Variational Matrix Product State Approach for Non-Hermitian System Based on a Companion Hermitian Hamiltonian Zhen Guo1Zheng-Tao Xu1Meng Li1Li You1 2 3 4and Shuo Yang1 2 3y
2025-05-06
0
0
837.41KB
12 页
10玖币
侵权投诉
Variational Matrix Product State Approach for Non-Hermitian System Based on a
Companion Hermitian Hamiltonian
Zhen Guo,1Zheng-Tao Xu,1Meng Li,1Li You,1, 2, 3, 4, ∗and Shuo Yang1, 2, 3, †
1State Key Laboratory of Low Dimensional Quantum Physics,
Department of Physics, Tsinghua University, Beijing 100084, China
2Frontier Science Center for Quantum Information, Beijing, China
3Hefei National Laboratory, Hefei, 230088, China
4Beijing Academy of Quantum Information Sciences, Beijing 100193, China
Non-Hermitian systems exhibiting topological properties are attracting growing interest. In this
work, we propose an algorithm for solving the ground state of a non-Hermitian system in the matrix
product state (MPS) formalism based on a companion Hermitian Hamiltonian. If the eigenvalues
of the non-Hermitian system are known, the companion Hermitian Hamiltonian can be directly
constructed and solved using Hermitian variational methods. When the eigenvalues are unknown,
a gradient descent along with the companion Hermitian Hamiltonian yields both the ground state
eigenenergy and the eigenstate. With the variational principle as a solid foundation, our algorithm
ensures convergence and provides results in excellent agreement with the exact solutions of the non-
Hermitian Su-Schrieffer-Heeger (nH-SSH) model as well as its interacting extension. The approach
we present avoids solving any non-Hermitian matrix and overcomes numerical instabilities commonly
encountered in large non-Hermitian systems.
Introduction.— The idea of using non-Hermitian
Hamiltonian to effectively describe open system back-
dates to the mid-1900s [1–3], not long after the birth
of Quantum Mechanics. Over the years, non-Hermitian
Hamiltonians have arisen in a variety of non-conservative
systems, both classical [4–10] and quantum [11–19]. In
systems exhibiting non-Hermitian skin effects, the con-
ventional bulk-boundary correspondence (BBC) is bro-
ken [20–26]. Significant efforts are being made to charac-
terize non-Hermitian BBC, such as defining BBC through
singular value gap [27], detecting BBC using entangle-
ment entropy [28–30], and understanding BBC by gener-
alized Bloch theory [31–33].
Most studies of non-Hermitian systems focus on single-
particle Hamiltonians, but not all properties of non-
Hermitian many-body states can be directly derived from
single-particle wave functions, even in the non-interacting
case [22]. Beyond the single-particle research, standard
methods for many-body non-Hermitian systems are of-
ten limited to small system size [25, 28, 34, 35] due to
the exponential growth of Hilbert space. Fortunately,
not all quantum states in the many-body Hilbert space
are equally important to the main physics. For Hermitian
systems, low energy states of realistic Hamiltonians are
constrained by locality and obey the entanglement area
law [36, 37]. Tensor network (TN) states can be con-
structed based on this property, allowing them to natu-
rally capture the most relevant states in Hilbert space. In
recent years, TN has emerged as a powerful tool to study
strongly correlated quantum many-body systems. When
solving the ground state of a one-dimensional (1D) Her-
mitian Hamiltonian, the variational matrix product state
∗lyou@tsinghua.edu.cn
†shuoyang@tsinghua.edu.cn
(VMPS) method [38, 39] and the equivalent density ma-
trix renormalization group (DMRG) algorithm [40] al-
ways converge, as guaranteed by the variational princi-
ple. For large strongly correlated non-Hermitian systems,
analogous principles and stable algorithms are desired.
Although DMRG has been successfully applied to cer-
tain non-Hermitian problems [41–45], debates remain re-
garding the choices of density matrices [41, 44–51] and
numerical difficulties remain for some approaches. For
instance, Ref. [51] points out that the convergence char-
acteristics of non-Hermitian algorithms are less favorable
than in the Hermitian case, with the storage and com-
puting time more than doubled due to the inequivalent
left and right eigenspaces. Reference [43] reports an in-
stability of non-Hermitian DMRG even after many it-
erations and attributes this to non-Hermiticity. Refer-
ences [45, 48, 52] introduce biorthonormal DMRG ap-
proaches capable of providing accurate results while re-
maining problematic at exceptional points. Our study
ascribes the latter to the biorthonormal condition itself
since nearly orthogonal left and right eigenvectors are
commonly found in non-Hermitian systems exhibiting
skin effect [53].
