V^{2}\over(96t^{2}-24(E-\Delta)^{2})\Delta^{4}}={t_{\text{v}}^{4}\over\Delta^{4}}\xi_{0}^{-1}(E-\Delta),$
(2)

in the limit when $\Delta\gg|V|$. Here $t_{\text{v}}$ is the inter-chain
coupling, and $\Delta$ is the energy shift between the free and disordered
chains. Thus localization is greatly suppressed when $t_{\text{v}}\ll\Delta$.
For instance, when $t_{\text{v}}=0.1t$ and $\Delta=10t$, the localization
length can be suppressed by eight orders of magnitude. We examine the above
conclusions using the transfer matrix method and the sparse matrix method with
system sizes $L\sim 10^{6}-10^{9}$. Our results show that the inter-chain
coupling, disorder strength, and energy shift are the three major factors
influencing the localization length of states in the un-overlapped regime. In
the regime when $\xi\gtrsim L$, we can understand the localization of wave
functions with the following general theorem based on the large amount of
researches; see review articles (Ishii, 1973; Beenakker, 1997; Belitz and
Kirkpatrick, 1994; Evers and Mirlin, 2008).

Figure 2: (a) (b) The averaged density of states $\rho(E)$ for the free and
disordered chains with $t_{\text{v}}=0$. Other parameters are (a)
$\Delta=-10$, $V=10$; and (b) $\Delta=-6$ and $V=10$. (c) (d) Logarithm of the
wave function amplitude with different energy against the lattice index $m$
for the coupled disordered model ($t_{\text{v}}=0.1$). The index $m<L$ ($m\geq
L$) is for Hilbert space of $H_{0}$ ($H_{1}$). Inset of (c) shows the detailed
wave function around its localized center. In panel (a) and (b), $L=10^{3}$;
(c) and (d) $L=10^{6}$.

Theorem: In 1d disordered systems with short-range hopping and uncorrelated
random potential, almost all states are localized in the thermodynamic limit
($L\rightarrow\infty$).

The exact proof of this theorem is still a great challenge at the present
stage; however, it can be understood intuitively from the observation by Mott
et al (Mott and Twose, 1961), the argument by Thouless (Thouless, 1974), and
the scaling argument by Abrahams et al, in which the $\beta$ function is
always negative (Abrahams  _et al._ , 1979). This theorem is also addressed by
the celebrated Dorokhov-Mello-Pereyra-Kumar equation (Beenakker, 1997) and the
non-linear sigma model (Efetov, 1983). Here, not all states are localized
because, in the 1d model with off-diagonal random potential, the state with
$E=0$ is extended while all the other states are localized (Theodorou and
Cohen, 1976). The mathematicians have great interest in this problem and have
proved this theorem with random potentials (Carmona  _et al._ , 1987; Kunz and
Souillard, 1980; Fröhlich and Spencer, 1983; Pastur, 1980), showing of no
continuous spectra for extended states. This theorem will play a determinative
role for localization when $\xi\gtrsim L$, which is beyond the capability of
the numerical simulations.

Physical model and methods: We consider the following coupled disordered model
(see Fig. 1)

$H=H_{0}+H_{1}+\sum_{m,\sigma}t_{\text{v}}a_{m,\sigma}^{\dagger}a_{m,\bar{\sigma}},$
(3)

where
$H_{\sigma}=\sum_{m}t_{\sigma}a_{m,\sigma}^{\dagger}a_{m+1,\sigma}+\mathrm{h.c.}+V_{m,\sigma}a_{m,\sigma}^{\dagger}a_{m,\sigma}$,
with $H_{0}$ for the free chain and $H_{1}$ for the disordered chain. Here
$V_{m,0}=\Delta$ is the energy shift of the free chain $H_{0}$ and $V_{m,1}\in
U[-V/2,V/2]$ is the random potential in the disordered chain $H_{1}$, and
$\bar{\sigma}=1-\sigma\in\\{0,1\\}$. When $t_{\text{v}}=0$, this model is
reduced to a free chain with all states extended, and a disordered chain with
all states localized. By changing $\Delta$, the energy spectra of these two
chains can be un-overlapped (Fig. 2 (a)) or overlapped (Fig. 2 (b)). By the
theorem, all states should be localized in the presence of inter-chain
coupling (with $t_{v}\neq 0$). The fundamental question is what are the
quantitative differences between the states in the overlapped regime and the
un-overlapped regime of the coupled model during localization?

