A Generalised Haldane Map from the Matrix Product State Path Integral to the Critical Theory of the $J_1$-$J_2$ Chain
F. Azad Adam J. McRoberts Chris Hooley A. G. Green







These authors contributed equally to this work
Electrical Engineering and Computer Science Department, Technische Universität Berlin, 10587 Berlin, Germany
London Centre for Nanotechnology, University College London, Gordon St., London, WC1H 0AH, United Kingdom


These authors contributed equally to this work
International Centre for Theoretical Physics, Strada Costiera 11, 34151, Trieste, Italy
Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany


Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany


London Centre for Nanotechnology, University College London, Gordon St., London, WC1H 0AH, United Kingdom




We study the J_1-J_2 spin-1/2 chain using a path integral constructed over matrix product states (MPS). By virtue of its non-trivial entanglement structure, the MPS ansatz captures the key phases of the model even at a semi-classical, saddle-point level, and, as a variational state, is in good agreement with the field theory obtained by abelian bosonisation. 
Going beyond the semi-classical level, we show that the MPS ansatz facilitates a physically-motivated derivation of the field theory of the critical phase: by carefully taking the continuum limit—a generalisation of the Haldane map—we recover from the MPS path integral a field theory with the correct topological term and emergent SO(4) symmetry, constructively linking the microscopic states and topological field-theoretic structures. Moreover, the dimerisation transition is particularly clear in the MPS formulation—an explicit dimerisation potential becomes relevant, gapping out the magnetic fluctuations.    

A Generalised Haldane Map from the Matrix Product State Path Integral 
to the Critical Theory of the J_1-J_2 Chain
    A. G. Green
    April 29, 2024
==================================================================================================================



 

Introduction
The J_1-J_2 chain has provided an archetypal model in which to investigate the role of topology in quantum systems, and harbours a rich phase diagram. Semi-classically, an incommensurate helimagnetic phase interpolates between the ferromagnetic and antiferromagnetic (Néel) phases, and a plethora of numerical and analytical approaches have revealed further features <cit.>. 
We will be most concerned with the transition in the quantum S = 1/2 chain between a critical antiferromagnetic phase and the dimer phase, where competing first- and second-neighbour antiferromagnetic terms drive adjacent spins into singlets.

Under abelian bosonisation <cit.>, this transition appears to be of Kosterlitz-Thouless (KT) character. In obscuring the SU(2) symmetry, however, this misses the spacetime topology of the quantum states, and the fact that topological defects in one phase are bound to the charges of the other—the boundary between two singlet covers has a single (delocalised) spin. Field-theoretically, this is encoded in additional topological Wess-Zumino (WZ) terms <cit.> (generalisations of the Haldane Θ-term <cit.>), which measure the winding of the joint Néel-singlet order parameter during the tunneling events (instantons) that drive the transition out of the dimer phase.

These ideas are central to the understanding of deconfined quantum criticality <cit.> and underpin a rich network of dualities between different two-dimensional quantum states. In the J_1-J_2 model, these terms may be derived via a mapping of the spins to fermionic degrees of freedom <cit.>. In this letter, we give an alternative derivation working with the original spin degrees of freedom, mirroring the derivation of the Θ-term in the antiferromagnet <cit.>.

A matrix product state (MPS) ansatz encompassing the key physics of the different phases is the basis of our approach—even as a variational wavefunction, it captures a continuous phase transition between the critical and dimer phases. Reparametrising in terms of the order parameters of these phases, we construct the path integral over the MPS ansatz <cit.>, and demonstrate how the resulting effective action recovers the non-linear sigma model (NLSM) of the critical phase, with the SO(4) WZ term (cf. <cit.>), explicitly linking these topological field-theoretic structures to the microscopic states. 
Further, the MPS field theory clarifies the nature of the dimerisation transition, with an explicit potential term which breaks the SO(4) symmetry and opens the gap.








Model and MPS ansatz
The Hamiltonian of the  chain consists of competing Heisenberg interactions between nearest and next-nearest neighbours,

    Ĥ = 
     ∑_i( 
     J_1 σ̂_i ·σ̂_i+1 
     +
     J_2 σ̂_i ·σ̂_i+2),

where σ̂ = (σ̂^x, σ̂^y, σ̂^z) is the vector of Pauli matrices. 
We introduce our variational state as an MPS ansatz with bond-dimension two,

|Ψ⟩    = ∑_{σ} i0.9⊗ ( A_[i]^σ_i |σ_i_i⟩ ),  
A_[i]^+ = [         cosψ_i              0;              0 i e^iξ_isinψ_i ], 

               A_[i]^- = e^iπ/4 [              0         cosψ_i; i e^iξ_isinψ_i              0 ],


where |⟩ represents a spin-coherent state polarised in the direction of the unit vector . (we discuss the gauge-fixing in the supplementary <cit.>). The angle ξ allows tuning between singlet and S=0 triplet configurations and will generally not appear in the following.

This state captures the key phases of the J_1-J_2 model. In particular, it can represent a product state antiferromagnet with ψ = 0 or π/2, and |_i⟩ = |(-1)^i ⟩, and the singlet covers are obtained by setting  
ψ_2i= π/4,  ψ_2i-1 = 0 (or vice-versa), with |_i⟩ = |(-1)^i⟩.






Saddle-point phase diagram
Before turning to the connections to the field theory, let us first discuss the phase diagram that results from the ansatz (<ref>). The finite, two-site correlation length implies that ⟨Ĥ⟩ can be calculated and manipulated analytically; Eq. (<ref>) is in left canonical form with left-environment 1 and right-environment R_i= (1 + σ^z cos 2ψ_i+1)/2.

