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PHYSICAL REVIEW D 103, 054033 (2021)

Hamiltonian models of lattice fermions solvable by the meron-cluster algorithm

Hanqing Liu ,1,* Shailesh Chandrasekharan ,1 and Ribhu K. Kaul2 1Department of , Box 90305, Duke University, Durham, North Carolina 27708, USA 2Department of Physics & Astronomy, University of Kentucky, Lexington, Kentucky 40506, USA

(Received 9 December 2020; accepted 10 March 2021; published 25 March 2021)

We introduce a half-filled Hamiltonian of spin-half lattice fermions that can be studied with the efficient meron-cluster algorithm in any dimension. As with the usual bipartite half-filled Hubbard models, the naïve C Uð2Þ symmetry is enhanced to SOð4Þ. On the other hand, our model has a novel spin-charge flip Z2 symmetry which is an important ingredient of free massless fermions. In this work we focus on one spatial dimension and show that our model can be viewed as a lattice-regularized two-flavor chiral- Gross- Neveu model. Our model remains solvable in the presence of the Hubbard coupling U, which maps to a combination of Gross-Neveu and Thirring couplings in one dimension. Using the meron-cluster algorithm we find that the ground state of our model is a valence bond solid when U ¼ 0. From our field theory analysis, we argue that the valence bond solid forms inevitably because of an interesting frustration C between spin and charge sectors in the group flow enforced by the Z2 symmetry. This state χ spontaneously breaks translation symmetry by one lattice unit, which can be identified with a Z2 chiral symmetry in the continuum. We show that increasing U induces a quantum to a critical phase described by the SUð2Þ1 Wess-Zumino-Witten theory. The quantum critical point between these two phases is known to exhibit a novel symmetry enhancement between spin and dimer. Here we verify the scaling relations of these correlation functions near the critical point numerically. Our study opens up the exciting possibility of numerical access to similar novel phase transitions in higher dimensions in fermionic lattice models using the meron-cluster algorithm.

DOI: 10.1103/PhysRevD.103.054033

I. INTRODUCTION Since our work here may be of interest to both The connections between lattice models of quantum communities, we explain our motivations from both many body physics and continuum field theories remain a viewpoints in turn. In particle physics, there is a great forefront topic of research in both high energy and interest in mechanisms through which fermions acquire . In high energy physics one . While fermion masses are usually thought to arise often starts with a continuum field theory, and then from fermion bilinear operators, it is well known that introduces a lattice to enable nonperturbative numerical strong four-fermion couplings can also generate masses computations. Even though the lattice is viewed as a through spontaneous symmetry breaking [1,2].Theroleof calculational artifice, it has led to many profound physical four-fermion couplings is still poorly understood since insights. In condensed matter physics, on the other hand, such couplings are irrelevant perturbatively at the free one starts with the lattice, which is derived from the fermion fixed point in two or more spatial dimensions and physical crystal structure, and the connection to continuum have to be of finite strength when they generate a mass, field theories emerges in the long distance physics close to requiring a nonperturbative analysis. From a Lorentz- a critical point. Despite these contrasting motivations, there symmetry point of view the interactions constructed with are recurring common themes across the two approaches scalar or pseudoscalar fermion bilinears are referred to as leading to a wonderful synergistic exchange of ideas. Gross-Neveu (GN) couplings [3], while those constructed with vector or pseudovector fermion bilinears are called

* Thirring couplings [4]. These relativistic four-fermion [email protected] field theories can be studied through the following Published by the American Physical Society under the terms of Lagrangian: the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to ’ the author(s) and the published article s title, journal citation, L χ¯ γμ∂ χ L L and DOI. Funded by SCOAP3. ¼ α μ α þ GN þ Thirring; ð1Þ

2470-0010=2021=103(5)=054033(19) 054033-1 Published by the American Physical Society LIU, CHANDRASEKHARAN, and KAUL PHYS. REV. D 103, 054033 (2021) where we use the Euclidean signature, and we will keep combination of many continuum four-fermion couplings, this convention through the paper. The case with two which makes it difficult to understand which continuum flavors of Dirac fermions that is relevant to our paper is model is being explored without careful fine-tuning. discussed in the Appendix B. The fixed point structure For example, recently it was recognized that the lattice of renormalization group (RG) flows in the space of Thirring model and the lattice GN models in 2 þ 1 these couplings is rich and reveals that the GN and dimensions seem to flow to the same continuum theory Thirring couplings can have very different physics at at the critical point [21]. strong couplings depending on the number of fermion The field has become more exciting recently due to new flavors especially in 2 þ 1 dimensions [5–7]. interest from condensed matter physics, which was origi- In condensed matter physics one usually works with nally inspired in part by the physics of graphene. Again the Hamiltonians instead of Lagrangians, which are motivated focus has been on mass terms generated by four-fermion more naturally by the physics of the underlying material. interactions on the honeycomb lattice . Here For many important systems, such as one-dimensional the motivation is to study the semimetal to antiferromag- metals and two- and three-dimensional semimetals, the netic transition and its universality class [22]. While the long distance physics is described by relativistic four- original work focused on SUð2Þ, generalizations to SUðNÞ fermion field theories. These Hamiltonians usually have have been carried out [23]. Interacting spinless lattice two parts: an electron hopping term that in certain cases can Hamiltonian models have also been demonstrated to be result in massless Dirac fermions and an electron-electron free of sign problems, which then allows one to study the interaction term that is modeled by a four-fermion inter- simplest fermion mass generation mechanism at a quantum action. These kinds of models are typified by the iconic critical point, the chiral Ising transition [24,25]. Perhaps the Hubbard model, most striking recent result is that strong four-fermion X X couplings may also create fermion masses without sym- − † – 2 1 H ¼ ðtijciαcjα þ H:c:ÞþU ni↑ni↓; ð2Þ metry breaking [26 29].In þ dimensions there is hiji i growing evidence that this mechanism of symmetric fermion mass generation may be a feature of continuum where ciα destroys an electron on lattice site i with spin α since it appears to be connected to an † – 3 1 and niα ¼ ciαciα. While the choice of the matrix elements exotic quantum critical point [30 33].In þ dimensions tij depends on the material to be modeled, by now many it has been suggested that this mechanism may be helpful to physically motivated cases give Dirac fermions whose construct lattice chiral gauge theories [34–36]. Another masslessness is protected by lattice symmetries or topology interesting scenario that has been proposed is the existence [8]. The topic of interest is how symmetries are sponta- of second order quantum phase transitions between differ- neously broken at strong interactions that convert the ent massive fermion phases [37,38]. While such transitions semimetal to an insulator and the nature of the quantum cannot easily be understood within the Landau-Ginzburg critical point [9]. These models, although motivated from paradigm, there is speculation that they may arise through a the physics of electrons in crystals, are natural Hamiltonian deconfined quantum critical point with emergent gauge discretizations of the four-fermion mass generation prob- fields [39]. It is also believed that some of them could even lem in Eq. (1). The complexities of the RG flows of the be driven by topological terms in the low energy bosonic continuum theory manifest themselves here in rich phase theory [30,40]. A four-fermion realization of this transition diagrams that are currently poorly understood, making this has also been proposed [41,42], where the existence of an intensely studied area of research. symmetry enhancement has also been studied at multi- Generally speaking the only available unbiased method critical points [43]. to study such strongly coupled four-fermion lattice models Despite the power of MC methods, two bottlenecks is the Monte Carlo (MC) method [10,11]. Given the great quickly arise. The first one is the sign problem that limits interest in this topic, there has been a large body of work the type of fermion models that can be studied. from both communities that has led to important progress Furthermore, even when sign problems are solved, most and also a number of unresolved issues. Most work found traditional MC studies of quantum critical points in in the high energy literature has been focused on under- fermionic systems can only be performed on rather small standing phase diagrams and fermion mass generation at lattice sizes, due to the poor scaling of the computational strong four-fermion couplings [12–17]. The value of the time with system size. Given these hurdles it is clearly of critical number of fermion flavors below which fermion great interest to find new four-fermion models that are sign bilinear condensates in the Thirring model has been an problem free and can be accessed with efficient MC interesting quantity to compute [18–20]. Lattice calcula- algorithms. Recently, alternate types of fermion MC tions with a local Lagrangian typically suffer from the methods have become available, which are able to reach fermion doubling problem. Due to this problem simple somewhat larger lattice sizes. For example, the recently lattice four-fermion couplings get mapped into a linear proposed fermion-bag approach [44–46] can study system

