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SUPPLEMENTARY INFORMATION
for
Phase transitions in the complex plane of physical parameters
Bo-Bo Wei, Shao-Wen Chen, Hoi-Chun Po & Ren-Bao Liu*
Department of Physics, Centre for Quantum Coherence, and Institute of Theoretical
Physics, The Chinese University of Hong Kong, Shatin, New Territories, China
*[email protected]
I. Full Methods and Supplementary Equations
A. Exact solution and Lee-Yang zeros of the1D Ising model
The Hamiltonian of the 1D Ising model at finite external magnetic field is
H  h     J  j j 1  h j .
N
(S1)
j 1
The partition function can be exactly obtained by the transfer matrix method as


Z  e N  J  cosh(  h)  sinh 2 (  h)  e 4  J

 
N
 cosh(  h)  sinh 2 (  h)  e 4  J

N

 .
(S2)
With z  exp  2 h  and x  exp  2 J  , the partition function is written as
2


1

z


N (  J  h) 1  z
2


Z e
 

x
z

 2

 2 


N
 
  1 z 
  2
1  
  1 z
  2 
 
N
2
 
 1 z 
2


 x z  

 2 
 .
2
1

z


2  

 x z  
 2 
 
(S3)
Therefore the Lee-Yang zeros are given by
1 z
 1 z 
2
 
 x z
2
 2 
2
1 z
 1 z 
2
 
 x z
2
 2 
2
 eikn ,
(S4)
1
where kn   (2n  1) / N , n  1, 2,
, N . After solving the above algebraic equation of z,
the Lee-Yang zeros for ferromagnetic coupling ( J  0 ) are
zn  cos  n  i sin  n
cos  n   x 2  (1  x 2 ) cos kn ,
(S5)
sin  n   (1  x 2 )[sin 2 kn  x 2 (1  cos kn ) 2 ].
For anti-ferromagnetic phase, J  0 ,
zn   x2  (1  x2 )cos kn  ( x2  1)[sin 2 kn  x2 (1  cos kn )2 ].
(S6)
For non-interacting case, J  0 ,
zn  1.
(S7)
B. 2D Ising model in a finite field
The Hamiltonian of the 2D square lattice Ising model with a finite magnetic field is
H   J  i j  h  j
i, j
(S8)
j
The 2D square lattice Ising model was solved exactly by Onsager in the zero field. For
non-zero field, there is no exact solution, but one can map the 2D classical spin model
to a 1D quantum spin model by the transfer matrix method:
  J  i1, j1 i , j i , j1   J  i1, j1 i , j i1, j   h  i1, j1 i , j 
Z (  , h)  Tr e



M ,N
=

 j , j 
M ,N
M ,N
C11 ' D1 ' 2  C2 2 ' D2 ' 3 
CM M ' DM ' 1 
(S9)
 Tr[(CD) M ],
where
C '  e
J
 j1  ( j )  '( j 1)   h  j1  '( j )
N
N
  ' and D '  e
J
 j1  ( j )  '( j )
N
 e Js1s1 e  Js2 s2
'
'
e  JsN sN .
'
Therefore, evaluation of the partition function of the 2D Ising model amounts to
solution of a 1D quantum spin chain:
2
Z (  , h)=Tr[(CD) M ],
C  exp   J  j 1 zj  zj 1   h j 1 zj  ,


D  (e  J  e   J  1x )(e  J  e   J  2x ) (e  J  e   J  Nx )
N
N
(S10)
N
 [2sinh(2 K )]N /2 exp  K *  j 1 xj  .


C. Exact RG flow equations for 1D Ising model
Denoting the dimensionless parameters K0   J and h0   h for the Hamiltonian in
equation (S1), one can evaluate the partition function as [12]

 N

Z ( J , h; N )  Tr exp    K 0 j j 1  h0 j   
 j 1
 

 N /2 
h

 
 Tr     exp  K 0 ( j j 1   j 1 j  2 )  0 ( j  2 j 1   j  2 )   

2

  
 j even   j1  1
 N /2
h


 Tr   exp  K1 j j  2  1  j   j  2   G1  
2


 j even
NG1 / 2
e
Z ( K1 , h1 ; N / 2),
(S11)
where we have assumed periodic boundary condition and N to be even for simplicity.
This gives the RG flow equations
K1 
1  cosh  2 K 0  h0  cosh  2 K 0  h0  
ln 

4 
cosh 2  h0 

1  cosh  2 K 0  h0  
h1  h0  ln 
; &
2  cosh  2 K 0  h0  
1
G1  ln 16 cosh 2  h0  cosh  2 K 0  h0  cosh  2 K 0  h0   .
4
(S12)
D. Approximate RG flow equations for 2D Ising model
Denoting the dimensionless parameters K0   J and h0   h for the Hamiltonian in
equation (S8). The partition function of 2D Ising model can be evaluated
3


N
N
N
Z (  , h)  Tr exp K 0  i , j 1 i , j i , j 1  K 0  i , j 1 i , j i 1, j  h0  j 1 i , j 


e
 NG1 /2



h1  p  p  K1  nn  p q


 ,
Tr exp

  K
   K 3  pqr  p q r K 4  pqrs  p q r s  

 2  nnn p q

(S13)
where nn denotes nearest neighbor spins and nnn denotes next nearest neighbor spins in
the renormalized lattice. The renormalized parameters are
1  cosh  4 K 0  h0  cosh  4 K 0  h0  
K1  ln 
 ,
8 
cosh 2  h0 