In order to simplify the optimisation, we restrict  to lie in the xy-plane with a constant pitch angle Δ∈ [0, π], such that _i ·_i+1 = cosΔ. 
Further, we assume that the entanglement parameters ψ_i will at most alternate between two values ψ_A on even sites and ψ_B on odd sites.
Now, the expectation value of the Hamiltonian (<ref>), over the MPS ansatz (<ref>) with these restrictions, is

E =    J_1 [ (cos^2  2ψ_A + cos^2  2 ψ_B)  cosΔ

            - sin[ 2(ψ_A+ψ_B)] ( 1 - cosΔ) ] 

   + J_2 [2 cos^2 2 ψ_A cos^22 ψ_B cos2 Δ

               + sin4 ψ_A sin4 ψ_B (1+ cos2 Δ)/4
].


The variational (saddle-point) phase diagram (Fig. <ref>) follows by minimising Eq. (<ref>) over ψ_A, ψ_B and Δ.
The pitch angle follows the classical result for most of the phase diagram—an (incommensurate) helimagnetic phase with Δ = arccos[-J_1/(4 J_2)] interpolates between ferromagnetic and Néel order at J_1/J_2 < -4 and J_1/J_2 > 4 respectively, dressed with some uniform entanglement. The helimagnetic order is suppressed, however, by the dimerised singlet phase for 0.1619 ≲ J_2/J_1 ≲ 1.317. 

The transition between the Néel and singlet phases occurs at J_2/J_1≈1/6. On the scales indicated in Figs. <ref>(a) & (b), the singlet phase appears to be formed at an abrupt first order discontinuity in the parameters of the MPS ansatz; however, zooming in on the region around this point reveals two continuous transitions (at saddle-point level): the first at J_2/J_1≈0.1619, where the translation symmetry is first broken; and the second at J_2/J_1 = 3-2 √(2)≈ 0.1716, where the singlet state is fully formed (see the supplementary <cit.> for the analytic details). 

This in-plane optimisation of the MPS ansatz invites a comparison with abelian bosonisation, which predicts a KT dimerisation transition around the same point J_2/J_1 ≈ 1/6 <cit.>. Whilst this picture will be modified by the topological terms, we still expect a universal jump in the spin stiffness ρ (which we derive in the supplementary <cit.>). The MPS ansatz (<ref>), on the other hand, furnishes us with an estimate of ρ much more straightforwardly: we simply use the dependence of the energy upon the pitch Δ to evaluate the resistance to inducing a twist in the magnetic order. That is,


    ρ = ⟨ N ⟩.∂^2 E/∂Δ^2|_Δ=π,


where ⟨ N⟩  = cos(2ψ_A)cos(2ψ_B) is the Néel order parameter. We show the spin stiffness in Fig. <ref>, and find that, even at saddle point level, the universal jump associated to the KT transition is visible—though it occurs between the split transitions, rather than discontinuously—and broadly agrees with the field-theoretic estimate [Optimising over the full bond-order two MPS manifold produces a better variational state, but the point of the MPS ansatz (<ref>) is that it can be manipulated analytically—allowing it to connect to the field theory.].

The rest of our treatment focuses on this dimerisation transition, for which later work <cit.> went beyond a purely KT treatment and showed the importance of WZ terms in the field theory—they encode the binding of spins to domain walls between singlet covers. We will show in the following that the MPS ansatz captures this physics directly, in the structure of the spin wavefunction.









SO(4) order parameter and continuum limit
The first step is to construct a local SO(4) order parameter that treats the Néel and dimer order on an equal footing.
In terms of the MPS, the Néel order parameter is

    N   =    (-1)^i⟨σ̂_i ⟩ 
    = cos (2 ψ_i)  cos( 2 ψ_i+1)  _i.


And, whilst the singlet states should formally be distinguished by a string order parameter, it suffices here to use a local singlet order parameter


D_i =  (-1)^i ⟨σ̂^+_i  σ̂^+_ i+1 +σ̂^-_i  σ̂^-_ i+1 - σ̂^+_i-1  σ̂^+_ i -σ̂^-_i-1  σ̂^-_ i ⟩

valid when _i ≈ -_i+1, where σ̂^+ and σ̂^- are spin raising and lowering operators in the basis {|⟩, |-⟩} (these are the usual raising and lowering operators if =). 
For states given by Eq. (<ref>), this is, explicitly, 


    D_i 
       =     (-1)^i  [
     - sin(2 ψ_i+1) cos (ψ_i-ψ_i+2) cos (ψ_i + ψ_i+2) 
     .
     
          .
      + sin(2 ψ_i) cos (ψ_i-1-ψ_i+1) cos (ψ_i-1 + ψ_i+1)
      ]


Now, if we allow entanglement on either only even or only odd bonds, we obtain  

   [even:   ψ_2i+1 = 0]        [odd:   ψ_2i=0 ]

   N_2i = cos(2ψ_2i)_2i 
      N_2i = cos(2ψ_2i+1)_2i 

   N_2i-1 = -cos(2ψ_2i)_2i-1 
      N_2i+1 = -cos(2ψ_2i+1)_2i+1

   D_2i = sin(2ψ_2i)       D_2i = -sin(2ψ_2i+1) 

   D_2i-1 = sin(2ψ_2i)       D_2i+1 = -sin(2ψ_2i+1).

such that in either case N^2+D^2=1, and (D, N) forms an SO(4) multiplet [Note that sin(2 ψ) and cos(2 ψ) are swapped compared to the usual polar decomposition of an SO(4) multiplet.].
We introduce a single angle χ_2i (defined on even sites), which interpolates between the two singlet orders: 2ψ_2i+1 =χ_2iΘ(χ_2i) and 2ψ_2i= -χ_2iΘ(-χ_2i), where Θ(x) is the Heaviside step function. That is, χ_2i > 0 denotes entanglement between sites 2i and 2i+1; χ_2i < 0 between 2i-1 and 2i. We show how χ rotates between the singlet and Néel orders in Fig. <ref>. 