054033-2 HAMILTONIAN MODELS OF LATTICE FERMIONS SOLVABLE … PHYS. REV. D 103, 054033 (2021) sizes involving up to 10,000 sites [47]. The meron-cluster excitations. This helps us uncover the phase diagram algorithm, a precursor to the fermion-bag approach, is even of our lattice model. In Sec. III C, we explain how the more efficient while being applicable only to a more symmetries of the lattice model are enhanced at the critical restricted class of models [48–50]. The main motivation point and derive expressions for the spin and dimer behind our current work is to design fermionic models that correlation functions near the critical point and in the can be studied on large lattices using the meron-cluster conformal phase. In Sec. IV we verify the theoretical algorithm. analysis against MC results obtained using the meron- Interestingly the new four-fermion model we introduce cluster algorithm and determine the critical point. We also in this work is not only amenable to efficient simulations, it confirm our estimate for the critical point using an exact also has some novel physical features. In brief we dem- diagonalization method. In Sec. V we summarize our onstrate below a novel mechanism by which the valence results and provide an outlook for the future. bond solid (VBS) state which breaks translations symmetry is realized in a one-dimensional fermionic system at II. OUR LATTICE MODEL arbitrarily weak coupling. It is well known that the Hubbard model Eq. (2) at half filling on bipartite lattices In this section we introduce our lattice model and discuss 2 2 its symmetries. We focus on models with two flavors (or has an explicit SUð Þs spin symmetry and a hidden SUð Þc charge symmetry [51]. The hopping term of the Hubbard spins) of lattice fermions which can be annihilated and created at each lattice site j by the usual fermionic Fock model has an additional spin-charge flip symmetry ZC 2 † α ↑ ↓ under which the two SUð2Þ symmetries are interchanged. operators cjα and cjα, where ¼ ; . The choice of our Indeed, as is well known when U is repulsive (attractive) model is constrained by the ability to use the meron-cluster the spin (charge) sector is favored resulting in antiferro- algorithm as discussed in [49]. While a variety of models magnetic (superconducting) correlations. It is interesting to fall in this class, here we focus on a particularly simple one ask, what is the fate of the Hubbard model when the whose Hamiltonian is interactions added preserve ZC? Here we show by field X 2 − theoretic arguments and explicit MC simulations that in one HJ ¼ J Hhi;ji↑Hhi;ji↓; ð3Þ C hi;ji spatial dimension, when the interactions preserve Z2 , the system releases the frustration between spin and charge where sectors by forming a VBS. This is a novel mechanism for the formation of a VBS in the one-dimensional Hubbard 1 1 1 model. From a field theory point of view, we argue that our − † † 2 − − − Hhi;jiα ¼ ðciαcjα þ cjαciαÞþ niα 2 njα 2 2 : model can be considered as a Hamiltonian lattice regulari- zation of the two-flavor chiral-mass GN model with a ð4Þ χ spontaneously broken Z2 chiral symmetry. Using the meron-cluster representation we observe that our model The symbol hi; ji refers to a bond between nearest neighbor is also related to the θ ¼ π phase of the CP3 model where sites i and j. Our model above remains solvable by the the charge conjugation symmetry is spontaneously broken meron-cluster algorithm in the presence of some other [52]. When U ≠ 0 a combination of GN and Thirring carefully chosen interactions, for example, the Hubbard couplings is introduced. These new couplings induce a interaction, χ quantum phase transition to a phase where Z symmetry is 2 X 1 1 1 restored, which turns out to be the SUð2Þ1 Wess-Zumino- − − HU ¼ U nj↑ 2 nj↓ 2 þ 4 : ð5Þ Witten (WZW) model perturbed by a marginally irrelevant j coupling. The universality class of the transition in our model has been studied earlier by Affleck et al. [53],who In this work we will study the Hamiltonian has argued that at the quantum critical point the marginal coupling vanishes, enhancing the symmetry of the theory. H ¼ HJ þ HU: ð6Þ This transition has also been well studied numerically within the context of the quantum spin-half chain [54,55]. This model and its variants can be defined on a bipartite Our paper is organized as follows. In Sec. II we introduce lattice in any dimension and remain solvable using the our lattice model and explain its symmetries. In Sec. III A meron-cluster approach. The constraints of this solvability we argue that our lattice model is naturally mapped into a typically make them strongly interacting and end up within continuum model with a variety of four-fermion couplings. massive phases. However, as we will show in this work, We discuss the continuum symmetries and relate them to they can still be useful to study phase transitions to the lattice symmetries. In Sec. III B we apply non-Abelian massless phases. Here we will study this interesting to the continuum four-fermion theory and quantum phase transition of Eq. (6) in one spatial dimen- rewrite the low energy physics in terms of bosonic sion as a function of U.

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Our model has a variety of interesting lattice symmetries multiple of four, as will be shown in Appendix A. In fact, that become manifest when written in terms of Majorana all energy levels are evenly degenerate, and in particular the γ1 γ2 operators. We define two such operators j and j for spin- ground state is doubly degenerate. up fermions on each lattice site through the relations In order to compute quantities in our lattice model with the meron-cluster algorithm we use the continuous time 1 1 γ1 − γ2 † γ1 γ2 formulation of the partition function, cj↑ ¼ ð j i j Þ;cj↑ ¼ ð j þ i j Þð7Þ 2 2 X Z X k Z ¼ dt dt1 J TrðH ðt ÞH ðt1ÞÞ; ð12Þ for even j and k bk k b1 k ½b 1 1 2 1 † 2 1 − c ↑ ¼ ðγ þ iγ Þ;c¼ ðγ − iγ Þð8Þ tHU tHU j 2 j j j↑ 2 j j where HbðtÞ¼e Hhi;ji↑Hhi;ji↓e is the bond operator associated with the bond b ¼hiji inserted at time t. Notice 1 − γ1γ2 1 for odd j. Note that nj↑ ¼ 2 ð i j j þ Þ in both cases. that here we use Tr to denote the trace over the full Hilbert γ3 γ4 space. Later we will use tr to denote the trace over local Similarly we define two more Majorana operators j and j using the spin-down fermions. In terms of the four Hilbert spaces or local fields. The integrals over Euclidean time are always assumed to be time ordered such that Majorana operators the two parts of the Hamiltonian take β 0 the form >tk > >t2 >t1 > . The choice of Hb leads to a simple formula for the trace in the fermionic Hilbert space. X Y4 In particular, it does not contain any determinants of large J μ μ − 1 γ γ matrices, as is the case in the traditional auxiliary field HJ ¼ 4 ð þ i i j Þ; ð9Þ hi;ji μ¼1 methods. Instead, it can be shown that X Y U μ ρ l ν σ TrðHb ðtkÞHb1 ðt1ÞÞ ¼ Wð iÞ; ð13Þ HU ¼ − εμνρσγ γ γ γ : ð10Þ k 96 j j j j i j l l … l 4 Zχ where f 1; 2; g is a set of loops and Wð iÞ is the weight Let us now argue that HJ is invariant under Oð Þ × 2 l 4 Zχ associated with the loop i [49]. The loops can be identified transformations, while HU is invariant under SOð Þ × 2. by introducing two parallel bonds for each H at the 4 b The SOð Þ symmetry becomes obvious when we realize appropriate imaginary time, as illustrated in Fig. 1. As can that the following six operators be seen from the figure, these bonds naturally divide the X lattice into disconnected loop clusters. When the trace Γμν γμγν ¼ i j j ð11Þ over the fermionic Hilbert space is performed, each cluster j −Utl=2 gets a weight WðlÞ¼2ð1 e Þ, where tl is the linear so 4 temporal size of the loop. The sign associated with each satisfy the ð Þ algebra and commute with H. In fact, the 2 1 μ −1 ntþnb= þ four operators γ transform as an SOð4Þ vector. We will loop is given by ð Þ , where nt is the number of j temporal winding of the loops and n is the number of argue in the next section that this symmetry is the vector b bonds in the loop. The fermionic nature of the problem is subgroup of the full chiral symmetry group in the con- hidden in this sign. Note that when U ¼ 0 clusters with a tinuum. In addition,P the HJ has a discrete symmetry ≔ γ1γ3γ4 ZC generated by C↑ i j j j j, which we denote as 2 for convenience. It is easy to verify that C↑ flips the sign of γ2 j but not the other three Majorana operators. Hence along with C↑ the four Majorana operators actually transform under the Oð4Þ group. Note that in the Dirac fermion −1 j † language, C↑cj↑C↑ ¼ð Þ cj↑ is the familiar particle-hole transformation on the spin-up fermions. Therefore it can also be viewed as the spin-charge flip transformation that is more familiar in condensed matter physics. On a regular lattice H is also invariant under translations by one lattice † site TacjαTa ¼ cjþ1α. As we will argue in the next section, χ there is a remnant discrete Z2 subgroup of the continuum 0 chiral symmetry group buried within Ta. When U ¼ , the FIG. 1. A configuration of bonds that naturally divides space- above lattice symmetries actually lead to an interesting time into loops. The fermionic trace is given as a product of degeneracy in the energy spectrum when the lattice size is a weights associated with each loop as described in the text.

