Holographic Matter : Deconfined String at Criticality
We derive a holographic dual for a gauged matrix model in general dimensions from a first-principle construction. The dual theory is shown to be a closed string field theory which includes a compact two-form gauge field coupled with closed strings in one higher dimensional space. Possible phases of the matrix model are discussed in the holographic description. Besides the confinement phase and the IR free deconfinement phase, there can be two different classes of critical states. The first class describes holographic critical states where strings are deconfined in the bulk. The second class describes non-holographic critical states where strings are confined due to proliferation of topological defects for the two-form gauge field. This implies that the critical states of the matrix model which admit holographic descriptions with deconfined string in the bulk form novel universality classes with non-trivial quantum orders which make the holographic critical states qualitatively distinct from the non-holographic critical states. The signatures of the non-trivial quantum orders in the holographic states are discussed. Finally, we discuss a possibility that open strings emerge as fractionalized excitations of closed strings along with an emergent one-form gauge field in the bulk.
Extracting dynamical information on strongly interacting critical states of matter is in general a hard problem in theoretical physics. Fortunately, there are classes of strongly coupled quantum field theories whose non-perturbative dynamics can be accessed through dual descriptions which become weakly coupled when the number of degrees of freedom is large.
One such dual description that has been extensively studied in condensed matter physics is the so-called slave-particle formulation(1); (2); (3). In this theory, a gauge redundancy is introduced in order to take into account dynamical constraints imposed by strong interactions. Unphysical states introduced in the redundant description is projected out by a dynamical gauge field. In the large limit, where is the number of flavor degrees of freedom, the dynamical gauge field becomes weakly coupled and emerges as a low energy collective excitation of the system.
The slave-particle theory may be viewed as a mere change of variables which allows one to compute dynamical properties conveniently, which could have been computed using a different set of variables albeit more complicated. However, the real power of the mathematical reformulation lies in the fact that it allows one to classify various novel phases of matter beyond the symmetry breaking scheme(4). In particular, those phases that support emergent gauge boson possess subtle quantum orders that make them qualitatively distinct from the conventional phases. Because of the non-trivial quantum orders, the phases with an emergent (deconfined) gauge boson can not be smoothly connected to the conventional phases. Signatures of the non-trivial quantum order include fractionalized excitations and protected gapless excitations (or ground state degeneracy on a space with a non-trivial topology).
The gauge-string correspondence is another type of duality(5); (6); (7). According to the duality, a class of -dimensional quantum field theories is dual to a -dimensional string theory. The question we would like to address in this paper is : Do those phases that admit holographic descriptions in one higher dimensional space possess non-trivial quantum orders ? If so, what we call holographic states that can be described in one higher dimensional space can not be smoothly connected to the conventional non-holographic states. We claim that the answer to this question is ‘yes’. The signatures of the non-trivial quantum order in holographic phases are the emergent space with an extra dimension, deconfined strings and the existence of an operator whose scaling dimension is protected from acquiring a large quantum correction at strong coupling in the large limit, even though the operator is not protected by any microscopic symmetry of the model.
The paper is organized in the following way. In Sec. II, we start by reviewing the slave-particle theory with an emphasis on quantum order in fractionalized phases. In Sec. III, we introduce a gauged matrix model which will be the focus of the rest of the paper. The model is general enough to include the gauge theory. In Sec. IV, through a first-principle derivation, we show that the matrix model in general dimensions is holographically dual to a closed string field theory in one-higher dimensional spaces. In Sec. V, it is shown that the partition function of the original matrix model can be interpreted as a transition amplitude between quantum many-loop states in the holographic description. In Sec. VI, we show that the holographic description has a gauge redundancy, and strings are coupled with a compact two-form gauge field in the bulk. Because of the compact nature of the two-form gauge field, topological defects for the two-form gauge field are allowed. In Sec. VII, we discuss possible states of the matrix model. Different states are characterized by different dynamics of topological defects in the bulk. If topological defects are gapped, strings are deconfined in the bulk, and the holographic state is stable. On the other hand, if topological defects are condensed, strings are confined, and the bulk description is not useful anymore. Suppressed topological defect in the holographic phase is responsible for a non-trivial quantum order which protects the scaling dimension of the phase mode of Wilson loop operators from acquiring a large quantum correction at strong coupling in the large limit. We discuss the differences between the holographic and non-holographic states. The holographic critical phases can be divided further into two different classes. In the first case, there exist only closed strings in the bulk. In the second case, there are both closed and open strings, where open strings emerge as fractionalized collective excitations of closed strings. The latter state has a yet another quantum order which supports an emergent one-form gauge field in the bulk. Finally, we close with speculative discussions on a possible phase diagram, a world sheet description of deconfined strings, and a continuum limit.
