ABSTRACT (original article): The notion of “half fire, half ice” was recently introduced to describe an exotic macroscopic ground-state degeneracy emerging in a ferrimagnet under the critical magnetic field, in which the “hot” spins are fully disordered on the sublattice with smaller magnetic moments and the “cold” spins are fully ordered on the sublattice with larger magnetic moments. Here, we further point out that this state has a twin named “half ice, half fire” in which the hot and cold spins switch positions. The new state is an excited state—thus hidden in the ground-state phase diagram—and is robust with respect to the interactions that destroy the half-fire, half-ice state. We demonstrate with exact results how this hidden state can drive phase switching at desirable finite temperature, even for the one-dimensional Ising model where phase transition at finite temperature is forbidden. We suggest that our findings may open a new door to the understanding and controlling of phase competition and transition in unconventional frustrated systems. CITATION (original article): Weiguo Yin and A. M. Tsvelik, Phase Switch Driven by the Hidden Half-Ice, Half-Fire State in a Ferrimagnet, Phys. Rev. Lett. 133, 266701 (2024). https://doi.org/10.1103/PhysRevLett.133.266701
Introduction
Introduction
The design of frustration-driven phase switch at finite temperature has been traditionally carried out by first computing the zero-temperature phase diagram, in which two phases with frustration-induced macroscopically different degeneracy have their own stable regions, and then placing the system near their phase boundary and on the side with less entropy. When heated up, the system will enter the other phase due to lower free energy gained from high entropy. Here, we demonstrate an alternative mechanism, where the sharp phase switch is driven by the high entropy of a hidden state (i.e., it is always an excited state everywhere in the zero-temperature phase diagram and thus invisible in the ground-state phase diagram).
Method
Method
Step 1: set up the 4 states of the two parents on a rung for the ladder or on a bond for the single chain
Step 1: set up the 4 states of the two parents on a rung for the ladder or on a bond for the single chain
This initial step defines the configurations for the “parent” spins:
We set the spin value to 1/2, indicating Ising spins which can take on values of +1 or -1. Then generates all possible configurations for two parent (type-a) spins, σi and σi+1, which form the “backbone” of the single chain in the minimal model. These are the green balls in Fig. 1(a). The four resulting states—(1, 1), (1, -1), (-1, 1), (-1, -1)—represent all combinations of two Ising spins.
We set the spin value to 1/2, indicating Ising spins which can take on values of +1 or -1. Then generates all possible configurations for two parent (type-a) spins, σi and σi+1, which form the “backbone” of the single chain in the minimal model. These are the green balls in Fig. 1(a). The four resulting states—(1, 1), (1, -1), (-1, 1), (-1, -1)—represent all combinations of two Ising spins.
The Ha part of the Hamiltonian in the article, −J ∑ σiσi+1 − hµa ∑ σi, describes the interaction between these backbone spins (σi) with a ferromagnetic (FM) interaction
J>0
(*Don'tchangethispart*)Clear[S,outer1,outer2,states,parent,nstatesp]S=1/2;outer1=1;outer2=2;(*states=Array[{#1-(S+1),#2-(S+1),#3-(S+1)}&,{sdim,sdim,sdim}];*)states[S_,M_]:=Module[{sdim,matDim,tmp},sdim=2*S+1;matDim=ConstantArray[sdim,M];tmp=Array[{##}&,matDim]-(sdim-S);tmp=Reverse[Flatten[tmp,M-1]];tmp=If[OddQ[2S],2*tmp,If[EvenQ[2S],tmp]]]parent=states[S,2];nstatesp=Length[parent];Print["parent states= ",parent//MatrixForm]Print[nstatesp," states in total"]
parent states=
1 | 1 |
1 | -1 |
-1 | 1 |
-1 | -1 |
4 states in total
Step 2: Set up the states for the M decorated sites
Step 2: Set up the states for the M decorated sites
This step defines the configurations for the “decorated” or “child” spins.