K2 
1  cosh  4 K 0  h0  cosh  4 K 0  h0  
ln 
 ,
16 
cosh 2  h0 

2
1  cosh  4 K 0  h0  cosh  2 K 0  h0  
K3  ln 
,
16  cosh  4 K 0  h0  cosh 2  2 K 0  h0  
(S14)
6
1  cosh  4 K 0  h0  cosh  4 K 0  h0  cosh  h0  
K 4  ln 
 ,
16 
cosh 4  2 K 0  h0  cosh 4  2 K 0  h0 

2
1  cosh  4 K 0  h0  cosh  2 K 0  h0  
h1  h0  ln 
.
4  cosh  4 K 0  h0  cosh 2  2 K 0  h0  
The RG equations are not closed as new parameters K2 , K3 , K4 emerge. It is necessary
to make some approximation to make the RG equations close. A truncation scheme [13]
is obtained by correcting the theory in an approximate way for the presence of the term
K2 , K3 , K4 . Since both K 2 and K 4 are positive, the emergent interaction associated
with them have the effect of increasing the alignment of spins. Hence a reasonable
approximation is to drop K2 , K3 , K4 but simultaneously increase K1 to a new value
so that the new alignment tendency remains the same. Thus we have the approximate
RG equations for 2D Ising model,
K1 
3  cosh  4 K 0  h0  cosh  4 K 0  h0  
ln 
 ,
16 
cosh 2  h0 

2
i  cosh  4 K 0  h0  cosh  2 K 0  h0  
h1  h0  ln 
.
4  cosh  4 K 0  h0  cosh 2  2 K 0  h0  
(S15)
4
E. Exact solution and Lee-Yang zeros of the one-dimensional transverse field Ising
model.
The Hamiltonian of the one-dimensional transverse-field Ising model is
H  1 H1  2 H 2 ,
N
N
j 1
j 1
H1    xj  xj 1 , H 2    zj ,
(S16)
which can be exactly solved [14]. By applying Jordan-Wigner transformation, Fourier
transformation and Bogoliubov transformation, the Hamiltonian is transformed to be a
free-fermion one as
H  1    k   bk†bk  1 2 ,
(S17)
k
with energy dispersion
2

 
 (k )  2  cos k  2   sin 2 k .
1 

(S18)
Therefore, at temperature T, the positions of the zeros of the partition function

Z  Tr e   H

are decided by
exp   1  k  2  exp  1  k  2  0,
(S19)
i.e.
2

 
21  cos k  2   sin 2 k  i  2n  1  ,
1 

(S20)
with n being an integer.
F. Physical realizations for the time-domain measurement.
The time-domain measurement may be implemented by coupling the system to a probe
spin through the probe-system coupling H SB     H  (t )     H  (t )
( S z       ) and measuring the probe spin coherence. If we initialize the
5
probe spin in a superposition state    and the system in a thermal equilibrium
state described by the density matrix,   exp   H  / Tr[exp   H ] , the coherence
function of the quantum probe is,
Sx  i S y 
Tr[e
i ( H  H )t   H i ( H  H )t
e e
Tr[e  H ]
]
,
(S21)
(1). If [ H I , H ]  0 , the time-domain measurement can be reduced to
L  t   Z 1Tr exp    H  exp  iH I t   , which can be implemented by probe spin
decoherence with a time-independent probe-bath coupling, i.e. H    H   H I / 2 .
(2). If [ H I , H ]  0,[[ H I , H ], H ] [[ H I, H], H I] 0 , the time-domain measurement can
be written as
L t  
Tr exp    H  iH I t  
Tr exp    H  

Tr exp    H / 2  iH I t / 2  exp    H / 2  iH I t / 2  

Tr ei t [ H , H I ]/4 e iH I t /2 e   H /2 e   H /2 e iH I t /2 e i t [ H , H I ]/4 

Tr exp  iH I t / 2  exp    H  exp  iH I t / 2  
Tr exp    H  
(S22)
Tr e   H 
Tr exp    H  
.
Comparing with equation (S21), we can implement the time-domain measurement by
probe spin decoherence with a time-independent probe-bath coupling
H  H I / 2  H & H  H I / 2  H .
(3). If [ H I , H ]  0,[ H , H I ]  isH I , with s a real number, the time-domain measurement
can be written as
6
L t  

Tr exp    H  iH I t  
Tr exp    H  
Tr exp    H / 2  iH I t / 2  exp    H / 2  iH I t / 2  
Tr exp    H  
Tr e  ( e
 

is
1) H I t /2 s   H /2   H /2  (1 e  is ) H I t /2 s
e
e
e
(S23)


Tr e   H 
Tr  exp( it sin( s) H I / 2 s) exp(   H ) exp( it sin( s ) H I / 2 s ) 
Tr exp    H  
.
Comparing with equation (S21), we can implement the time-domain measurement by
probe spin decoherence with a time-independent probe-bath coupling
H   sin( s) H I / 2s  H & H    sin( s) H I / 2s  H .
(4). In the more general cases where [ H I , H ]  0 , the time-domain measurement in

equation(1) of the main text can be written as L  t   Tr e   H e  itH I  / Tr e   H  , where


 
e2  H   T exp  
 2

1
1

 
H  u  du  T exp  

 2

1
1

H  u  du  ,

exp  itH I   exp   H  exp   H  itH I  , H  u   exp  iutH I  H exp  iutH I  and
T and T are the time-ordering and anti-ordering operators, respectively. If one
initializes the bath in a canonical state   exp   H  /Tr[exp   H ] , the
time-domain measurement can be implemented by probe spin decoherence with a
probe-bath coupling H  H I  / 2  H  & H   H I  / 2  H  up to a normalization
factor
L t  
Tr exp    H  iH I t  
Tr exp    H  

Tr e  H eitH I  Tr e  H  
 
.
 
 H 
 H

Tr e
Tr e 
(S24)
7