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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 i1, j1 i , j i , j1 J i1, j1 i , j i1, j h i1, j1 i , j Z ( , h) Tr e M ,N = j , j M ,N M ,N C11 ' D1 ' 2 C2 2 ' D2 ' 3 CM M ' DM ' 1 (S9) Tr[(CD) M ], where C ' e J j1 ( j ) '( j 1) h j1 '( j ) N N ' and D ' e J j1 ( 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 j1 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 21 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 e2 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 eitH I Tr e H . H H Tr e Tr e (S24) 7