First we signify that there is only one decorated Ising spin per unit cell. Like the parent spins, these are also spin-1/2 Ising spins (=±1). We then generate the two possible states for a single decorated spin: (1) and (-1). These type-b spins are the gray balls in Fig. 1(a).
b
i
The Hb part of the Hamiltonian, −J1 ∑ (σi + σi+1)bi − hµb ∑ bi, describes the antiferromagnetic (AF) interaction J1 < 0 between the decorated spins and the backbone spins, bridging σi and σi+1
In[]:=
Clear[Sc,M,child,nstatesc]M=1;Sc=1/2;Print["M= ",M]child=states[Sc,M];nstatesc=Length[child];Print["decorated states= ",child//MatrixForm]Print[nstatesc," states in total"]
M= 1
decorated states=
1 |
-1 |
2 states in total
Step 3 : Build the transfer matrix with the decorated sites being exactly summed out, cf . Equations (S1) and (S2)
This is where the core interactions and parameters of the ferrimagnetic Ising model are incorporated to construct the transfer matrix.
◼
J2*T is the interaction between decorating (type - b) spins
◼
J1*T is the coupling between decorating and backbone spins
◼
a*T is , b*T is , is the magnetic field.
μ
a
μ
b
h
◼
J2 is set to 0 for as there’s only one decorated spin per unit cell, meaning no interaction between decorated spins. See https://doi.org/10.1103/PhysRevB.109.214413 for
M=1
M=2
◼
a () and b () are the magnetic moments of type-a and type-b spins, respectively. The condition > > 0 is crucial for defining the ferrimagnetism of the system
μ
a
μ
b
μ
b
μ
a
◼
childOneConf and childAllConf functions calculate the statistical mechanical contribution of the decorated spins for each configuration of the parent spins. This effectively “sums out” the decorated sites, simplifying the problem for the transfer matrix method.
◼
pp, pm, mp, mm: These variables represent the contribution of the decorated spins for the four possible configurations of the parent spins. pm == mp implies a symmetry in the model
◼
Ym and Yp: These terms are derived from the contributions of the decorated spins and are essential for constructing the transfer matrix. Ym specifically relates to the “frustration function”.
◼
transferM This is the transfer matrix, a key component in statistical mechanics for calculating the partition function and, subsequently, all thermodynamic properties of a 1D system. The elements aa, bb, and cc are functions of the interaction parameters (J, J1) and magnetic field (h). The model considered is a minimal decorated single-chain Ising model, which, notably, contains no geometric frustration in the conventional sense because the AF J1 and FM J interactions can be satisfied simultaneously.
Clear[childAllConf,childOneConf,gMat,yMat,muMat,J1,J2,a,b,h,pp,mm,pm,mp]J2=If[M<=1,0,J2];gMat=Table[If[m1≠m2,J2,0],{m1,1,M},{m2,1,M}];yMat=Table[ConstantArray[J1,M],{m,1,2}];muMat=ConstantArray[b,M];Print["yMat= ",yMat//MatrixForm]Print["gMat= ",gMat//MatrixForm]Print["muMat= ",muMat//MatrixForm]childOneConf[k_,i_]:=Module[{sumM,sumMM},(*k_=1...4:parent'sspin1&2configuration,i_=1...2^M:children'sspinconfiguration*)sumM=0;sumMM=0;Do[sumM=sumM+child[[i,m1]]*(yMat[[outer1,m1]]*parent[[k,outer1]]+yMat[[outer2,m1]]*parent[[k,outer2]]+hmuMat[[m1]]),{m1,1,M}];Do[sumMM=sumMM+gMat[[m1,m2]]*child[[i,m1]]*child[[i,m2]],{m1,1,M},{m2,1,M}];Exp[sumM+sumMM/2]]childAllConf[k_]:=Module[{sum,Mdim},If[M==0,Return[1]];Mdim=nstatesc;sum=0;Do[sum=sum+childOneConf[k,i],{i,Mdim}];sum]pp=childAllConf[1];pm=childAllConf[2];mp=childAllConf[3];mm=childAllConf[4];pmmpPrint["The decoration's contribution= ",{{pp,pm},{mp,mm}}//MatrixForm]Clear[Ym,Yp,aa,bb,cc,transferM]Ym=Exp[ah]pp-Exp[-ah]mm;Yp=Exp[ah]pp+Exp[-ah]mm;Print["The frustration function= ",Ym]aa=Exp[J+ah]pp;bb=Exp[J-ah]mm;cc=Exp[-J]pm;transferM={{aa,cc},{cc,bb}};Print["transfer Matrix= ",transferM//MatrixForm]
yMat=
J1 |
J1 |
gMat= (
)
0 |
muMat= (
)
b |
Out[]=
True
The decoration's contribution=
-bh-2J1 bh+2J1 | -bh bh |
-bh bh | bh-2J1 -bh+2J1 |
The frustration function= -(+)+(+)
-ah
bh-2J1
-bh+2J1
ah
-bh-2J1
bh+2J1
transfer Matrix=
ah+J -bh-2J1 bh+2J1 | -J -bh bh |
-J -bh bh | -ah+J bh-2J1 -bh+2J1 |
Step 4 : Solve the 2 x2 transfer Matrix, cf . Equations (S3) - (S8)
Step 4 : Solve the 2 x2 transfer Matrix, cf . Equations (S3) - (S8)
This section extracts the thermodynamic properties of the system from the transfer matrix.
