Insulators and metals with topological order and discrete symmetry breaking

TL;DR

This paper develops a theory where topological order coexists with discrete symmetry breaking in insulators and metals, explaining pseudogap phenomena and symmetry behaviors in high-temperature cuprates.

Contribution

It extends existing topological order theories to include coexistence with discrete symmetry breaking, providing a unified explanation for pseudogap and symmetry phenomena in cuprates.

Findings

Translationally-invariant states with topological order coexist with Ising-nematic order.

The theory explains why broken symmetries vanish without a pseudogap at high doping.

Optimal doping criticality relates to the loss of topological order.

Abstract

Numerous experiments have reported discrete symmetry breaking in the high temperature pseudogap phase of the hole-doped cuprates, including breaking of one or more of lattice rotation, inversion, or time-reversal symmetries. In the absence of translational symmetry breaking or topological order, these conventional order parameters cannot explain the gap in the charged fermion excitation spectrum in the anti-nodal region. Zhao et al. (1601.01688) and Jeong et al. (arXiv:1701.06485) have also reported inversion and time-reversal symmetry breaking in insulating Sr2IrO4 similar to that in the metallic cuprates, but co-existing with Neel order. We extend an earlier theory of topological order in insulators and metals, in which the topological order combines naturally with the breaking of these conventional discrete symmetries. We find translationally-invariant states with topological order co-existing with both Ising-nematic order and spontaneous charge currents. The link between the discrete broken symmetries and the topological-order-induced pseudogap explains why the broken symmetries do not survive in the confining phases without a pseudogap at large doping. Our theory also connects to the O(3) non-linear sigma model and CP1 descriptions of quantum fluctuations of the Neel order. In this framework, the optimal doping criticality of the cuprates is primarily associated with the loss of topological order.

Tables

Table 1. Table 1: Symmetry signatures of various fields under time reversal ( 𝒯 𝒯 \mathcal{T} ), translation by a square lattice spacing along the x 𝑥 x ( T x subscript 𝑇 𝑥 T_{x} ) and y 𝑦 y ( T y subscript 𝑇 𝑦 T_{y} ) directions, and reflections about a square lattice site involving x → − x → 𝑥 𝑥 x\rightarrow-x ( I x subscript 𝐼 𝑥 I_{x} ) or y → − y → 𝑦 𝑦 y\rightarrow-y ( I y subscript 𝐼 𝑦 I_{y} ).

	$𝒯$	$T_x$	$T_y$	$I_x$	$I_y$
$\overrightarrow{n}$	$-$	$-$	$-$	$+$	$+$
$\overrightarrow{L}$	$-$	$+$	$+$	$+$	$+$
$e_x$	$+$	$-$	$-$	$-$	$+$
$e_y$	$+$	$-$	$-$	$+$	$-$
$b$	$-$	$-$	$-$	$-$	$-$
$J_x$	$-$	$+$	$+$	$-$	$+$
$J_y$	$-$	$+$	$+$	$+$	$-$

Table 2. Table 2: Quantum numbers of the matter fields in the SU(2) Lattice gauge theory. The transformations under the SU(2)’s are labelled by the dimension of the SU(2) representation, while those under the electromagnetic U(1) are labeled by the U(1) charge. The spin correlations are characterized by Φ Φ \Phi in Eq. ( 16 ). The Higgs field is the transform of Φ Φ \Phi into a rotating reference frame via Eq. ( 26 ).

Field	Symbol	Statistics	SU(2)gauge	SU(2)spin	U(1)e.m.charge
Electron	$c$	fermion	$\mathrm{𝟏}$	$\mathrm{𝟐}$	-1
Spin magnetic moment	$\Phi$	boson	$\mathrm{𝟏}$	$\mathrm{𝟑}$	0
Chargon	$\psi$	fermion	$\mathrm{𝟐}$	$\mathrm{𝟏}$	-1
Spinon	$R$ or $z$	boson	$\overline{\mathrm{𝟐}}$	$\mathrm{𝟐}$	0
Higgs	$H$	boson	$\mathrm{𝟑}$	$\mathrm{𝟏}$	0

Equations

$$\int d^{2}r⟨\mathbf{J}(\mathbf{r})⟩=0,$$

$$\sum _{\rho }⟨{\mathbf{J}}_{\rho }⟩=0.$$

$$⟨J_{\rho p}⟩=K_{p{p}^{\prime }}^{\rho }⟨O_{p^{\prime }}⟩,$$

$$\sum _{\rho }{K}_{ij}^{\rho }=0,$$

$${\mathcal{N}}_1=O_x^2-O_y^2,{\mathcal{N}}_2=O_x{O}_y.$$

$${\mathcal{S}}_{\vec{n}}=\frac{1}{2g}\int d^{2}rd\tau {\left({\partial }_{\mu }\vec{n}\right)}^2,$$

$$\mathbf{O}=\vec{L}\cdot (\vec{n}\times \nabla \vec{n}).$$

$$\vec{n}=z_{\alpha }^{\ast }{\vec{\sigma }}_{\alpha \beta }{z}_{\beta }^{},$$

$${\mathcal{S}}_z=\frac{1}{g}\int d^{2}rd\tau |({\partial }_{\mu }-i{a}_{\mu })z_{\alpha }{|}^2.$$

$$\mathbf{e}={\partial }_{\tau }\mathbf{a}-\nabla a_{\tau },b=\hat{\mathbf{z}}\cdot (\nabla \times \mathbf{a}),$$

$$\mathbf{e}=\frac{1}{2}\vec{n}\cdot \left({\partial }_{\tau }\vec{n}\times \nabla \vec{n}\right),b=\frac{1}{2}\vec{n}\cdot \left({\partial }_x\vec{n}\times {\partial }_y\vec{n}\right).$$

$$\mathbf{O}=\mathbf{e}\times (b\hat{\mathbf{z}}).$$

$${\mathcal{S}}_a=\int d^{2}rd\tau \left[\frac{{\gamma }_1}{2}({\partial }_{\tau }{a}_i-{\partial }_i{a}_{\tau }{)}^2+\frac{{\gamma }_2}{2}({\partial }_x{a}_y-{\partial }_y{a}_x{)}^2+i{\Gamma }_i({\partial }_{\tau }{a}_i-{\partial }_i{a}_{\tau })({\partial }_x{a}_y-{\partial }_y{a}_x)\right],$$

$$\mathbf{\Gamma }\propto \hat{\mathbf{z}}\times ⟨\mathbf{O}⟩;$$

$${\mathcal{H}}_c=-\sum _{i,\rho }{t}_{\rho }\left(c_{i,\alpha }^{†}{c}_{i+{\mathbf{v}}_{\rho },\alpha }^{}+c_{i+{\mathbf{v}}_{\rho },\alpha }^{†}{c}_{i,\alpha }^{}\right)-\mu \sum _i{c}_{i,\alpha }^{†}{c}_{i,\alpha }^{}+{\mathcal{H}}_{\mathrm{int}}$$

$${\mathcal{H}}_{\mathrm{int}}=-\lambda \sum _i{\Phi }^{\ell }(i)c_{i,\alpha }^{†}{\sigma }_{\alpha \beta }^{\ell }{c}_{i,\beta }^{}+V_{\Phi }$$

$$\left(\begin{array}[]{c}c_{i\uparrow}\\ c_{i\downarrow}\end{array}\right)=R_{i}\left(\begin{array}[]{c}\psi_{i,+}\\ \psi_{i,-}\end{array}\right),$$

$$R_i^{†}{R}_i=R_i{R}_i^{†}=1.$$

$$\displaystyle\left(\begin{array}[]{c}\psi_{i,+}\\ \psi_{i,-}\end{array}\right)$$

$$R_i$$

$${\mathcal{H}}_{\mathrm{int}}=-\lambda \sum _i{H}^a(i){\psi }_{i,s}^{†}{\sigma }_{s{s}^{\prime }}^a{\psi }_{i,s^{\prime }}^{}+V_H$$

$$H^a(i)=\frac{1}{2}{\Phi }^{\ell }(i)\text{mbox}Tr\left[{\sigma }^{\ell }{R}_i^{}{\sigma }^a{R}_i^{†}\right],$$

$${\Phi }^{\ell }(i)=\frac{1}{2}{H}^a(i)\text{mbox}Tr\left[{\sigma }^{\ell }{R}_i^{}{\sigma }^a{R}_i^{†}\right].$$