The present construction is beyond the level of identifying the equations of motion in the bulk with the beta function of the boundary theory. We construct a full quantum theory of string in the bulk that is dual to the boundary theory. The construction of the dual theory makes use of the fact that loop variables associated with Wilson loops become classical objects in the planar limit of matrix models(8); (9); (10); (11). The current construction of the string field theory is directly based on the earlier works(12); (13). Compared to the the previous work on the gauge theory(13), the present construction has two major improvements. First, the extra dimension generated out of the renormalization group flow is continuous, while the earlier construction produces a discrete extra dimension. The infinitesimally small parameter associated with a continuously increasing length scale allows one to write the bulk action in a compact form in this formalism. As a result, one can readily take a continuum limit starting from a boundary theory defined on a lattice. Second, the earlier construction involves infinitely many loop fields in the bulk associated with multi-trace operators, which makes the theory highly redundant. In the present construction, the relation between single-trace operators and multi-trace operators are explicitly implemented. As a result, the dual theory can be written only in terms of the loop fields for single-trace operators. Because of these improvements, the dual theory takes a much simpler form, and this transparency allows one to uncover deeper structures in the theory.
There also exist alternative approaches to derive holographic duals for general quantum field theories(14); (15); (16); (18); (17); (19); (20). All these constructions including the present one are based on the notion that the extra dimension in the holographic description is related to the length scale in the renormalization group flow(21); (22); (23); (24); (25).
Ii Quantum order in fractionalized phase
In this section we review some of the key features of the slave-particle theory(1); (2); (3) using a pedagogical model introduced in Ref. (26). We consider a model defined on the four-dimensional Euclidean hypercubic lattice,
Here ’s describe phase fluctuations of
boson fields defined at site .
Each boson carries one flavor index
and one anti-flavor index
represents nearest neighbor bonds of the lattice.
We assume that the phases satisfy the constraints
In the weak coupling limit (), the model describes weakly coupled bosons. As the strength of the kinetic term is increased, there is a phase transition from the disordered phase to the bose condensed phase. In the disordered phase, all excitations are gapped. In the condensed phase, there are Goldstone modes. (At the special point of , there are Goldstone modes due to the enhanced symmetry).
In the strong coupling limit (), the large potential energy imposes an additional set of dynamical constraints, which is solved by a decomposition,
Here ’s are boson fields which parameterize the low energy manifold. Note that these fields carry only one flavor quantum number contrary to the original boson fields. The new bosons are called slave-particles (or partons). The low energy effective action for the slave-particles becomes
Note that this theory has a gauge symmetry,
This is due to the redundancy introduced in the decomposition in Eq. (2). Because of the gauge symmetry, the slave-particles can not hop by themselves. However, these particles can move in space by exchanging their positions with other particles. For example, in Eq. (3), the particle with flavor can hop from site to as the particle with flavor hops from to . In this sense, they can move only through the help of other slave-particles. One can introduce a collective hopping field to characterize the amplitude of this mutual hopping. If we use this collective field, Eq. (3) can be written as
The magnitude of the collective field characterizes the strength of hopping, and the phase plays the role of the U(1) gauge field to which the slave-particles are coupled electrically. This mapping from Eq. (3) to the U(1) gauge theory can be made more rigorous, by using the Hubbard-Stratonovich transformation(26). Although the gauge field does not have the usual Maxwell’s term, the kinetic energy is generated once high energy modes of the boson fields are integrated out, which renormalizes the gauge coupling from infinity to . It is clear that slave-particles can propagate coherently in space only when the hopping field is ‘condensed’, and provides a smooth background. Since the hopping field is not a gauge invariant quantity, we need to be careful when we say that the hopping field is condensed. This notion can be sharply characterized by examining dynamics of topological defect.
