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Under the conditions of superfluid–Mott-insulator criticality in two dimensions, the trapping centers are generically characterized by an integer chargecorresponding to the number of trapped particles or holes. Varying the strength of the center leads to a transition between two competing ground states withcharges differing by ±1. The hallmark of the transition scenario is a splitting of the number density distortion into a half-integer core and a large halo carryingthe complementary charge of ±1/2. The sign of the halo changes across the transition and the radius of the halo diverges on the approach to the criticalstrength of the center.

TransitionBetweenDifferentIntegerStates

Trapping Centers at the Superfluid–Mott-insulator Criticality: Transition between Charge-quantized States

Yuan Huang1,2, Kun Chen1,2, Youjin Deng1,2, Boris Svistunov1 1 University of Massachusetts, Amherst

2 University of Science and Technology of China

Charge properties of impurities in bosonicsystems

Massive impurities static potential)

�n(r)

n̄

�V n̂

Charge quantization:

is integer⇠ ⌘Rddr�n(r)

Superfulid Mott-insulator⇠

V

⇠

V

1

2

�1

�2

compressible gappedno quantization quantized

Method

Worm-algorithm path-integral quantum MonteCarlo method for Bose Hubbard model at 2D

H = �P

hiji b†i bj +

U2

Pi ni(ni � 1)� µ

Pi ni + V n0

Asymptotic behavior far away from the center

dictated by the linear-response function

featuring the universal critical behavior

The integral of distorted density profile

GenericQuantization

�n(r) / �(r) (r ! 1)

�(r) =R �0 d⌧ [hn(0, 0)n(r, ⌧)i � |hn(0, 0)i|2]

I(r) ⌘P

rir(ni � 1) = ⇠ ± const

r (r ! 1)

Numerical result for V=3.5 at QCP in canonical ensemble with total particle number N=L2

As approaching the transition point, thedensity profile develops bimodal distributionincluding a half-integer charged core and amacroscopic halo with charge±1/2.The size of halo is characterized by r0 .

�(r) / 1r3

r0

⇠ 1r1.43

⇠ 1r3

a r

�n

±1/2

�1/2

Conclusion

ContinuousTransition

Continuous quantum phase transition betweentwo states with different quantized charge

The healing length r0 diverges as

Use the integral of density profile I(r) over acertain macroscopic range as an observable

The integral in the Hubbard model as a function of rescaled strength of the center at different system sizes. The simulation is performed in the grand canonical ensemble.

I(L/2p2)

Vc

net charge = 0 net charge = -1

⌫̃ = 2.33(5)r0 ⇠ |V � Vc|�⌫̃

v generic charge quantization at 2D SF-MI criticality

v continuous transition between different charged states

• half-integer charge at the transition point

• bimodal distribution at the critical regime: half-integer core surrounded by half-integer halo with diverging radius

v halo with divergent size will induce non-pairwise long range interactions which can be utilized to engineer new quantum phases out of impurity gas

ChargeoftheCenteratSF-MICriticality:Quantizedornot?

arxiv:1608.02232

SF-MI quantum criticality in d>1:• incompressible• no gap