We note that this order parameter cannot completely characterise the uniform entanglement saddle-points; those states directly include significant quantum fluctuations which, in the field theory, are encoded as instantons between the two singlet covers.  






Effective action
Now, armed with this joint order parameter, we can construct the field theory as an MPS path integral <cit.>,


= ∫Ψe^-S[Ψ],    S = ∫dτ⟨Ψ||_τΨ⟩ + ,

where Ψ=χξcos^2χ is the functional Haar measure over the restricted MPS ansatz <cit.>. We note that, despite the enforced staggering of ψ_i, we are still taking advantage of the fact that the MPS path integral captures entanglement in its saddle-points (the singlet covers). Now, the effective action S has three relevant contributions: the kinetic and topological terms, which arise from the Berry phase; and the Hamiltonian term. Let us start with the action in the form that results from assuming ψ_2i+1 = 0; the same continuum limits are recovered in the opposite case. 
We have the Berry phase

S_B = ∫dτ⟨Ψ||∂_τΨ⟩ = ∫dτ∑_2i [  i ξ̇_2i/2( 1-cosχ_2i ) 

+ i/2 cosχ_2i ( ϕ̇_2i cosθ_2i + ϕ̇_2i-1 cosθ_2i-1 )


and the Hamiltonian

   = J_1 ∑_2i [ -w^x_2i + (1 + w_2i^x)_2i-1·_2i 
 
                      + w^z_2i w^z_2i+2 _2i·_2i+1 ] 

   + J_2 ∑_2i w^z_2i w^z_2i+2(_2i·_2i+2 + _2i-1·_2i+1 ),


where w^x=sinχcosξ, w^z=cosχ are components of an O(3) vector parametrising the entanglement. 
To proceed, we apply Haldane's mapping <cit.> to the spin degrees of freedom,

_i = (-1)^i _i √(1 - _i^2) + _i,


where  is the slowly-varying Néel field, and  captures the fast fluctuations of the magnetisation, such that

_i^2 = 1,     _i·_i = 0,     _i^2 ≪1.

We apply this mapping to the Berry phase (<ref>) and Hamiltonian (<ref>) above, expanding around the saddle points χ = ±π/2 (the singlet covers), and retaining terms up to second order in the fluctuations  and ξ, and in the derivatives of the slow fields  and w. We thus obtain the action (<ref>)
of the SO(4) NLSM with a topological term and dimerisation potential.

We will give here all of the essential points, and all of the approximations used (see the supplementary <cit.> for full details). We begin with the topological term: inserting Eq. (<ref>) into Eq. (<ref>), we obtain a term

iΩ   = ∫dτ∑_2i i/2cosχ_2i ( ϕ̇_2i cosθ_2i -  ϕ̇_2i-1 cosθ_2i-1 ) 

   ∼∫dτdx  i/4cosχ(_τϕ _xcosθ- _xϕ _τcosθ)


(where θ and ϕ are now the angular co-ordinates of the Néel field ), which is the polar coordinate form of the SO(4) topological term identified in Ref. <cit.> (see also the supplementary <cit.>),


    Ω   =   2 π k/2 π^2∫_0^1 d ζ∫ dτ dx
     ϵ_abcd 
    u^a ∂_ζ u^b ∂_τ u^c ∂_x u^d


where k∈ℤ is odd. The factor of 2 π^2 in the denominator is the area of S^3, and u(ζ, τ, x) is an arbitrary extension of the field u(τ,x) = (sinχ, cosχ m) ≡ (u^0,  u) satisfying 


    u(ζ=0,τ,x)    =    (1,0,0,0) 
    
    u(ζ=1,τ,x)    =    u(τ,x).


In the MPS treatment, the internal structure of the ansatz constrains domain walls between different singlet covers to contribute to the overall Berry phase as a free spin; the topological term imposes this same constraint in the field theory.

The kinetic terms follow by integrating out the fast fields, ξ and . Integrating out ξ is straightforward after expanding Eq. (<ref>) to quadratic order. To integrate out  we neglect any terms ∼cos^2χ ^2, but otherwise it proceeds as in the standard Haldane mapping <cit.> (see also the supplementary <cit.>). We obtain kinetic terms (_τ w^z)^2 and (w^z)^2 (_τ)^2, respectively, with the same prefactor 1/(32J_1).

To deal with the Hamiltonian, we simply expand in the gradients of  and w. We note, however, that fluctuations of  are heavily suppressed on the entangled bonds—this means that gradients of , like the gradients of w^z, occur over two lattice spacings, not one. Accounting for this fact ensures that both terms, (_x w^z)^2 and (w^z)^2(_x)^2, appear with the same prefactor, J_1 - 2J_2.

Finally, we switch to the standard angular co-ordinates on S^3, i.e., we define α = π/2 - χ, in terms of which (w^z)^2 = sin^2α and ( w^z)^2 ∼ (α)^2. We have, then,

S    = iΩ+ ∫dτdx [ 1/32J_1((_τα)^2 + sin^2α(_τ)^2 ) 

   + (J_1 - 2J_2) ((_xα)^2 + sin^2α(_x)^2 ) + V(α) ],


where the dimerisation potential is

V(α) = -3J_1/2 + J_2/2 - J_2/2cos2α.


This form of the action, derived directly from an MPS parametrisation of the spin states (<ref>), makes the physics of the transition particularly transparent: if J_2 < J_2^c, V(α) is irrelevant, the SO(4) symmetry emerges in the infrared, and the topological term ensures the theory remains gapless; if J_2 > J_2^c, V(α) is relevant, and the SO(4) symmetry is broken in favour of dimer order.







Discussion and outlook
In this letter, we have introduced an MPS ansatz that captures the key physics of the J_1-J_2 chain, encompassing ferromagnetic, Néel, spiral, and dimer orders. Even at saddle-point level, it reproduces the essential features of the deconfined Néel-singlet transition at J_2/J_1≈1/6, including the universal jump in the spin stiffness (Fig. <ref>) [Indeed, if one restricts to helimagnetic phases with uniform entanglement and the pure singlet covers, then the transition is first order at exactly J_2/J_1=1/6].