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X negative sign (merons) are naturally forbidden. On the ð4Þ † † † † H J − c c ↑ c c ↑ c c ↓ c c ↓ other hand, when U is very large, all clusters are allowed, J ¼ ð i↑ j þ j↑ i Þð i↓ j þ j↓ i Þ and from the cluster representation our model becomes hi;ji X 1 1 identical to the Heisenberg spin-half chain [56,57]. − − þ niα 2 njα 2 ð17Þ α III. CONTINUUM FIELD THEORY ANALYSIS ð6;8Þ In this section, we review the results for the non-Abelian is a four-fermion interaction. Higher order terms HJ are bosonization of the Hubbard model pioneered by Affleck irrelevant perturbatively near the free fermion fixed point, et al. [53,58] and the field theoretic picture for the phase and therefore their exact forms will not affect our dis- transition from critical to VBS in one dimension. While this ð2Þ ð4Þ cussion. By focusing on HJ and HJ we can identify the scenario is now standard, a new aspect of our model is the continuum model and also map the lattice symmetries to ZC spin-charge flip symmetry 2 which is present in HJ at a the continuum. Our lattice model with ε ¼ 1 will be lattice level even beyond the quadratic level. This extra regarded as merely another lattice discretization of the symmetry appears in an interesting way in the continuum same continuum model. description, the RG flow, and the resulting phase diagram. In order to analyze the low energy physics near the free fermion fixed point, we expand our lattice fields cjα in A. Connection to the Gross-Neveu-Thirring model terms of smooth fields ψ α;LðajÞ and ψ α;RðajÞ near the two ∓ π 2 In this section we will identify the (1 þ 1)-dimensional Fermi momenta kF, where kF ¼ = a and a is the lattice continuum quantum field theory which is described by our spacing. Without loss of generality we choose lattice model in one spatial dimension, given in Eq. (6). 1 Since the model is strongly interacting, in principle such an ik aj −ik aj pffiffiffi c α ≈ e F ψ α ðajÞþe F ψ α ðajÞð18Þ identification can be questionable. Still we can try to a j ;L ;R perform a tree level analysis by expanding the lattice Hamiltonian in terms of modes near the Fermi points of α ð2Þ the free Hamiltonian and then include interactions pertur- for both . Inserting Eq. (18) into HJ , we can expand in batively. Such an analysis will teach us how the lattice powers of lattice spacing a to obtain the leading low energy symmetries are embedded within the continuum ones [59]. effective Hamiltonian in the continuum as In particular, we show below that the HJ term can be Z X identified with the strongly coupled lattice-regularized two- ð2Þcont − ψ † d ψ − ψ † d ψ HJ ¼ aJ dx α;Li α;L α;Ri α;R : flavor chiral-mass GN model, while the HU term introduces α dx dx a combination of the usual GN coupling and a Thirring coupling. More details of these couplings and the mapping ð19Þ are explained in Appendix B. In order to perform the tree level analysis described This is the Hamiltonian for two-flavor free massless above, we introduce a small parameter ε and deform Dirac fermions. It is easy to verify that the above free → ε O 4 O 4 HJ HJ as follows: Hamiltonian is invariant under ð ÞL × ð ÞR which we refer to as the full continuum chiral symmetry. The SOð4Þ X Y4 J μ μ transformation in each chiral sector can be decomposed Hε ¼ − ð1 þ iεγ γ Þ: ð14Þ 2 2 J 4ε i j into spin SUð Þs and charge SUð Þc transformations, hi;ji μ¼1 which becomes explicit if we introduce a 2 × 2 matrix of operators, Clearly all lattice symmetries discussed in the previous ε ! section are maintained order by order in . Expanding in ψ ψ † powers of ε, we get ↑;LðRÞ ↓;LðRÞ Ψ ¼ ; ð20Þ LðRÞ ψ −ψ † X4 ↓;LðRÞ ↑;LðRÞ ε ≕ εn−1 ð2nÞ HJ HJ ; ð15Þ 1 Ψ ↦ n¼ and define the spin and charge transformations as LðRÞ † S Ψ Q , where S and Q are SUð2Þ and where the leading term LðRÞ LðRÞ LðRÞ LðRÞ LðRÞ s 2 SUð Þc matrices in each chiral sector. When written in X Ψ ð2Þ J † † terms of LðRÞ, the Hamiltonian H ¼ − c αcjα þ c αciα ð16Þ J 2 i j Z hi;jiα 1 ð2Þcont − Ψ† d Ψ − Ψ† d Ψ HJ ¼ aJ dx tr Li L Ri R ð21Þ describes free fermions, and 2 dx dx

054033-5 LIU, CHANDRASEKHARAN, and KAUL PHYS. REV. D 103, 054033 (2021) is clearly invariant under the above transformation. Coming to the Hubbard interaction HU in Eq. (5), the − − However, since ð SLðRÞ; QLðRÞÞ should be identified with corresponding continuum Hamiltonian was already derived ðSLðRÞ;QLðRÞÞ, the symmetry group of the continuum by Affleck et al. [60] and is given by ð2Þcont 2 2 Z ≅ Z Hamiltonian HJ is actually ðSUð Þs × SUð ÞcÞ= 2 1 SO 4 in each chiral sector. These transformations are cont ≈ J i J i − J i J i ð Þ HU 2 aU dx cL cR sL sR generated on the Hilbert space by the spin current operators 1 − J i J i J i J i ψ ð sL sL þ sR sRÞ ð27Þ 1 † † ↑;LðRÞ 3 J i ðxÞ¼ ðψ ; ψ Þσi ð22Þ sLðRÞ 2 ↑;LðRÞ ↓;LðRÞ ψ ↓;LðRÞ up to irrelevant pieces. Note that HU preserves all the symmetries of HJ except the spin-charge flip symmetry and the charge current operators χ which means the Oð4Þ × Z2 symmetry is reduced to † χ 1 ψ SOð4Þ × Z2. The last two terms in Eq. (27) only renorm- i † iT ↑;LðRÞ J ðxÞ¼ ðψ ↑ ; ψ Þσ : ð23Þ cLðRÞ 2 ;LðRÞ ↓;LðRÞ ψ alize the speed of light and do not introduce any inter- ↓;LðRÞ actions, which is clearer in the language of conformal In addition, the Hamiltonian is also invariant under field theory (CFT). Throwing away these chiral terms, and normalizing the kinetic term to have a coefficient of one, two independent spin-charge flip transformations ZC : 2LðRÞ we obtain the final continuum Hamiltonian of our lattice Ψ ↦ Ψ† J i J i LðRÞ LðRÞ, under which sLðRÞ and cLðRÞ exchange model as with each other. Including them the free Hamiltonian is Z X indeed invariant under the Oð4Þ × Oð4Þ symmetry as cont † d † d L R H ¼ dx −ψ i ψ α þ ψ i ψ α stated above. α;L dx ;L α;R dx ;R α We know from Eq. (18) that under lattice translation Ta, ψ α ↦ iψ α ψ α ↦ −iψ α T λ J i J i λ J i J i L L and R R. This means a, correspond- þ s sL sR þ c cL cR ; ð28Þ † π σ3 ing to QL ¼ QR ¼ expði 2 Þ in the continuum, generates Z 2 a 4 subgroup of the chiral SUð Þc group, under which λ 2 ε − U λ 2 ε U 1 2 1 2 where s ¼ ð 4 Þ and c ¼ ð þ 4 Þ. Assuming the J i ↦J i , J ; ↦ −J ; , and J 3 ↦ J 3 . J J sLðRÞ sLðRÞ cLðRÞ cLðRÞ cLðRÞ cLðRÞ Hð6;8Þ terms do not modify the above analysis except Note that here since Q2 ¼ Q2 ¼ −1, T2 belongs to the L R a perhaps to change the couplings λs and λc, we argue that vector subgroup of the chiral symmetry group, and there- Eq. (6) is a lattice regularization of Eq. (28). In Appendix B Z fore Ta is effectively a 2 subgroup of the chiral symmetry we construct the Lagrangian of Eq. (28) and identify it with Zχ group, hence denoted by 2. a linear combination of GN-Thirring couplings. Using a similar analysis as above we can identify the In summary the above discussion shows that the Oð4Þ ð4Þ following continuum operators that corresponds to HJ : symmetry of our lattice model in Eq. (6) can then be Z identified with the diagonal Oð4Þ subgroup of the con- ð4Þcont 2 2 tinuum Hamiltonian Eq. (28). The translation by one site HJ ¼ aJ dxMðxÞ ; ð24Þ symmetry Ta can be identified as the remnant chiral χ symmetry Z2 discussed above. While it is known that where lattice translation is part of the continuum chiral symmetry, X we have shown explicitly how it is embedded along with ψ † ψ − ψ † ψ MðxÞ¼i ð α;L α;R α;R α;LÞ: ð25Þ the flavor symmetries. α