In this work, we introduce two practical VMPS ap-
proaches for a non-Hermitian system based on a compan-
ion Hermitian Hamiltonian. The Hermitian variational
principle consequently guarantees convergence, and we
avoid all numerical difficulties by not directly solving any
non-Hermitian matrix.
Variational principle.— The VMPS method, like
other Hermitian variational methods, relies on the fact
that a ground state has the lowest energy. To go beyond
this Hermitian paradigm, the first step is to define the
ground state of a non-Hermitian system and its energy
gradient. Two common definitions of ground states are
used, depending on whether the real or imaginary parts
arXiv:2210.14858v1 [quant-ph] 26 Oct 2022
2
of energy are minimized, with the state denoted by |SRi
or |SIi, respectively. For convenience, the ground state
mentioned below refers to the right-eigenvector |riunless
otherwise specified, i.e., H|ri=e|ri. To ensure numeri-
cal stability, we impose normalization conditions on the
left and right eigenstates separately, such that hl|li= 1
and hr|ri= 1. A normalized right-eigenvector has the
form |ri=|xi/phx|xi, and the corresponding expec-
tation energy becomes e(|xi) = hx|H|xi/hx|xi. Because
e(|xi)is not a holomorphic function, the derivative is not
well-defined or cannot be used for straightforward opti-
mization [54]. Nevertheless, if the real and imaginary
parts of the energy are treated as two real-valued func-
tions of complex variables, the Wirtinger derivative [55]
∂hx|,1
2(∂<{|xi} +i∂={|xi})can be adopted instead [53]:
∂hx|<{e(|xi)}=(H+H†)|xi
2hx|xi−hx|(H+H†)|xi|xi
2hx|xi2.(1)
Naively, one expects to use Eq. (1) for gradient de-
scent to find the |SRiground state of a non-Hermitian
system. The gradient ∂hx|<{e(|xi)}should become zero
at the end of iterations, resulting in (H+H†)|xi ∝ |xi.
However, such a condition shows that the converged |xi
is an eigenvector of H+H†rather than H. The states
consequently obtained are usually not eigenstates of the
non-Hermitian Hamiltonian H. Therefore, the conven-
tional variational principle is no longer directly suitable
for non-Hermitian systems, unless a proper cost function
is available.
Algorithm.— We propose to take the eigenvector
residual norm
N(|xi),|H|xi − e(|xi)|xi|2(2)
as the cost function. It is worth noting that any eigen-
state of H, not just the ground state, fulfills N(|xi) =
0, and N(|xi)is always real and non-negative. The
Wirtinger derivative of N(|xi)is simplified to [53]
∂hx|N(|xi) =[H†−e∗(|xi)][H−e(|xi)]|xi
,G(H, e(|xi))|xi.(3)
Our goal is then reduced to finding the lowest-energy
state satisfying G(H, e(|xi))|xi= 0.
Before attempting to solve the above equation, we
first establish a relation between the companion Hermi-
tian Hamiltonian G(H, ε)and the original non-Hermitian
Hamiltonian H. Here G(H, ε)is non-negative definite
and Hermitian for any arbitrary ε. Using singular value
decomposition
H−ε=USV †, U †U=V†V=I, (4)
G(H, ε)is decomposed to
G(H, ε) = (H†−ε∗)(H−ε) = V S2V†,(5)
where singular values siin Sare sorted in descending or-
der. We can see that G(H, ε)has the smallest eigenvalue
s2
n= 0 if and only if Hhas an eigenvalue ε. Furthermore,
the vector Vnis a shared eigenstate of Hand G(H, ε),
with eigenvalues εand s2
n= 0, respectively.
For non-interacting systems, the energy eof a many-
body ground state can be obtained from summing up
single-particle energies. Finding the ground state of a
non-Hermitian Hamiltonian Htherefore reduces to find-
ing the zero-energy ground state of the companion Hermi-
tian Hamiltonian G(H, e), for which the powerful VMPS
approach can be employed. In practice, given a finite
virtual bond dimension Dof MPS, the ground energy
of G(H, e)after convergence will retain a tiny non-zero
value η, and sn=√ηmeasures whether Dis large
enough. Hereafter, we will refer to the aforementioned
algorithm with supplied eigenenergies as the Hermitian-
ized variational matrix product state (HVMPS) method,
which is guaranteed to converge according to the stan-
dard VMPS method.
When eigenvalues are not provided or unknown, a
gradient descent method facilitated by the companion
Hermitian Hamiltonian can determine the ground en-
ergy and the corresponding eigenstate simultaneously,
and this will be called the gradient variational matrix
product state (GVMPS) method. For simplicity, we illus-
trate GVMPS using a parity-time (PT ) symmetric sys-
tem since its many-body ground energy is always real.