Because the U(1) gauge field is compact, monopole is allowed as a topological defect in the theory. The mass of monopole is for a large . Whether the slave-particles arise as low energy excitations of the theory depends on the dynamics of monopole. One can consider the following three different phases.
For a small and small , monopoles are light, and slave-particles are heavy. If monopoles are condensed, strong fluctuations of the phase mode of the hopping field confine the slave-particles. Only gauge neutral composite particles, which are nothing but the original bosons in Eq. (1), appear as low energy excitations. In this phase, all excitations are gapped. This phase is adiabatically connected to the disordered phase in the weak coupling limit.
This is the phase which is electromagnetically dual to the confining phase. The slave-particles are condensed when is large. As a result of the condensation of charged fields, monopoles and anti-monopoles are connected by vortex lines which produce a linearly increasing potential : monopoles and anti-monopoles are confined. One slave-particle is eaten by the massive U(1) gauge boson, and gapless bosons are left. These modes are the Goldstone modes. This phase is smoothly connected to the bose condensed phase in the weak coupling limit.
Fractionalized (Coulomb) phase
For a large , the mass of monopole is large. When both the slave-particles and monopoles are gapped, the Coulomb phase is realized. In this phase, slave-particles are deconfined, and arise as (gapped) excitations of the system. They are fractionalized modes because they carry only half the flavor quantum number of the original bosons. Moreover, the U(1) gauge field arises as a gapless excitation. It is noted that the gapless excitation in this phase is not a Goldstone mode. It is not protected by any microscopic symmetry. Saying that there is a gapless gauge boson in a gauge theory may sound trivial. However, we have to remember that the gauge boson is nothing but a collective excitation of the original boson fields. The existence of a collective excitation which remains gapless without a fine tuning is actually something remarkable : someone who does not use the language of gauge theory would find the origin of the gapless collective excitation mysterious. It turns out that the gapless mode is protected by a subtle order which is not characterized by any symmetry breaking scheme. This order, dubbed as quantum order(4), is associated with suppression of topological excitation, monopole in the long distance limit. Formally, this order can be expressed as the emergence of the Bianchi identity in the long distance limit, where is the field strength for the emergent gauge field. The key features of the non-trivial quantum order is the presence of the fractionalized excitations and the emergent gauge field. Note that slave-particles are not gauge invariant objects. However, ’s become ‘classical’ in the large limit where non-perturbative fluctuations of the hopping field are suppressed. In this regard, fractionalization is associated with the emergence of an ‘internal’ space.
|slave-particle||monopole||low energy excitations|
|Coulomb phase||deconfined||gapped||, monopole, gauge boson|
|Higgs phase||condensed||confined||Goldstone bosons|
Table. I summarizes the physics in each phase of the boson model. Now, we switch gear to discuss about a matrix model and its possible phases. We will draw a close analogy between the quantum order present in the Coulomb phase of the boson model and a quantum order present in the holographic phase of the matrix model. We will see that the holographic phase has a distinct quantum order associated with the emergence of an ‘external’ space.