From this ansatz, we have directly identified a joint Néel-dimer order parameter, and constructed the field theory from an MPS path integral <cit.>. Whilst this is, mechanically, somewhat similar to the construction of the field theory of the Heisenberg antiferromagnet (J_2 = 0) as a coherent state path integral, the use of MPS-valued fields allows us to recover the correct topological term and NLSM <cit.>.
Moreover, the nature of the dimerisation transition is remarkably clear in the MPS field theory, where the potential term of an explicit entanglement field flows either to weak- or strong-coupling.

The MPS resums instantons of the coherent state theory and encodes their topological structure locally in the MPS fields. Whilst this connection has been noted previously <cit.>, the point here is that we have provided a constructive link: we have shown how the MPS faithfully encodes these topological features, and how this leads directly to the corresponding topological terms in the field theory. 

It is intriguing to speculate that this connection might be used more generally as a method to resum instantons in favour of a higher bond-dimension MPS path integral—the topological term, after all, arises independently of the Hamiltonian. Moreover, it would be desirable if these methods could be extended to two dimensions to capture similar physics in, say, the J-Q model <cit.>; whilst generic projected entangled pair states (PEPS) are not efficiently contractible <cit.>, rendering the action for a PEPS path integral non-local <cit.>, it may be possible to circumvent this difficulty using sequential circuit ansätze <cit.>.






This work was in part supported by the Deutsche Forschungsgemeinschaft under grants SFB 1143 (project-id 247310070) and the cluster of excellence ct.qmat (EXC 2147, project-id 390858490), by the EPSRC under EP/S021582/1 and EP/I031014, and by the ERU under `Perspectives of a Quantum Digital Transformation'. We acknowledge fruitful discussions in workshops funded by EP/W026872/1.



















  
  
  
  

    Supplementary Material
  

















In this Supplementary Material, we give a more detailed analysis of the saddle points of the MPS ansatz; derive an estimate for the spin stiffness from abelian bosonisation, which we compare to the estimate from the MPS ansatz; discuss the dynamical gauge-fixing of the spin coherent states, and show that this makes only a negligible contribution to the Berry phase; and give a more detailed account of the derivation of the non-linear sigma model from the MPS path integral, including how the polar co-ordinate and Wess-Zumino forms of the topological term are equivalent.

















§ SADDLE-POINT ANALYSIS OF THE DIMERISATION TRANSITION



 There are three saddle-point equations corresponding to the two-site energy density given in Eq. (<ref>). Two corresponding to derivatives with respect to the entanglement parameters are given by


    ∂ E/∂ψ_A   =    
    2 J_1(cosΔ - 1) cos(2(ψ_A + ψ_B)) 
    
    
    - 
     2 (J_1 cosΔ + 
        2 J_2 cos2Δcos^2 2ψ_B) sin4ψ_A 
     
     
     + 
     2 J_2 cos^2Δcos4ψ_A sin4ψ_B = 0, 
     
    ∂ E/∂ψ_B   =    2 J_1(cosΔ - 1) cos(2(ψ_A + ψ_B)) 
     
     
     - 
     2 (J_1 cosΔ + 
        2 J_2 cos2Δcos^2 2ψ_A) sin4ψ_B
     
     
     + 
     2 J_2 cos^2Δcos4ψ_B sin4ψ_A = 0,


and one from the derivative with respect to the pitch angle is given by


    ∂ E/∂Δ   =   
     -J_1/2sinΔ[2 + cos4ψ_A + cos4ψ_B + 
        2sin2(ψ_A + ψ_B)] 
     
          
     -J_2sin2 Δcos2ψ_A cos2ψ_B  [3cos2(ψ_A - ψ_B) +  cos2(ψ_A + ψ_B)] = 0.


It is clear that either a ferromagnetic Δ = 0 or an antiferromagnetic Δ = π will solve Eq. (<ref>), regardless of the values of ψ_A and ψ_B (solutions involving an incommensurate pitch angle are more involved, and will not be discussed here). Let us focus upon the antiferromagnetic regime, where the Néel-dimer transition occurs.

With this value for Δ, the saddle-point equations for the entanglement parameters reduce to 


    ∂ E/∂ψ_A   =    -4 J_1 cos(2(ψ_A + ψ_B)) + 
     2 (J_1 - 
        2 J_2 cos^2 2ψ_B) sin4ψ_A + 
     2 J_2 cos4ψ_A sin4ψ_B = 0, 
     
    ∂ E/∂ψ_B   =    -4 J_1 cos(2(ψ_A + ψ_B)) + 
     2 (J_1 -
        2 J_2 cos^2 2ψ_A) sin4ψ_B + 
     2 J_2 cos4ψ_B sin4ψ_A = 0.


We seek solutions to these reduced saddle-point equations (<ref>). We first note that the two singlet covers, (ψ_A, ψ_B) = (π/4, 0) and (ψ_A, ψ_B) = (0, π/4), are degenerate solutions with E_S = -3J_1. 

Deep in the Néel phase, we expect that the entanglement structure in the ground state will not break any lattice symmetries. Setting ψ_A = ψ_B = ψ, both equations reduce, after some simplification, to


    ∂ E/∂ψ = -4 J_1 cos4ψ + 
     2 (J_1 - J_2) sin4ψ = 0.


The uniform entanglement solution is, therefore,


    ψ = 1/4arctan( 2J_1/J_1 - J_2),


with energy


    E_N = -(J_1 - J_2)(1 + √(1 + (2J_1/J_1 - J_2)^2)).