We apply the transfer matrix and sparse matrix methods, whose available size
is $L\sim 10^{6}-10^{9}$, to understand the localization of wave functions in
these two regimes. In the transfer matrix method (Hof, ), the Lyapunov
exponent $\gamma(E)=\xi(E)^{-1}$ is defined as the smallest positive
eigenvalue of the matrix

$\Gamma(E)=\lim_{L\rightarrow\infty}\frac{1}{2L}\ln(T_{1}^{\dagger}\dots
T_{L}^{\dagger}T_{L}\dots T_{1}),$ (4)

where $T_{i}$ is the transfer matrix at the $i$-th site. From the Oseledets
ergodic theorem (Oseledets, 1968; Raghunathan, 1979), the above multiplication
of transfer matrices is converged when $L\rightarrow\infty$. When
$\gamma(E)\neq 0$, the state with eigenvalue $E$ is localized. In the sparse
matrix method, we use the shift-invert method (Pietracaprina  _et al._ , 2018;
Luitz  _et al._ , 2015) to obtain about $N_{E}=20$ eigenstates
$|\psi_{E_{i}}\rangle$ with eigenvalues $E_{i}$ around a given $E$ and define
the averaged inverse participation ratio (IPR) as (Evers and Mirlin, 2008)

$\langle\text{IPR}\rangle_{E}={1\over
N_{E}}\sum_{i=1}^{N_{E}}\sum_{m=0}^{2L-1}|\psi_{E_{i}}(m)|^{4}.$ (5)

For the extended state, $\langle\text{IPR}\rangle_{E}\propto L^{-1}$ and for
the localized state, $\langle\text{IPR}\rangle_{E}$ is finite. Furthermore, we
can define the fractal dimension
$\tau_{2}(E,L)=-\ln(\langle\text{IPR}\rangle_{E})/\ln(L)$ and its limit
$\tau_{2}(E)=\lim_{L\rightarrow\infty}\tau_{2}(E,L)$. We have $\tau_{2}(E)=0$
for localized states, $\tau_{2}(E)=1$ for extended states, and
$0<\tau_{2}(E)<1$ for critical states, respectively (Hiramoto and Kohmoto,
1989; Evers and Mirlin, 2008; Lin  _et al._ , 2022). We note that the IPR
should be proportional to the Lyapunov exponent $\gamma(E)$ for an
exponentially localized state $\psi_{m}\sim e^{-|m|/\xi}$, with
$\text{IPR}\propto\xi^{-1}=\gamma$, in the limit $L\gg\xi$.

Physics in the overlapped and un-overlapped regimes: Although all states are
expected to be localized in our model, the effect of inter-chain coupling in
the overlapped regime and un-overlapped regime should be different, leading to
distinct localization behavior. We consider two different cases, which are
shown in Fig. 2 (a) and (b). In the first case, the spectra in the two chains
are un-overlapped to avoid the resonant coupling between the extended and
localized states; while in the second case, resonant coupling is induced in
their overlapped regime. Furthermore, we present their typical wave functions
in these regimes in Fig. 2 (c) and (d) with $t_{\text{v}}=0.1$. The results
show that the wave functions in the un-overlapped regime of the disordered
chain are exponentially localized with localization length around unity (see
the state with $E=0$ in Fig. 2 (c) and (d)). The wave functions in the
overlapped regime are also exponentially localized, however, with localization
length $\xi\sim 0.05L$, which is much larger than the localized states in the
un-overlapped regime of the disordered chain. Strikingly, the wave functions
in the un-overlapped regime of the free chain are extended even when the
system size $L=10^{6}$. Similar features can be found when $L$ is increased to
$L\sim 10^{9}$ for smaller $t_{\text{v}}$. This seems to contradict with the
general theorem. This dilemma is the major concern of this work.

Next, we investigate the asymptotic behavior of the wave function using the
transfer matrix method. In Fig. 3 (a) and (b), we present the Lyapunov
exponent $\gamma(E)$ against the energy $E$ for a system with size $L=10^{9}$.
When $t_{\text{v}}=0$, the states in the free chain are extended and the
states in the disordered chain are localized. When $t_{\text{v}}=0.1$, all
states become localized with $\gamma(E)>L^{-1}$. However, there are three
distinct energy regimes for $\gamma(E)$, corresponding to the overlapped and
un-overlapped regimes. In the overlapped regime, we have $\gamma(E)\sim
10^{-4}$, while in the un-overlapped regime, we have $\gamma(E)\sim 10^{0}$
(disordered chain) or $\gamma(E)\sim 10^{-7}$ (free chain). These distinct
behaviors are unique features of the coupled disorder model, which should not
be regarded as some kind of phase transition between extended and localized
states; see below.