Iii Matrix model
We start with a matrix model defined on the D-dimensional Euclidean hypercubic lattice,
with the action,
Here , are site indices in the lattice with lattice spacing , and is complex matrix defined on the nearest neighbor bond . is Wilson line defined on the closed oriented loop ,
where the product is ordered along the path. is a function of Wilson loop operators,
which is, in general, non-linear in the presence of multi-trace operators. Here ’s are loop dependent coupling constants. This theory is invariant under the gauge transformation : . Eq. (6) may be viewed as the partition function for a -dimensional quantum matrix model in the imaginary time formalism.
To see that this model includes the usual gauge theory, we consider the following quartic action in Eq. (7) as an example,
where represents unit plaquettes on the lattice. Here , , . We assume that is sufficiently large compared to . The relative magnitude of and determines the shape of the potential for the matrix field. For small , is the minimum, and the system is fully gapped. For large , the low energy manifold is spanned by the matrices that satisfy with . In this case, the low energy effective theory becomes the lattice gauge theory with the ’t Hooft coupling . This theory can be viewed as a ‘linear sigma model’ for the gauge theory. Presumably, the gapped phase in the small limit is smoothly connected to the confinement phase of the gauge theory. As is increased further, the system can go through a phase transition to the deconfinement phase at a critical coupling , depending on the dimension. If the phase transition is continuous, we can take the continuum limit by taking and such that the confining scale is fixed.
Iv General Construction
In this section, we construct a holographic dual for the matrix model in Eq. (7) with general potential in general dimensions. We will follow the idea introduced in Ref. (12) where coupling constants are lifted to dynamical fields in the bulk space where the extra dimension corresponds to the length scale of the renormalization group flow. In the presence of multi-trace operators, this formalism becomes rather complicated(13) because one has to introduce independent fields for infinitely many multi-trace operators that are generated along the renormalization group flow. This issue is present even though multi-trace operators are not turned on initially, because they are generated at low energy scales in any case. To avoid this complication, here we express multi-trace operators in terms of single-trace one, by introducing a complex auxiliary field for each loop (see Appendix A),
Here we dropped a multiplicative numerical factor in the partition function, which is not important. It is noted that is well defined although is not bounded from below as a function of and . This is because is complex and contributions from large negative is canceled because of rapid oscillation in phase. The repeated indices and are understood to be summed over nearest neighbor links and closed loops, respectively.
Here is the number of links in the lattice, and
is an action for . We change the variables as
where is a positive constant, is an infinitesimally small parameter, and
In terms of the new variables, the partition function becomes
Here , and , where is the length of the loop . The field with the large mass has taken away a small amount of quantum fluctuations from the original field , which leaves an action for with smaller couplings . Therefore, we can interpret ’s as low energy fields and ’s as high energy fields.
Fluctuations of renormalize the (dynamical) couplings for the low energy field . Integrating over , we obtain
to the linear order in , where
with . In the third and the fourth terms, runs over all nearest neighbor links, and , are understood to run over all possible loops including null loops with the convention , and for null loops, where refers to the null loop at site . Here we regard null loops at different sites as different loops. By this, we can keep the combinatorics simpler. In the third term, is a form factor that tells whether or not two loops and are ‘nearest neighbors’ : if and can be merged into one loop by adding the link and rejoining the loops, and otherwise. denotes the loop that is made of and with the addition of the link . When both and are non-trivial loops, the third term describes a process where a loop splits into two loops (Fig. 1 (a)). When one of the two loops is a null loop, it describes a process where a loop becomes shorter by eliminating a self-retracting link (Fig. 1 (b)). When both are null loops, it describes a self-retracting link disappearing (Fig. 1 (c)). In the fourth term, is a form factor that tells whether or not two loops and are sharing the link : if and can be merged into one loop by removing the shared link , and otherwise. denotes the loop that is made by merging and by removing the shared link . The fourth term describes a process where two loops merge into one loop (Fig. 1 (d)). In the small limit, , and we can replace with in the third and fourth terms of the action to the linear order in .