In order to understand the symmetries of this term it is B. Bosonization and phase diagram useful to express it in terms of spin and charge currents In order to understand the physics of our lattice model 2 J i J i J i J i − 1 using the relation MðxÞ ¼ð sL sR þ cL cRÞ , we begin by discussing the RG flows of the continuum which implies model Eq. (28) near the free fermion fixed point. The one- Z loop β functions in the continuum are given by [60] ð4Þcont 2 J i J i J i J i HJ ¼ aJ dxð sL sR þ cL cRÞ: ð26Þ λ2 dλs c s c ð Þ ¼ − ð Þ ; ð29Þ d log μ 2π In this form it is easy to see that this term breaks the chiral 4 Oð Þ symmetry of the free theory down to the diagonal which shows that the spin and charge currents are com- 4 Zχ Oð ÞL¼R subgroup. However, the 2 group generated by pletely decoupled in the low energy theory; i.e., terms ð4Þcont 4 Zχ Ta survives. Therefore HJ has Oð Þ × 2 symmetry. involving both spin currents and charge currents and

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g and h are SUð2Þ group-valued fields, describing the spin and charged sectors, respectively, and Z 1 2 ∂ −1∂μ SWZW½g¼16π d xtrð μg gÞ Z i 3 εμνρ −1∂ −1∂ −1∂ þ 24π d x trðg μgg νgg ρgÞ ð31Þ

is the action for the WZW theory in the Euclidean signature. While the first term in SWZW½g involves an integration over 2-sphere spacetime, the second term involves an integration over a 3-disk whose boundary is J i J i the 2-sphere. The currents sLðRÞ and cLðRÞ are given by

i J j ∂ −1σj sL ¼ 4π trðð gÞg Þ; ð32Þ FIG. 2. Phase diagram in the λs − λc plane. Both of the couplings are relevant when positive and irrelevant when neg- i ative. Our model lives on the U axis. When U ¼ 0, our model is J j −1∂¯ σj sR ¼ 4π trðg g Þ; ð33Þ on the λs ¼ λc line due to the spin-charge flip symmetry and is in a massive phase described by a lattice regularized chiral-mass GN Zχ i model with a spontaneously broken 2 symmetry. When U J j ∂ −1σj cL ¼ trðð hÞh Þ; ð34Þ increases to Uc, λs ¼ 0 and our model is at the critical point of a 4π second order phase transition, whose low energy physics is described by the SUð2Þ1 WZW model. When U>U, λ < 0 j i −1 ¯ c s J ¼ trðh ∂hσjÞ; ð35Þ and a marginally irrelevant coupling would modify the low cR 4π energy WZW theory mildly. ¯ where ∂ ¼ ∂x þ i∂t and ∂ is the complex conjugate of ∂. respecting the SOð4Þ symmetry are irrelevant. The RG Note that although in the Euclidean space those currents flow diagram based on this β function is shown in Fig. 2. are usually called holomorphic and antiholomorphic and λ 0 J J¯ The couplings are relevant when sðcÞ > and irrelevant denoted by and , here we keep the convention from the λ 0 Minkowski space and still call the currents as left- and when sðcÞ < . right-handed for convenience. The bosonized action Our lattice model falls somewhere on the ðλs; λcÞ plane. For example, when U ¼ 0, the model must be on the Eq. (30) enjoys almost the same symmetries as Eq. (28), except for the fact that ðg; hÞ and ð−g; −hÞ should be λc ¼ λs line due to the spin-charge flip symmetry, which is identified with the chiral-mass GN model with the well- identified. This constraint comes from the fact that we have 2 2 known physics of asymptotic freedom. Here we expect written the bosonized theory with an SUð Þs × SUð Þc χ symmetry rather than SOð4Þ. This means that observables hMðxÞi ≠ 0 due to the spontaneous breaking of the Z2 chiral symmetry [3], which is related to the dimer order in the fermionic theory will always be mapped into parameter and verified by our MC results presented later. terms that do not violate this identification. For example, When U>0, our model moves away from the spin-charge trg itself is not an observable. Besides, the chiral sym- 4 symmetric axis toward the conformal phase in the second metries of SWZW in each sector are SOð Þ instead of 2 2 quadrant where λ < 0 and λ > 0. This implies the SUð ÞL × SUð ÞR. Also note that under spin-charge flip s c ↔ existence of a quantum phase transition at some critical symmetry g h. value U U Armed with these bosonization results we see that when ¼ c, which we find to be second order. 0 The most convenient way to understand the physics of U> the charge excitations have higher energies than the conformal phase is through the language of non- the spin excitations. Thus, the long distance physics of Abelian bosonization [61], where Eq. (28) is mapped to the conformal phase in the second quadrant must be the following nonlinear : described by Z Z 2 2 λ J i J i λ J i J i S ¼ SWZW½gþ s d x sLðgÞ sRðgÞ: ð36Þ S½g; h¼SWZW½gþ s d x sLðgÞ sRðgÞ Z λ 2 J i J i This long distance physics can also be seen directly in þ SWZW½hþ c d x cLðhÞ cRðhÞ: ð30Þ the lattice model, where the charge sector becomes

054033-7 LIU, CHANDRASEKHARAN, and KAUL PHYS. REV. D 103, 054033 (2021) energetically less favorable when U>0 and the low distances. Here we derive the form of these corrections and energy physics is described by a Heisenberg spin-half use them in our analysis. chain. In a famous theorem, Lieb, Shultz, and Mattis It was explained in [58,69] that the symmetry enhance- showed that spin-half chains can be in only one of two ment is visible in correlation functions of spin and dimer possible ground states [62]: a dimerized phase where the operators on the lattice defined through the relations translation symmetry is broken or a critical phase. In 1 fermionic viewpoint, the dimerized phase is the chirally z j S ðtÞ¼ð−1Þ ðn ↑ − n ↓ÞðtÞ; ð38Þ broken GN phase while the critical phase is the CFT j 2 j j described by Eq. (36) with λ ≤ 0. This implies the s 1 universality class of the transition at U ¼ U in our model j z z z z c DjðtÞ¼ð−1Þ ðS S 1 − S −1S ÞðtÞ: ð39Þ can also be understood from simpler spin-half chain 2 j jþ j j models. One such example is The leading continuum terms of these two operators are the X 3 primary fields φS ¼ trhtrðgiσ Þ and φD ¼ trhtrg of the H − J1S · S 1 J2S · S 2 37 J1 J2 ¼ f i iþ þ i iþ gðÞ WZW model. In the fermionic language these operators i φ ∝ ψ †ασ3 ψ β φ ∝ are given by S L αβ R þ H:c: and DðxÞ MðxÞ 0 0 defined in Eq. (25). We see that φD can be used as an with both J1;J2 > . When J2 ¼ , this model is the usual χ Heisenberg spin chain and is well known to be described order parameter on the lattice for Z2. Since φS and φD the WZW conformal phase. Also, as Majumdar and Ghosh transform into each other under the SOð4Þ chiral trans- pointed out long ago, when J2=J1 ¼ 0.5, the ground state is formations, their correlation functions will be related doubly degenerate due to dimerization, suggesting that the to each other. Using the well-known techniques in CFT theory at those couplings is in a different phase [63,64]. [70–72], their conformal dimensions have been computed 1 Previous studies have shown that the phase transition to be hS ¼ hD ¼ 2, and the two point correlation functions between the two phases occurs at J2=J1 ¼ 0.241167ð5Þ of the spin and dimer are [55,65]. Although the phase transition has also been studied −2hi without a sign problem in a one-dimensional spin-half GiðrÞ¼hφiðrÞφið0Þi ∝ r ; ð40Þ chain using the J-Q idea [66,67], our model provides a route to the one-dimensional WZW critical phase to VBS for large values of r at the critical point U ¼ Uc. quantum phase transition starting from a lattice fermionic In order to be able to fit our MC data away from the Hubbard model Hilbert space instead of a spin-half chain critical point we have to include the logarithmic corrections Hilbert space. A very narrow region of VBS was also to Eq. (40), due to the presence of the marginal coupling λs. reported in an extended Hubbard model [68]. We begin with the β-function of the marginal coupling λs,

dλ λ2 C. Symmetry enhancement and correlation functions βðλ Þ¼ s ¼ s þ Oðλ3Þ: ð41Þ s dlnr 2π s Since the charge sector decouples when U>0, the long distance physics of the lattice model seems to only have the This is the same as Eq. (29) except that we have replaced 2 Zχ diagonal SUð Þ spin symmetry and the remnant 2 chiral the energy scale μ with a length scale r, which reverses the 3 3 symmetry, under which g ↦ iσ giσ . These symmetries sign of the beta function. Integrating to first order, we can be observed in the continuum low energy theory obtain described by Eq. (36). However, if we set λs ¼ 0 the 1 1 1 continuum theory is invariant under the enhanced chiral − − r ↦ † λ λ ¼ 2π ln ; ð42Þ transformation g SLgSR where SL and SR are two sðrÞ 0 r0 independent SUð2Þ matrices implementing the left and right spin symmetries of the fermion model, except that where λ0 is the coupling at some short distance length scale λ SL ¼ SR ¼ −1 needs to be quotiented out. Based on the r0. Note that the solution sðrÞ above is meaningful for very RG flow diagram shown in Fig. 2 we see that by tuning to large values of r only when λ0 < 0 (i.e., when U>Uc the critical point U ¼ Uc we can indeed set λs ¼ 0, and and the theory is in the conformal phase), and therefore thus we must see the enhanced chiral symmetry there. the coupling λsðrÞ is marginally irrelevant. When λ0 > 0, 2π=λ0 In our work we look for this symmetry enhancement, the coupling λsðrÞ diverges at r ¼ r0e , which is the although observing it could be nontrivial since there will remnant of the Landau pole of perturbation theory and always be higher dimensional operators that break the signals new physics at long distances due to the formation symmetry on the lattice. Also, when U ≠ Uc but close to it, of a fermion mass. This means in the broken phase the since λs is a marginal coupling, the logarithmic corrections above form of λsðrÞ will only be valid close to the critical due to it would be visible in the lattice MC data at long point and small values of r.