Figure 3:  (a) (b) The Lyapunov exponent $\gamma(E)$ against the energy $E$
for $t_{\text{v}}=0$ and $t_{\text{v}}=0.1$, with system size $L=10^{9}$. The
shift is (a) $\Delta=-10$; (b) $\Delta=-6$. The vertical dashed lines in (a)
denote $E=-8$ and $E=-6.4$, and in (b) denote $E=-6.4$ and $E=-4$. The red
dashed lines are estimated by Eq. 2. (c) (d) The fractal dimension
$\tau_{2}(E,L)$ versus energy $E$ for $t_{\text{v}}=0.1$. The system size is
$L=2^{11}$ (blue); $L=2^{14}$ (green); $L=2^{16}$ (yellow); $L=2^{18}$ (red).
The inset present the results $\tau_{2}(E,L)$ against $1/\ln(L)$ for different
energy $E$.

We further characterize the three energy regimes using IPR and fractal
dimension $\tau_{2}(E,L)$, with results presented in Fig. 3 (c) and (d). It is
found that $\tau_{2}(E,L)\rightarrow 0$ for states in the overlapped regime
and un-overlapped regime of the disordered chain, indicating localization.
However, $\tau_{2}(E,L)\rightarrow 1$ for states in the un-overlapped regime
of the free chain, contradicting the previous theorem at first sight. This
contradiction is actually a finite-size effect since $L=2^{18}\sim
10^{5}<\xi\sim 10^{7}$. Thus, it is expected that all these states will be
localized from the general theorem in the thermodynamic limit ($L\gg\xi$, or
equivalently, $L\gamma(E)\gg 1$). This also clarifies the previous
disagreement presented in Fig. 2.

Origin of the suppressed localization length: The above results raise some
fundamental questions that need to be addressed much more carefully. In the
overlapped regime, the localized states and extended states are coupled
through resonant coupling because their energy is close to each other. From
perturbation theory Anderson (1958), all the higher-order terms will become
important, leading to significant modification of the wave functions for
localization. In the un-overlapped regime of the free chain, wave function
localization is greatly suppressed by a different mechanism.

To this end, we first consider the localization in the following minimal model

$\displaystyle
H=H_{0}+(t_{\text{v}}a_{m,0}^{\dagger}a_{m,1}+\mathrm{h.c.})+V_{m,1}a_{m,1}^{\dagger}a_{m,1}.$
(6)

As compared with Eq. 3, here we set $t_{1}=0$ and $t_{0}=t=1$ (see Fig. 4
(a)). When $t_{\text{v}}=0$, the eigenstates of component $\sigma=1$ are fully
localized at one site with eigenvalue distributed in the interval
$[-V/2,V/2]$. On the other hand, the eigenstates of component $\sigma=0$ are
$\psi(m)\propto e^{ikm}$ with energy spectra in $[\Delta-2t,\Delta+2t]$. Thus,
the energy spectra of each component are well separated when
$|\Delta|>|V/2|+2t$. We focus on the physics in this regime (with
$t_{\text{v}}\neq 0$) for the suppressed localization.

The central problem is to verify the major result of Eq. 2. We employ the
sparse matrix method to examine the physic in Eq. 6, and the results for
$E=\Delta$ are presented in Fig. 4. It is found that
$\langle\text{IPR}\rangle_{E}\propto t_{\text{v}}^{4}$ with intermediate
$t_{\text{v}}$ with $\xi<L$. In the small inter-chain coupling limit, the IPR
will be saturated to $L^{-1}$ when $\xi\gtrsim L$ due to the finite size
effect. Thus the relation of $\text{IPR}\propto t_{\text{v}}^{4}$ always holds
in the thermodynamical limit. In Fig. 4 (c), we also examine the dependence of
IPR as a function of energy shift $\Delta$, finding that
$\text{IPR}\propto\Delta^{-4}$ in the large $\Delta$ limit. However, when
$t_{\text{v}}/\Delta\ll 1$ we have $\xi\gtrsim L$ and saturation of IPR is
found again for the same reason as Fig. 4 (b). Combining these two power-law
dependence will yields $\xi^{-1}\propto(t_{\text{v}}/\Delta)^{4}$ in Eq. 2,
using IPR$\propto 1/\xi$.