◼
λ: This calculates the largest eigenvalue of the transfer matrix. In the transfer matrix method, the largest eigenvalue dominates the partition function in the thermodynamic limit, allowing for the calculation of extensive thermodynamic quantities.
◼
free[t_]: This computes the free energy f per unit cell (-t Log[λ]), where t represents kBT (temperature, with Boltzmann constant kB set to 1)
◼
entropy[t_]: This calculates the entropy S from the free energy (-∂f/∂T). A sharp jump in entropy of about kB ln 2 per unit cell within the ultranarrow crossover region is a key signature resembling a genuine first-order phase transition.
◼
Cv[t_]: This calculates the specific heat (T ∂S/∂T).
◼
magnetization[t_]: This computes the total magnetization m = µa⟨σi⟩ + µb⟨bi⟩ of the system per unit cell (-∂f/∂h).
◼
maa[t_] (equivalent to ⟨σi⟩): This calculates the magnetization of the type-a (backbone) spins. This is used as the order parameter for identifying the ultranarrow phase crossover (UNPC). The ⟨σi⟩ = 0 line for defines the crossover temperature T0.
h≠0
◼
mbb[t_] (equivalent to ⟨bi⟩): This calculates the magnetization of the type-b (decorated) spins. For J=0, ⟨bi⟩ is always positive. However, for sufficiently large J > 0, ⟨bi⟩ can become negative for 0 < h < hf, indicating that decorated spins flip from the ground state. At h = hf, ⟨bi⟩ = 0, signifying the Half-Ice, Half-Fire (HIHF) state where type-b spins are decoupled.
◼
ThisTc[t_] and OP[t_]: ThisTc is essentially Ym, and OP is a normalized version of Ym, representing the order parameter ⟨σi⟩.
◼
dT[t_]: This calculates the crossover width 2δT. A key finding is that 2δT can be exponentially narrowed by J > 0 (e.g., 2δT(J) = e^(-2β0J) 2δT(J=0)), leading to the ultranarrow phase crossover (UNPC). This means that T0 and 2δT can be independently controlled by different model parameters, which is a highly desirable feature for applications like temperature sensing or refrigeration.