As a check of the quality of this variational ansatz, we can compare this state's energy at J_2 = 0, E_N = -(1 + √(5))J_1 ≈ -3.23601, to the exact (Bethe ansatz) ground state at this point, -2J_1(1 - 4ln2) ≈ -3.54518.

Now, the uniform entanglement state's energy crosses that of the singlet state, E_S = E_N, at precisely J_2/J_1 = 1/6. This is not, however, where the MPS ansatz (<ref>) predicts the transition to occur. Rather, there is another saddle-point solution to Eqs. (<ref>) which interpolates between the uniform and singlet solutions: the transition splits in twain, the entanglement parameters evolve continuously (though not differentiably) as a function of J_2/J_1, and the energy is continuously differentiable throughout.

There are two degenerate such solutions, obtained via mathematica, interpolating to either of the singlet covers. The first is


    ψ_A    =   1/2arcsin(
      1/2(
       1 -6J_2 - 4J_2^2 - 10J_2^3 + 3J_2^4 + 
           4 J_2√(3J_2^4 - 4J_2^3 + 14J_2^2 + 4J_2 - 1)/(J_2 - 1)^3 J_2)^1/2), 
           
    ψ_B    =    -1/2arctan(
      3J_2 - 1/1 + J_2(
       1 -2J_2 - 16 J_2^2 + 2 J_2^3 - J_2^4 + 
           4 J_2√(3J_2^4 - 4J_2^3 + 14J_2^2 + 4J_2 - 1)/-1 + 6J_2 + 4 J_2^2 + 10 J_2^3 - 3 J_2^4 - 
           4 J_2√(3J_2^4 - 4J_2^3 + 14J_2^2 + 4J_2 - 1))^1/2),

where we have set J_1 = 1, and the second interchanges ψ_A and ψ_B. These transition solutions are only valid (ψ_A, ψ_B ∈ℝ) between the two transition points, J_2^(-) and J_2^(+), where their energy crosses E_N and E_S, respectively. Over this region, however, the transition solutions are the lowest energy saddle-point states. 


The upper transition point, explicitly, is J_2^(+) = 3 - 2√(2) > 1/6. The analytic expression for the lower transition point, whilst it can be expressed using radicals, is much lengthier – the most concise way of stating it is that J_2^(-) is the (unique) positive real root of 3x^4 - 4x^3 + 14x^2 + 4x - 1, which gives J_2^(-)≈ 0.1619 < 1/6.









§ ABELIAN BOSONISATION AND SPIN STIFFNESS



An alternative estimate of the spin stiffness may be obtained from a bosonised field theory. We begin with the linearised J_1-J_2 Hamiltonian from Ref. <cit.>,


Ĥ = ∫dx/2π  : i J_1∑_η η ψ̂^†_η _x ψ̂_η 
+ 2J_1/2π∑_ηη' ρ̂_η ρ̂_η'
- 4J_2/2π∑_ηη' ηη' ρ̂_η ρ̂_η'
+ J_1 - 6J_2/2π ∑_η (ψ̂^†_η_x ψ̂^†_η)(ψ̂_-η_x ψ̂_-η)  : ,

where ψ̂_R, ψ̂_L are chiral fermion fields, the indices η = (+, -) correspond to (R, L), the dots :...: denote normal ordering (necessary because the linearisation introduces a Dirac sea of negative energy fermion states), and ρ̂_η is the density of η-fermions. The final term is the umklapp term which induces the quantum phase transition in this formulation. Note that some of the coefficients have extra factors of 2π compared to Ref. <cit.>, because we are following the conventions of Ref. <cit.>.

To these chiral fermions we associate chiral boson fields ϕ̂_η, in terms of which the fermion densities are given by ρ̂_η = _x ϕ̂_η, and the fermion fields by vertex operators ψ̂_η∼ e^iηϕ̂_η. These chiral bosons have the algebra

[ϕ̂_η(x), _xϕ̂_η'(x')] = 2πi ηδ_ηη' δ(x - x').


Defining the total density and current fields,

ϕ̂ = ϕ̂_R + ϕ̂_L/2,   
θ̂ = ϕ̂_R - ϕ̂_L/2,

such that ϕ̂ is canonically conjugate to 1/π_xθ̂(x'), i.e., 

[ϕ̂(x), 1/π_xθ̂(x')] = iδ(x - x'),

the Hamiltonian becomes

Ĥ = ∫dx/2π  : (J_1 + 4J_1/π)(_x ϕ̂)^2 + (J_1 - 8J_2/π)(_x θ̂)^2 + J_1 - 6J_2/2πa^2cos4ϕ̂  : ,


where we have explicitly reinstated the lattice spacing in the umklapp term. Now, the bare value of the fermion charge stiffness is just the coefficient of the current fluctuations (_x θ̂)^2. However, unlike the saddle-point of the MPS ansatz, the bosonised field theory includes ultraviolet contributions from arbitrarily high momentum; we thus identify the physical value of the spin stiffness with the renormalised (infrared) charge stiffness.

We introduce the usual Luttinger liquid parameters u and K, where u is the analogue of the Fermi velocity and K is dimensionless. In the Hamiltonian (<ref>), uK is the coefficient of (_x θ̂)^2, and u/K is the coefficient of (_x ϕ̂)^2. 

To obtain the flow equations, consider the (imaginary time) partition function, 

Z = ∫𝒟ϕ𝒟θ  e^- S[ϕ, θ],    S = ∫dτdx [-i/π(∂_τϕ)(∂_xθ) + ℋ ],

and integrate out the conjugate field θ, leaving the effective sine-Gordon action

S = ∫dx dτ1/2πK ( 1/u (∂_τϕ)^2 + u (∂_x ϕ)^2 ) + g/2πa^2 cos4ϕ.