More details and its application to non-PT symmetric
models are given in Supplemental Material.
The Wirtinger derivative of sn(ε)with respect to ε
reads [53]
∂ε∗sn=ε−V†
nHVn
2sn
.(6)
One may employ the gradient descent method to find
the smallest εopt that minimizes sn(ε), such that εopt be-
comes the ground state energy of Hand the correspond-
ing eigenvector Vnis obtained at the same time. How-
ever, when sn(ε)approaches its minimum, the gradient is
usually close to zero, which slows down the convergence
and even results in instabilities [53]. This can be avoided
by manually setting the gradient to ε−V†
nHVnand us-
ing an adaptive learning rate [53]. Regarding the initial
value of ε, we note that
<{e(|xi)}=hx|(H+H†)|xi
2hx|xi≥τ, (7)
where τis the smallest eigenvalue of the Hermitian ma-
trix (H+H†)/2. Therefore, all eigenvalues of Hhave
real parts greater than τ, making τa good starting point
for determining the eigenvalue of |SRi. Since the gradi-
ent descent only depends on one scalar parameter εand
there is no other local minimum from the initial value τ
to the desired energy, the GVMPS method is found to
be efficient and well-converged.
Both the HVMPS and GVMPS approaches we pro-
pose avoid directly solving non-Hermitian problems, and
3
the companion Hermitian Hamiltonian helps to prevent
typical numerical instabilities of non-Hermitian systems.
Some of our benchmark results are presented below.
Non-interacting model.— We first test for non-
interacting systems using the 1D nH-SSH model, which
exhibits interesting topological properties and has re-
cently received a lot of attention [20, 29, 56–58]. The
Hamiltonian takes the following form
H0=X
ih(t+γ/2) a†
ibi+ (t−γ/2) b†
iai
+b†
iai+1 +a†
i+1bii,
(8)
with one unit cell composed of two sites. The intra-cell
hopping is non-reciprocal and characterized by a non-
Hermitian strength γ, while the inter-cell hopping is Her-
mitian and set as unit strength. High-order exceptional
points for this model [29, 59–61] appear at |t|=|γ/2|.
Since MPS methods are more accurate and efficient for
finding low entanglement states that satisfy the area law,
it is necessary to investigate the entanglement properties
throughout the parameter space to determine which re-
gions are more favourably described by MPS. Under pe-
riodic boundary condition (PBC), previous study [62] on
the bi-orthogonal entanglement entropy (EE) [29, 30, 62–
65] of the nH-SSH model finds its ground state obeys the
area law at t= 1 and γ > 4. Since our algorithms only
focus on the right eigenstate and the bi-orthogonal condi-
tion may induce extra numerical difficulties [53], we will
use the EE of the right eigenstate |riinstead. The bipar-
tite EE between subsystem Aand its complementary part
¯
Ais given by SA=−Tr(ρAlnρA), where ρA= Tr ¯
Aρrr ,
ρrr =|rihr|/hr|ri, and the length of Ais LA. Usually,
SAreaches its maximum when LAis half of the total
length, and we call it the maximum EE. As shown in the
Supplemental Material, this definition of EE reveals the
same area-law behavior for t= 1 and γ > 4.
Now we apply HVMPS to investigate the entanglement
behavior of the |SRiground state of the 25-unit-cell nH-
SSH model under open boundary condition (OBC), with
the many-body energy supplied by the sum of single-
particle energies. As shown in Fig. 1(a), we find three
regions with different entropy distributions for γ= 1,
roughly separated by t= 0.5and 1.5. The solid lines
in regions I and III display the maximum EE as a func-
tion of t, and their convergence is confirmed by increasing
the virtual bond dimension D. Typical EE in these re-
gions as a function of LAis shown in Fig. 1(d), where the
plateau in the middle region indicates area-law behavior.
In most parts of region II, a stable entropy distribution
is not reached for D= 300. Near the boundaries of re-
gion II, area-law violations are observed and shown in
Figs. 1(b) and (c). As indicated by the background color
in Fig. 1(a), the logarithm of the converged ground en-
ergy ηof G(H, ε)in region II is a few orders of magnitude
larger than in regions I and III. In fact, a bond dimension
D∼100 is enough to reach η < 10−13 in regions I and
(a)
I II III
0.5 1 1.5
0
0.5
1
t
EE
−14
−12
−10
−8
log10 η
0 25 50
0.7
0.8
t= 0.54
(b) LA
EE
0 25 50
0.4
0.6
0.8
1
t= 1.41
(c) LA
0 25 50
.26
.28
t= 1.69
(d) LA
(e)
0 0.511.5
0
1
2
3
4
t
γ
(f)
0 0.511.5
t
−14
−12
−10
−8
−6
−4
log10 η
Figure 1. (Color online) (a) Solid lines show the maximum
EE of |SRicalculated for the OBC nH-SSH model at γ= 1 for
various t. The missing region indicates no convergence found
with bond dimension D= 300. The background color repre-
sents the logarithm of the converged ground energy of G(H, ε),
i.e., log10 η. (b-d) Bipartite EE as a function of the subsystem
length LA. (e-f) log10 ηin the (t, γ)parameter space for the
ground states |SRi(e) and |SIi(f), calculated with bond di-
mension D= 100. The blue regions obey the area law. Black
dashed (solid) lines are the topological phase boundaries de-
fined by energy gap closing points for OBC (PBC). Dotted
lines indicate exceptional points, beneath which the energy
spectra are real.