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With the caveats discussed above, we can use λsðrÞ in Inserting these values of bi and λsðrÞ from Eq. (42) into Eq. (42) in both phases to derive the corrections for GiðrÞ. Eq. (45) we can compute γiðλsðrÞÞ. Substituting this into Noting that it depends on r both explicitly and through Eq. (44) we get λsðrÞ, we can write GiðrÞ ≕ GiðλsðrÞ;rÞ to be more precise. 1 We note that it satisfies the following RG equation: G ðλðrÞ;rÞ r0 λ0 r 2 S ¼ 1 − ln ; ð47Þ GSðλ0;r0Þ r 2π r0 λ dlnGið sðrÞ;rÞ −2 γ λ ¼ ðhi þ ið sðrÞÞÞ; ð43Þ −3 dlnr G ðλðrÞ;rÞ r0 λ0 r 2 D ¼ 1 − ln : ð48Þ GDðλ0;r0Þ r 2π r0 where hi is the conformal dimension of φi at the critical γ λ φ point and ið sðrÞÞ is the anomalous dimension of i Similar expressions have been derived earlier in the context induced by the marginal operator. Integrating this equation of spin chains [55,74] and verified numerically [55]. Note we get again that for a fixed value of λ0 but large values of r, the Z above expressions make sense only when λ0 ≤ 0, i.e., when G λ r ;r r ið sð Þ Þ −2 γ λ 0 0 we are in the conformal phase as already mentioned above. ln λ ¼ ðhi þ ið sðr ÞÞÞdlnr ; ð44Þ λ → 0 Gið 0;r0Þ r0 On the other hand, when r is in a fixed range but 0 , the above expressions give us the leading corrections to from which we can derive the corrections due to the conformal behavior on both sides of the critical point. presence of the marginal operator once we know the We could do a similar analysis in the massive phase. γiðλsðrÞÞ. However, in the massive phase the anomalous dimen- First, let us focus on the conformal phase and near the sion Eq. (45) obtained in the conformal phase cannot be β critical point, i.e., λ0 ≲ 0. In two-dimensional (2D) CFT the completely correct. Besides, as the -function, it was conformal dimension hi of the operator φi can be calculated derived in perturbation theory and so there could be some using the finite size energy Ei ¼ 2πhi=L of the state jφii nonperturbative corrections to it due to the presence of the through the state-operator correspondence [73]. Then the mass scale M. In particular, if we assume the correlation −α ∼ iMr anomalous dimension γi of the operator is related to the function to the leading order is of the form GiðrÞ e , change in this energy due to the presence of the marginal this would contribute additional nonperturbative terms γ λ operator. In our case, to leading order in λs, this leads to the to the anomalous dimension ið sÞ. For example, if we expression [53] assume 2π b γ λ i λ λ2 λsðrÞ¼− ð49Þ ið sÞ¼2π s þ Oð s Þ; ð45Þ ln Mr as a possible definition for the RG invariant mass scale M, where then it is easy to see that Z j j α b ¼ dxhφ jJ J jφ i bi i −2π=λ 2 i i sL sR i γ ðλ Þ¼ λ þ e s þ Oðλ Þ: ð50Þ Z i s 2π s 2 s 1 φ J j J j 2 − J j J j − J j J j φ ¼ 2 dxh ijð sL þ sRÞ sL sL sR sRj ii Since we are only deriving corrections to GðrÞ due to the 1 marginal coupling perturbatively, in this work we ignore 1 − 1 − 1 ¼ 2 ðstotðstot þ Þ sLðsL þ Þ sRðsR þ ÞÞ: ð46Þ such nonperturbative effects. Then, Eqs. (47) and (48) can still be used in the massive phase near the critical point λ 0 1 2 except that we will find 0 > and r is constrained to be In the above formula sL ¼ sR ¼ = for both spin and λ 2π 1 φ such that 0= lnðr=r0Þ < . dimer states, while stot depends on the state j ii. The values of bi can now be calculated for both spin and dimer fields and are tabulated in Table I below. IV. NUMERICAL RESULTS A. Monte Carlo results We have used the meron-cluster algorithm to compute i TABLE I. hi and bi for the lattice fields S , D. the equal time spin and dimer correlation functions defined through the expressions Lattice field WZW field stot sL sR hi bi i φ σi 1 2121214 S S ¼ trhtrg 1 = = = = 1 ˜ −β D φ ¼ trhtrg 0 1=21=21=2 −3=4 G ðrÞ¼ Trðe HO ðrÞO ð0ÞÞ; ð51Þ D i Z i i

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FIG. 3. Spin correlation functions GSðr˜ðrÞÞ and dimer correlation functions GDðr˜ðrÞÞ as functions of r at L ¼ 128 for U ¼ 0, 0.5, 1.0, and 1.5. Both correlation functions decrease exponentially (with possible multiplicative power corrections) in r. In addition, the dimer correlation functions exhibit long range orders.

where OiðrÞ, i ¼ S, D are the spin and dimer operators account, we make the following ansatz for the lattice defined in Eqs. (38) and (39). The symbol r is used for the correlation functions at U ≳ Uc on a finite lattice: spatial lattice site j at some fixed time slice. In our work we ˜ 1 r choose β ¼ L where L is the size of our spatial lattice in A1 λ0 2 A2ð−1Þ G ðr˜Þ¼ 1 − ln r˜ − ; ð53Þ lattice units. Based on our estimate for the speed of light S r˜ 2π r˜2 this leads to a small effective temperature. In particular, we have evidence that physical temperatures and physical ˜ −3 B λ0 2 ≈ 0 25 −1 G r˜ 1 − ln r˜ ; 54 length scales are related by Tphys . ðLaÞ in our Dð Þ¼ ˜ 2π ð Þ ˜ r lattice model. We compute GiðrÞ at various lattice sizes 0 ˜ and U. In Fig. 3 we illustrate some of our results at U ¼ , where we have introduced a single fit parameter λ0 ¼ 0.5, 1, and 1.5 on a lattice size of L ¼ 128. For clarity we λ0 λ0=ð1 þ ln r˜0Þ, which is an RG invariant quantity and focus on the region of 8 ≤ r ≤ 40. The figure clearly shows 2π numerically equals the coupling measured at r˜0 ¼ 1 in the a nonzero expectation value for the dimer order parameter λ˜ 0 hD ðtÞi at U ¼ 0 while there is no such expectation value lattice unit. The critical point is obtained when 0 ¼ . j We have performed a combined fit for both G ðr˜Þ and for the spin operator. This is consistent with our expect- S G ðr˜Þ to the form Eqs. (53) and (54) at each fixed value of ations that for small values of U the lattice model breaks the D χ U ≥ 1.5, which is tabulated in the combined fit column of Z2 chiral symmetry as explained in Sec. III B. Another Table II. Each of these is a four parameter fit involving A1, important point to note is that the spin and dimer correlation ˜ A2, B, λ0 and uses all data from L 64, 80, 96, 128, and functions also behave very differently at least for small ¼ 12 ≤ r ≤ 40. As can be seen from these results, our data fit values of U but slowly begin to become similar by U ≈ 1.5. well to the form Eqs. (53) and (54) for all couplings in the Assuming we are in the vicinity of the critical point range 1.5 ≤ U ≤ 2.0. Since at the critical point we expect around U ≳ 1.5, we want to fit our MC data to Eqs. (47) λ˜ ≈ 0 and (48). However, since we work on a finite lattice with 0 , applying conservative systematic errors related to 1 75 5 periodic boundary conditions, the correlation functions our fitting procedures we estimate Uc ¼ . ð Þ. In order receive finite size corrections. Fortunately, in the conformal to visualize that the correlation functions indeed obey the phase the finite size corrections can be obtained using the power law near the critical point, we plot them in log scale map from an infinite plane to a cylinder, which results in in Fig. 4. We only plot at even r in order to avoid the replacing r by distracting oscillation. The quality of the combined fit column in Table II π becomes poor for U>2.0, and changing the fitting range → ˜ L r r r ¼ π sin ; ð52Þ of r does not seem to improve the fits much. In fact, the fit L 2 seems to have a very bad reduced chi-squared χν ¼ 9.68 where L is the spatial size [73,74]. Furthermore, the spin when U → ∞. This seems a bit surprising since the forms correlation functions clearly show oscillating behavior described by Eqs. (53) and (54) must work well in the entire due to the higher order operators in the observable with conformal phase. One reason is that our data are very ferromagnetic behavior. Taking these corrections into precise at U ¼ ∞, where H is the same as the Heisenberg