$${\sigma }^a{H}^a(i)=R_i^{†}{\sigma }^{\ell }{\Phi }^{\ell }(i)R_i^{}.$$

$$H^a(i)\to \frac{1}{2}{H}^b(i)\text{mbox}Tr\left[{\sigma }^a{V}_i{\sigma }^b{V}_i^{†}\right],$$

$${\sigma }^a{H}^a(i)\to V_i{\sigma }^b{H}^b(i)V_i^{†}.$$

$${\mathcal{H}}_{\psi }=-\sum _{i,\rho }\left(w_{\rho }{\psi }_{i,s}^{†}{U}_{s{s}^{\prime }}^{\rho }(i){\psi }_{i+{\mathbf{v}}_{\rho },s^{\prime }}^{}+\text{mbox}H.c.\right)-\lambda \sum _i{H}^a(i){\psi }_{i,s}^{†}{\sigma }_{s{s}^{\prime }}^a{\psi }_{i,s^{\prime }}^{}-\mu \sum _i{\psi }_{i,s}^{†}{\psi }_{i,s}^{}$$

$$U^{\rho }(i)\to V_i^{}{U}^{\rho }(i)V_{i+{\mathbf{v}}_{\rho }}^{†}.$$

$$U^{\rho }(i)$$

$${\sigma }^a{H}^a(i)$$

$$V_i=\exp \left(-\frac{i}{2}(\mathbf{Q}\cdot {\mathbf{r}}_i){\sigma }^z\right).$$

Full text

I Introduction

Experimental studies of the enigmatic high temperature ‘pseudogap’ regime of the hole-doped cuprate compounds have reported numerous possible discrete symmetry breaking order parameters Ando et al. (2002); Fauqué et al. (2006); Hinkov et al. (2008); Li et al. (2008); Xia et al. (2008); Daou et al. (2010); Li et al. (2010); Lawler et al. (2010); Lubashevsky et al. (2014); Mangin-Thro et al. (2015); Zhao et al. (2016, 2017); Jeong et al. (2017). There is evidence for lattice rotation symmetry breaking, interpreted in terms of an Ising-nematic order Kivelson et al. (1998), and for one or both of inversion and time-reversal symmetry breaking, usually interpreted in terms of Varma’s current loop order Varma (1997); Simon and Varma (2002, 2003). Both of these orders have the full translational symmetry of the square lattice, and cannot, by themselves, be responsible for gap in the charged fermionic spectrum near the ‘anti-nodal’ points ($(\pi,0)$ and $(0,\pi)$) of the square lattice Brillouin zone.

An interesting and significant recent development has been the observation of inversion Zhao et al. (2016) and time-reversal Jeong et al. (2017) symmetry breaking in the iridate compound Sr2Ir1-xRhxO4; Ref. Wang and Senthil, 2011 has shown that this iridate is described by a one-band Hubbard model very similar to that for the cuprates. The inversion symmetry breaking is strongest in the insulator at $x=0$ where it co-exists with Néel order; at non-zero $x$, both orders persist, but the discrete order is present at higher temperatures. Motivated by the similarities in the light and neutron scattering signatures between the cuprate and iridate compounds, we will present here a common explanation based upon the quantum fluctuations of antiferromagnetism.

Long-range Néel order (which breaks translational symmetry) can clearly be the origin of a gap in the charged fermionic spectrum at the anti-nodes. In the traditional spin density wave theory of the quantum fluctuations of the Néel order Hertz (1976), there is a transition to a state without Néel order, with full translational symmetry, a large Fermi surface, and no anti-nodal gap. However, the anti-nodal gap can persist into the non-Néel phase Chubukov and Sachdev (1993) when the resulting phase has topological order Senthil et al. (2003, 2004a); Paramekanti and Vishwanath (2004); Punk et al. (2015); Punk and Sachdev (2012) (see footnote111 Topological order is defined by the presence of ground state degeneracy of a system on a torus. More precisely, on a torus of size $L$, the lowest energy states have an energy difference which is of order $\exp(-\alpha L)$ for some constant $\alpha$. Topological order can also be present in gapless states, including those with Fermi surfaces Senthil et al. (2003, 2004a). In such states, the non-topological gapless excitations have an energy of order $1/L^{z}$ (for some positive $z$) above the ground state on the torus, and so can be distinguished from the topologically degenerate states. Topological order is required for metals to have a Fermi surface volume distinct from the Luttinger volume Senthil et al. (2004a), and hence to have a ‘pseudogap’. for the precise definition of topological order, and the review in Ref. Sachdev and Chowdhury, 2016). We shall use topological order as the underlying mechanism for the pseudogap. Moreover, early studies of spin liquid insulators with $\mathbb{Z}_{2}$ topological order showed that there can be a non-trivial interplay between topological order and the breaking of conventional discrete symmetries. The $\mathbb{Z}_{2}$ spin liquid obtained in Refs. Read and Sachdev, 1991; Sachdev and Read, 1991 co-existed with Ising-nematic order: this was a consequence of the $p$-wave pairing of bosonic spinons. A similar interplay with time-reversal and inversion symmetries was discussed by Barkeshli et al. Barkeshli et al. (2013), using higher angular momentum pairing of fermionic spinons. Here we shall use the formalism of Refs. Sachdev et al., 2009; Chowdhury and Sachdev, 2015; Sachdev and Chowdhury, 2016 to generalize the state Read and Sachdev (1991); Sachdev and Read (1991) with $\mathbb{Z}_{2}$ topological order to also allow for the breaking of inversion and time-reversal symmetries, both in the insulator and the metal. We will find states with spontaneous charge currents (see Fig. 2) and topological order, one of which (Fig. 2a) also has the Ising-nematic order observed in experiments Ando et al. (2002); Hinkov et al. (2008); Daou et al. (2010); Lawler et al. (2010).

The association between topological order and discrete broken symmetries implies that the broken symmetries will not be present in the confining phases at larger doping. This is an important advantage of our approach over more conventional excitonic condensation theories of broken symmetries. In the latter approaches there is no strong reason to connect the disappearance of the pseudogap with vanishing of the symmetry order parameter.

We will begin in Section II by a semi-classical treatment of the quantum fluctuations of antiferromagnetism Chakravarty et al. (1988, 1989); Haldane (1988) using the O(3) non-linear sigma model. In the insulator, this approach has been successfully used to describe the thermal fluctuations of the Néel order, and also the adjacent quantum phase without Néel order; the latter was argued to have valence bond solid (VBS) order Read and Sachdev (1989, 1990), and is accessed across a deconfined quantum critical point Senthil et al. (2004b, c). Here, we will identify an order parameter, ${\bm{O}}$, for inversion and time-reversal symmetry breaking in terms of the fields of the O(3) sigma model.

It is also useful to formulate the semi-classical treatment using the $\mathbb{CP}^{1}$ model for bosonic, fractionalized spinons coupled to a U(1) gauge field. In these terms, an order parameter ${\bm{O}}$ for inversion and time-reversal symmetry breaking turns out to be the cross product of the emergent U(1) electric and magnetic fields. The $\mathbb{CP}^{1}$ formulation yields an effective gauge theory, in Eq. (13) for quantum phases with spontaneous charge currents, but without Néel order.

Formally, a model expressed in terms of spins alone has no charge fluctuations, and so has vanishing electromagnetic charge current, ${\bm{J}}=0$. However, in practice, every spin model arises from an underlying Hubbard-like model, in which states suppressed by the on-site repulsion $U$ are eliminated by a canonical transformation. If we undo this canonical transformation, we can expect that a suitable multi-spin operator will induce a non-zero ${\bm{J}}$ at some order in the $1/U$ expansion. As ${\bm{O}}$ has the same symmetry signature as ${\bm{J}}$, we can expect that a state with $\langle{\bm{O}}\rangle$ non-zero will also have $\langle{\bm{J}}({\bm{r}})\rangle$ non-zero. We will examine states in which $\langle{\bm{O}}\rangle$ is independent of ${\bm{r}}$ in the continuum limit, so that translational symmetry is preserved. However, by Bloch’s theorem Bohm (1949); Ohashi and Momoi (1996), we must have

[TABLE]

and so ${\bm{J}}$ cannot be ${\bm{r}}$ independent. If we want to preserve translational symmetry, the resolution is that there will be intra-unit cell variations in $\langle{\bm{J}}({\bm{r}})\rangle$ to retain compatibility with Bloch’s theorem. In a tight-binding model with one site per unit cell, we label each unit cell by a site label, $i$, and a link label $\rho$ so that the combination $(i,\rho)$ identifies the complete set of lattice links, with no double counting. So from each lattice site $i$, there are set of vectors ${\bm{v}}_{\rho}$ connecting $i$ to its neighboring sites, and both ${\bm{v}}_{\rho}$ and $-{\bm{v}}_{\rho}$ are not members of this set: see Fig. 1.