The Wilsonian renormalisation procedure of dividing the field into fast modes ϕ_> and slow modes ϕ_<, and successively integrating out the ϕ_>, may now be performed. Following Ref. <cit.>, we have the flow equations

dK/dl = -A K^2 g^2,     dg/dl = (2 - 4K)g,

for some constant A > 0. We can read off the critical value K_c = 1/2, below which g always flows to strong coupling. The Luttinger velocity u does not flow under renormalisation – the action (<ref>) is Lorentz covariant, and u is its light speed. 

In principle, we should now compute the infrared values of the couplings. In fact, this is not necessary – for all J_2 < J_2^c, the isotropic J_1-J_2 model lies at a transition between the easy-plane spin-fluid and easy-axis Néel state (see Fig. 2 of  Ref. <cit.>). It follows, then, that the bare values of K and g must lie on the separatrix between the strong and weak-coupling phases, and so K(l →∞) → K_c = 1/2, g(l→∞)→ 0. The transition to the dimer state is marked by the bare value of K falling below K_c = 1/2, which happens at J_2 > J_2^c = 3π - 4/32≈ 1/6, whereupon K(l→∞)→ 0; this universal discontinuity in the infrared behaviour of K is the quantum analogue of the universal jump in the spin stiffness in classical Kosterlitz-Thouless transitions. 

With these considerations, the bosonisation estimate for the spin stiffness of the J_1-J_2 model is

ρ= u K(l→∞) =  1/2 √((J_1 + 4J_1/π)(J_1 - 8J_2/π))     J_2 < J_2^c ≈J_1/6, 
 0     J_2 > J_2^c ≈J_1/6. 

We show a comparison of the spin stiffness estimates obtained from the MPS ansatz and bosonisation in Fig. <ref> of the main text. The two methods are in reasonable agreement with each other, though neither is exact. Moreover, the MPS estimate is considerably easier to obtain, both conceptually and computationally.








§ GAUGE FIXING



The spin coherent states used in the parametrisation of the MPS ansatz (<ref>) have apparent U(1) gauge  freedom corresponding to rotations of the tangent space basis of the point on S^2 specified by . Making this gauge freedom explicit, we have

   |+⟩ = e^iμ/2( cosθ/2|↑⟩ + sinθ/2e^iϕ|↓⟩ ) 

   |-⟩ = e^-iμ/2( sinθ/2|↑⟩ - cosθ/2e^iϕ|↓⟩ ),

where the unit vector parametrising the states is

= (sinθcosϕ, sinθsinϕ, cosθ).

We also define

= _θ/|_θ| = (cosθcosϕ, cosθsinϕ, -sinθ),     = _ϕ/|_ϕ| = (-sinϕ, cosϕ, 0),


which provide a basis for the tangent space such that {, , } forms a right-handed set. We denote the complex combinations of the tangent space vectors by =  + i and ^* =  - i. These combinations show up in the matrix elements of the Pauli matrices with respect to the coherent states,

⟨||⟩    = 

⟨||-⟩    = -e^iμ^* 

⟨-||⟩    = -e^-iμ

⟨-||-⟩    = -,

where = (σ^x, σ^y, σ^z) is the vector of Pauli matrices. 
Identifying = '= '+i '=  -e^iμ^*, we see that μ in Eq. (<ref>) is the angle of rotation between ,  and ', '.

This redefinition of the tangent space at  allows us to rewrite the expectation value of the Hamiltonian so that the explicit dependence is only upon  and not upon . Taking the expectation value of the Hamiltonian that results from setting ψ_2i+1 = 0 (as in the main text), we have


   = J_1 ∑_2i [_2i-1·_2i + cosχ_2i cosχ_2i+2 _2i·_2i+1 

              + 1/2sinχ_2i(e^-iξ_2i - iμ_2i-1 - iμ_2i_2i-1·_2i + e^iξ_2i + iμ_2i-1 + iμ_2i^*_2i-1·^*_2i ) ] 

   + J_2 ∑_2i cosχ_2i cosχ_2i+2(_2i·_2i+2 + _2i-1·_2i+1).



We now perform a series of manipulations that remove the explicit appearance of  and ^* from this expression.

First we note that the appearance of μ_2i-1-μ_2i in Eq,(<ref>) reveals that it is in fact not a local gauge invariance when taking account of entangled configurations. We may absorb μ_2i and μ_2i-1 into ξ_2i, and note that it parametrises the relative phase between |↑↓⟩ and |↓↑⟩ across a bond—and so determines the degree of singlet or triplet correlation. 

Next, we fix ξ to its optimum value. This  can be found by maximising the singlet content across each bond. Geometrically, this optimisation corresponds to  a rotation such that '_2i-1 = '_2i∝_2i-1×_2i, in which case _2i-1·_2i = _2i-1·_2i, and we obtain the form (<ref>) for the energy given in the main text. A similar construction is clearly possible if we instead began with ψ_2i = 0. 

It is instructive to view this simplification of Eq. (<ref>) from a complementary perspective.  The change of tangent space basis is equivalent to identifying a rotated polar axis ẑ'_2i=ẑ'_2i-1 = _2i-1 = '_2i∝_2i-1×_2i, and the coherent states have a simple expression in terms of spin-up and down states (| ↑_ẑ'⟩ and | ↓_ẑ'⟩, respectively) relative to this polar axis:
| _2i⟩ = (e^-i ϕ'_2i/2 | ↑_ẑ'⟩ + e^- ϕ'_2i/2 | ↓_ẑ'⟩)/√(2). In this case, the expectation values of the spin operators that comprise the middle line in Eq. (<ref>) simplify to 
⟨ | σ |-⟩·⟨ -' | σ |'⟩  = 1 + ·'.
Crucially, the overlap between coherent states also simplifies to ⟨ | ' ⟩ =  √( (1 + ·')/2). The complex phase factor that is found in calculating this overlap from the coherent states, as defined in Eq. (<ref>), is absent, having been absorbed into ξ_2i. 