Note that double trace operators are generated for . Another set of auxiliary fields is introduced to express the double-trace operator in terms of single-trace operators as
What is the physical meaning of the auxiliary fields ? In the last line of Eq. (26), we note that acts as a source for the low energy matrix field at scale . The key difference from the standard renormalization group procedure is that the source fields are dynamical fields rather than fixed constants at each scale(12). On the other hand, the equation of motion for implies that
Therefore, the conjugate field describes the Wilson loop operator. As we will see below, and are conjugate fields which satisfy a non-trivial commutation relation : sources and operators are conjugate to each other.
Finally, we integrate out to obtain
Here is the effective potential given by
For a future use, we define
which can be computed using the strong coupling expansion,
Here the delta function is defined as
where is the U(1) charge defined on link associated with the flux of loop (13). If the loop passes through the link from to (from to ) times, . The first, second and third terms are from a self retracting loop (Fig.2 (a)), two loops (Fig.2 (b)) and three loops (Fig.2 (c)), respectively. Higher order terms can be obtained similarly. Now we take and limits with fixed. Then, the partition function is written as
Since the partition function is independent of , we can take . From now on, we will interpret the scale parameter as an imaginary ‘time’. The dual description becomes a -dimensional field theory of closed loop. Although the action is written in terms of continuous , one should go back to the discrete version whenever there is an ambiguity, e.g., when extracting boundary conditions by taking variations with respect to boundary fields.
As is the case for matrix models, there are two important parameters that are independent with each other. The first is which controls the strength of quantum fluctuations of the loop fields : the whole action including the boundary actions scales as . The second is the ’t Hooft coupling. In this theory, there is no unique ’t Hooft coupling. Instead there is a set of couplings defined in the space of loops, which scales as the inverse of the ’t Hooft coupling. Since we could have scaled out by redefining in Eq. (7), the theory depends only on the combination . The small limit is equivalent to the large limit, which corresponds to the strong coupling limit of the matrix model where one expects to have the confinement phase. The set of ’s sets the magnitudes of loop fields in the bulk. We will see that background loop fields, in turn, control the size of strings which describe small fluctuations of the loop fields.
V Hamiltonian picture
v.1 Partition function as a transition amplitude between many-body loop states
The partition function can be viewed as an imaginary-time transition amplitude between many-body loop states. To see this, we will use a rescaled loop variable in this sub-section,
The bulk action in the new variable becomes
The action has the form for canonical bosonic fields, where () corresponds to the coherent field associated with the annihilation (creation) operator defined in the space of closed loops. The annihilation and creation operators , satisfy the standard commutation relation
where is a Kronecker-delta function defined in the space of loops. Then the partition function can be written as an imaginary-time transition amplitude,
between the initial (UV) state at ,
and the final (IR) state at ,
Here and are the wavefunctions of loops written in the coherent state basis,
The first term in the Hamiltonian describes a tension of closed loops. The second and the third terms are the interaction terms which describe the processes where one loop splits into two loops, and two loops merge into one loop, respectively, as is shown in Fig. 1. We use the convention , for null loops. Similar loop Hamiltonians that describe joining and splitting processes of loops were considered in matrix models(30); (31).
This is an exact mapping between the -dimensional matrix model (-dimensional quantum matrix model) and the -dimensional loop model (or -dimensional quantum loop model). Several remarks are in order. First, the Hamiltonian in Eq. (45) is a many-body Hamiltonian that governs the quantum dynamics of loops along the scale which is interpreted as an imaginary time. It is noted that the Hamiltonian is not Hermitian. Due to the cubic interaction term, the Hamiltonian is unbounded from below. However, the transition amplitude in Eq. (39) is well defined because eigenvalues of the Hamiltonian are complex. Eigenvalues with a large negative real part in general come with a large imaginary part, and their contributions cancel with each other due to oscillation in phase. Second, the bulk Hamiltonian is universal, and it is independent of the details of the matrix model. All informations pertaining to the specifics of the matrix model are encoded in the initial wavefunction at . Third, the strength of the interaction between loops is order of , and loops are weakly interacting in the large limit. Therefore, the theory becomes classical in the large limit. Fourth, does not have any hopping term such as with different and . This fact will become important for gauge symmetry, which will be discussed in Sec. V. For earlier works on string field theories formulated without quadratic action, see Ref. (32); (33).