III, whereas the required Dquickly exceeds 300 once en-
tering region II. Together with the sudden increase of the
maximum EE near the boundaries shown in Fig. 1(a), we
conclude that area law is violated in region II.
Similarly, we sweep the entire parameter space for |SRi
and |SIiground states using log10 ηas a criterion, and the
area-law-obeyed regions are painted in blue in Figs. 1(e)
and (f). Remarkably, the OBC area law boundaries coin-
cide with the PBC topological phase boundaries, which
are solid lines in Figs. 1(e) and (f). The BBC is bro-
ken for the nH-SSH model, and the energy gap closing
points under OBC (dashed lines in Figs. 1(e) and (f))
can no longer be taken as good indicators for bulk phase
boundaries [30]. Nevertheless, the EE of ground states
under OBC contains bulk phase information like in the
Hermitian case, which helps to restore BBC.
According to Ref. [66], a local non-Hermitian Hamil-
tonian with real spectra can be mapped to a non-local
Hermitian one by a similar transformation, with area
law no longer necessarily valid. In our case, the ground
摘要：
展开>>
收起<<
VariationalMatrixProductStateApproachforNon-HermitianSystemBasedonaCompanionHermitianHamiltonianZhenGuo,1Zheng-TaoXu,1MengLi,1LiYou,1,2,3,4,andShuoYang1,2,3,y1StateKeyLaboratoryofLowDimensionalQuantumPhysics,DepartmentofPhysics,TsinghuaUniversity,Beijing100084,China2FrontierScienceCenterforQuantumI...
声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！
相关推荐
-
【词汇变形总汇】2025高考词汇变形总汇 - 教师版VIP免费
2024-12-06 3 -
【超简37页】新课标高考英语考纲3500词汇VIP免费
2024-12-06 4 -
《高考英语3500词详解》（WORD版）VIP免费
2024-12-06 13 -
《高考英语3500词详解》VIP免费
2024-12-06 11 -
高中英语-[教师版]80天通关高考3500词汇VIP免费
2024-12-06 12 -
高中人教选修7课文逐句翻译VIP免费
2024-12-06 7 -
高中人教选修7课文原文及翻译VIP免费
2024-12-06 13 -
高中人教必修4课文逐句翻译VIP免费
2024-12-06 8 -
高中人教必修4课文原文及翻译VIP免费
2024-12-06 13 -
高考英语核心高频688词汇VIP免费
2024-12-06 10
分类：图书资源
价格：10玖币
属性：12 页
大小：837.41KB
格式：PDF
时间：2025-05-06
作者详情
-
Voltage-Controlled High-Bandwidth Terahertz Oscillators Based On Antiferromagnets Mike A. Lund1Davi R. Rodrigues2Karin Everschor-Sitte3and Kjetil M. D. Hals1 1Department of Engineering Sciences University of Agder 4879 Grimstad Norway10 玖币0人下载
-
Voltage-controlled topological interface states for bending waves in soft dielectric phononic crystal plates10 玖币0人下载
相关内容
-
山西大学附属中学2023-2024学年高三上学期10月月考 数学答案
分类：中学教育
时间：2025-05-11
标签：无
格式：PDF
价格：10 玖币
-
山西大学附属中学2023-2024学年高三上学期10月月考 数学
分类：中学教育
时间：2025-05-11
标签：无
格式：PDF
价格：10 玖币
-
山西大学附属中学2023-2024学年高三上学期10月月考 生物答案
分类：中学教育
时间：2025-05-11
标签：无
格式：PDF
价格：10 玖币
-
山西大学附属中学2023-2024学年高三上学期10月月考 生物
分类：中学教育
时间：2025-05-11
标签：无
格式：PDF
价格：10 玖币
-
山西大学附属中学2023-2024学年高三上学期10月月考 历史答案
分类：中学教育
时间：2025-05-11
标签：无
格式：PDF
价格：10 玖币
渝公网安备50010702506394