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TABLE II. Parameters in Eqs. (53) and (54) obtained by fitting the MC data for U ≳ Uc ¼ 1.75ð5Þ. Combined fit Spin fit Dimer fit ˜ 2 ˜s 2 ˜d 2 UA1 A2 B λ0 χν A1 A2 λ0 χν B λ0 χν 1.5 0.0816(2) 0.0241(4) 0.00747(4) 0.333(7) 0.55 0.0832(3) 0.0245(4) 0.409(11) 0.20 0.00770(5) 0.294(9) 0.32 1.6 0.0836(2) 0.0247(4) 0.00766(4) 0.230(7) 0.69 0.0845(3) 0.0250(4) 0.273(12) 0.73 0.00780(5) 0.204(9) 0.46 1.7 0.0838(1) 0.0248(4) 0.00818(3) 0.066(6) 1.10 0.0854(2) 0.0252(4) 0.145(12) 0.31 0.00828(4) 0.047(6) 1.44 1.745 0.0839(1) 0.0249(4) 0.00847(3) −0.019ð6Þ 1.23 0.0858(3) 0.0253(4) 0.082(14) 0.97 0.00857(4) −0.037ð7Þ 1.02 1.8 0.0852(1) 0.0253(3) 0.00846(3) −0.048ð6Þ 1.10 0.0851(2) 0.0252(4) −0.054ð13Þ 0.75 0.00845(4) −0.046ð7Þ 0.30 1.9 0.0854(2) 0.0253(3) 0.00890(5) −0.195ð9Þ 1.32 0.0867(2) 0.0255(3) −0.123ð13Þ 1.21 0.00922(7) −0.254ð13Þ 0.99 2.0 0.0871(2) 0.0257(3) 0.00894(5) −0.258ð10Þ 0.78 0.0875(3) 0.0258(3) −0.234ð15Þ 0.32 0.00904(7) −0.278ð14Þ 1.20 2.5 0.0911(2) 0.0262(3) 0.00982(6) −0.670ð11Þ 1.85 0.0929(2) 0.0265(3) −0.560ð14Þ 2.03 0.01054(10) −0.812ð18Þ 0.56 4.0 0.0997(2) 0.0268(3) 0.01129(6) −1.345ð13Þ 1.90 0.1014(2) 0.0271(3) −1.222ð15Þ 1.74 0.01264(13) −1.628ð27Þ 0.31 ∞ 0.1110(2) 0.0269(2) 0.01322(5) −1.975ð10Þ 9.68 0.1243(2) 0.0276(2) −1.683ð11Þ 3.94 0.01601(11) −2.548ð21Þ 0.31

˜ 1=2 2 3 ˜ −3=2 spin chain, and therefore the results become sensitive to 2πA1ð−λ0Þ ≈ 1.06, 4π A2 ≈ 1.06, and 8π Bð−λ0Þ ≈ higher order terms in the lattice model which are ignored in 1.18. While the first two constraints are not far from the theoretical analysis. In this limit the meron clusters have expectations, the last one seems quite a bit off. We want to ˜ the same weight as the nonmerons, and therefore we do not check how the fits change if we allow for λ0 to be different need to check merons and the algorithm is much more in the spin and dimer correlation functions. Therefore efficient. However, we believe there is more to the story we performed separate fits of our data, which are tabulated here. Fortunately, there are precise asymptotic results for in the spin fit and dimer fit columns in Table II, and the spin and dimer correlation functions in the Heisenberg plotted in Fig. 5. Note that in the vicinity of the critical spin chain [74–76], point both values tend to become small as expected. → ∞ 1 Focusing on the U limit, the separate fits now suggest ðln r˜Þ2 ð−1Þr ˜s 1=2 2 3 ˜d −3=2 ˜ → ∞ − 2πA1ð−λ0Þ ≈ 1.01, 4π A2 ≈ 1.09, and 8π Bð−λ0Þ ≈ GSðr Þ¼ 3=2 2 2 ; ð55Þ ð2πÞ r˜ ð2πÞ r˜ 0.98. Although the deviation in A2 still seems to be large, this is understandable because A2 is a higher order −3 2 ðln r˜Þ 2 ð−1Þrðln r˜Þ correction. Now they are in much better agreement with G ðr˜ → ∞Þ¼ þ ; ð56Þ D 2π 3=2r˜ 6π4r˜4 the above constraints, suggesting that for some reason that ð Þ ˜ we do not yet understand the values of λ0 for spin and dimer and indeed they are consistent with Eqs. (53) and (54) begin to drift apart as we go away from the critical point. in the r˜ → ∞ limit. More precisely in the limit U → ∞,we must observe the following constraints among the fit B. Exact diagonalization results ! ! ! ˜ 1=2 2 3 ˜ −3=2 parameters 2πA1ð−λ0Þ ¼ 4π A2 ¼ 8π Bð−λ0Þ ¼ 1. In order to confirm that our estimate for the critical point However, results from the combined fit give us is reasonable, we also use an alternate idea outlined in [54],

˜ ˜ ˜ FIG. 4. Spin correlation functions GSðrÞ and dimer correlation functions GDðrÞ as functions of r at even r and L ¼ 128 for U ¼ 1.6, 1.7, 1.745, 1.8, and 1.9. Both correlation functions decrease algebraically in r˜ with log corrections.

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FIG. 5. MC results for the spin and dimer correlation functions GSðr˜ðrÞÞ (left) and GDðr˜ðrÞÞ (right) obtained at U ¼ ∞ are shown as a function of r. The solid line that passes through the data are fits to Eqs. (53) and (54). based on the low energy physics of the WZW model degenerate. When L is finite, the critical coupling where the Eq. (36) that emerges at the critical point. Since the WZW energies of these two total spin sectors cross can be referred model is invariant under the enhanced chiral transforma- to as a pseudocritical point UcðLÞ. This point turns into the tions with two independent SUð2Þ (“left” and “right”) true critical point as L becomes large. symmetries, the energy eigenstates can be labeled with spin In order to implement the above idea we have com- quantum numbers ðsL;sRÞ. It is known that the ground state puted the first five eigenenergies by diagonalizing the has ðsL;sRÞ¼ð0; 0Þ with momentum 0, while the first Hamiltonian on small lattices with the coordinate descent 1 1 excited state has ðsL;sRÞ¼ð2 ; 2Þ with momentum π [53]. method [77,78]. The behavior of the lowest five states as a However, since the lattice model is only invariant under function of U at L ¼ 14 and L ¼ 16 is plotted in Fig. 6.We the diagonal SUð2Þ subgroup, the energy eigenstates on observe that in the broken phase (small U) the ground state the lattice will only be labeled by the total spin s . The and the first excited state turn out to be spin singlets with tot 0 fourfold degeneracy requires fine-tuning to the critical stot ¼ . The next three degenerate excited states form a 0 1 1 point where the singlet (stot ¼ ) and the triplet (stot ¼ ) triplet with stot ¼ . In contrast, when U>Uc, the triplet 1 1 states have lower energy as compared to the singlet state. states together form an ðsL;sRÞ¼ð2 ; 2Þ state. This suggests an alternative method to determine the critical point: we can We thus can determine UcðLÞ as the coupling where the compute the lowest five energy eigenvalues as a function of triplet and singlet states cross, which are tabulated in 1 2 U and L using an exact diagonalization method and locate Table III and plotted in Fig. 7 as a function of =L . ≈ 1 746 1 the coupling where the first excited state becomes fourfold Based on Fig. 7 we can estimate UcðLÞ . ð Þ, which

FIG. 6. The lowest five energy eigenvalues obtained using an exact diagonalization method as a function of U at L ¼ 14 and L ¼ 16. Note that only three eigenvalues are shown because the s ¼ 1 states are threefold degenerate.