It is this phase factor (with a phase angle proportional to the solid angle between , ' and ẑ) that ultimately generates the coherent state contribution to the Berry phase. It is transferred, therefore, to a Berry phase arising from the dynamical fixing of the rotation ξ_2i≡ξ_2i(_2i-1(t)×_2i(t)) (see the first term in Eq. (<ref>) below). In particular, we note that fixing ξ_2i to its optimal value in this way does not alter the topological contribution to the Berry phase.









§ DERIVATION OF THE NON-LINEAR SIGMA MODEL




The central result of this letter is to show how the non-linear sigma model action and the Wess-Zumino topological term describing the long-wavelength physics of the J_1-J_2 chain can be connected to the spin wavefunction via the MPS ansatz.
Although we have given all of the essential steps in the main text, we include a more detailed derivation here. As mentioned in the main text, in order to map the dynamics of the MPS ansatz to the field theory, we need to temporarily restrict the entanglement to only half of the bonds (though the possibility of entanglement appearing on the other set of bonds is restored in the continuum limit, in the sense that both singlet covers are accessible).

Again, we express the action in imaginary time, and, as in the main text, we begin with the form that results from assuming ψ_2i+1 = 0. 
That is,


S = ∫_0^β dτ⟨Ψ||∂_τΨ⟩ + ,


where


= J_1 ∑_2i (-w^x_2i + (1 + w_2i^x)_2i-1·_2i + w^z_2iw^z_2i+2_2i·_2i+1 )
+ J_2 ∑_2i w^z_2iw^z_2i+2(_2i-1·_2i+1 + _2i·_2i+2),


and


⟨Ψ||∂_τΨ⟩    = ∑_2i [ iξ̇_2i/2(1 - cosχ_2i )
+ i/2cosχ_2i
( ϕ̇_2i cosθ_2i +  ϕ̇_2i-1 cosθ_2i-1 )
] 

   = ∑_2i [ iξ̇_2i/2(1 - cosχ_2i ) + i/2cosχ_2i ((_2i)·_2i + (_2i - 1)·_2i-1 ) ],



where () = -cosθ/sinθ is a vector potential which generates the spin Berry phase. 





 §.§ Haldane's mapping and continuum limit


To derive the continuum limit of this action, we adapt Haldane's mapping <cit.>, and write


_i = (-1)^i _i √(1 - _i^2) + _i,     _i^2 = 1,     _i·_i = 0,     _i^2 ≪1.



where _i is the slowly-varying Néel field, and _i is the transverse canting field which captures any fast fluctuations of _i. 
Now, to second-order in , we have


_i·_j ∼(-1)^i + j_i·_j + _i·_j - 1/2(-1)^i + j(_i^2 + _j^2)
+ (-1)^j _i·_j + (-1)^i_i·_j + 𝒪(^3).


We will, shortly, approximate the differences of the slow -field by derivatives. We cannot take the continuum limit of the fast -field, though the cross-terms turn out to be negligible in the Hamiltonian <cit.>. At this order, then (and dropping the cross terms), the Hamiltonian becomes

   = ∑_2i -J_1w^x_2i -J_1(1 + w^x_2i)_2i-1·_2i - J_1 w_2i^z w_2i+2^z _2i·_2i+1 + J_2 w_2i^z w_2i+2^z (_2i-1·_2i+1 + _2i·_2i+2) 

   + ∑_2i [ J_1(1 + w_2i^x)(_2i-1^2/2 + _2i-1·_2i + _2i^2/2) + J_1 w_2i^z w_2i+2^z (_2i^2/2 + _2i·_2i+1 + _2i+1^2/2) 

             + J_2w_2i^z w_2i+2^z( - _2i-1^2/2 + _2i-1·_2i+1 - _2i+1^2/2 - _2i^2/2 + _2i·_2i+2 - _2i+2^2/2) ].


The first line in the above contains only the slow fields, and so we can proceed to the continuum limit. There is a subtlety, however, regarding the appropriate continuum limit for the Néel field—there is a much higher stiffness against gradients of  on the entangled bonds. Essentially, this causes the magnetisation to fluctuate on a scale of two lattice spacings—the same scale as for the entanglement. 

More explicitly, let us, temporarily, transform to sum and differences of the Néel fields across the entangled bonds,

_2i^(+) = _2i-1 + _2i/2,     _2i^(-) = _2i-1 - _2i.

But, since the entangled bonds are stiff, we have, approximately,

_2i-1 ≈_2i^(+),     _2i ≈_2i^(+),     _2i^(-) ≈0,     |_2i^(+)|^2 ≈1.
 
This implies the following continuum limits:

w_2i^z w_2i+2^z    ∼(w^z)^2 + 2w^z _x^2 w^z ↦(w^z)^2 - 2(_x w^z)^2 

_2i-1·_2i    ∼^(+)_2i·^(+)_2i ∼1, 
 
_2i·_2i + 1    ∼^(+)_2i·^(+)_2i + 2 ∼1 - 2(_x ^(+))^2 ∼1 - 2(_x )^2 

_2i - 1·_2i + 1    ∼_2i·_2i + 2 ∼^(+)_2i·^(+)_2i + 2 ∼1 - 2(_x ^(+))^2 ∼1 - 2(_x )^2,

where we convert from _x^2 (·) ↦ -(_x ·)^2 using integration by parts, and the lattice spacing has been set to unity. Then the part of Eq. (<ref>) that contains only the slow fields becomes

...    ∼∫dx [-J_1/2 - J_1 w^x - (J_1/2 - J_2 ) w_z^2 + (J_1 - 2J_2) ( (_x w^z)^2 + w_z^2 (_x )^2 )].