To more accurately describe localization length as a function of eigenvalue
$E$, we then derive an effective Hamiltonian as

$\displaystyle
H_{\text{eff}}=\sum_{m}ta_{m+1,0}^{\dagger}a_{m,0}+\mathrm{h.c.}+\sum_{m}(\Delta+W_{m})a_{m,0}^{\dagger}a_{m,0},$
(7)

where $W_{m}=t_{\text{v}}^{2}/(\Delta-V_{m,1})$ for states with $E\sim\Delta$.
A direct calculation shows that

$\displaystyle\langle W_{m}\rangle$ $\displaystyle={1\over
V}\int_{-V/2}^{V/2}W_{m}dV_{m,1}=\frac{t_{\text{v}}^{2}}{V}\ln(\frac{2|\Delta|/V+1}{2|\Delta|/V-1}),$
$\displaystyle\langle W_{m}W_{n}\rangle$
$\displaystyle=\frac{4t_{\text{v}}^{4}}{4\Delta^{2}-V^{2}}\delta_{m,n},$ (8)

with $\langle\cdot\rangle$ represents its disorder averaged value. The
variance of $W_{m}$ can be written as

$v^{2}=\langle W_{m}^{2}\rangle-\langle
W_{m}\rangle^{2}=t_{\text{v}}^{4}(\frac{4}{4\Delta^{2}-V^{2}}-f(\Delta,V)),$
(9)

with $f(\Delta,V)=(\frac{1}{V}\ln(\frac{2|\Delta|/V+1}{2|\Delta|/V-1}))^{2}$.
It yields

$v^{2}=\frac{t_{\text{v}}^{4}V^{2}}{12\Delta^{4}}+\frac{11t_{\text{v}}^{4}V^{4}}{360\Delta^{6}}+\mathcal{O}(\Delta^{-6}),$
(10)

when $\Delta\gg V$. The leading term yields the localization length in Eq. 2
with a suppressed prefactor of $t_{\text{v}}^{4}/\Delta^{4}$, which is
numerically confirmed in Fig. 4 in a large system. This completes the proof of
Eq. 2.

Figure 4:  (a) The schematic of the model in Eq. 7. The free chain $H_{0}$
(red) is coupled to a disordered chain $H_{1}$ (blue) with $t_{1}=0$. (b)
$\log_{10}(\text{IPR})$ against $\log_{10}(t_{\text{v}})$ for different energy
shift $\Delta$ and lattice size with $E=\Delta$. The colors represent
$\Delta=-20$ (red); $\Delta=-10$ (green); $\Delta=-7$ (blue). The system size
is $L=10^{4}$ (square), $10^{5}$ (circular), and $10^{6}$ (cross). The two
black dashed lines are linear fitting with
$\log_{10}(\langle\text{IPR}\rangle_{E})\sim 4\log_{10}(t_{\text{v}})$. (c)
$\log_{10}(\text{IPR})$ against $\log_{10}(|\Delta|)$ for different
$t_{\text{v}}$. The meaning of symbols is the same as (b). The colors
represent $t_{\text{v}}=1.5$ (red); $t_{\text{v}}=1.0$ (green); and
$t_{\text{v}}=0.5$ (blue), and the black dashed lines denote
$\log_{10}(\langle\text{IPR}\rangle_{E})\sim-4\log_{10}(|\Delta|)$.

The previous conclusion is based on the minimal model with $t_{1}=0$. We then
move to examine the effect of hopping $t_{1}$ in the disordered chain, which
is expected to extend the wave functions and hence suppressed the localization
length in the free chain. Thus it is crucial to ask to what extent $t_{1}$ can
influence the localization length $\xi$. To this end, we fix $\Delta$ and
$t_{\text{v}}$ and change the value of $t_{1}$, and the results of IPR against
$t_{1}$ are presented in Fig. 5. We find that the IPR is slightly decreased
with the increasing of $t_{1}$, indicating that $t_{1}$ is not the essential
term for the localization of the free chain. Therefore, we expect that Eq. 2
serves as a good approximation for the localization length even with finite
$t_{1}$, which accounts for the excellent agreement between the numerical and
theoretical results in Fig. 3 (a) and Fig. 6. Finally, we present the relation
between the localization length as a function of energy $E$ in a single
disordered chain and in a coupled disordered model in Fig. 6, which further
confirms the empirical formulas of Eq. 1 and Eq. 2. In Fig. 3 (b), we have
used

$\xi^{-1}(E)=(\frac{4}{4\Delta^{2}-V^{2}}-f(\Delta,V))\frac{t_{\text{v}}^{4}}{8t^{2}-2(E-\Delta)^{2}},$
(11)

which yields Eq. 2 in the large $\Delta$ limit. We point out that for finite
$\Delta$, the higher-order term $\mathcal{O}(\Delta^{-4})$ in $f(\Delta,V)$ is
important. Therefore, while the localization length in the overlapped and un-
overlapped spectra exhibits distinct behaviors with all wave functions being
localized, it is not a phase transition process.