In this setup, Bloch’s theorem states that

[TABLE]

where ${\bm{J}}_{\rho}$ is the current along the ${\bm{v}}_{\rho}$ direction. Note that Eq. (2) is a stronger statement than current conservation because the sum over $\rho$ does not include all links connected to site $i$, only half of them. Eq. (2) is equivalent to the statement that there are current ‘loops’, and these are clearly possible even in a single-band model Stanescu and Phillips (2004); Berg et al. (2008). In the presence of a ${\bm{r}}$-independent ${\bm{O}}$ condensate, we can write by symmetry that (to linear order in the broken symmetry)

[TABLE]

where $p,p^{\prime}=x,y$ are spatial indices, and $K_{pp^{\prime}}^{\rho}$ is a response function obtained in the $1/U$ expansion which respects all square lattice symmetries. Compatibility with Bloch’s theorem requires that

[TABLE]

and there are no conditions on the value of $\langle{\bm{O}}\rangle$.

We will turn to an explicit treatment of the charged excitations, and a computation of ${\bm{J}}_{\rho}$ in Section III: our results there do obey Eqs. (1) and (4). Section III will present a lattice formulation in which the U(1) gauge field of the $\mathbb{CP}^{1}$ model is embedded in a SU(2) lattice gauge theory Sachdev et al. (2009); Chowdhury and Sachdev (2015); Sachdev and Chowdhury (2016); Sachdev et al. (2016). This lattice gauge theory has the advantage of including all Berry phases and charged fermionic excitations, and also for allowing an eventual transition into a conventional Fermi liquid state at high enough doping. Our interest here will be in insulating and metallic states at lower doping, which have topological order and a gap to charged fermionic excitations in the anti-nodal region. At the same time we shall show that, with an appropriate effective action, there can be a background modulated gauge flux under which gauge-invariant observables remain translationally invariant but break one or more of inversion, time-reversal and lattice rotation symmetries. Our computations will demonstrate the presence of spontaneous charge currents obeying Eq. (2) in states which break both inversion and time-reversal, but preserve translation. The two classes of spontaneous current patterns we find are shown in Fig. 2. Note that the product of time-reversal and inversion is preserved in these states. The state in Fig. 2a has $\langle{\bm{O}}\rangle\sim(-1,0)$, while the state in Fig. 2b has $\langle{\bm{O}}\rangle\sim(1,-1)$. Both states belong to separate quartets of equivalent states (with $\langle{\bm{O}}\rangle\sim(\pm 1,0),(0,\pm 1)$ and $\langle{\bm{O}}\rangle\sim(\pm 1,\pm 1)$) which can be obtained from them by symmetry operations. Both states also break an Ising-nematic symmetry. In general, on the square lattice, we can define two Ising-nematic order parameters, which are invariant under both inversion and time-reversal, but not under lattice rotation symmetries. In terms of ${\bm{O}}$, these order parameters are

[TABLE]

The state in Fig. 2a has only $\langle\mathcal{N}_{1}\rangle\neq 0$, while the state in Fig. 2b has only $\langle\mathcal{N}_{2}\rangle\neq 0$. We note that the state in Fig. 1b of Simon and Varma Simon and Varma (2002) in a two band model has the same symmetry as the state in our Fig. 2b, and also the spontaneous current states considered in Refs. Stanescu and Phillips, 2004; Berg et al., 2008. The state in our Fig. 2a appears to not have been considered earlier: it has the same Ising-nematic order observed in experiments in the cuprates Ando et al. (2002); Hinkov et al. (2008); Daou et al. (2010); Lawler et al. (2010).

Finally, we note that our results are also easily extended to states with long-range antiferromagnetic order by condensing the spectator bosonic spinons.

II O(3) non-linear sigma and $\mathbb{CP}^{1}$ models

The familiar O(3) model describes quantum fluctuations of the unit vector $\vec{n}({\bm{r}},\tau)$, representing the local antiferromagnetic order, with action over space, ${\bm{r}}$, and imaginary time $\tau$

[TABLE]

where $\mu$ extends over the 3 spacetime indices, and $g$ is a coupling constant. For our purposes, we need the symmetry transformation properties of the operator $\vec{n}$ and its canonically conjugate angular momentum $\vec{L}$; the latter is also interpreted as the conserved ferromagnetic moment Chakravarty et al. (1989). We list these transformations properties in Table 1.

Table 1 also shows the symmetry transformations of charge current ${\bm{J}}$. Formally, a model expressed in terms of spins alone has no charge fluctuations, and so we will have ${\bm{J}}=0$.

However, in practice, every spin model arises from an underlying Hubbard-like model, in which states suppressed by the on-site repulsion $U$ are eliminated by a canonical transformation. If we undo this canonical transformation, we can expect that a suitable multi-spin operator will couple linearly to ${\bm{J}}$ at some order in the $1/U$ expansion; naturally, we need this multi-spin operator to have the same symmetry signature as ${\bm{J}}$. We therefore use Table 1 to find the simplest such combination of $\vec{n}$ and $\vec{L}$; the needed operator turns out to be

[TABLE]

We will therefore be interested in states in which $\langle{\bm{O}}\rangle$ is non-zero and independent of ${\bm{r}}$. In magnetically ordered states, $\langle{\bm{O}}\rangle$ is non-zero for a ‘canted spiral’ in which the spins precess around the base of a cone along a fixed spatial direction Yoshida et al. (2012): a non-zero $\vec{n}\times{\bm{\nabla}}\vec{n}$ corresponds to a spin spiral, which must cant to introduce a non-zero $\vec{L}$. However, our interest here is in states in which $\langle{\bm{O}}\rangle$ is non-zero without long-range magnetic order, in which $\langle\vec{n}\rangle=0$ and $\langle\vec{L}\rangle=0$. For example, we can add to $S_{\vec{n}}$ an effective potential $V({\bm{O}})$ which is invariant under all symmetries, and a suitable $V({\bm{O}})$ will induce an ${\bm{O}}$ condensate. In Section III, we will present specific lattice models for which such condensates arise. In any such state with an ${\bm{O}}$ condensate, we can also expect that $\langle{\bm{J}}({\bm{r}})\rangle$ is also non-zero, and obeys Eqs. (3) and (4).

Let us now turn to the $\mathbb{CP}^{1}$ model. This is expressed in terms of bosonic spinons, $z_{\alpha}$, with $\alpha=\uparrow,\downarrow$ and $|z_{\uparrow}|^{2}+|z_{\downarrow}|^{2}=1$, related to the antiferromagnetic order by

[TABLE]

where $\vec{\sigma}$ are the Pauli matrices. The action for the $\mathbb{CP}^{1}$ model has an emergent U(1) gauge field $a_{\mu}=(a_{\tau},{\bm{a}})$:

[TABLE]

We define the associated emergent electric and magnetic fields by, as usual by

[TABLE]

where $\hat{\bm{z}}$ is a unit vector orthogonal to the square lattice in the $x$-$y$ plane. These gauge-invariant fields are connected to the topological charge of the O(3) order parameter $\vec{n}$ via

[TABLE]

We can now use (11) to deduce the symmetry signatures of ${\bm{e}}$ and $b$, and the results were shown in Table 1. Finally, as in the O(3) formulation, we now search for a combination of ${\bm{e}}$ and $b$ which has the same symmetry signature as the charge current; the simplest possibility is

[TABLE]

Note that the operators in Eqs. (7) and (12) are not equal to each other: they are distinct representations with the same symmetry signature. The connection between ${\bm{O}}$ and the charge current ${\bm{J}}$ in Eq. (3) also applies to Eq. (12). Also at this order, ${\bm{O}}$ is equal to the conserved Poynting vector of the gauge field, but we do not expect possible higher order terms in ${\bm{O}}$ to yield a conserved quantity. Also, as for the O(3) model, we can add a suitable potential $V({\bm{O}})$ to the $\mathbb{CP}^{1}$ action $\mathcal{S}_{z}$ in Eq. (9) and induce a phase with an ${\bm{O}}$ condensate.