 §.§ Berry phase and kinetic terms


We now turn to the Berry phase terms. Again, applying Eq. (<ref>), we have

⟨Ψ||∂_τΨ⟩ ∼   ∫dx/2 [ i/2_τξ(1 - cosχ) + i/2 cosχ(_τϕ _xcosθ- _xϕ _τcosθ)] 

   + ∑_2i i/2cosχ_2i [(_2i ×_τ_2i) ·_2i + (_2i-1 ×_τ_2i-1) ·_2i-1 ],

where θ and ϕ are the standard spherical co-ordinates for . The second term in the above is the SO(4) topological term, which we will discuss shortly. It remains only to obtain the kinetic terms, for which we need to integrate out the remaining fast fields, ξ and . We start with ξ, for which the corresponding part of the action is

S_ξ   = ∫dτdx [i/4(1 - cosχ)(_τξ) - J_1 sinχcosξ] 

   ∼∫dτdx [+i/4(_τcosχ)ξ- J_1|sinχ| + J_1/2|sinχ|ξ^2 ] 

   = ∫dτdx [ J_1/2|sinχ|( ξ+ i/4 J_1 |sinχ| (_τcosχ))^2 + 1/32J_1|sinχ|(_τcosχ)^2 - J_1|sinχ| ],

which, after completing the Gaussian integrals over the transformed ξ-field, and approximating |sinχ| ≈ 1 in the coefficient of the entanglement field, contributes

∫dτdx [-J_1|sinχ| + 1/32J_1(_τcosχ)^2 ]

to the total action.

We now turn to the fast fluctuations of the magnetisation, . Since we are expanding around the singlet covers, we neglect the terms ∼cos^2χ ^2 in Eq. (<ref>). We thus obtain:

S_l    = ∫dτ∑_q i/2cosχ(×_τ)_q ·_-q + J_1 ω_q f(χ) _q·_-q 

   = ∫dτ∑_q [ J_1ω_q f(χ) (_q + i/4J_1ω_q f(χ)cosχ(×_τ)_q )·(_-q + i/4J_1ω_q f(χ)cosχ(×_τ)_-q ) 

                     + 1/16J_1ω_q f(χ) cos^2χ(×_τ)_q·(×_τ)_-q ],

where f(χ) = 1 + |sinχ| and ω_q = 1/2(1 + cos q), and we have assumed that χ varies much more slowly than , such that its momentum-dependence can be neglected when taking the Fourier transform of . Then, completing the integrals of the transformed -field, approximating |sinχ| ≈ 1 in the coefficient, and replacing ω_q by its zero momentum value (again, justified because the variation of  is much faster than that of ), we have

∫dτdx   1/32J_1cos^2χ(_τ)^2,

where we have made use of the identity (×)^2 = ()^2. 

Finally, we switch to the standard angular co-ordinates on S^3. As in the main text, we define α = -χ + π/2, in terms of which (w^z)^2 = sin^2α and ( w^z)^2 ∼ (α)^2. The full action, then, becomes

S = iΩ+ ∫dτdx [ 1/32J_1((_τα)^2 + sin^2α(_τ)^2) + (J_1 - 2J_2)( (_x α)^2 + sin^2α(_x)^2 ) + V(α) ],

with the dimerisation potential

V(α) = -J_1/2 - J_1|cosα| - (J_1/2 - J_2)sin^2α≈-3J_1/2 + J_2/2 - J_2/2cos2α
(the approximation preserves V(α) to second order around the singlet covers α = 0 or π). If V(α) is relevant, we are in the dimer phase; if V(α) is irrelevant we are in the critical phase, with the SO(4) symmetry of the joint-order parameter emergent in the infrared, and the topological term protecting the gaplessness. 






 §.§ Manipulating the topological term




Finally, let us show how the polar co-ordinate form of the topological term (<ref>) is equivalent to the standard formulation of the Wess-Zumino (WZ) term. 

We write the WZ term (<ref>) with an anti-symmetrisation over the derivatives, 
∂_ζ  ...  ∂_τ   ...   ∂_x ... = 1/6ϵ_μνρ∂_μ  ...   ∂_ν   ...   ∂_ρ ..., and insert u =  (sinχ, cosχ m ), with m = ( sinθcosϕ, sinθsinϕ, cosθ), into Eq. (<ref>). The topological term becomes


    Ω   =   
    -2 π k/2 π^2∫_0^1 d ζ∫ dτ dx  cos^2 χ sinθ|
    [  _ζχ  _τχ _x χ;  _ζθ  _τθ _x θ;  _ζϕ  _τϕ _x ϕ ]|
    
       =   
    - 2 π k/2 π^2∫_0^1 d ζ∫ dτ dx 
    |
    [    ∂_ζ f    ∂_τ f    ∂_x f;        0  ∂_τcosθ ∂_x cosθ;        0     ∂_τϕ    ∂_x ϕ ]|
    
       =   2 π k/2 π^2∫_0^1 d ζ∫ dτ dx 
     
    (∂_ζ f)  (
    ∂_x ϕ ∂_τcosθ  - ∂_τϕ ∂_x cosθ)
    
       =   2 π k/2 π^2∫ dτ dx 
     
    f(χ) ·(
    ∂_x ϕ ∂_τcosθ   - ∂_τϕ ∂_x cosθ),


where we have assumed without loss of generality that only χ depends upon the extension coordinate ζ, such that _ζϕ = _ζθ = 0 (the extension is arbitrary, up to the boundary conditions), and defined 

f(χ) = ∫_0^1 dζ ∂_ζf = ∫_π/2^χd χ' cos^2 χ' = 1/4 (2χ- π+ sin2χ).

Now, the form given in Eq. (<ref>) contains a factor of (cosχ)/2 in the integrand instead of f(χ)/π. However, since this is a topological term, deformations of f(χ) that preserve the values at the poles of the sphere give the same integral. Here, f(χ=0) = -π/4, and f(χ=π) = π/4. Therefore, replacing f ↦  -π/4cosχ gives the same integral and recovers 
Eq. (<ref>).