Clear[λ,paraT,free,entropy,Cv,magnetization,Chi,Caa,Cbb,ThisTc,OP,dT,maa,mbb,C12,C13,t,J,J1,J2,h,a,b]paraT[t_]:={JJ/t,J1J1/t,J2J2/t,hh/t};λ=1/2Exp[J](Yp+Sqrt[Ym^2+4Exp[-4J]pm^2]);free[t_]:=-tLog[λ]/.paraT[t]Print["Free energy free[t]= ",free[t]]entropy[t_]:=-D[free[ttt],ttt]/.{tttt}entropyDiff[t_,hh_]:=(entropy[t]/.{h->hh})-(entropy[t]/.{h->0})Cv[t_]:=tD[entropy[ttt],ttt]/.{tttt}magnetization[t_]:=-D[free[t],h]Chi[t_]:=D[magnetization[t],h]Chih0[t_]:=Exp[2J/t]/2(Exp[2J1/t](a+b)^2+Exp[-2J1/t](a-b)^2)/t-Sinh[2J/t]b^2/(2Cosh[2J1/t]+2Exp[-2J/t])/tCaa[t_]:=-D[free[t],a]Cbb[t_]:=-D[free[t],b]maa[t_]:=-D[free[t],a]/hmbb[t_]:=-D[free[t],b]/hC12[t_]:=-D[free[t],J]C13[t_]:=-D[free[t],J1]/2ThisTc[t_]:=Ym/.paraT[t];Print["ThisTc[t]= ",ThisTc[t]]OP[t_]:=Ym/Sqrt[Ym^2+4Exp[-4J]pm^2]/.paraT[t];dT[t_]:=1/D[OP[ttt],ttt]/.{tttt}Print["order parameter OP[t]= ",OP[t]]
Step 5 : Verification by using the effective interaction method, cf . Equations (S9) - (S13)
Step 5 : Verification by using the effective interaction method, cf . Equations (S9) - (S13)
This section implements an alternative theoretical approach, the effective Hamiltonian method, to verify the results obtained from the direct transfer matrix calculations.
heff, Jeff, Aeff, λeff, freeEff, entropyEff, CvEff, maaeff, mbbeff, ThisTcEff, OPeff, dTeff: These variables represent the same thermodynamic quantities and parameters calculated using the effective method. The purpose of this step is to ensure the rigorousness and accuracy of the findings by confirming that both methods yield consistent results for T0 and δT. The article emphasizes that exact results are used to demonstrate the phase switching.
Step 6: Use the two (direct and effective Hamiltonian) methods to calculate T0 and dT, cf. Equations (S7)(S8) and (S12)(S13). Confirmed
Step 6: Use the two (direct and effective Hamiltonian) methods to calculate T0 and dT, cf. Equations (S7)(S8) and (S12)(S13). Confirmed
This final step leverages the defined functions to compute and display the crossover temperature (T0) and crossover width (dT or 2δT) for various parameters.
◼
GetTc and GetdT functions: These are wrappers that use both the direct and effective methods to calculate T0 and 2δT. The output confirms that GetTc[...]= ... GetTcEff[...] and GetdT[...]= ... GetdTeff[...], indicating that the calculations are confirmed across both methods.
◼
The results show that T0 is independent of J.
◼
Crucially, as J increases, GetdT (the crossover width) decreases exponentially, becoming extremely small (e.g., 2.73809×10^-42 for J=100). This demonstrates the ultranarrow phase crossover (UNPC) driven by J > 0 at a fixed finite temperature.
◼
The code then generates plots (Fig. 2 and Fig. 3 in the article) to visualize the density plots of ⟨σi⟩, ⟨bi⟩, and total magnetization, as well as the temperature dependence of ⟨σi⟩, entropy, and specific heat. These plots visually confirm the theoretical predictions, such as the ⟨σi⟩ = 0 line determining T0, the ⟨bi⟩ = 0 line associated with the HIHF state, and the entropy jump.
Test GetTc[r,h],GetdT[J,r,h] for h -> 0. Passed
Illustration of the definition of dT, cf. Equations (S8) and (S13). The slope is verified
Results
Results
Define colors and precision for visualization
Fig. 2
Fig. 2
Fig. 3
Fig. 3
Fig . 3 (d) . The density plot in the h/hc − μa/μb plane of ΔS the entropy jump at T0 in the unit of kBln2 for J = 100
Fig . 3 (d) . The density plot in the h/hc − μa/μb plane of ΔS the entropy jump at T0 in the unit of kBln2 for J = 100
Fig. S2
Fig. S2
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Phase switch driven by the hidden half-ice, half-fire state in a ferrimagnet
by Weiguo Yin & A. M. Tsvelik
Wolfram Community, STAFF PICKS, August 13, 2025
https://community.wolfram.com/groups/-/m/t/3530595
by Weiguo Yin & A. M. Tsvelik
Wolfram Community, STAFF PICKS, August 13, 2025
https://community.wolfram.com/groups/-/m/t/3530595