The advantage of the $\mathbb{CP}^{1}$ formulation is that we can now write down an effective action for the phase without Néel order, where the $z_{\alpha}$ spinons are gapped. We integrate out the $z_{\alpha}$ spinons and generate an effective action for the U(1) gauge field $a_{\mu}$ in the state where ${\bm{O}}$ is condensed; using gauge invariance and symmetries, the imaginary time action has the form

[TABLE]

where $\gamma_{1,2}$ are coupling constants. The novel feature is the last term which has a co-efficient proportional to $\langle{\bm{O}}\rangle$

[TABLE]

this term leads to a relatively innocuous modification of the gauge field propagator from the familiar relativistic form. By itself, the U(1) gauge theory $\mathcal{S}_{a}$ is unstable to confinement by the proliferation of monopoles and the appearance of VBS order Read and Sachdev (1989). However, topological order can be stabilized if there are Fermi surfaces of U(1) charged fermions Hermele et al. (2004); Kaul et al. (2008a) which suppress monopoles. Alternatively, $\mathbb{Z}_{2}$ topological order can be stabilized Read and Sachdev (1991); Sachdev and Read (1991); Wen (1991) by condensing a Higgs scalar with U(1) charge 2. We will meet both mechanisms in the model of Section III. The resulting state has co-existing topological order and spontaneous charge currents.

III SU(2) lattice gauge theory

This section will extend the SU(2) gauge theory of Refs. Sachdev et al., 2009; Chowdhury and Sachdev, 2015; Sachdev and Chowdhury, 2016; Sachdev et al., 2016 to obtain lattice model realizations of the physics sketched in Section II. The SU(2) gauge theory was initially proposed as a convenient reformulation of particular theories of topological order in insulators Read and Sachdev (1989, 1990, 1991); Sachdev and Read (1991) and metals Kaul et al. (2007, 2008a, 2008b), which also allowed one to recover the large Fermi surface Fermi liquid at large doping. For our purposes, it also turns out to be a convenient setting in which to realize the states discussed in Section II. The theory explicitly includes charged fermionic excitations, and so it is possible to obtain a gap near the antinodes, and also directly compute the charged currents.

We start with electrons $c_{i\alpha}$ on the square lattice with dispersion

[TABLE]

As discussed above Eq. (2), we label half the links from site $i$ by the index $\rho$ and the vector ${\bm{v}}_{\rho}$: to avoid double-counting the vectors ${\bm{v}}_{\rho}$ do not contain any pair that add to 0. With first, second, and third neighbors, ${\bm{v}}_{\rho}$ ranges over the 6 vectors $\hat{\bm{x}}$, $\hat{\bm{y}}$, $\hat{\bm{x}}+\hat{\bm{y}}$, $-\hat{\bm{x}}+\hat{\bm{y}}$, $2\hat{\bm{x}}$, and $2\hat{\bm{y}}$.

We represent the interactions between the electrons in a ‘spin-fermion’ form Hertz (1976) using an on-site field $\Phi^{\ell}(i)$, $\ell=x,y,z$, which is conjugate to the spin moment on site $i$:

[TABLE]

where $\sigma^{\ell}$ are the Pauli matrices. We leave the effective action for $\Phi$ in $V_{\Phi}$ unspecified - different choices for $V_{\Phi}$ allow us to tune between the phases discussed below.

The key to obtaining insulators and metals with topological order (and hence a pseudogap without breaking translational symmetry) is to transform the electrons to a rotating reference frame Sachdev et al. (2009); Chowdhury and Sachdev (2015); Sachdev and Chowdhury (2016); Sachdev et al. (2016) along the local magnetic order, using a SU(2) rotation $R_{i}$ and (spinless-)fermions $\psi_{i,s}$ with $s=\pm$,

[TABLE]

where

[TABLE]

Note that this representation immediately introduces a SU(2) gauge invariance (distinct from the global SU(2) spin rotation)

[TABLE]

under which the original electronic operators remain invariant, $c_{i\alpha}\rightarrow c_{i\alpha}$; here $V_{i}(\tau)$ is a SU(2) gauge-transformation acting on the $s=\pm$ index. So the $\psi_{s}$ fermions are SU(2) gauge fundamentals, carrying the physical electromagnetic global U(1) charge, but not the SU(2) spin of the electron: they are the fermionic “chargons” of this theory, and the density of the $\psi_{s}$ is the same as that of the electrons. The bosonic $R$ fields also carry the global SU(2) spin (corresponding to left multiplication of $R$) but are electrically neutral: they are the bosonic “spinons”. We will relate them below to the spinons, $z_{\alpha}$, of the $\mathbb{CP}^{1}$ model in Eq. (9). A useful summary of the gauge and global symmetry quantum numbers of the various fields is in Table 2.

Inserting the parameterization in Eq. (17) into $\mathcal{H}_{\rm int}$, we can write Eq. (16) as

[TABLE]

We have introduced here the on-site Higgs field $H^{a}(i)$, where $a=x,y,z$ and $\sigma^{a}$ are the Pauli matrices. This is the spin magnetic moment transformed into the rotating reference frame, and is related to $\Phi^{\ell}(i)$ via

[TABLE]

and the inverse relation

[TABLE]

These relations can also be written as

[TABLE]

The Higgs field transforms as an adjoint under the SU(2) gauge transformation, but does not carry spin or charge (see Table 2)

[TABLE]

or equivalently

[TABLE]

We recall in Fig. 3 an earlier mean-field phase diagram Sachdev and Chowdhury (2016) obtained by condensing $R$ or $H$ or both. Our interest here will be primarily in phase C, which has $\mathbb{Z}_{2}$ topological order because the condensation of the Higgs field breaks the SU(2) invariance down to $\mathbb{Z}_{2}$.

We focus here on the effective Hamiltonian for the ‘chargons’, the $\psi$ fermions in phase C. This is motivated by our aim of eventually computing the charge currents. To obtain the Hamiltonian, we insert the parameterization in Eq. (17) into the hopping terms in $\mathcal{H}_{c}$, and decouple the resulting quartic terms. Such an effective Hamiltonian has the form

[TABLE]

The magnitudes of the bare hoppings of the $\psi$ fermions are determined by the real numbers $w_{\rho}$; for simplicity, we fix these hopping parameters at their bare values $w_{\rho}=t_{\rho}$. We have also included a SU(2) matrix on every link, $U^{\rho}(i)$, which represents the gauge connection used by the $\psi$ fermions to hop between sites. This clearly transforms under the gauge transformation in Eq. (23,24) via

[TABLE]

The previous analyses of this model Sachdev et al. (2009); Chowdhury and Sachdev (2015); Sachdev and Chowdhury (2016); Sachdev et al. (2016) only examined the unit SU(2) matrix case $U^{\rho}=\mathbb{I}$. Below, we will describe other choices for $U^{\rho}$, and show that they can lead to states with spontaneous charge currents: this is the main new proposal in this paper for the SU(2) lattice gauge theory.

We will work with a translationally invariant ansatz Wen (2002) for the SU(2) gauge-charged fields, $U^{\rho}(i)$ and $H^{a}(i)$, which can be taken to be independent of $i$. However, to make contact with earlier formulations in which the SU(2) is broken down to a U(1) or $\mathbb{Z}_{2}$ gauge theory Sachdev et al. (2009); Chowdhury and Sachdev (2015); Sachdev and Chowdhury (2016), it is useful to sometimes perform a gauge transformation to a spatially-dependent ansatz. The spatially dependent form cannot be gauge transformed back to the translationally invariant form using only the U(1) or $\mathbb{Z}_{2}$ gauge transformations, and so the spatial dependence is not optional in the U(1) or $\mathbb{Z}_{2}$ gauge theories. We choose the space dependence of the SU(2) gauge fields in the following form

[TABLE]

where

[TABLE]

The background gauge and Higgs fields are fully specified by the wavevector ${\bm{Q}}$, the 3 real numbers $\Theta^{a}$, the angle $\theta_{\rho}$ and the unit vector $\ell^{a}_{\rho}$ ($\sum_{a=x,y,z}(\ell^{a}_{\rho})^{2}=1$) on each $\rho$ link. Note that the ${\bm{r}}_{i}$ dependence is purely in fields performing the gauge transformation so all gauge-invariant combinations will be translationally invariant. In component form, we can write Eq. (33) as

[TABLE]

where $\sigma^{+}=(\sigma^{x}+i\sigma^{y})/2$.

