Interaction-induced non-Hermitian topological phases from a dynamical gauge eld W. N. Faugno1and Tomoki Ozawa1 1Advanced Institute for Materials Research WPI-AIMR Tohoku University Sendai 980-8577 Japan

2025-05-05 0 0 433.07KB 7 页 10玖币

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Interaction-induced non-Hermitian topological phases from a dynamical gauge ﬁeld

W. N. Faugno1and Tomoki Ozawa1

1Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Sendai 980-8577, Japan

(Dated: October 5, 2022)

We present a minimal non-Hermitian model where a topologically nontrivial complex energy

spectrum is induced by inter-particle interactions. Our model consists of a one-dimensional chain

with a dynamical non-Hermitian gauge ﬁeld with density dependence. The model is topologically

trivial for a single particle system, but exhibits nontrivial non-Hermitian topology with a point gap

when two or more particles are present in the system. We construct an eﬀective doublon model to

describe the nontrivial topology in the presence of two particles, which quantitatively agrees with

the full interacting model. Our model can be realized by modulating hoppings of the Hatano-Nelson

model; we provide a concrete Floquet protocol to realize the model in atomic and optical settings.

Non-Hermitian Hamiltonians have been found to host

a rich variety of topological phases [1–14]. While some

non-Hermitian phases are direct analogues of Hermitian

phases, there are many uniquely non-Hermitian phases.

These result when the system has a point gap in the com-

plex energy plane, which allows for non-trivial winding of

the energy spectrum. Non-Hermitian topology manifests

in an analogous bulk-boundary correspondence known as

the skin eﬀect wherein a macroscopic number of states lo-

calize at the boundary [15–19]. These phenomena have

been primarily investigated as single particle eﬀects.

Comparatively little work has been done to understand

the role of correlations and interactions in non-Hermitian

topological phases [20–24]. Interactions have played an

important role in Hermitian topological physics, giving

rise to many paradigmatic phases including the fractional

quantum Hall eﬀect and quantum spin liquids. Such

strongly interacting systems have led to many advance-

ments in physics, including developments in gauge theo-

ries. Given the richness of Hermitian interacting systems,

it remains to be seen how analogous non-Hermitian inter-

actions can enrich the topology of open systems. Exper-

imentally, two body loss terms are ubiquitous in optical

lattices and photonics with an increasing degree of con-

trol, further motivating investigations of the topology of

many body open systems.

In this letter, we report on a minimal 1D non-

Hermitian model exhibiting interaction induced topology.

We demonstrate that our model is topologically trivial for

a single particle, but gains a non-trivial winding number

in the complex energy plane for two or more particles.

We characterize the spectrum by the clustering proper-

ties of the eigenstates. This leads us to derive an eﬀective

SSH model of Doublons with an emergent sublattice sym-

metry, which quantitatively captures the complex energy

ring of the full spectrum. The winding number of the

interacting model corresponds with the winding of this

eﬀective model analogous to interaction induced topology

in Hermitian systems [25–27]. We conclude by proposing

a two-frequency Floquet protocol that realizes our model

as an eﬀective Hamiltonian. As an intermediate step,

this Floquet protocol realizes a Hatano-Nelson model.

Model.— Our model consists of bosons populating a

1D chain with hoppings dependent on the gradient of the

density, which can be interpreted as a density-dependent

synthetic dynamical gauge ﬁeld. The Hamiltonian of our

system is

H=X

a†

j+1−t+iγR(nj+1 −nj)aj

+a†

j−t+iγL(nj−nj+1)aj+1 (1)

where ajand a†

jare bosonic annihilation and creation

operators, respectively, tis the single particle hopping

parameter, njis the density operator a†

jajon the jth

site, and γR/L are the couplings to the gauge ﬁeld for

right and left hoppings. To realize a non-Hermitian

model, we take γR6=γ∗

Lin similar fashion to the

Hatano-Nelson model [28,29]. Description in terms of

non-Hermitian Hamiltonians can be obtained through a

post-selection procedure on quantum trajectories [21,30].

Later we present a concrete experimental protocol com-

bining quantum trajectory and Floquet theory to real-

ize this Hamiltonian. In the present model, when there

is only one particle present in the system, the density

terms are identically zero. Therefore, for a single parti-

cle the Hamiltonian is Hermitian and corresponds to a

free boson. The single particle spectrum is shown in Fig.

1a which reproduces the free boson result for a periodic

chain of length L= 20.