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TABLE III. Pseudocritical couplings UcðLÞ as a function of L TABLE IV. Pseudocritical couplings UcðLÞ as a function of L obtained using the exact diagonalization method. obtained using the exact diagonalization method for the model defined by Eq. (14) at ε ¼ 0.1 and J ¼ 1. LUcðLÞ LUcðLÞ LUL LUL 4 2.53982388 6 1.26699259 cð Þ cð Þ 8 1.86035876 10 1.68834326 4 0.399755 6 0.195927 12 1.75726171 14 1.73529271 8 0.397655 10 0.310813 16 1.74508577 18 1.74308979 12 0.396644 14 0.347239 20 1.74502594 22 1.745311 16 0.396076 18 0.3636 24 1.745996 26 1.746393

fermionic nature of our model, since there is an intrinsic is consistent with our estimate in the previous section and extra minus sign when the system is translated by one site with the estimate using finite size scaling in the conference when L ¼ 2n. proceeding published earlier [59]. While it is not easy to extend our meron-cluster Looking more closely at Fig. 6 we observe some algorithm for the more general Hamiltonian with the ε peculiarities. For example, at U ¼ 0 we note that the parameter introduced in Eq. (14), we can extend ground state is degenerate at L ¼ 16 but not at L ¼ 14. the above exact diagonalization method to determine the We have explained the reason for this degeneracy in critical point for arbitrary ε. For example, when ε ¼ 0.1 and 1 ε Appendix A. In fact, we show that there is an interesting J ¼ , UcðLÞ is tabulated in Table IV. When is small, the difference in the energy spectrum when the lattice size is perturbative analysis should be a good guide, and we 4 ε 0 4 L ¼ 4n versus when it is L ¼ 4n − 2, where n ¼ 1; 2; …. obtained at leading order Uc ¼ J ¼ . in Sec. III A, Moreover, this difference permeates even away from U ¼ 0 which is in good agreement with Table IV. and is observed in the dramatically different values of Using the exact diagonalization method we have also 0 U ðLÞ when L is small. The difference, however, decreases confirmed that our model at U ¼ is indeed in a massive c Zχ rapidly as L increases, and both of them approach the true phase with spontaneously broken 2 symmetry. In such a critical point as expected. Another peculiarity comes from phase the energy gap to the first excited state is expected to the momentum quantum number k for the first five energy become exponentially small as L becomes large, while the eigenstates. We note that k flips between π and 0 when gap to the second excited state remains nonzero at L → ∞. comparing L ¼ 4n with 4n þ 2. This is as expected We plot these gaps in Fig. 8, where the solid lines are because similar phenomena are also observed in the exponential fits. These features are qualitatively visible, Heisenberg spin chain, with k ¼ 0 for L ¼ 4n and k ¼ π along with the peculiar degeneracy in our model. for L ¼ 4n þ 2 [79]. The extra π phase in our model compared to the Heisenberg spin chain is due to the

FIG. 8. Scaling of energy gaps of the first and second excited states with the ground state at U ¼ 0. The gaps of the first excited states close at L → ∞ for both L ¼ 4n and 4n þ 2. At finite L, FIG. 7. The behavior of the pseudocritical coupling UcðLÞ as a the former is identically zero, while the latter closes exponen- function of 1=L2. The values of the couplings for this plot are tially. The gaps of the second excited states stay open for both given in Table III. L ¼ 4n and 4n þ 2. The curves show exponential fits.

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V. SUMMARY AND CONCLUSIONS Xin Zhang for helpful discussions. The material presented here is supported by the U.S. Department of Energy, Office In this work we constructed a strongly correlated lattice of Science, Nuclear Physics program under Award No. DE- fermion Hamiltonian that was solvable by the meron- FG02-05ER41368. R. K. K. was supported in part by NSF cluster algorithm. We were able to add the Hubbard Grant No. DMR-1611161. coupling U to our model without losing this property. χ This combined lattice model had SOð4Þ × Z2 symmetry for Zχ APPENDIX A: DEGENERACY all values of U, and an extra spin-charge flip symmetry 2 WITH LATTICE FERMIONS at U ¼ 0. While the lattice model can be formulated in any dimension, here we studied it in one spatial dimension. In In this Appendix we discuss a curious symmetry of a order to study the phase diagram of the model as a function class of lattice fermion Hamiltonians in one dimension of U we used non-Abelian bosonization to relate our model that is generalizable to higher dimensions, which makes to two decoupled SUð2Þ1 WZW models with marginal the degeneracy of all energies, including that of the ground couplings that depend on U. In particular, we discovered state, to be an even number when the lattice size is a that the model undergoes a quantum phase transition multiple of four. We can formulate this result as the χ between a massive phase where the Z2 symmetry is following theorem: spontaneously broken and a conformal phase without such Theorem Degeneracy of all energy eigenvalues of a symmetry breaking. The massive phase is the well-known spin-half fermion system on a periodic lattice with trans- asymptotically free chiral-mass GN model with massive lation symmetry, spin and charge symmetries, spin-charge fermions and spontaneous symmetry breaking, while the flip symmetry, and parity symmetry will be an even number conformal phase is the single SUð2Þ1 WZW model. Using when the lattice size is a multiple of four. the meron-cluster method and exact diagonalization we Proof. From the assumptions in the theorem, the lattice provided further evidence of this scenario. Hamiltonian H commutes with Ta, We view the current study as just a first step in X demonstrating that meron-cluster algorithms can be useful − † † Ta ¼ exp ia kðck↑ck↑ þ ck↓ck↓Þ ; ðA1Þ to study phase diagrams of lattice fermion systems where k fermions become massive, while the low energy bosonic physics is still interesting due to either frustration or the where presence of topological terms. Extending our model to 1 X 2πn 2 þ 1 dimensions would be very interesting. For example, ffiffiffiffi ikaj ckα ¼ p cjαe ;k¼ ;n¼ 1; …;L; in [42] it was argued that in an extended Hubbard model L j aL very similar to ours, there is a direct second order phase transition between a VBS phase and an antiferromagnetic ðA2Þ 5 phase, and an enhanced SOð Þ symmetry is expected to 2 1 emerge at the transition. The critical exponents at this and the spin and charge SUð Þ generators Si;Qi;i¼ ,2,3, exotic transition were computed in the earlier studies, but X 1 † † unfortunately they do not match those obtained from a loop S1 c c ↑ c c ↓; ¼ 2 j↓ j þ j↑ j gas formulation of the same phase transition [80]. On the j X   other hand, the critical exponents of the fermionic reali- 1 † † S2 i c c ↑ − c c ↓ ; zation seem consistent with the bounds obtained from ¼ 2 j↓ j j↑ j conformal bootstrap [81]. As far as we know, this con- j X troversy remains unresolved, partially because the fer- 1 † † S3 ¼ c c ↑ − c c ↓; mionic model was studied on rather small lattices with 2 j↑ j j↓ j j less than 500 lattice sites, due to difficulties associated with 1 X   j † † auxiliary field fermion MC algorithm. We are currently Q1 −1 c c c ↑c ↓ ; ¼ 2 ð Þ j↓ j↑ þ j j exploring if our model and its extensions allow us to study j this transition more efficiently using the meron-cluster 1 X   j † † Q2 −1 i c c − c ↑c ↓ ; approach. ¼ 2 ð Þ j↓ j↑ j j j X ACKNOWLEDGMENTS 1 † † Q3 c ↑c − c c ↓: A3 ¼ 2 j j↑ j↓ j ð Þ We thank Emilie Huffman and Hersh Singh for helpful j discussions and pointing us to the literature on interesting fermionic quantum critical behavior. S. C. thanks Uwe-Jens Further, H is also assumed to commute with the spin- † † Wiese for discussions. H. L. thanks Zhe Wang for sharing charge flip operator C↑ ¼ C↑ and parity operator P ¼ P . his computer codes on the coordinate descend method and These operators can be conveniently represented on