The remaining task before us is to describe the physical properties of the phases obtained for different values of the parameters $\Theta^{a}$, $\theta_{\rho}$, $\ell^{a}_{\rho}$ which are determined by minimizing a suitable free energy. We will do this first for the previously studied phases in Section III.1, and then in Section III.2 for the new phases obtained here. We will find that almost all ansatzes break the SU(2) gauge symmetry to a smaller gauge group: this Higgs phenomenon is accompanied by the appearance of topological order and the gapping of the fermionic spectrum to yield a pseudogap state.

Before turning to this task, we note the transformations of the $\psi_{i}$, $R_{i}$, $H^{a}(i)$ and $U^{\rho}(i)$ under symmetries in Table 1. The simplest choice is to assign the transformations so that they commute with SU(2) gauge transformations. Then the transformations under spatial symmetries ($T_{x}$, $T_{y}$, $I_{x}$, and $I_{y}$) are equal to the identity in SU(2) space, and simply given by the transformations on the spatial indices. More non-trivial are the transformations under time reversal, $\mathcal{T}$; these we assign as

[TABLE]

along with the anti-unitary complex conjugation.

We close this discussion by pausing to recall the reasoning Chubukov et al. (1994); Sachdev (2003); Sachdev et al. (2016) for the presence of $\mathbb{Z}_{2}$ topological order in the Higgs state C in which the SU(2) gauge invariance has broken down to $\mathbb{Z}_{2}$, and why such a state can have small Fermi pockets and a pseudogap even in the presence of translational symmetry. To break SU(2) down to $\mathbb{Z}_{2}$, the configuration of Higgs and link fields, $\Theta^{a}$ and $\ell_{\rho}^{a}$, must transform under global SU(2) transformations like a SO(3) order parameter. Because $\pi_{1}(SO(3))=\mathbb{Z}_{2}$, there are vortex line defects with single-valued Higgs and link fields. Such a defect must also correspond to a single-valued vortex configuration of the antiferromagnetic order. Now we imagine undoing the vortex configuration by choosing $R$ such that the $\psi$ fermions observe a locally constant background in $\mathcal{H}_{\psi}$. Then we will find that $R$ is double-valued, with $R\rightarrow-R$ upon encircling a loop around the vortex. Consequently, the $\psi$ fermions acquire a Berry phase of $\pi$ around the vortex, and the $\psi$ fermions and vortex excitations (the ‘visons’ Senthil and Fisher (2000)) are relative semions. These vortices will be suppressed in the Higgs-condensed ground state, and in such a ground state we can globally transform to a rotating reference frame in which the $\psi$ fermions are described by $\mathcal{H}_{\psi}$. The ${\bm{Q}}$ dependent configuration of Higgs and link fields in Eq. (35) can then reconstruct the $\psi$ Fermi surface into pockets.

III.1 Previously studied phases

III.1.1 Insulators with Néel or VBS order

These are obtained from the saddle point with ${\bm{Q}}=(\pi,\pi)$ and $\Theta^{a}=(\Theta,0,0)$, while all the $\theta_{\rho}=0$ so that $U^{\rho}=\mathbb{I}$. The Higgs field has two sublattice order polarized the $x$ direction with $H^{a}(i)=\eta_{i}(\Theta,0,0)$ where $\eta_{i}=\pm 1$ on the two sublattices.

The dispersion of the $\psi$ fermions is the same as that of electrons in the presence of Néel order, and we obtain the needed fermionic gap in the anti-nodal regions of the Brillouin zone. Note however that Eq. (27) implies that the appearance of physical Néel order requires the condensation of $R$. We assume $\Theta$ is large, and choose the chemical potential to lie within the band gap which has magnitude $|\Theta|$. Consequently, the $\psi$ fermions form a band insulator, and the charge gap is of order $|\Theta|$ which we assume is of order the $U$ of the underlying Hubbard model.

We now argue that fluctuations about this ‘band insulator’ saddle point are described by the $\mathbb{CP}^{1}$ model of Eq. (9). A key observation is that presence of the Higgs condensate $H^{a}(i)=\eta_{i}(\Theta,0,0)$ breaks the SU(2) gauge invariance down to U(1). Such a Higgs condensate is invariant under residual U(1) gauge transformations about the $x$ axis. So we parameterize the the fluctuations of the link fields by

[TABLE]

then ${\bm{a}}$ transforms like the spatial component of a U(1) gauge field under the residual gauge transformation. To obtain the spinons $z_{\alpha}$ in Eq. (9), we need to parameterize $R$ in terms $z_{\alpha}$ so that Eq. (24) implies that $z_{\alpha}$ have unit gauge charge under the gauge transformation $V=\exp(i\sigma^{x}\zeta)$, where $\zeta$ generates the gauge transformation. This is obtained from

[TABLE]

under which $z_{\alpha}\rightarrow z_{\alpha}e^{-i\zeta}$.

We note here a subtlety in identifying the $z_{\alpha}$ and ${\bm{a}}$ above with the fields of the $\mathbb{CP}^{1}$ model of Eq. (9): the symmetry assignments discussed near Eq. (36) for the SU(2) gauge theory do not map under Eq. (37) to the symmetry assignments in Table 1 and Ref. Balents and Sachdev, 2007. The difference is present for transformations $T_{x}$, $T_{y}$ and $\mathcal{T}$, under which the Higgs field $\Theta\rightarrow-\Theta$ in the SU(2) formulation for ${\bm{Q}}=(\pi,\pi)$. In the $\mathbb{CP}^{1}$ formulation, it is implicitly assumed that the Higgs field is invariant under all transformations. To remedy this, we need to combine the SU(2) gauge transformation $V=\exp(-i(\pi/2)\sigma^{z})$ with the operations of $T_{x}$, $T_{y}$, and $\mathcal{T}$ in the SU(2) gauge theory.

Beyond the fluctuations described by the $\mathbb{CP}^{1}$ model, we have to consider the non-perturbative role of monopoles in the U(1) gauge field Read and Sachdev (1989, 1990). In the earlier works, the spin liquid was described using Schwinger bosons with a unit boson density per site. In the presence of monopoles, this background density of bosons contributed a net Berry phase Read and Sachdev (1990). In the present formulation, we have a background of a filled band of the $\psi$ fermions. The monopole Berry phase computation of Ref. Read and Sachdev, 1990 (Section III.A) carries over with little change to the fermion case, and we obtain the same monopole Berry phases.

The remaining analysis of the $\mathbb{CP}^{1}$ model augmented with monopole Berry phases is as before Read and Sachdev (1989, 1990); Senthil et al. (2004b, c). The phase with $\langle z_{\alpha}\rangle\neq 0$ has Néel order, while the strong coupling phase $\langle z_{\alpha}\rangle=0$ is initially a U(1) spin liquid which eventually confines at the longest scales to a VBS; the transition between these phase is described by the critical $\mathbb{CP}^{1}$ model.

III.1.2 Insulators with spiral spin order or $\mathbb{Z}_{2}$ topological order

The saddle point is similar to that in Section III.1.1, except that ${\bm{Q}}$ is incommensurate. So we have $\Theta^{a}=(\Theta,0,0)$, while all the $\theta_{\rho}=0$ so that $U^{\rho}=\mathbb{I}$. Eq. (33) implies that the spatial dependence of the Higgs field is specified by

[TABLE]

For generic ${\bm{Q}}$, there is no residual U(1) gauge invariance left by such a condensate. Instead, the only residual gauge invariance is $\mathbb{Z}_{2}$, associated with the choice $V_{i}=\pm 1$. Consequently, the spin liquid described by this Higgs condensate has $\mathbb{Z}_{2}$ topological order. Again, to obtain an insulator we assume that the chemical potential is within the gap of the $\psi$ bands.