Exact Diagonalization.— Let us contrast this result

with the two particle spectrum shown in Fig 1b. We

consider ﬁxed particle number as the Hamiltonian has

U(1) symmetry. Physically, particle number is conserved

between quantum jumps during which the non-Hermitian

Hamiltonian description is valid. Here we ﬁnd that the

energy spectrum consists of a sector where energies are

nearly real and a sector consisting of a ring of complex

energies. We project each eigenstate into the subset of

basis states where particles lie on the same site or adja-

cent sites, and ﬁnd that the states with complex energies

largely lie in this subspace while those with nearly real

energies have almost zero weight in this subspace. To

arXiv:2210.01572v1 [quant-ph] 4 Oct 2022

FIG. 1. Summary of key results. Panels a and b are energy

spectra for periodic boundary conditions plotted in the com-

plex plane for 1 and 2 particles, respectively. For one particle,

the spectrum is real while for two particles, the spectrum is

complex with a point gap. The coloring in panel b is obtained

projecting each eigenstate onto the subspace of basis states

where particles lie on the same site or adjacent sites. The

red line is the spectrum obtained from the eﬀective doublon

model described below. Panel c and d are the real space pro-

ﬁles of the eigenstates in the open boundary geometry for 1

and 2 particles respectively. For one particle, eigenstates are

typical standing waves while for two particles we observe a

skin eﬀect. The chosen parameters are t= 1, γL= 1.5 and

γR= 0 for a lattice with 20 sites.

further characterize this separation we calculate the cor-

relator a†

ja†

kajakfor each eigenstate. Representative cor-

relators for complex energy states and nearly real energy

states are shown as a function of jfor a ﬁxed kin Fig

2. This correlator conﬁrms that the states with complex

energy occur when the particles cluster while those with

nearly real energies occur when the particles separate.

FIG. 2. Four point correlator ha†

ja†

kajakifor k= 10 with

periodic boundary conditions. The solid line is representa-

tive of states with corresponding energies on the ring while

the dashed line is representative of states with nearly real

eigenenergies.

The energy spectrum has a point gap indicating the

presence of nontrivial topology. To verify the nontrivial

FIG. 3. Plot demonstrating the nontrivial winding number.

A jump from 1 to -1 increases the winding by 1 while a jump

from -1 to 1 descreases the winding by 1.

topology, we calculate the winding number following the

ﬂux insertion procedure outlined in Ref [31]. We deﬁne

H(φ) by multiplying e−iφ (eiφ) to the boundary hopping

term in the ﬁrst (second) term of Eq.(1), where φis the

strength of the inserted magnetic ﬂux. We then calculate

π=∂φln det[H(φ)−δI](2)

as a function φ, where =[·] stands for the imaginary part,

and Iis the identity matrix. The signed number of jumps

in this quantity give the winding of the phase about the

point δ, chosen to lie within the point gap, in the complex

plane. In Fig 3, we plot this quantity versus the ﬂux, φ,

and clearly see that it jumps twice as the ﬂux is tuned

from 0 to 2π, giving a winding number of 2 and con-

ﬁrming that the system is topologically non-trivial when

there are two particles.

A non-trivial winding number implies the existence of

the skin eﬀect in the open boundary geometry. The open-

boundary eigenstates are plotted in Figs 1c and 1d for

the one and two particle systems respectively. The one

particle eigenstates are exactly those obtained for a free

Boson model while the two particle eigenstates demon-

strate a clear skin eﬀect. In fact, all states will localize

on the edge for strong enough gauge coupling. For com-

pleteness, we present the energy spectrum in the open

boundary geometry in Fig 4, which shows that the spec-

trum does not cleanly separate along particle clustering

properties as both particles localize on the edge. Unlike

the single-particle Hatano-Nelson model, the spectrum

in the open boundary condition does not lie on the real

axis only; we discuss below the origin of this complex

spectrum.

Eﬀective Doublon Model.— To understand the topol-

ogy of the system, we derive an eﬀective doublon model

that captures the physics of the complex energy ring. We

obtain this by restricting our basis to states where the

particles lie on the same site or on adjacent sites. The

eﬀective doublon model consists of two sublattices; we

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Interaction-inducednon-HermitiantopologicalphasesfromadynamicalgaugeeldW.N.Faugno1andTomokiOzawa11AdvancedInstituteforMaterialsResearch(WPI-AIMR),TohokuUniversity,Sendai980-8577,Japan(Dated:October5,2022)Wepresentaminimalnon-Hermitianmodelwhereatopologicallynontrivialcomplexenergyspectrumisinducedb...

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Interaction-induced non-Hermitian topological phases from a dynamical gauge eld W. N. Faugno1and Tomoki Ozawa1 1Advanced Institute for Materials Research WPI-AIMR Tohoku University Sendai 980-8577 Japan

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