054033-14 HAMILTONIAN MODELS OF LATTICE FERMIONS SOLVABLE … PHYS. REV. D 103, 054033 (2021) fermion creation and annihilation operators in the two states jE; k; ls;s3;lq;q3; αi and jE; k þðs3 − q3 þ following way: 2 1 π α L= þ Þ a ;lq;q3;ls;s3; i are different but have the same energy irrespective of the value of k. Thus, the −1 i † † C↑ci↑C↑ ¼ð Þ ci↑;C↑ck↑C↑ ¼ cπ=a−k↑; only situation where an energy eigenstate could remain nondegenerate is when k ¼ 0 or π=a, ls ¼ lq ¼ j, and PciαP ¼ cðLþ1−iÞα;PckαP ¼ c−kα: ðA4Þ s3 ¼ q3 ¼ m. In this case Eq. (A8) mixes the states E; k; j; m; j; m; α and E; k L=2 1 π=a;j;m;j;m;α . Note that the particle number N is not an independent j i j ð þ þ Þ i Indeed, when L ¼ 4n − 2 these two states are identical and operator since N ¼ L − 2Q3. An example lattice C↑ cannot be used to pair degenerate states. However, when Hamiltonian with the above symmetries is H defined J 4 0 α π in Eq. (3). L ¼ n, C↑ mixes jE; ;j;m;j;m; i with jE; =a; j; m; 2 2 j; m; αi. This means the whole spectrum has an even Since H, Ta, S , S3, Q , and Q3 commute with each degeneracy when L ¼ 4n. ▪ other, we can label the energy eigenstates by jE; k; ls; iak s3;lq;q3; αi, where E; e ;lsðls þ 1Þ;s3;lqðlq þ 1Þ;q3 are 2 2 eigenvalues of H, Ta, S , S3, Q , Q3, and α denotes APPENDIX B: LAGRANGIAN OF THE possible additional quantum numbers. Since P satisfies CONTINUUM MODEL † the relationships PTaP ¼ Ta, PSiP ¼ Si, and PQiP ¼ Qi, In this Appendix we construct the Euclidean Lagrangian we have for the continuum Hamiltonian given in Eq. (28).For this purpose we define two flavors of two-component α ∝ − α PjE; k; ls;sz;lq;qz; i jE; k; ls;sz;lq;qz; i: ðA5Þ Grassmann variables χα and χ¯α, using the following map to Fock space operators: This implies that the pair of states with k k0 but all ¼ other quantum numbers being the same will be degen- ψ α;L † † 1 ≠ 0 π χα ≔ ; χ¯α ≔ ψ ; ψ γ ; B1 erate as long as k0 , =a. This is easily understand- ψ ð α;L α;RÞ ð Þ able since a state with a fixed lattice momentum will α;R have a partner with a negative momentum and both will 1 1 2 2 3 1 2 3 and choose γ ¼ σ , γ ¼ −σ , γ ¼ iγ γ ¼ σ , where σi have the same energy as long as parity is a symmetry of are the Pauli matrices. The free Hamiltonian theory is then the theory. mapped to the theory with the Lagrangian Interestingly, we will now show that there are additional pairs of degenerate states due to the C↑ operator, which has μ L0 ¼ χ¯αγ ∂μχα: ðB2Þ the following properties: P † † In order to construct the interaction terms of the −ia kðC↑c C↑C↑c ↑C↑þc c ↓Þ C↑T C↑ ¼ e k k↑ k k↓ k Lagrangian, we normal order the interaction terms in a P † † Eq. (28). A little bit of algebra gives −ia kðcπ − ↑c þc c ↓Þ ¼ e k =a k π=a−k↑ k↓ k P † † 1 1 1 −i ðπ−akÞc ↑c þakc c ↓ i i 3 2 2 μ 2 ¼ e k k k↑ k↓ k ∶J J ≔ ðχ¯αγ χαÞ − ðχ¯αχαÞ − ðχ¯ αγ χαÞ ; ðB3Þ P sL sR 2 2 4 † † −i ðak−πÞðc c ↑−1Þþakc c ↓ ¼ e k k↑ k k↓ k 1 1 1 ∶J i J i ≔ χ¯ γ3χ 2 χ¯ χ 2 χ¯ γμχ 2 S3−Q3þL=2þ1 cL cR ð α αÞ þ ð α αÞ þ ð α αÞ : ðB4Þ ¼ð−1Þ Ta; ðA6Þ 2 2 4 and Therefore the interaction Hamiltonian in Eq. (28) Z C↑S C↑ Q ;C↑Q C↑ S : A7 λ J i J i λ J i J i i ¼ i i ¼ i ð Þ Hint ¼ dx s sL sR þ c cL cR ðB5Þ Using these relations it is easy to show that is mapped to the interaction Lagrangian α C↑jE; k; ls;s3;lq;q3; i g1 g2 g3  L χ¯ γ3χ 2 χ¯ χα 2 χ¯ γμχα 2 int ¼ ð α αÞ þ ð α Þ þ ð α Þ ; ðB6Þ π 2 2 2 ∝ E; k þðs3 − q3 þ L=2 þ 1Þ ;l ;q3;l ;s3; α : a q s where g1 ¼ λs þλc ¼ 4ε and g2 ¼ 2g3 ¼ −λs þ λc ¼ U=J. ðA8Þ Therefore our lattice model contains all the three allowed four-fermion couplings in a relativistic two-flavor fermion This relation shows that additional pairs of state can have model: the chiral-mass GN coupling g1, the normal GN the same energy. For example, when ls ≠ lq or s3 ≠ q3, the coupling g2, and the Thirring coupling g3. Our model is a

054033-15 LIU, CHANDRASEKHARAN, and KAUL PHYS. REV. D 103, 054033 (2021) lattice regularized chiral-mass GN model, while the Hubbard coupling U introduces a linear combination of GN and Thirring terms. A natural question now is whether we can find three independent current couplings and three independent lattice couplings, which map to the three GN-Thirring couplings. Note that we know the massless Thirring model is 2 2 conformal and invariant under ðSUð ÞsL × SUð ÞsRÞ × 1 1 ðUð ÞcL × Uð ÞcRÞ. Therefore in order to introduce a 2 Thirring coupling, we must break the charge SUð Þc 1 symmetry down to Uð Þc, while preserving all the other J 3 J 3 chiral symmetries. Clearly the term cL cR is what we need, and indeed maps to the Thirring coupling in the Lagrangian language, 1 ∶J 3 J 3 ≔ χ¯ γμχ 2 cL cR 4 ð α αÞ : ðB7Þ FIG. 9. RG flow in the three-dimensional four-fermion cou- pling space of two-flavor massless Dirac fermions in 2D invariant 2 Zχ When analyzing the RG flow it turns out to be more under Uð Þ × 2 flavor symmetries. All the axes are eigen directions of the RG flow. In addition, the red line of the Thirring convenient if we introduce the three independent current coupling g3 is a line of fixed points. couplings as follows:

L λ ∶J i J i ∶ λ ∶J i J i ∶ above, we need a lattice interaction which breaks SU 2 int ¼ s sL sR þ c cL cR ð Þc down to U 1 while preserving particle hole symmetry. λ˜ ∶−J 1 J 1 − J 2 J 2 J 3 J 3 ∶ ð Þc þ c cL cR cL cR þ cL cR ; ðB8Þ The most straightforward way to do this is to include the interaction because those are the eigen directions of the RG flow. The X beta functions of the three couplings are given by HV ¼ V ðni − 1Þðnj − 1ÞðB11Þ dλ λ2 hiji s ¼ − s ; d log μ 2π in the lattice Hamiltonian, where ni ¼ ni↑ þ nj↓ is the total dλ λ − λ˜ fermion number operator at the site i. At the tree level in the c ¼ − c c λ ; d log μ 2π c continuum limit we can show that λ˜ λ − λ˜ Z d c c c λ˜ 1 ¼ c: ðB9Þ cont J i J i − J i J i 4J 3 J 3 d log μ 2π HV ¼ 2 aV dxð sL sR cL cR þ cL cRÞ; The RG flow diagram is shown in Fig. 9. ðB12Þ Using the relations given by Eqs. (B3), (B4), and (B7) we see that these couplings are related to g1, g2, and g3 via which maps to the Lagrangian the linear transformation 0 1 0 10 1 TABLE V. Various interaction couplings and the subgroup of g1 11−1 λs B C B CB C the full chiral symmetry group of the free theory that they @ g2 A ¼ @ −11−1 A@ λc A: ðB10Þ preserve. The free theory is invariant under the chiral symmetry 4 2 −11 1 λ˜ group OLð Þ × ORðLÞ while a generic point in the three- g3 c dimensional space is invariant under 2 1 Z ðSUsðL¼RÞð Þ × UcðL¼RÞð ÞÞ= 2. Notice that the pure Thirring coupling g3 is obtained when ˜ Coupling Symmetry λs ¼ λc − λc ¼ 0. This direction is special because the beta λ 2 2 2 Z function vanishes for all the couplings and we obtain a line s ðSUð ÞsðL¼RÞ × SUð ÞcL × SUð ÞcRÞ= 2 of fixed points. In Table V we summarize the symmetries λ 2 2 2 Z c ðSUð ÞsL × SUð ÞsR × SUð ÞcðL¼RÞÞ= 2 that are preserved by the various couplings. λ˜ 2 2 2 Z c ðSUð ÞsL × SUð ÞsR × SUð ÞcðL¼RÞÞ= 2 In our lattice model we introduced two independent 2 2 Z Zχ ε g1 ðSUð ÞsðL¼RÞ × SUð ÞcðL¼RÞÞ= 2 × 2 couplings H and HU. In order to explore the full three- χ J g2 ðSUð2Þ × SUð2Þ Þ=Z2 × Z dimensional space discussed above we will need one more sðL¼RÞ cðL¼RÞ 2 g3 ðSUð2Þ × SUð2Þ Þ × ðUð1Þ × Uð1Þ Þ independent coupling. Using the same symmetry argument sL sR cL cR

054033-16 HAMILTONIAN MODELS OF LATTICE FERMIONS SOLVABLE … PHYS. REV. D 103, 054033 (2021) 1 μ α 2 α 2 observe that when both U and V are positive, HU favors the LV ¼ aV ðχ¯αγ χ Þ − ðχ¯αχ Þ : ðB13Þ 2 spin sector while HV favors the charge sector, and the frustration between them gives the conformal Thirring Therefore we see that HU þ HV gives the Thirring coupling, model. Furthermore, HV can also be included in the while HU − HV gives the usual GN model. It is interesting to meron-cluster algorithm for range couplings.

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