The phases obtain by Eq. (39) are precisely those described in Refs. Read and Sachdev, 1991; Sachdev and Read, 1991; Chatterjee et al., 2016, and the earlier analyses can be applied directly here. The phase with $R$ condensed has spiral spin order, while the phase with $R$ gapped is a $\mathbb{Z}_{2}$ spin liquid.

III.1.3 Metals with topological order

A key advantage of the present SU(2) gauge theory formulation is that the results obtained in Sections III.1.1 and III.1.2 are immediately generalized from insulators to metals. One only has to change the chemical potential $\mu$ so that one of the $\psi$ bands is partially occupied, and we obtain a Fermi surface of $\psi$ chargons.

For the $U(1)$ gauge theory in Section III.1.1, the $\psi$ Fermi surface can suppress the monopoles, and the U(1) topological order survives in an ‘algebraic charge liquid’ (ACL) Kaul et al. (2008a). The $\mathbb{Z}_{2}$ topological order was already stable in the insulator in Section III.1.2, and it continues to survive in the presence of the $\psi$ Fermi surface.

It is also possible that the ACL becomes a ‘fractionalized Fermi liquid’ (FL*) Senthil et al. (2003, 2004a); Paramekanti and Vishwanath (2004). This appears when the $\psi$ fermions bind with the $R$ spinons to form ‘small’ Fermi surfaces of electron-like quasiparticles Kaul et al. (2007, 2008a); Punk and Sachdev (2012); Punk et al. (2015) while retaining the topological order.

III.2 States with SU(2) gauge fields on links

We turn to our new results on the SU(2) lattice gauge theory. We will examine saddle points with non-zero Higgs condensate $\langle H\rangle\neq 0$ (as above) and also a non-trivial background gauge flux $U^{\rho}\neq\mathbb{I}$. We will find that such saddle points can break time-reversal and inversion symmetries in gauge-invariant observables, and that is sufficient to induce charge currents. Ising-nematic order can also be present, as found previously, but it can also co-exist with spontaneous charge currents. This subsection will report results in the gauge ${\bm{Q}}=0$. Recall that the value of ${\bm{Q}}$ is merely a gauge choice in the full SU(2) gauge theory (but not in U(1) or $\mathbb{Z}_{2}$ gauge theory formulations). In this gauge, the Higgs field is $i$ independent with $H^{a}(i)=\Theta^{a}$.

Formally, we should integrate out the fermions in $\mathcal{H}_{\psi}$ in Eq. (31), and then minimize the resulting action functional for the Higgs and gauge fields. However, this is computationally demanding, and the structure assumed in $\mathcal{H}_{\psi}$ is phenomenological anyway. So we will be satisfied by minimizing a phenomenological gauge-invariant functional of the Higgs and gauge fields, consisting of short-range terms that can be constructed out of a single plaquette. In metallic states, the fermion determinant can also induce longer-range terms with a power-law decay, but we will not include those here: in our simple treatment, we assume that the dominant energy arises from the short-range terms.

The effective potential also has terms contributing to a Higgs potential $V_{H}$ which arise from $V_{\Phi}$ in Eq. (16) via Eq. (27). As we will not specify $V_{H}$, we assume that this potential has already been minimized to yield the values of $\Theta^{a}$. So we will only consider the remaining free energy, $\mathcal{F}$, which is a function only of the $U^{\rho}$.

The following gauge-invariant link variables are useful ingredients in constructing the free energy

[TABLE]

These link variables are even under the time-reversal operation described in Eq. (36). In terms of these link variables, we can define the nematic order parameters in Eq. (5) by

[TABLE]

In writing the free energy, it is useful to change notation and write the link variables via

[TABLE]

We minimized the free energy

[TABLE]

In the above expressions we assume that all terms obtained from the pictured symbols by square lattice symmetry operations have been summed over. This free energy depends upon 9 parameters $K_{1-9}$, and a priori they are free to take arbitrary values. We used the residual SU(2) gauge degree for freedom to set $\Theta^{a}=(\Theta,0,0)$, and then with 4 possible values of the link variable $\rho$, the free depends upon 12 real numbers which determine the $U^{\rho}$.

We characterized the free energy minima by their values of the nematic order parameters $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$. We also need gauge-invariant observables which are odd under time-reversal; for this we evaluated the combinations defined on right triangles, $ijk$:

[TABLE]

The spatial patterns of the $\mathcal{P}_{ijk}$, along with the values of $\mbox{Tr}\left(U_{ij}U_{jk}U_{ki}\right)$, yield much information on the nature of time-reversal and inversion symmetry breaking. Note that the $\mathcal{P}_{ijk}$ are non-zero only if the Higgs field is non-zero—this is a consequence of the transformation in Eq. (36). So time-reversal symmetry can only be broken in states in which the SU(2) gauge invariance is also broken.

Another important characterization of the states is provided by the values of the physical charge current. We used the values of the link variables obtained by the minimization of $\mathcal{F}$, and inserted them into the Hamiltonian $\mathcal{H}_{\psi}$ in Eq. (31). We then determined the current on each link by evaluating the expectation value of the current operator

[TABLE]

in the fermion state specified by the Hamiltonian $\mathcal{H}_{\psi}$ at a low temperature. As shown in Appendix A, for the background field configurations in Eq. (35), $\langle{\bm{J}}_{\rho}(i)\rangle$ turns out to be independent of $i$ for general values of the variational parameters $\Theta^{a}$, $\theta_{\rho}$, $\ell^{a}_{\rho}$ and ${\bm{Q}}$ in the Hamiltonian. This is as expected from our arguments that Eq. (35) implies that all gauge-invariant observables should be translationally invariant. Moreover, we find that the value of $\langle{\bf J}_{\rho}\rangle$ always obeys Bloch’s theorem in Eq. (2); this is true in our numerics, and a general proof is in Appendix A.

It is also useful to examine local gauge-invariant operators which have the same symmetry signatures as the physical current ${\bm{J}}_{\rho}$. Such operators will be realizations of the operator ${\bm{O}}$ characterizing states with broken inversion and time-reversal symmetry. We obtained expressions using the symmetry transformations described near Eq. (36), and one set of operators is presented in Fig. 4. A derivation based upon a large $|H^{a}(i)|$ expansion is presented in Appendix C, along with other sets of possible operators. For the translationally invariant solution and ${\bm{Q}}=0$ gauge being considered here, Fig. 4 yields these expressions for the order parameters $O_{\rho}$ along the directions ${\bm{v}}_{\rho}$:

[TABLE]

These will be connected to ${\bm{J}}_{\rho}$ via an expression like Eq. (3). Note that the $O_{\rho}$ can only be non-zero when the Higgs condensate is non-zero, because only the Higgs field is odd under time-reversal in Eq. (36). An explicit demonstration that a non-zero charge current requires a non-zero Higgs field is in Appendix B.

Turning to the minimization of $\mathcal{F}$ in Eq. (43), we did not perform an exhaustive search of different classes of states over the 9 parameters, $K_{1-9}$ in $\mathcal{F}$. Rather we explored a few values to yield representative minima, and will describe a few of the typical states in the subsections below. All of the minimization was performed with the Higgs field oriented along the $x$ direction, $\Theta^{a}=(\Theta,0,0)$.

III.2.1 Symmetric state

This state preserves all square lattice symmetries and time reversal. We obtained such a minimum at $K_{1}=1$, $K_{2}=-1$, $K_{3}=2$, $K_{4}=2$, $K_{5}=2$, $K_{6}=2$, $K_{7}=0.1$, $K_{8}=0$, $K_{9}=0$. In the ${\bm{Q}}=0$ gauge, the link fields take the values:

[TABLE]

where $\theta=0.344\pi$. All the $O_{\rho}$, $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ order parameters, and the currents ${\bm{J}}_{\rho}$, vanish in this state. The SU(2) gauge invariance is broken down to $\mathbb{Z}_{2}$ because the $U^{\rho}$ and $\Theta^{a}$ have no common orientation, and so this state has $\mathbb{Z}_{2}$ topological order.