The fact that the partition function is independent of has a remarkable consequence. By taking the derivative of Eq. (39) (for a finite ) with respect to , we obtain
Since physical states are singlets of , the Hamiltonian can be viewed as a generator of a ‘gauge transformation’. The gauge transformation corresponds to a reparameterization of . It is based on the fact that one can choose different speed of renormalization group flows at different scales without affecting the physics. By choosing the parameter to be -dependent, the reparameterization symmetry can be made explicit(12). Here becomes the lapse function. Reparameterizations of form a subgroup of the full diffeomorphism in the -dimensional space. It would be interesting to formulate the theory where the full diffeomorphism can be made explicit in the bulk. Here we proceed with the present formalism where we choose specific time slices along the direction.
v.2 Wilson loop operator
The physical picture for the transition amplitude is the following. At the UV boundary (), a condensate of loops are emitted and propagate in under the evolution governed by . The amplitude of the condensate is . This can be seen from the fact that the action for the unscaled loop fields has as an overall prefactor, which implies . Loops can join and split through the interactions as is illustrated in Fig. 3. A loop and its anti-loop , the loop with the opposite orientation, can get pair-annihilated through a series of interactions as is shown in Fig. 4 (a). Moreover, a self-retracting loop can become a loop and an anti-loop as is shown in Fig. 4 (b). As it will be shown in Sec. VII. A, loop fields for self-retracting loops have non-zero vacuum expectation values in the bulk. Therefore, a pair of loop and anti-loop can be created out of vacuum. This means that two loops with the opposite orientations act as particle and anti-particle in a relativistic field theory. Finally, those loops emitted at the UV boundary are absorbed at the IR boundary. In this sense, the UV boundary is a source of loops, and the IR boundary is a sink.
Now let us consider a Wilson loop operator for a loop which is much larger than the size of Wilson loops for which sources are turned on at the UV boundary. The expectation value of the Wilson loop operator is given by the one-point function,
If is large, loops propagate independently in the bulk. To the zeroth order in , the loop propagate to the sink along the straight path. However, this configuration vanishes as in the large limit because of the tension. In order for the expectation value to survive, the large loop should absorb other smaller loops from the condensate to disappear before it reaches the IR boundary. Then the evolution of the Wilson loop forms a world-sheet in the bulk. One such configuration is shown in Fig. 5. Then the expectation value is given by the sum over all world-sheets of the Wilson loop.
Since the interaction between loops is , loops become classical in the large limit. This implies factorization of Wilson loop operators in the large limit,
Vi Gauge symmetry
The absence of the hopping term in the Hamiltonian has a deep origin : the loop field theory has a gauge symmetry. Note that this gauge symmetry is not related to the gauge symmetry of the original matrix model. Loop fields are singlets for the gauge symmetry. In this section, we examine the consequences of the new gauge symmetry carefully. From now on, we return to the unscaled loop variable .
The bulk action in Eq. (35) is invariant under the time-independent transformation generated by at each link
where is summed over all sites, is summed over directions of nearest neighbor links, and is a time-independent angle defined on the link . The IR boundary action respects the symmetry, but the UV action does not. This is because the UV potential
includes sources which explicitly break the symmetry. It is useful to view as an expectation value of another dynamical loop field. Then, the full theory is invariant if we allow the UV source to transform as
This time-independent symmetry can be lifted to a full space-time gauge symmetry by introducing temporal components of a two-form gauge field in the bulk with ,
where with are the temporal components of the two-form gauge field defined at each spatial link. This two-form gauge field is the Kalb-Ramond gauge field(29). Now the full theory is invariant under the space-time dependent gauge transformation with