Figure 5:  The logarithm of IPR versus the intra-chain hopping $t_{1}$, with
$\Delta=-10$ and $t_{\text{v}}=1$ for fixed energy $E=-10$. The black dashed
lines are linear fitting with
$\log_{10}(\langle\text{IPR}\rangle_{E})\sim\nu\log_{10}(t_{1})$, with
$\nu=-0.0026$ for $L=10^{4}$, $\nu=-0.0066$ for $L=10^{5}$, and $\nu=-0.0075$
for $L=10^{5}$ Figure 6:  The logarithm of the Lyapunov exponent $\gamma(E)$
versus the energy $E$. (a) Results in a single chain with different disorder
strengths $V$, with corresponding localization length by Eq. 1. (b) Results in
the un-overlapped regime in the coupled disordered model in Eq. 3 with
different energy shifts $\Delta$ with $t_{0}=t_{1}=t$. The localization length
is given by Eq. 11. In the transfer matrix calculation we have used $L=2\times
10^{9}$.

Conclusion and discussion: In this work, we present a coupled disordered model
by coupling a disordered chain with a free chain, where the localization
lengths in the overlapped and un-overlapped regimes differ by several orders
of magnitude. In the overlapped regime, the states from the free chain are
localized by resonant coupling between the localized and extended chains.
However, in the un-overlapped regime of the free chain, while the states are
still localized by the general theorem, they exhibit suppressed localization
with a prefactor of $t_{\text{v}}^{4}/\Delta^{4}$. We find that the inter-
chain coupling, disorder strength, and energy shift play a leading role in
localization, yet the effect of intra-chain hopping $t_{1}$ in the disordered
chain is not significant. The results presented in this work are in 1d
disordered models, and extending this research to higher dimensional models
(Tarquini  _et al._ , 2017; Janiš and Kolorenč, 2005; Avishai and Meir, 2002)
and many-body models (Nandkishore and Huse, 2015; Alet and Laflorencie, 2018;
Abanin  _et al._ , 2019) is also intriguing, in which we expect the overlapped
regime and un-overlapped regime will also exhibit totally different behaviors
(Lin  _et al._ , 2023). Furthermore, for a system with a large localization
length in the higher-dimensional models, it may call for much more advanced
numerical methods.

These results can be readily confirmed in the states-of-art experiments with
ultracold atoms (Roati  _et al._ , 2008; Gross and Bloch, 2017; Darkwah Oppong
_et al._ , 2022; Mandel  _et al._ , 2003; Yang  _et al._ , 2017), in which the
two chains can be realized by the hyperfine states. The inter-chain coupling
can be realized by Raman coupling and their potential shift is a natural
consequence of detuning and Zeeman field. In these systems, the wave functions
in each chain can be independently realized in the limit $t_{v}\sim 0$, and
their localization can be measured individually using the time-of-flight
imaging technique. In recent years, AL in disordered systems has been an
important direction in ultracold atoms and huge progress has already been
achieved (Roati  _et al._ , 2008; Skipetrov  _et al._ , 2008; Kohlert  _et
al._ , 2019; An  _et al._ , 2021; Wang  _et al._ , 2022; White  _et al._ ,
2020; Dikopoltsev  _et al._ , 2022) and we expect the experimental
confirmation of these results can provide perspicuous evidences for the
dilemma of (I) and (II).

Finally, it is necessary to emphasize that the disordered potential (with
short-range correlation) has totally different features from the
incommensurate potential. In the coupled free chain and the incommensurate
chain, without the guarantee of the general theorem, one can realize a
critical phase in the overlapped spectra Lin  _et al._ (2022), in which the
overlapped and un-overlapped spectra also exhibit distinct behaviors in
localization. The similar critical phase by coupling of extended and localized
states in the Floquet model with incommensurate potential has also been
presented by Roy et al in Ref. (Roy  _et al._ , 2018). Here, we present a
much-simplified model, which can be solved analytically in the limiting
condition, in the hope that these intriguing results to be found in the more
complicated coupled many-body models and coupled random matrices Lin .

Acknowledgments: This work is supported by the National Natural Science
Foundation of China (NSFC) with No. 11774328, and Innovation Program for
Quantum Science and Technology (No. 2021ZD0301200 and No. 2021ZD0301500).

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