III.2.2 Ising-nematic order

This state preserves time reversal and inversion, but breaks a square lattice rotation symmetry. We obtained such a minimum at $K_{1}=0.5$, $K_{2}=0.5$, $K_{3}=-1$, $K_{4}=0.25$, $K_{5}=0$, $K_{6}=0$, $K_{7}=0$, $K_{8}=0$, $K_{9}=5$. In the ${\bm{Q}}=0$ gauge the link fields take the values:

[TABLE]

where $\theta_{1}=0.672\pi$ and $\theta_{2}=0.427\pi$. All the $O_{\rho}$, and the currents ${\bm{J}}_{\rho}$, vanish in this state. However the nematic order $\mathcal{N}_{1}\neq 0$, while $\mathcal{N}_{2}=0$. Note that the $U^{\rho}$ are oriented along a common $z$ direction, while the Higgs field $\Theta^{a}$ is oriented along the distinct $x$ direction. So SU(2) gauge invariance is broken down $\mathbb{Z}_{2}$, and $\mathbb{Z}_{2}$ topological order is present. In the insulator, this state has the same properties as the “$(\pi,q)$ SRO” state of Refs. Read and Sachdev, 1991; Sachdev and Read, 1991.

III.2.3 State with broken time-reversal

Now we present a state which breaks time-reversal but not inversion. So this state has no spontaneous currents, and $O^{\rho}=0$ and ${\bm{J}}^{\rho}=0$. Nevertheless, time reversal is broken as signaled by the non-zero values of some of the $\mathcal{P}_{ijk}$. Roughly speaking, such a state has spontaneous currents along different directions in the gauge group, but the net electromagnetic current vanishes. We obtained such a minimum at $K_{1}=0.5$, $K_{2}=0.5$, $K_{3}=1$, $K_{4}=0.667$, $K_{5}=0$, $K_{6}=0$, $K_{7}=1$, $K_{8}=5$, $K_{9}=5$. In the ${\bm{Q}}=0$ gauge the link fields take the values:

[TABLE]

where $\theta_{1}=0.446\pi$ and $\theta_{2}=0.497\pi$. Again, SU(2) gauge invariance is broken down $\mathbb{Z}_{2}$, and $\mathbb{Z}_{2}$ topological order is present.

III.2.4 States with spontaneous charge currents.

Finally, we turn to a description of the states presented in Fig. 2.

First, we present a state with the symmetry of Fig. 2a. Such a state was obtained for $K_{1}=0.5$, $K_{2}=0.5$, $K_{3}=-1$, $K_{4}=0.25$, $K_{5}=0$, $K_{6}=0$, $K_{7}=0$, $K_{8}=2$, $K_{9}=5$. In the ${\bm{Q}}=0$ gauge the link fields take the values

[TABLE]

where $\theta_{1}=0.410\pi$, $\phi_{1}=0.5063\pi$, $\phi_{2}=0.558\pi$, $\theta_{2}=0.387\pi$. This state has the $O^{\rho}$ and ${\bm{J}}^{\rho}$ non-zero, along with a non-zero Ising-nematic order $\mathcal{N}_{1}\neq 0$, but $\mathcal{N}_{2}=0$. So it has the full generic symmetry structure of Fig. 2a. The gauge field configuration shows that SU(2) is broken down to $\mathbb{Z}_{2}$, and so $\mathbb{Z}_{2}$ topological order is present. States in this class were the most common in our search over the parameters $K_{1-9}$ among those that broke time-reversal symmetry.

Among states with a residual U(1) gauge invariance, we found global minima with the symmetry of Fig. 2a only when we restricted the search to states in which the Higgs and link fields were collinear in the gauge SU(2) space. We can parameterize the fluctuations about such a saddle point by multiplying the $U^{\rho}$ by the factor in Eq. (37), and then we obtain a theory of a gapless U(1) photon $a_{\mu}$. Because of the presence of the breaking of inversion and time-reversal symmetries, this action will take the form in Eq. (13), including the term proportional to ${\bm{\Gamma}}$. As in Section III.1.1, we have to consider the non-perturbative effects of monopoles: such a state can be stable against monopole proliferation only in the presence of gauge-charged Fermi surfaces.

Next, we present a state with the symmetry of Fig. 2b. Such a state was obtained for $K_{1}=0.5$, $K_{2}=0.5$, $K_{3}=1$, $K_{4}=0.667$, $K_{5}=0$, $K_{6}=0$, $K_{7}=0$, $K_{8}=5$, $K_{9}=-1$. In the ${\bm{Q}}=0$ gauge the link fields take the values

[TABLE]

where $\theta_{1}=0.451\pi$, $\theta_{2}=0.503\pi$, $\theta_{3}=0.379\pi$. The order parameters $O^{\rho}$ and the currents ${\bm{J}}^{\rho}$ are non-zero, and are consistent with the pattern in Fig. 2a. There is also a non-zero Ising-nematic order $\mathcal{N}_{2}\neq 0$, but $\mathcal{N}_{1}=0$. The non-collinear alignment of the gauge and Higgs fields indicates the presence of $\mathbb{Z}_{2}$ topological order.

IV Conclusions

We have presented computations showing that emergent background gauge connections, and associated Berry phases, arising from the local antiferromagnetic spin correlations can induce spontaneous charge currents, while preserving translational symmetry. The main requirement on the gauge theory is that gauge-invariant observables break time-reversal and inversion, but preserve translation. At the same time, the topological order associated with the emergent gauge fields can account for the anti-nodal gap in the charged fermionic excitations.

The specific model we used for a stable pseudogap metal had $\mathbb{Z}_{2}$ topological order. We employed a SU(2) lattice gauge theory with a Higgs field to realize such a phase. Going beyond earlier work on this theory, we allowed the SU(2) gauge fields on the links to acquire non-trivial values in the saddle point of the Higgs phase. These link fields had two important consequences. First, it became possible to obtain $\mathbb{Z}_{2}$ topological order even under conditions in which the proximate magnetically ordered phase had collinear spin correlations at $(\pi,\pi)$; earlier realizations Read and Sachdev (1991); Sachdev and Read (1991); Sachdev et al. (2009); Chowdhury and Sachdev (2015); Sachdev and Chowdhury (2016) required non-collinear spiral spin correlations. Second, the gauge-invariant combinations of the SU(2) gauge fluxes and the Higgs field could break time-reversal and inversion symmetries without breaking translational symmetry. This allowed the appearance of spontaneous charge currents and Ising-nematic order in the Higgs phase. Linking the discrete broken symmetries to the presence of the Higgs condensate also explains why the broken symmetries do not survive in the confining phases at large doping.

An attractive features of our results is that the charge currents, and the anti-nodal gap, continue largely unmodified across transitions to states with long-range antiferromagnetic order, but without topological order. This is consistent with recent experiments Zhao et al. (2016); Jeong et al. (2017) on Sr2Ir1-xRhxO4 showing co-existence of Néel order and inversion and time reversal breaking over a certain range of doping and temperature.

The possible patterns of symmetry breaking in the translationally-invariant states with broken time-reversal and inversion (but not their product) are illustrated in Fig. 2. Both states also break a lattice rotation symmetry, and so they also have Ising-nematic order. The state in Fig. 2b has the same pattern of symmetry breaking as states considered earlier Simon and Varma (2002); Stanescu and Phillips (2004); Berg et al. (2008). However, the state in Fig. 2a does not appear to have been described previously in the literature. The Fig. 2a state has the attractive feature that its Ising-nematic order is precisely that observed in other experiments Ando et al. (2002); Hinkov et al. (2008); Daou et al. (2010); Lawler et al. (2010). The onset of Ising-nematic order and time-reversal and inversion symmetry breaking could happen at the same or distinct temperatures, as we also found states in Section III.2.2 with Ising-nematic order but no charge currents. However, if a particular symmetry is broken in the pseudogap phase (phase C in Fig. 3), it must be restored when the Higgs condensate vanishes in the over-doped regime (phases D and B in Fig. 3).

The existing experiments Zhao et al. (2016); Jeong et al. (2017) do not contain the polarization analysis needed to distinguish between the states in Fig. 2a and b, and we hope such experiments will be undertaken.

We placed our results in the context of a global phase diagram for antiferromagnetism and superconductivity in two dimensions in Fig. 3. In particular, we noted that this phase diagram Sachdev et al. (2009); Chowdhury and Sachdev (2015); Sachdev and Chowdhury (2016) is in accord with experiments exploring the hole-doped cuprates over a range of carrier density. Badoux et al. Badoux et al. (2016) observe pseudogap metal at low $T$ and large doping, without any charge density wave order: this is a candidate for our phase C. We note recent theoretical works Eberlein et al. (2016); Chatterjee et al. (2017) which studied electrical and thermal transport across the phase transition C $\rightarrow$ D in Fig. 3, and found results in good accord with observations Badoux et al. (2016); Laliberte et al. (2016); Collignon et al. (2016); Michon et al. (2017). Cooper et al. Cooper et al. (2009) observe an extended overdoped regime of linear-in-$T$ resistivity when the superconductivity is suppressed by a magnetic field: our phase D could be such a critical phase. And Božović et al. Božović et al. (2016) see strong deviations from BCS theory in the doping and temperature dependence of the superfluid stiffness in the overdoped regime: this could also be described by phase D.

Acknowledgements

We thank E. Berg, P. Bourges, D. Hsieh, S. Kivelson, L. Taillefer, C. Varma, and A. Vishwanath for valuable discussions. This research was supported by the NSF under Grant DMR-1360789 and the MURI grant W911NF-14-1-0003 from ARO. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. SS also acknowledges support from Cenovus Energy at Perimeter Institute.

Appendix A Momentum space

This appendix presents a few expressions from Section III in momentum space. These expressions were used for our numerical computation.

The momentum space form of the electron dispersion in Eq. (15) is

[TABLE]

The Hamiltonian for the $\psi$ fermions is obtained from Eqs. (31) and (35) (we have set $\lambda=-1$)):

[TABLE]

The average kinetic energy and current on each bond can be evaluated from

[TABLE]

Note that the result is explicitly independent of the site $i$. The kinetic energy is twice the real part of the result, while the current, ${\bm{J}}_{\rho}$ in Eq. (45), is $-{\bm{v}}_{\rho}$ times the imaginary part.

From the expression in momentum space, it is straightforward to see that the value of $\langle{\bf J}_{\rho}\rangle$ always obeys Bloch’s theorem. The Hamiltonian $H_{\psi}$ can be re-written in momentum space in terms of a 2-component spinor $\chi_{{\bm{k}}}$ as follows:

[TABLE]

The minimal coupling to an external electromagnetic gauge field $\bm{A}$ corresponds to a transformation ${\bm{k}}\rightarrow{\bm{k}}-\bm{A}$ in the momentum space Hamiltonian $h_{{\bm{k}}}$. The operator for the net current ${\bf J}$ in any given direction is the negative of the derivative with respect to $\bm{A}$ at $\bm{A}=0$, which can be recast as a derivative with respect to ${\bm{k}}$:

[TABLE]

where the last step uses the Feynman-Hellman theorem and periodicity in the Brillouin zone.

Appendix B Relation between loop currents and Higgs condensate

We show that a non-zero current necessarily requires a Higgs condensate. To do so, we need an operator which reverses the current, and is a symmetry of the Hamiltonian only if the Higgs condensate is absent. Consider the following anti-unitary operator $T$ that leaves the Higgs field unchanged:

[TABLE]

Note that $T$ is not equivalent to the physical time-reversal $\mathcal{T}$ defined earlier in Eq. (36), which always leaves $H_{\psi}$ invariant. Rather, as we show below, $T$ leaves the Hamiltonian invariant only if the Higgs condensate is absent.

Under $T$, we find the following transformation of the Hamiltonian $H_{\psi}$:

[TABLE]

Therefore, the Hamiltonian $H_{\psi}$ commutes with $T$ when $\hat{\Theta}=0$. One can also show that the charge current operator $J_{\rho}(i)$ is odd under $T$, i.e,

[TABLE]

Therefore, when $\hat{\Theta}=0$, we can use the symmetry of $H_{\psi}$ under $T$ to find that:

[TABLE]

The physical content of the above equation is that current loop order cannot arise if all the SU(2) gauge bosons are deconfined, but can possibly arise when a Higgs condensate reduces the gauge group to U(1) or $\mathbb{Z}_{2}$.

Appendix C Real space perturbation theory for current in presence of large Higgs field

We consider the limit where the Higgs field $H^{a}(i)$ is much larger compared to the hopping matrix elements of the $\psi_{\pm}$ fermions, characterized by $w_{\rho}U_{i,i+{\bm{v}}_{\rho}}$. In the $|H_{i}|\rightarrow\infty$ limit, the Hamiltonian has only on-site terms, and therefore there is no current on the links. In this section, we perform a perturbation series expansion in $1/|H_{i}|$ to find an expression for the current. Recall that the charge gap of the SU(2) lattice gauge theory is determined by $|H_{i}|$, and so this is similar to a $1/U$ expansion in the underlying Hubbard model.

We define a lattice Green’s function in imaginary time in the standard fashion:

[TABLE]

The Matsubara Green’s function in the bare limit, $G^{0}_{i,n}$ is diagonal in real space (we set $\lambda=-1$):

[TABLE]

where we have set $\mu=0$ for convenience (it does not modify our conclusion). The Dyson equation for the Green’s function in real space is given by:

[TABLE]

where $w_{ij}=w_{\rho}$ is the hopping along the link $\langle i,j\rangle=\langle i,i+{\bm{v}}_{\rho}\rangle$. Recall that the current operator on the link $\langle i,i+{\bm{v}}_{\rho}\rangle$ is given by:

[TABLE]

Therefore, we can write the its expectation value in terms of the Green’s function defined above as follows:

[TABLE]

The lowest order term in $1/|H_{i}|$ corresponds to $G_{ij}=G^{0}_{i}\delta_{ij}$, which gives zero current consistent with our expectations. To the next order in $1/|H_{i}|$, we find that the forward and backward currents exactly cancel and therefore the current is equal to zero to this order as well (via the cyclic property of the trace).

[TABLE]

This exemplifies the importance of requiring non-nearest neighbor coupling for a non-zero current on the nearest neighbor bonds, albeit in a large Higgs field limit.

The next term in the perturbation series, coming from $G^{(2)}$, gives us a non-zero current. To be more specific, let us label the sites as in Fig. 4, and compute the current from $m$ to $j$. It involves all triangles consisting of $m$, $j$ and a third site connected to both by a non-zero hopping (for simplicity we consider only nearest neighbor and next nearest neighbor hoppings ).

[TABLE]

where $w_{1},w_{2}$ are the nearest and next nearest neighbor hopping respectively. Note that the second term is just the Hermitian conjugate of the first term. We now convert to Matsubara Green’s functions and evaluate the frequency summation (for simplicity we assume that $H_{i}$ are different on each site). The eigenstates at site $i$ have energy $\pm|H_{i}|$, therefore in the $T=0$ limit only the negative energy eigenstates contribute to the current.

Since the contribution of all triangular plaquettes to the current are similar, we only evaluate the contributions to the current by the first term in Eq. (67) and its Hermitian conjugate (corresponding to the triangular plaquette $\triangle_{jlm}$).

[TABLE]

Using the unitarity of U and $U_{\alpha\beta}=U^{\dagger}_{\beta\alpha}$, we can show quite generally that $\operatorname{Tr}(U_{jm}U_{ml}U_{lj})=\operatorname{Tr}(U_{jl}U_{lm}U_{mj})$, so the term without any Higgs field $H_{\alpha}^{a}\sigma^{a}$ for some site $\alpha$ cancels with the contribution from the second line in Eq. (67). The terms with two Higgs fields of the form $H_{\alpha}^{a}\sigma^{a}$ also cancel out with their Hermitian conjugates for the same reason. Therefore, we are left with two kinds of terms, both of which fall off as $|H_{i}|^{-2}$. The contribution to the current from this particular triangular plaquette can be written as:

[TABLE]

where $f(|H|)$ and $g(|H|)$ are scalar functions of the Higgs fields (invariant under all symmetry operations), given by:

[TABLE]

We chose a particular set of terms, coming from $\triangle_{jlm}$ and the three other triangles related to it by reflection symmetries, as our order parameter $O_{jm}$ in Fig. 4. The other three contributions to the current on the link $\langle j,m\rangle$ in Eq. (67) may be obtained by replacing the third vertex $l$ of the triangle by that of the triangle under consideration ($i,k$ and $n$). The net result for the current therefore contains the expressions presented in Fig. 4, along with three other expressions which can also serve as valid representations of ${\bm{O}}$.